THOMPSON RIVERS UNIVERSITY

Robust Regression Models with Missing and Censored Data
via the EM Algorithm
By

Minoli Randika Munasinghe
A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE
REQUIREMENTS FOR THE DEGREE OF
Master of Science in Data Science
KAMLOOPS, BRITISH COLUMBIA

September, 2024

SUPERVISORS
Mateen Shaikh & Erfanul Hoque

© Minoli Randika Munasinghe, 2024

ABSTRACT
This study addresses the challenge of fitting a simple linear regression model to
data with missing or censored values by presenting a novel approach based on the
Expectation–Maximization algorithm. The proposed approach is developed to estimate the parameters of models under various incomplete data scenarios, including
missing, left censored, right censored, and interval censored observations in bivariate
normal data. Extensive simulation studies evaluate the performance of the proposed
approach across varying sample sizes, incomplete proportions, and correlations between variables. The study compares the proposed approach with the existing models,
which ignores incomplete observations in model fitting. Evaluation metrics including
bias, variance, Root Mean Squared Error, Coverage Probability, and Relative Root
Mean Squared Error are used to assess the estimates from both approaches. The
proposed method is applied to real datasets to validate its effectiveness against naive
estimates. Standard error, confidence interval, and interval width assess the precision
and accuracy of parameter estimates for both models, while the Akaike Information Criterion selected the best–fitting model. The results show that the proposed
approach provides more accurate and precise parameter estimates compared to the
naive approach in both simulation studies and real data applications.
Keywords: Bivariate normal distribution, Expectation–Maximization Algorithm,
Interval Censoring, Left Censoring, Missing Data, Regression; Right Censoring

ii

ACKNOWLEDGEMENTS
I am incredibly grateful to my supervisors, Dr. Mateen Shaikh and Dr. Erfanul Hoque from the Department of Mathematics and Statistics at Thompson Rivers
University, for their continuous guidance throughout my academic journey. Their
expertise, encouragement, and valuable feedback have been crucial in shaping the
direction of this thesis. I also thank Dr. Robin Kleiv, my committee member, for his
contributions.
I acknowledge the guidance of Dr. Mohammed Tawhid, the MSc Data Science
program coordinator, for directing me through the program requirements. I am also
thankful to the faculty members of the Department of Mathematics and Statistics,
specifically Dr. Jabed Tomal, Dr. Mila Kwiatkowska, and Dr. Roger Yu, former
program coordinator, for their contributions to my education in various ways.
I am grateful for the financial support provided by Dr. Mateen Shaikh and Dr.
Erfanul Hoque for this research project, which was funded by the Natural Sciences and
Engineering Research Council of Canada (NSERC) Discovery Grant and the Research
Accelerate Grant from Thompson Rivers University.
Special appreciation goes to my beloved husband, Muditha Ekanayake, whose
selfless encouragement and endless support have inspired me throughout my academic
journey. I am also deeply thankful to my parents and brothers for their support in
every situation. I would like to extend my thanks to my friends and peers for their
encouragement and moral support; their companionship and shared experiences have
made this journey more enjoyable. Lastly, I express my deepest gratitude to the Sri
Lankan free education system for laying a strong foundation for my academic studies.

iii

Contents

1 Introduction

1

1.1

Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3

1.2

Research Steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.3

Significance of the Study . . . . . . . . . . . . . . . . . . . . . . . . .

4

1.4

Organization of the Thesis . . . . . . . . . . . . . . . . . . . . . . . .

5

2 Literature Review

6

2.1

Missing Data Patterns . . . . . . . . . . . . . . . . . . . . . . . . . .

6

2.2

Censoring Data Patterns . . . . . . . . . . . . . . . . . . . . . . . . .

7

2.3

Introduction to Regression . . . . . . . . . . . . . . . . . . . . . . . .

10

2.3.1

Simple Linear Regression (SLR) . . . . . . . . . . . . . . . . .

10

2.3.2

Least Squares (LS) Method . . . . . . . . . . . . . . . . . . .

12

2.3.3

Maximum Likelihood (ML) Method . . . . . . . . . . . . . . .

13

2.4

Techniques for Handling Missing Data . . . . . . . . . . . . . . . . .

14

2.5

Expectation–Maximization (EM) Algorithm . . . . . . . . . . . . . .

17

iv

CONTENTS
2.5.1

2.6

2.7

v
Applications of the EM Algorithm in the Presence of Incomplete
Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

18

Univariate Normal Distribution . . . . . . . . . . . . . . . . . . . . .

20

2.6.1

Right Censored Univariate Normal Data . . . . . . . . . . . .

21

2.6.2

Left Censored Univariate Normal Data . . . . . . . . . . . . .

21

2.6.3

Interval Censored Univariate Normal Data . . . . . . . . . . .

22

Bivariate Normal (BVN) Distribution . . . . . . . . . . . . . . . . . .

23

3 Methodology

27

3.1

EM Algorithm for the BVN Distribution . . . . . . . . . . . . . . . .

27

3.2

EM Algorithm for Missing Observations . . . . . . . . . . . . . . . .

30

3.2.1

E–Step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

30

3.2.2

M–step . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

34

Performance Metrics to Evaluate the Regression Estimates . . . . . .

34

3.3

4 Simulation Studies

40

4.1

Simulation Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

40

4.2

Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

4.2.1

Results in the Presence of Missing Observations . . . . . . . .

43

4.2.2

Results in the Presence of Right Censored Observations . . . .

46

4.2.3

Results in the Presence of Left Censored Observations

. . . .

50

4.2.4

Results in the Presence of Interval Censored Observations . .

53

4.2.5

Summary of the Simulation Results . . . . . . . . . . . . . . .

56

CONTENTS

vi

5 Data Applications

57

5.1

Determining the Effect of Solar Radiation on Ozone Concentration–An
Application to the Presence of Missing Observations . . . . . . . . . .

5.2

57

Exploring the Relationship Between Blood and Feather Lead Concentrations in Black–Crowned Night Herons (Nycticorax nycticorax)–An

5.3

Application to the Presence of Left Censored Observations . . . . . .

62

Summary of the Data Applications . . . . . . . . . . . . . . . . . . .

66

6 Discussion

67

6.1

Limitations of the Study . . . . . . . . . . . . . . . . . . . . . . . . .

68

6.2

Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

A EM Algorithm for Different Censoring Types

77

B Additional Simulation Results

96

List of Figures

2.1

Scatter plots of BVN data in x and y variables, depicting different types
of censored observations. Vibrant blue–green points denote censored
data, including left censored, right censored, and interval censored observations. Black points represent uncensored data points. The red
vertical line indicates the threshold in the x variable, while the green
horizontal line denotes the threshold in the y variable. . . . . . . . . .

2.2

Traditional and modern missing data handling methods [Shylaja and
Kumar, 2018] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.1

16

Boxplots for estimates of intercept (βˆ0 ) and slope (βˆ1 ), n = 100 and
ρ = 0.2 over different missing proportions of 0.1, 0.3, and 0.5 . . . . .

4.2

9

43

Line charts for asymptotic variance of intercept (βˆ0 ) and slope (βˆ1 )
when ρ = 0.2 over different missing proportions of 0.1, 0.3, and 0.5 and
sample sizes 50, 100, and 500 . . . . . . . . . . . . . . . . . . . . . .

4.3

44

Boxplots for estimates of intercept (βˆ0 ) and slope (βˆ1 ), when n = 100
and ρ = 0.2 over different right censoring proportions of 0.1, 0.3, and 0.5 47

4.4

Line charts for asymptotic variance of intercept (βˆ0 ) and slope (βˆ1 )
when ρ = 0.2 over different right censoring proportions of 0.1, 0.3, and
0.5 and sample sizes 50, 100, and 500 . . . . . . . . . . . . . . . . . .

4.5

48

Boxplots for estimates of intercept (βˆ0 ) and slope (βˆ1 ), n = 100 and
ρ = 0.2 over different left censoring proportions of 0.1, 0.3, and 0.5 . .
vii

50

LIST OF FIGURES
4.6

viii

Line charts for asymptotic variance of intercept (βˆ0 ) and slope (βˆ1 )
when ρ = 0.2 over different left censoring proportions of 0.1, 0.3 and
0.5 and sample sizes 50, 100, and 500 . . . . . . . . . . . . . . . . . .

4.7

51

Boxplots for estimates of intercept (βˆ0 ) and slope (βˆ1 ), when n = 100
and ρ = 0.2 over different interval censoring proportions of 0.1, 0.3,
and 0.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.8

53

Line charts for asymptotic variance of intercept (βˆ0 ) and slope (βˆ1 )
when ρ = 0.2 over different interval censoring proportions of 0.1, 0.3,
and 0.5 and sample sizes 50, 100, and 500

. . . . . . . . . . . . . . .

54

5.1

Scatter plot of Ozone concentration (ppb) over Solar radiation (Lang)

59

5.2

Histogram and normal Q–Q plot for solar radiation . . . . . . . . . .

60

5.3

Histogram and normal Q–Q plot for Ozone concentration . . . . . . .

60

5.4

Histogram and normal Q–Q plot for log–transformed Ozone concentration 61

5.5

Scatter plot of lead concentration in blood and feather of Herons . . .

64

List of Tables

3.1

Data structure for variables x and y . . . . . . . . . . . . . . . . . . .

28

3.2

Scenarios with observed and missing data for x and y . . . . . . . . .

31

4.1

Simulation details for different sample sizes and correlations for a single
proportion of x and y. . . . . . . . . . . . . . . . . . . . . . . . . . .

4.2

41

Bias, asymptotic RMSE, 95% CP, and relative RMSE of intercept (βˆ0 )
for ρ = 0.2 over sample sizes 50, 100, and 500 and missing proportions
0.1, 0.3 and 0.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.3

45

Bias, asymptotic RMSE, 95% CP, and relative RMSE of slope (βˆ1 ) for
ρ = 0.2 over sample sizes 50, 100, and 500 and missing proportions 0.1,
0.3 and 0.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.4

45

Bias, asymptotic RMSE, 95% CP, and relative RMSE of intercept (βˆ0 )
for ρ = 0.2 over sample sizes 50, 100, and 500 and right censoring
proportions of 0.1, 0.3, and 0.5 . . . . . . . . . . . . . . . . . . . . . .

4.5

49

Bias, asymptotic RMSE, 95% CP, and relative RMSE of slope (βˆ1 )
for ρ = 0.2 over sample sizes 50, 100, and 500 and right censoring
proportions 0.1, 0.3, and 0.5 . . . . . . . . . . . . . . . . . . . . . . .

4.6

49

Bias, asymptotic RMSE, 95% CP, and relative RMSE of intercept (βˆ0 )
for ρ = 0.2 over sample sizes 50, 100, and 500 and left censoring proportions of 0.1, 0.3, and 0.5 . . . . . . . . . . . . . . . . . . . . . . .

ix

52

LIST OF TABLES
4.7

x

Bias, asymptotic RMSE, 95% CP, and relative RMSE of slope (βˆ1 ) for
ρ = 0.2 over sample sizes 50, 100, and 500 and left censoring proportions of 0.1, 0.3, and 0.5 . . . . . . . . . . . . . . . . . . . . . . . . .

4.8

52

Bias, asymptotic RMSE, 95% CP, and relative RMSE of intercept (βˆ0 )
for ρ = 0.2 over sample sizes 50, 100, and 500 and interval censoring
proportions of 0.1, 0.3, and 0.5 . . . . . . . . . . . . . . . . . . . . . .

4.9

55

Bias, asymptotic RMSE, 95% CP, and relative RMSE of slope (βˆ1 )
for ρ = 0.2 over sample sizes 50, 100, and 500 and interval censoring
proportions of 0.1, 0.3, and 0.5 . . . . . . . . . . . . . . . . . . . . . .

5.1

55

Comparison of model coefficients, standard errors, Lower Confidence
Limit (LCL) and Upper Confidence Limit (UCL), and interval width
of naive model and proposed model . . . . . . . . . . . . . . . . . . .

5.2

62

Comparison of model coefficients, standard errors, Lower Confidence
Limit (LCL) and Upper Confidence Limit (UCL), and interval width
of naive model, alternate model, and proposed model . . . . . . . . .

65

A.1 Scenarios with observed and right-censored data for x and y . . . . .

78

A.2 Scenarios with observed and left censored data for x and y . . . . . .

84

A.3 Scenarios with observed and interval censored data for x and y . . . .

89

Abbreviations
BVN Bivariate Normal
CDF Cumulative Density Function
CI Confidence Interval
CLS Complete Least Squares
CP Coverage Probability
EM Expectation–Maximization
FIM Fisher Information Matrix
LD Listwise Deletion
LS Least Squares
MCAR Missing Completely At Random
ML Maximum Likelihood
MLE Maximum Likelihood Estimate
MSE Mean Squared Error
PDF Probability Density Function
RMSE Root Mean Squared Error
RRMSE Relative Root Mean Squared Error
RSS Residual Sum of Squares
SLR Simple Linear Regression

x

Chapter 1
Introduction
The quality and comprehensiveness of datasets play a vital role in data–driven research to ensure reliable and accurate analytical outcomes. However, the reality is
that obtaining complete and perfectly recorded data for analysis is nearly impossible.
Many datasets remain incomplete due to the lack of significant information about the
observation of interest. This incompleteness can complicate the analysis into a complex problem that needs to be addressed effectively through an advanced approach.
Missing and censored data are two major types of incomplete data. Missing
data occurs due to multiple factors, including non–responsiveness of the respondents,
technical limitations, errors in data collection, and drop–offs from the studies. For
example, data collected from a healthcare study over a period of time may have missing observations due to patient dropouts or missed follow–up appointments. Censoring arises in numerous applications for reasons such as limitations in measuring
instruments or experimental design. For example, air pollutant measurements can be
censored due to the detection limits of the measuring instruments.
In each case of missing and censoring, incomplete data can pose significant challenges, resulting in statistical inconvenience, increased problem complexity, biased
estimates, and inaccurate analysis and decision–making. The impact can be serious
depending on the pattern, percentage, and mechanism of incompleteness. In some
cases, the missing data can be random, causing less impact on the results. In other

1

cases, the data can be missing in a way that introduces significant bias, creating
misleading results. Moreover, censored data often produces higher bias and variance
than missing data. Therefore, understanding and identifying the type, origin, and
implications of incomplete data is crucial for researchers, as it plays a vital role in
developing robust methodologies to mitigate its adverse effects.
This study addresses the problem of fitting Simple Linear Regression (SLR) models to Bivariate Normal (BVN) data with missing or censored observations. It is motivated by the audiometric study conducted by De Lima Taga and Singer [2016], which
explored the potential to predict behavioural thresholds based on electro–physiological
thresholds. They utilized a Maximum Likelihood (ML) estimation approach for a SLR
model, where the dependent and independent variables were subjected to interval censoring via the Newton–Raphson optimization algorithm. In addition, several studies
proposed new methodologies for regression modeling while handling missing and censored data in different distributions. For instance, Miller [1976] estimated parameters
of a SLR model by utilizing the Kaplan–Meier estimator for right censored response
data. Guzmán et al. [2020] developed linear regression models on skew scale mixtures
of normal distributions where response variable is left censored via the Monte Carlo
Expectation–Maximization (MCEM) algorithm. Austin and Hoch [2004] estimated
the regression coefficients for censored covariates using ordinary LS, partial LS, and
full likelihood methods. In addition, Buckley and James [1979], Koul et al. [1981], and
Ritov [1990] proposed various other procedures to obtain regression models for right
censored data. However, to our knowledge, previous research has not incorporated
the EM algorithm for fitting a SLR model when missing or censoring occurs in data.
Several researchers also explored the applications of the EM algorithm across
diverse data types, incomplete data scenarios, and distributions. Atkinson [1992] proposed a hybrid EM algorithm to estimate the parameters of a normal mixture model
in the presence of right censoring. McLachlan and Jones [1988] applied EM algorithms to fit mixture models to truncated and grouped data, Lee and Scott [2012]
developed multivariate Gaussian mixture models capable of handling both truncated
and censored data, and Josse et al. [2018] explored the EM approach for estimating
the parameters of a BVN distribution, specifically focusing on the scenario where

2

data is missing for one of the two variables. To the best of our knowledge, previous research has not specifically addressed the challenge of fitting SLR models for
missing or censored BVN data. Moreover, our study emphasizes joint modeling for
observations where both variables are censored. Therefore, this study aims to design
an EM–based approach that effectively integrates these complexities for obtaining
regression parameter estimates.

1.1

Problem Statement

Traditional statistical methods frequently encounter challenges in effectively addressing various types of incomplete data, including missing observations, left censoring,
right censoring, and interval censoring in regression modeling. While the EM algorithm has demonstrated flexibility in managing incomplete data across various distributions, existing research primarily focuses on obtaining only MLEs when incompleteness affects a single variable. This study aims to address the gap in the literature
by developing and validating an EM–based approach designed for regression modeling
with missing or censored BVN continuous data, where both variables may experience
missing data or censoring in a unified framework. To the best of our knowledge, this
specific problem has not been adequately addressed in a common framework, and
it is crucial for fields requiring precise estimation of regression parameters, such as
epidemiology, finance, and environmental sciences.
Therefore, this study will focus on enhancing the accuracy and reliability of regression parameter estimates for BVN data. The proposed method will integrate the
complexities of missing and censoring in both variables. This research intends to provide robust solutions that improve existing methods and contribute significantly to
advancing statistical modeling in complex data scenarios.

3

1.2

Research Steps

1. Develop an EM based approach to estimate regression parameters of a SLR
model in various incomplete data scenarios accurately. These scenarios include
missing, left censored, right censored, and interval censored observations within
both variables (x independent and y dependent variables) of BVN datasets.
2. Assess the performance of the parameter estimation scheme using the estimates
obtained from the proposed method under varying data scenarios, such as different sample sizes, proportions of missing and censoring, and correlations between
variables.
3. Compare the regression estimates derived from the proposed approach, which
handle missing and censored data, with those from naive models that only consider fully observed data for fitting regression models.
4. Apply the proposed method to real datasets.

1.3

Significance of the Study

This study extends previous research by addressing a variety of incomplete data scenarios in regression models related to the BVN distribution. It examines left censored
data, where observations are below a minimum threshold; right censored data, where
observations exceed a maximum threshold; interval censored data, where observations
lie within a specified range; and missing data in varying combinations within a BVN
distribution.
Simulation studies are conducted to obtain parameter estimates of SLR models
under varying data scenarios, such as different proportions of incompleteness, sample
sizes, and correlations. The regression estimates obtained from the proposed approach
are evaluated across these scenarios to determine its effectiveness in each scenario. The
naive models obtained by fitting regression models on fully observed data are used
to compare the performance of the proposed models. Real–data applications are in4

corporated to assess the applicability of the approach on real–data, demonstrating its
potential to solve real–world problems. Therefore, this research contributes valuable
insights for researchers dealing with regression modeling with missing or censored data
arising from limitations in measurement equipment, where the measures are linearly
dependent on each other.

1.4

Organization of the Thesis

The remainder of the thesis is organized into the following chapters. Chapter 2 briefly
explains regression analysis and applications of the EM algorithm in statistical modeling. It also covers essential concepts related to univariate and BVN distributions and
their properties. Chapter 3 details the research methodology, focusing on adapting
the EM algorithm for fitting SLR models to BVN data with missing or censored values. It further elaborates on the performance metrics used to evaluate the regression
parameters. Chapter 4 presents the design and outcomes of comprehensive simulation
studies. Chapter 5 shifts to real–world applications, applying the proposed method
to actual datasets to validate its practical utility and effectiveness. Finally, Chapter
6 summarizes each chapter, discusses the limitations of the proposed method, and
proposes directions for future work.

5

Chapter 2
Literature Review
This chapter discusses the most common missing data patterns, censoring data patterns, an overview of regression, existing techniques for handling incomplete data, a
brief explanation on the EM algorithm, existing literature on its applications, univariate normal distribution and the bivariate normal distribution.

2.1

Missing Data Patterns

It is crucial to understand the nature and forms of missing data to effectively address
the complexities of incomplete data. Rubin [1976] classified missing data into three
major categories based on the nature of the missingness.
Missing Completely At Random (MCAR) refers to the observations that are
missing completely at random without any influence of the observed or missing values.
It lacks a systematic pattern and occurs randomly. It is easier to handle missing data
when the MCAR assumption is satisfied, and the results can be obtained without
introducing significant bias into the analysis due to the complete randomness. For
example, when recording videos on the phone, the phone randomly fails and restarts,
but successfully records some videos. However, others are completely missing due to
the failure. This is considered MCAR, as specific videos lost do not depend on the
content of the video itself or any other factor related to the recording process.
6

Missing At Random (MAR) defines that the probability of missing is related
to the observed data, but not directly related to the missing information itself. The
MAR data has a specific pattern to the observed variables, which is also called ignorable missingness. When capturing videos on a phone with a low battery level of 1%,
the recording often fails, resulting in missing video data. This scenario is an example
of MAR. The failure of the video recording is more likely to occur when the battery
is low (an observed factor), and the failure (missing data) itself does not influence the
battery level.
Missing Not At Random (MNAR) is the most challenging form of missing
data. This means that the probability of missing is related to missing data itself, but
not to the observed data. It is also called non–ignorable missingness. The MNAR
observations occur due to unobserved variables. As an example, the phone camera
fails to record videos more often when the battery is low, but does not keep track of
the exact battery level that triggers the issue; the missing video data is MNAR as
the missing information (critical battery threshold) directly influences the likelihood
of missing videos.

2.2

Censoring Data Patterns

A censored datum refers to a value known to lie within a specific interval. This occurrence is most frequent in survival analysis, clinical trials, environmental studies, and
social sciences. In classical inference, the limitations in measuring instruments and
specific reporting protocols are two major causes of censoring. Generally, measuring instruments have minimum and maximum measurement thresholds beyond which
they cannot provide accurate measurements. This can cause censoring by reporting only the threshold instead of an exact measurement. Censoring has three major
types, and understanding them is crucial in order to provide meaningful and accurate
results.
Suppose an observation falls below the minimum measurement threshold of an
instrument. In that case, the minimum threshold is recorded as the measured value

7

and is considered as a Left Censored (LC) observation. Let x and y be random
variables of a BVN distribution. The observations of x and y take the form




lc
yi if yi ≥ clc
xi if xi ≥ cx
y
o
o
yi =
xi =


clc if yi < clc
clc if xi < clc
y
y
x
x
where i = 1, 2, ...n observations, ”o denotes observed data, with clc
x for threshold in x
and clc
y for threshold in y for left censored data. The left censored observations are
visually observable in the first scatter plot of Figure 2.1. The red and green vertical
and horizontal lines, respectively represent censoring thresholds for variables x and y,
set at -2. Observations falling below these thresholds (indicated by vibrant blue–green
points) are considered left censored, meaning their exact values are unknown, but are
at the threshold. For example, any observation below -2 for x or y is recorded as -2.
Suppose an observation falls above the maximum measurement threshold of an
instrument; it is considered as a Right Censored (RC) observation and this maximum threshold is recorded as the measured value as described below.




xi if xi ≤ crc
yi if yi ≤ crc
x
y
o
o
yi =
xi =


crc if xi > crc
crc if yi > crc
x
x
y
y
rc
where i = 1, 2, ...n observations with crc
x for threshold in x and cy for threshold in y

for right censored data. The right censored observations are visually observable using
the second scatter plot in Figure 2.1. The censoring thresholds for variables x and y
are set at 5, and the observations falling above the thresholds (indicated by vibrant
blue–green points) are considered right censored. For example, any observation above
5 for x and y is recorded as 5.

8

Figure 2.1: Scatter plots of BVN data in x and y variables, depicting different types of
censored observations. Vibrant blue–green points denote censored data, including left
censored, right censored, and interval censored observations. Black points represent
uncensored data points. The red vertical line indicates the threshold in the x variable,
while the green horizontal line denotes the threshold in the y variable.

Similar to one–sided censoring described above, measuring instruments may lack
the precision needed for accurate measurements. In such instances, the instrument can
only provide a range within which the true value falls rather than providing an exact
measurement. We refer to this situation as an Interval Censored (IC) observation,
where both the lower and upper bounds of the true value are known. Interval censoring
can also occur due to specific reporting protocols. The simplest example of interval
censoring is through a limited number of recorded digits: an instrument may record
the value of the mass of an object to be 0.7kg, which means the true value is some
unknown value between 0.65kg and 0.75kg. It is more formally described below.




ic
ic
ic
ic
ic
ic
[lyic , uic
[lx , ux ] if lx ≤ xi ≤ ux
y ] if ly ≤ yi ≤ uy
o
o
xi =
yi =


x i
y i
otherwise
otherwise
where i = 1, 2, ...n observations with lxic and uic
x for lower and upper limits of interval
9

in x and lyic and uic
y for lower and upper limits of interval in y for interval censored
data. A scatter plot is illustrated in the third plot of Figure 2.1. The lower and upper
limits of x and y are set at [2.5,4.5] and [0,2], respectively. The observations that fall
within the limits (indicated in vibrant blue–green) are considered interval censored
and only the lower and upper limits are recorded.

2.3

Introduction to Regression

Regression is a fundamental technique used in statistical modeling with two primary
objectives. First, regression modeling can be used to approximately quantify a relationship between dependent and independent variables. Second, regression models
can be used to predict or estimate future trends and outcomes in machine learning
[Maulud and Abdulazeez, 2020]. Regression analysis comprises different types of regression models, and the choice of the model depends on the nature of the data,
research questions, and assumptions.
Linear regression is often further characterized as either Simple Linear Regression (SLR) or Multiple Linear Regression (MLR). SLR involves a single independent
variable and a single dependent variable. It explains the linear relationship between
the independent variable and the dependent variable. MLR involves multiple independent variables and a single dependent variable. It explains the linear relationship
between the multiple independent variables and single dependent variable [Maulud
and Abdulazeez, 2020].

2.3.1

Simple Linear Regression (SLR)

The SLR model assumes a linear relationship between x and y which can be expressed
mathematically as
y = β0 + β1 x + ϵ, ϵ ∼ N (0, σ 2 )
where y represents the dependent variable, x represents the independent variable, β0
represents the intercept term, β1 represents the slope term and ϵ represents the error
10

term. The intercept provides the value of y when x is 0 and the slope indicates the
change of y for a unit change of x. The predicted value of y(ŷ) can be obtained
by βˆ0 + βˆ1 x for a given x, using βˆ0 and βˆ1 , which are the parameter estimates of
the model. The error term ϵ provides the difference between the observed value of y
and the ŷ [Maulud and Abdulazeez, 2020]. Moreover, linear regression modeling has
four assumptions and violating these assumptions can lead to significant issues in the
analysis. Kim [2019] explained them as follows.

1. Linearity: There should be an approximate linear relationship between the independent and dependent variables. It can be visually inspected using a scatter
plot between two variables, and if it follows a linear pattern, the linearity assumption is satisfied.
2. Independence: Observations should be independent of each other, meaning that
the residuals from one observation should not be correlated with residuals from
another observation. We can use a “residuals vs. fitted values” plot to examine
whether this assumption is met. Here, residuals are defined as the differences y
and ŷ, while fitted values represent the ŷ for each observation. If the scatter plot
has a randomly scattered pattern around zero, this suggests that the assumption
is satisfied. Cohen et al. [2003] also have identified that this assumption can be
satisfied if the sampling method is truly random.
3. Homoscedasticity: The variance of the residuals should be constant over fitted
values. This assumption is met if the “residuals vs. fitted values” plot shows a
constant spread around zero, without any pattern.
4. Normality of errors: The residuals are assumed to be normally distributed.
This can be examined using a Normal Q–Q plot of residuals. The assumption is
considered to be met if the residuals lie close to the diagonal line in the Normal
Q–Q plot.

11

2.3.2

Least Squares (LS) Method

The LS method estimates parameters of a model by minimizing the sum of the squared
differences between y and ŷ. In SLR, the coefficients of the regression model β0 and
β1 can be obtained by minimizing the RSS, which is given by
RSS =

n
X

e2i =

i=1

n
X
i=1

n
X
(yi − ŷi ) =
[yi − (β0 + β1 xi )]2
2

i=1

where x represents the independent variable and y represents the dependent variable
for i = 1, 2, ..., n observations [Maulud and Abdulazeez, 2020]. The parameter estimates of β0 and β1 can be determined by taking the partial derivative of the RSS
with respect to the parameters β0 and β1 and setting it to zero as given by
n
∂RSS
∂ X
=
[yi − (β0 + β1 xi )]2 = 0
∂β0
∂β0 i=1

and

n
∂RSS
∂ X
=
[yi − (β0 + β1 xi )]2 = 0.
∂β1
∂β1 i=1

It gives the following formulas
Pn
(x − x̄)(yi − ȳ)
Sxy
Pn i
=
β̂1 = i=1
2
Sxx
i=1 (xi − x̄)
β̂0 = ȳ − β̂1 x̄
to obtain the LS slope coefficient and LS intercept coefficient, respectively. Mamat
et al. [2019] expressed the formulas for Sxx and Sxy as
P
n
n
2
X
X
( ni=1 xi )
2
2
Sxx =
(xi − x̄) =
xi −
n
i=1
i=1
P
P
n
n
X
X
( ni=1 xi ) ( ni=1 yi )
Sxy =
(xi − x̄)(yi − ȳ) =
xi y i −
.
n
i=1
i=1
The best fitting line for the SLR model based on the RSS objective can be represented
as ŷ = βˆ0 + βˆ1 x.
Austin and Hoch [2004] studied the influence of a censored independent variable
on the estimation of regression coefficients. They used the LS method to obtain the
regression parameters and compared the performance of the method with the partial
12

LS method and full–likelihood models. Monte Carlo simulations were conducted to
assess the bias associated with each regression method. The findings revealed that
the naive LS method, which ignored the censoring observations for model fitting
introduced a significant bias in estimating the regression coefficient related to the
variable. They highlighted the importance of considering censoring when estimating
regression coefficients to obtain more accurate and unbiased results.
The study of Haitovsky [1968] explored two different approaches to address the
problem of missing data in regression analysis. The first approach involved applying
the LS method to complete data by discarding all incomplete data points. The second
approach incorporated the covariance between pairs of variables in which values are
available for both variables. Simulations were conducted on eight different regression
datasets, and various deletion patterns were applied to them. The estimates obtained
from the two approaches were compared with the true regression parameters. They
identified that the LS method performed superiorly in the majority of cases. However,
significant bias in the estimates was observed when the proportion of missing data
was high, and the missing data pattern was highly non–random. In addition, the LS
method has been extended by many authors such as Buckley and James [1979], Miller
[1976], Koul et al. [1981] to address the challenge of censoring in data.

2.3.3

Maximum Likelihood (ML) Method

The parameters of the model that maximize the likelihood of observing the given
data is employed in the ML estimation for linear regression [Gustavsson, 2012]. Soch
et al. [2023] assumed normally distributed independent data and defined the likelihood
function of a linear regression model as
n
Y


1
2
√
L(β0 , β1 , σ ) =
exp − 2 (yi − β0 − β1 xi ) .
2σ
2πσ 2
i=1
2

1



(2.1)

The log likelihood function is written as
l(β0 , β1 , σ 2 ) = log L(β0 , β1 , σ 2 )
n
n
= − log(2π) − log(σ 2 )
2
2

1
− 2 RSS.
2σ

(2.2)

13

The derivatives of the log–likelihood function in equation 2.2 with respect to β0 and
β1 are expressed as
n

∂l(β0 , β1 , σ 2 )
1 X
= 2
(yi − β0 − β1 xi ) = 0
∂β0
σ i=1
n

∂l(β̂0 , β1 , σ 2 )
1 X
= 2
(xi yi − β̂0 xi − β1 x2i ) = 0
∂β1
σ i=1
and obtained the following parameter estimates of the SLR model by setting it to
zero [Soch et al., 2023].
Pn
(x − x̄)(yi − ȳ)
Sxy
Pn i
β̂1 = i=1
=
2
Sxx
i=1 (xi − x̄)

(2.3)

β̂0 = ȳ − β̂1 x̄

(2.4)

Gustavsson [2012] performed regression analysis on log–normal heteroscedastic
data using a ML approach and compared its performance with the LS method and
the Weighted LS (WLS) method. They obtained unbiased estimates from all three
methods, but with smaller standard errors for the ML and WLS methods compared
to the LS method. Additionally, the ML method provided the highest power for tests
of regression coefficients. The CIs for the estimated response were calculated, and it
was observed that the ML and WLS methods provided correct CIs.
Hussain et al. [2022] conducted a study on estimating the parameters of a linear
regression truncated model using the ML method. They aimed to estimate the model
parameters and the error variance accurately using this method. The dependent
variable is assumed to follow a truncated t–distribution, with truncation occurring on
both sides. They found that the ML method provides more accurate results compared
to the LS method, based on the MSE.

2.4

Techniques for Handling Missing Data

Various techniques exist to handle missing data, which can be mainly classified into
two categories: traditional approaches and advanced (modern) approaches, as shown
in Figure 2.1. Traditional approaches, such as listwise deletion (LD), pairwise deletion,
14

mean imputation, and regression imputation, are simple and commonly used methods
that handle missing data depending on the missing data mechanism.
LD is a method used to handle missing data by eliminating observations with
missing values and considering only the complete observations. This is also known as
complete case analysis. This approach is well known for providing unbiased parameter
estimates only if the MCAR assumption is satisfied [Kang, 2013]. However, it is not
a reasonable strategy for small sample sizes or if the data is not MCAR. The pairwise
deletion method independently evaluates each feature in a dataset, considering all the
observed values within each feature while disregarding any missing values [Strike et al.,
2001]. The pairwise deletion method may result in less information loss compared to
the LD method due to a reduction in the number of cases lost [Shylaja and Kumar,
2018].
Mean imputation, regression imputation are basic methods of missing data replacement. This approach replaces the unobserved data depending on the observed
data. The mean imputation method involves replacing a missing value with the average of the observed values in the variable. Although the mean imputation method
can preserve the sample size of the original data set, it can introduce bias if the data
are not MCAR and if there is any relationship between the variables [Jadhav et al.,
2019]. The regression imputation method is a better approach for missing imputation
when compared with the mean imputation method, and it fits a statistical model, a
regression model using the observed data and predicts the missing values using the
fitted model [Annas et al., 2021]. This technique provides unbiased estimates for
MAR data with fewer missing observations. Moreover, having few complete observations can lead to overfitting of the model, ultimately providing inaccurate predictions.
These existing approaches are well–known for providing unbiased estimates only if the
MCAR assumption is satisfied under small sample sizes [Annas et al., 2021, Hardt
et al., 2012, Jadhav et al., 2019, Kang, 2013].

15

Figure 2.2: Traditional and modern missing data handling methods [Shylaja and
Kumar, 2018]

Multiple Imputation (MI), Maximum Likelihood (ML) method, and EM algorithm are advanced statistical techniques that effectively handle missing data. The
MI method involves generating multiple complete datasets using techniques such as
Bootstrap and Monte Carlo Markov Chain (MCMC) and replacing the missing values
with plausible values. Each complete dataset is then used for the analysis, and the
results are pooled together to obtain a combined result [Austin et al., 2021]. This
approach effectively addresses the uncertainty surrounding the actual values of the
imputed variables, providing a more comprehensive understanding of the potential
range of outcomes.
ML estimation is a method used to estimate the parameters of a statistical model
by maximizing the likelihood function, and this method provides consistent and
asymptotically efficient estimates similar to the multiple imputation method. Expressions for log–likelihood function in the presence of complete data and missing
data are explained by Allison [2012].

16

2.5

Expectation–Maximization (EM) Algorithm

Incomplete data makes the likelihood function complex, and the presence of missing
data, censored data, and data originating from mixture distributions are some reasons
for the complications in the likelihood function [Ng et al., 2011]. Dempster et al. [1977]
proposed this EM algorithm, and it is identified as another ML–based method that
can be used when the likelihood function becomes complicated. It estimates the
parameters directly by maximizing the complete data log–likelihood function. It does
so by iterating between the E–step and M–step of the algorithm. More information
about the history of the EM algorithm is included in McLachlan et al. [2004]. The
EM algorithm in the presence of incomplete data is defined as follows in Ng et al.
[2011]. θ represents a vector of model parameters, fi (xi , yi ) represents the joint PDF
of x and y for each observation i and complete data log–likelihood function l(θ; x, y)
if x and y are completely observed variables is expressed as
l(θ; x, y) =

n
X

logfi (xi , yi |θ).

i=1

Step 1: Initialization: Start the process by initializing the model parameters θ(0) .
This can be set based on random values, prior information, or parameters obtained
from the observed data. The initial model parameters are crucial as they directly
affect the convergence of the algorithm and the accuracy of the parameter estimates.
Step 2: Expectation step (E–step): Computes the expected value of the complete data log–likelihood function denoted as l(θ; x, y) with respect to the conditional
distribution of the unobserved data conditioned on the observed data and current
parameter estimates θ(k) , where k is the iteration number in the algorithm and expressed as
Q(θ|θ(k) ) = Eθ(k) l(θ|x, y, θ(k) ).
Step 3: Maximization step (M–step): Maximize the expected complete data log–
likelihood to update the parameters computed in the E–step of the algorithm and
expressed mathematically as
θ(k+1) = argmax(Q(θ|θ(k) )).
17

Step 4: Iteration: The E step and M step are iterated repeatedly until the convergence. The convergence in the sequence can be estimated using a stopping rule
such as θ(k+1) − θ(k) < τ where τ is a threshold value which is a smaller number.

2.5.1

Applications of the EM Algorithm in the Presence of
Incomplete Data

Applications of the EM algorithm have been established under different distributions, such as Exponential distributions, Poisson distributions, Gaussian distributions, Gaussian Mixture Models (GMM), and survival analysis models, such as Cox
proportional hazards models. In addition, it has been used to handle different types
of incomplete data, including missing data, truncated data, censored data, and both
truncated and censored data. Although the term EM algorithm was proposed by
Dempster et al. [1977], it has been identified that certain theoretical foundations of
the algorithm were introduced by Orchard and Woodbury [1972] and by Reynolds
and Sundberg [1976]. Dempster et al. [1977] provided a general definition for the
EM algorithm when the complete data belong to the regular and curved exponential
families. They discussed missing data in the multivariate normal, normal linear, and
multinomial models.
Atkinson [1992] proposed a hybrid EM and standard EM algorithm for a finite
mixture of two univariate normal distributions when the data were right censored
with varying degrees. Although it is commonly believed that the EM algorithm is
computationally intensive and inaccurate, Atkinson [1992] found it to provide accurate
and efficient results. He also observed failures in convergence under extreme right
censoring, but the proposed hybrid EM algorithm demonstrated superior performance
at any level of censoring.
Lee and Scott [2012] fitted multivariate Gaussian Mixture Models (GMM) while
handling truncated and censored data together using the EM algorithm. They implemented both the standard EM algorithm and a version of the EM algorithm adapted
for truncation and censoring. This was illustrated in both the simulation studies and

18

the analysis of flow cytometry data. A flow cytometer is a tool used to diagnose
diseases such as malignant lymphoma and acute leukemia. It has limitations in identifying the range of signal strength, which causes truncation and censoring in the data.
The properties of truncated multivariate normal distributions were incorporated into
the computation, and the simulation results showed that the proposed algorithm corrected the bias introduced by truncation and censoring, outperforming the standard
EM algorithm.
Ng et al. [2002] also conducted a study on obtaining parameter estimates for
progressively censored log–normal and Weibull lifetime distributions using the EM
algorithm. They observed that the EM algorithm converged more slowly compared
to the Newton–Raphson method, as highlighted by Little and Rubin [2002]. However,
Ng et al. [2002] emphasized the advantage of the EM algorithm in incorporating
information from censored data in a seamless manner, a feature not observed in the
Newton–Raphson method. Additionally, they highlighted its simple generalization to
other types of censoring. Furthermore, Ferreira and Silva [2017] investigated a similar
study on the Weibull distribution using the EM method, but with right censored data.
Park and Lee [2012] developed a parameter estimation method using the EM and
Monte Carlo Expectation–Maximization (MCEM) algorithms for normal, Laplace,
and Rayleigh distributions in the presence of right censored data. The performance
of the EM method for mixtures of elliptical distributions, assuming MCAR or MAR
assumptions, was investigated by Mouret et al. [2023]. The simulation study demonstrated that the proposed method is robust to noise and non–Gaussian data. In
addition, it has been applied to real–world datasets and concluded that the algorithm
is highly competitive compared to traditional imputation methods.
Hybrid censoring is a combination of both Type I and Type II censoring. Dube
et al. [2011] conducted a study on hybrid censored log–normal distributions and obtained the parameter estimates of the model using the EM method. The asymptotic
CIs were obtained using the FIM, and a real–world dataset was used to illustrate
the applicability of the method. Moreover, they extended the study to address the
problem of determining the optimum censoring scheme, which involves selecting from
all possible hybrid censoring schemes.
19

The parameter estimates of a quasi–Lindley distribution were obtained by Kayid
and Al-Maflehi [2022] for both right censored and uncensored data using the EM
algorithm. They performed a simulation study, which resulted in better estimates
from the EM method compared to the MLEs for both data scenarios. Additionally,
a dataset related to waiting times of customers in banks was used to illustrate the
method. Josse et al. [2018] estimated the mean of a BVN dataset in the presence
of missing data in one of the two variables. This was implemented for a synthetic
dataset under the MAR assumption. Furthermore, they designed the method to
handle missing data in logistic regression models using a Monte Carlo version of the
EM algorithm.

2.6

Univariate Normal Distribution

Burkardt [2023] mathematically defined the general univariate normal distribution
using the standard normal distribution. The x random variable with mean µx and
x
standard deviation σx can be transformed to a standard normal variable as z = x−µ
σx

where z ∼ N (0, 1). Let the random vector x = (x1 , x2 , ..., xn )T follows a normal
distribution with mean µx and standard deviation σx . Then the PDF is


(x − µx )2
1
exp −
fx (x; µx , σx ) = p
2σx2
2πσx2
where x ∈ R, µx ∈ R is a location parameter and σx > 0 is a scale parameter. The
first and second moments of the univariate normal distribution are written as
Z ∞
x · fx (x; µx , σx2 ) dx
E(x) =
−∞

E(x2 ) =

Z ∞

x2 · fx (x; µx , σx2 ) dx

−∞

and the variance, which is a measure of the mean squared difference between a randomly selected observation and the mean of a distribution, is written as
Z ∞
V ar(x)
(x − µx )2 · fx (x; µx , σx2 ) dx.
−∞

The CDF or the probability that a random value is less than or equal to xu is expressed
as

P (x ≤ xu ) = Φ

xu − µ x
σx



1
=p
2πσx2

Z xu



(x − µx )2
exp −
dx.
2σx2
−∞
20

The probability that a random value is greater than xl is expressed as




Z xu
1
xu − µ x
(x − µx )2
=p
dx
P (x ≤ xu ) = Φ
exp −
σx
2σx2
2πσx2 −∞
and the probability that a random value falls between xl and xu is expressed as




xu − µ x
xl − µ x
P (xl < x < xu ) = Φ
−Φ
σx
σx


Z xu
(x − µx )2
1
exp −
dx
=p
2σx2
2πσx2 xl
where Φ(.) is the CDF of the standard normal distribution. The PDF of standard
normal distribution can be denoted as ϕ(.)

2.6.1

Right Censored Univariate Normal Data

Ng et al. [2002] implemented the EM algorithm for right censored data in one variable and obtained the MLEs by incorporating first and second moments of the left
truncated normal distribution, as derived by Balakrishnan and Cohen [1991]. They
showed that the conditional distribution of right censored data follows a truncated
distribution from the left. Let x be a random variable that follows normal distribution
with mean µx and standard deviation σx , where observations are either right censored
or left truncated at the value x = crc
x .
The first moment of a random variable x right censored at crc
x , where αx =

cx −µx
is expressed by Balakrishnan and Cohen [1991] according to the right cenσx

 rc

sored normal distribution as
E(x|x > crc
x , µx , σx ) = µx + σx
and the conditional variance is obtained using
"
2
V ar(x|x > crc
x , µx , σx ) = σx

2.6.2

ϕ(αx )
1 − Φ(αx )

ϕ(αx )
1 + αx
−
1 − Φ(αx )



ϕ(αx )
1 − Φ(αx )

2 #
.

Left Censored Univariate Normal Data

Rueda and Garcia [2018] performed a study on estimating the process parameters with
highly censored data. They have assumed that the variable x is normally distributed
21

with mean µx and variance σx2 with some observations being left censored at the value
clc
x . Also, Steiner and Mackay [2000] determined that the formulas for left censoring
are similar for right censoring. The probability of x less than the censored point clc
x
is expressed as
 lc

cx − µ x
=P
=Φ
.
σx
 lc

cx −µx
The conditional expected value for left censored x where γx =
is expressed as
σx
P (x < clc
x) = P



clc − µx
x − µx
< x
σx
σx





clc − µx
Z≤ x
σx

E(x|x < clc
x ) = µx − σ x



ϕ(γx )
.
Φ(γx )

The conditional variance of left censored x is mostly similar to conditional variance
of right censored x and can be expressed as
"
2
V ar(x|x < clc
x , µx , σx ) = σx

ϕ(γx )
1 − γx
−
Φ(γx )



ϕ(γx )
Φ(γx )

2 #
.

The conditional expectation of x2 can be derived using the conditional expectation
and conditional variance of x.

2.6.3

Interval Censored Univariate Normal Data

Canavire-Bacarreza et al. [2023] modeled the probability that an observation falls
within a range of two values. Let x be a random vector that is normally distributed
with mean µx and standard deviation σx , with some observations recorded within the
interval [lxic , uic
x ]. The probability of an observation in x that falls within the range
[lxic , uic
x ] is expressed as
 ic

lx − µx
x − µx
uic
x − µx
≤
≤
σx
σx
σx
 ic

 ic

u x − µx
lx − µx
=Φ
−Φ
σx
σx

P (lxic ≤ x ≤ uic
x) = P

where Φ(.) is the CDF. The conditional mean of interval censored x where ηx =

 ic

 ic
lx −µx
ux −µx
and
δ
=
is expressed as
x
σx
σx
E(x|lxic < x < uic
x ) = µx − σx

ϕ(δx ) − ϕ(ηx )
.
Φ(δx ) − Φ(ηx )
22

Similarly, the conditional variance of interval censored x is expressed as
"

2 !#
δx ϕ(δx ) − ηx ϕ(ηx )
ϕ(δx ) − ϕ(ηx )
2
V ar(x|lxic < x < uic
−
.
x ) = σx 1 −
Φ(δx ) − Φ(ηx )
Φ(δx ) − Φ(ηx )
These equations can be used to obtain the second moment of the interval censored
observations.

2.7

Bivariate Normal (BVN) Distribution

The BVN distribution describes the joint distribution of two correlated normal random
variables. Let (x, y) be a pair of random variables following a BVN distribution with
means µx , µy ∈ R, standard deviations σx , σy > 0, and correlation ρ ∈ [−1, 1]
[Roussas, 2003]. The joint PDF is given by


1
1
(x − µx )2
p
fX,Y (x, y) =
exp −
2(1 − ρ2 )
σx2
2πσx σy 1 − ρ2

(x − µx )(y − µy ) (y − µy )2
−2ρ
+
σx σy
σy2
where x, y ∈ R. Roussas [2003] presented following properties of the BVN distribution.
Z ∞Z ∞
fx,y (x, y) dx dy = 1
−∞

−∞

x ∼ N (µx , σx2 ),

and y ∼ N (µy , σy2 )

If x and y follows a BVN distribution with the parameters µx , µy , σx , σy , ρ, the
conditional distribution of y|x = x is also normally distributed and the conditional
mean and conditional variance of y given x can be found in Chacko and Mathew
[2020] and Roussas [2003] as follows.
E(y|x = x) = µy + ρ

σy
(x − µx )
σx

V ar(y|x = x) = σy2 (1 − ρ2 ).
Thus, the distribution of y conditioned on x is
y|(x = x) ∼ N (µy + ρ

σy
(x − µx ), σy2 (1 − ρ2 )).
σx

(2.5)

23

The second moment of the distribution of y conditioned on x can be determined using
the above conditional mean and variance. The conditional distribution of x|y = y can
be obtained using
E(x|y = y) = µx + ρ

σx
(y − µy )
σy

V ar(x|y = y) = σx2 (1 − ρ2 ).
Thus, the distribution of x conditioned on y is
x|(y = y) ∼ N (µx + ρ

σx
(y − µy ), σx2 (1 − ρ2 )).
σy

(2.6)

The second moment of the distribution of x conditioned on y can be determined using
the conditional mean and variance. Roussas [2003] presented the joint PDF using the
marginal density of x and conditional density of y|x as
2

(y−bx)
(x−µx )2
−
1
1
−
2
fx,y (x, y) = p
e 2σx2 · q
e 2(σy (1−ρ2 ))
2
2
2
2πσx
2π(σy − ρ )

−

y
where b = µy + ρσ
(x − µx ) = fx (x) · fy|x (y|x) and fx (x) = √ 1 2 e
σx

2πσx

(x−µx )2
2
2σx

, fy|x (y|x) =

(y−bx)2

−
2 (1−ρ2 ))
1
2(σy
e
. It is incorporated to derive the formula for covariance as
2π(σy2 −ρ2 )

√

Z ∞Z ∞
E(XY ) =

xyfX,Y (x, y) dx dy

Z ∞
yfY |X (y|x) dy dx
xfX (x)
=
−∞
−∞


Z ∞
σy
=
xfX (x) µy + ρ (x − µx ) dx
σx
−∞
Z ∞
xfX (x) (µy + ρσy (x − µx )) dx
=
Z−∞
∞

−∞

−∞

= µx µy + ρσx σy .
Since E(x) = µx , E(y) = µy , V ar(x) = σx2 , V ar(y) = σy2 , Roussas (2003) have obtained the covariance between x and y by substituting it as
Cov(x, y) = E(xy) − E(x)E(y) = µx µy + ρσx σy − µx µy = ρσx σy .
In addition, Roussas [2003] found the MLEs of the model parameters as follows. Set
θ = (µx , µy , σx , σy , ρ) for the convenience in writing and the likelihood function is

24

written as follows for the BVN distribution of x = (x1 , ..., xn ), y = (y1 , ..., yn ) for
sample size n.
!n
!
n
1
1X
p
L(θ|x, y) =
qi
exp −
2 i=1
2πσx σy 1 − ρ2

2

2 
yi −µy
2ρ(xi −µx )(yi −µy )
xi −µx
1
where qi = 1−ρ2
+
. The log–likelihood function
−
σx
σx σy
σy
is written as
n
n
n
n
n
1X
2
2
2
l(θ|x, y) = − log(2π) − log σx − log σy − log(1 − ρ ) −
qi .
2
2
2
2
2 i=1

(2.7)

The partial derivatives of the l(θ; x, y) with respect to the the parameters µx , µy , σx , σy , ρ
are written as
∂l(θ|x, y)
n
nρ
= 2
(x̄ − µx ) −
(ȳ − µy ),
2
∂µx
σx (1 − ρ )
σx σy (1 − ρ2 )
∂l(θ|x, y)
n
nρ
= 2
(ȳ − µy ) −
(x̄ − µx ),
2
∂µy
σy (1 − ρ )
σx σy (1 − ρ2 )
Pn
P
2
ρ ni=1 (xi − µx )(yi − µy )
∂l(θ|x, y)
n
i=1 (xi − µx )
=− 2 +
−
,
∂σx2
2σx
2σx4 (1 − ρ2 )
2σx3 σy (1 − ρ2 )
Pn
P
2
ρ ni=1 (xi − µx )(yi − µy )
n
∂l(θ|x, y)
i=1 (yi − µy )
=− 2 +
−
,
∂σy2
2σy
2σy4 (1 − ρ2 )
2σx σy3 (1 − ρ2 )
P
P
ρ ni=1 (xi − µx )2 ρ ni=1 (yi − µy )2
∂l(θ|x, y)
nρ
=
−
−
∂ρ
1 − ρ2
σx2 (1 − ρ2 )
σy2 (1 − ρ2 )
P
(1 + ρ)2 ni=1 (xi − µx )(yi − µy )
.
+
σx σy (1 − ρ2 )2
The formulas for ML estimates for µx and µy can be obtained by setting ∂l(θ|x,y)
=
∂µx
∂l(θ|x,y)
= 0 as
∂µy
n
P

µ̂x = x̄ =

n
P

xi

i=1

n

and µ̂y = ȳ =

yi

i=1

n

.

(2.8)

The ML estimates for σx , σy , and ρ can be obtained by setting ∂l(θ|x,y)
= ∂l(θ|x,y)
=
∂σ 2
∂σ 2
x

y

∂l(θ|x,y)
= 0 and substituting the expressions obtained for µx and µy .
∂ρ

v
 n 2
uP
P
v
u n 2
u n
x
u
i
X
u1
 i=1 xi 
u i=1
2
t


σ̂x =
(xi − x̄) = u
t n − n 
n

(2.9)

i=1

v
 n 2
uP
P
v
u n 2
u n
y
u
i
u1 X
 i=1 yi 
u i=1
2
t

u
σ̂y =
(yi − ȳ) = t
−
 n 
n i=1
n

(2.10)

25

n
P

σ̂xy
ρ̂ =
σ̂x σ̂y

where σ̂xy =

n
P

xi y i

i=1

n

−

n
P

xi

i=1

n

·

yi

i=1

n

(2.11)

Although closed–form solutions for the ML estimates exist, estimating them is not
feasible due to their dependence on x and y, which contain missing or censored observations in the data. To overcome this problem, we are seeking the EM based
procedure.

26

Chapter 3
Methodology
This chapter details the design of the EM algorithm to fit an SLR model for BVN data
under various incomplete data scenarios. The first section offers a general overview
for any type of incomplete data, while the second section focuses specifically on the
missing data scenario.

3.1

EM Algorithm for the BVN Distribution

The log–likelihood function provided in 2.7 based on the complete data (x, y) can be
rewritten as
n
n
n
n
log(2π) − log σx2 − log σy2 − log(1 − ρ2 )
2
2
2
 2n
n
P 2
P
2
 i=1 xi − 2µx i=1 xi + µx
1

−
2(1 − ρ)2 ) 
σ2

l(θ; x, y) = −

x

2ρ(

n
P

x i y i − µy

i=1

−

n
P

xi − µ x

i=1

σx σy

2
2
yi − 2µy
yi + µy 
i=1
i=1
.
+

σ2
n
P

n
P

y

27

n
P
i=1

y i + µx µy

(3.1)

Pn
The sufficient statistics of the log–likelihood function in equation 3.1 are
i=1 xi ,
Pn 2 Pn
Pn 2
Pn
i=1 xi ,
i=1 yi ,
i=1 yi , and
i=1 xi yi . However, it is challenging to estimate the
parameter estimates of the model due to the presence of incomplete data in x and
y. In particular, some x or y values are observed while some are missing or censored.
Hence, the EM algorithm is incorporated to determine the optimal parameters by
maximizing E(l(θ; x, y)). It achieves this by replacing the missing values with their
respective expected values.
The E–step of the algorithm compute conditional expected values for the incomplete observations based on the specific type of incompleteness present in the data.
Specifically, E(x), E(x2 ) E(y), E(y 2 ), E(xy) are computed considering the distributional assumptions, missing nature, and current parameter estimates. Suppose
we have a BVN dataset with (xi , yi ) for (i = 1, 2, .., n) , where the data is categoinc
o
) to (xop , ypinc ) have
) are fully observed; (xom+1 , ym+1
rized as follows: (xo1 , y1o ) to (xom , ym
inc o
o
observed x and incomplete y; (xinc
p+1 , yp+1 ) to (xq , yq ) have incomplete x and obinc
inc inc
served y; and (xinc
q+1 , yq+1 ) to (xn , yn ) have incomplete x and incomplete y data,

where 0 < m < p < q < n, “o” denotes observed data and “inc” denotes incomplete data, which can be either missing or censored. The initial parameters
(0)

(0)

(0)

(0)

θ(0) = (µx , µy , σx , σy , ρ(0) ) are computed in the first step by ignoring the incomplete cases in the dataset.
Index

1

···

m

m+1

···

p

p+1

···

q

q+1

···

n

x

xo1

···

xom

xom+1

···

xop

xinc
p+1

···

xinc
q

xinc
q+1

···

xinc
n

y

y1o

···

o
ym

inc
ym+1

···

ypinc

ypo

···

yqo

inc
yq+1

···

yninc

Table 3.1: Data structure for variables x and y

Therefore, the E–step at the k th iteration compute
(k)

s1 = Eθ(k)

n
X

!
inc
xi xoi , yio , xinc
i , yi

(3.2)

i=1
(k)

s2 = Eθ(k)

n
X

!
inc
yi xoi , yio , xinc
i , yi

(3.3)

i=1

s21 (k) = Eθ(k)

n
X

!
inc
x2i xoi , yio , xinc
i , yi

(3.4)

i=1

28

n
X

s22 (k) = Eθ(k)

!
inc
yi2 xoi , yio , xinc
i , yi

(3.5)

i=1
(k)

s12 = Eθ(k)

n
X

!
inc
xi yi xoi , yio , xinc
.
i , yi

(3.6)

i=1
(k)

(k)

Here s1 denotes the expected value of sum of x statistic, s2 denotes the expected
value of sum of y statistic, s21 (k) denotes the expected value of sum of x2 statistic,
(k)

s22 (k) denotes the expected value of sum of y 2 statistic, and s12 denotes the expected
value of the sum of the product of x and y statistics in the k th iteration.
Maximization is the next step of the algorithm and the sufficient statistics calculated in the E–step, equations 3.2–3.6 are maximised. The parameters are then
updated accordingly. The formulas for MLEs of complete data are obtained from
formulas 2.8–2.11 in section 2.7.
n
P

µ̂x =

n
P

xi

i=1

yi

i=1

, µ̂y =
,
n
n
v
v
n
n
uP
uP
u x2
u y2
t
t
i
i
σ̂x = i=1 − µ̂2x , σ̂y = i=1 − µ̂2y ,
n
n
n
P
xi y i
i=1
σ̂xy =
− µ̂x µ̂y
n

(3.7)

Since the complete data sufficient statistics in equation 3.7 cannot be computed directly due to the unobserved observations from (m + 1, ..., n), the expectations in
equations 3.2–3.6 obtained in the E–step are substituted as the complete data sufficient statistics to compute the new estimates at the next iteration.
(k)
s1
(k+1)
µ̂x
=

n

(3.8)

(k)

s
µ̂(k+1)
= 2
y
n
r
s21 (k)
(k+1)
σ̂x(k+1) =
− (µx )2
n
r
s22 (k)
(k+1) 2
σ̂y(k+1) =
− (µy
)
n

(3.9)
(3.10)
(3.11)

(k)

(k+1)
σ̂xy
=

s12
− µ(k+1)
· µy(k+1)
x
n

(3.12)

29

(k+1)

The estimate for ρ is computed using σ̂xy
3.12

(k+1)

, σ̂x

(k+1)

, and σ̂y

in equations 3.10 -

(k+1)

σ̂xy

ρ̂(k+1) =

(k+1) (k+1)
σ̂y

(3.13)

σ̂x

The E–step computes the sufficient statistics, starting with the initial parameters
θ(0) . M–step substitutes the new parameter values to equations 3.8–3.13 and calculate
the new parameter values, etc. This iterates until convergence and the estimates of
intercept (β0 ) and slope (β1 ) of the SLR model derived in equations 2.3 and 2.4 of
section 2.3.3 is obtained using the maximized parameters.
(k+1)

β̂1 =

3.2

σ̂xy

2(k+1)
σ̂x

and β̂0 = µ̂(k+1)
− β̂1 µ̂(k+1)
x
y

(3.14)

EM Algorithm for Missing Observations

This section explains the computation of the E–step and the M–step when the data
contain missing observations. The computations for right censoring, left censoring,
and interval censoring are detailed in Appendix A.

3.2.1

E–Step

The E–step of the EM algorithm requires estimating the expected values of the randomly missing observations conditioned on the observed data and the current parameter estimates. Datasets with BVN random variables (x, y) are consisted of 4 scenarios,
where “o” denotes observed data and “mis” denotes missing data as shown in Table
mis
3.2. The scenario 4, observations of (xmis
) are ignored from the computation as
i , yi

they does not provide any information about the data.

Scenario 1 : xi observed, yi observed
The E–step involves computing the sufficient statistics expressed in equations 3.2–3.6
for a set of observations in which xi and yi both are observed. It can be expressed
30

Scenario

Index

x

y

1

1, . . . , m

xoi

yio

2

m + 1, . . . , p

xoi

yimis

3

p + 1, . . . , q

xmis
i

yio

4

q + 1, . . . , n

xmis
i

yimis

Table 3.2: Scenarios with observed and missing data for x and y
directly as summations of the observed values as
!
m
m
X
X
o o o
Eθ(k)
xoi
xi xi , yi =
m
X

Eθ(k)

!
yio xoi , yio

=

m
X

i=1

Eθ(k)

m
X

x2i (o) xoi , yio

=

x2i (o)

(3.17)

yi2 (o)

(3.18)

xoi .yio .

(3.19)

m
X

!
yi2 (o) xoi , yio

=

m
X

i=1

Eθ(k)

(3.16)

i=1

m
X

m
X

yio

i=1

!

i=1

Eθ(k)

(3.15)

i=1

i=1

i=1

!
xoi yio xoi , yio

m
X

=

i=1

i=1

Scenario 2 : xi observed, yi missing
The E–step involves computing the sufficient statistics expressed in equations 3.2–3.6
when xi is observed while yi is missing in a set of observations. The expected value
of xi when it is observed is the value itself. However, the expected value of yimis need
to be calculated conditioned on the value of xoi . The computation of the E–step is
expressed as
Eθ(k)

p
X

!
xoi xoi , yimis

=

i=m+1

Eθ(k)

p
X
i=m+1

Eθ(k)

xoi

i=m+1

!
yimis xoi , yimis

p
X

=

p
X

E(yimis |xoi = x, θ(k) )

(3.20)

i=m+1
m
X
i=1

!
x2i (o) xoi , yimis

=

p
X

x2i (o)

i=m+1

31

m
X

Eθ(k)

=

Eθ(k)

E(yi2 (mis) |xoi = x, θ(k) )

(3.21)

i=m+1

i=1
m
X

p
X

!
yi2 (mis) xoi , yimis
!

xoi yimis xoi , yimis

=

p
X

xoi · E(yimis |xoi = x, θ(k) ).

i=m+1

i=1

E(yimis |xoi = x, θ(k) ) and E(yi2 (mis) |xoi = x, θ(k) ) in equations 3.20 and 3.21 can be
computed for each missing observation using the conditional distribution of BVN
distribution. The conditional mean and variance provided by Chacko and Mathew
(2020) (equation 2.5) are incorporated to obtain the expectations of yi conditioned
on xi .

(k)
mis o
(k)
(k)
(k) σy
E(yi |xi = x, θ ) = µy + ρ
(xoi − µ(k)
x )
(k)
σx

(3.22)

Var(yimis | xoi = x, θ(k) ) = σy(k)2 (1 − ρ(k)2 )

(3.23)

The second moment of yimis conditioned on xoi based on current parameter estimates θ(k) can be obtained using equations 3.22 and 3.23 as
E(yi2 (mis) |xoi = x, θ(k) ) = (E(yimis |xoi = x, θ(k) ))2 + V ar(yimis |xoi = x, θ(k) ).

Scenario 3: xi missing and yi observed
This is similar to scenario 2 and the sufficient statistics need to compute when xi is
missing while yi is observed in a set of observations. The expected values of yi when it
is observed is the yi itself. The missing xi is replaced with the expected values of xmis
i
conditioned on the value of yio . The computation of the E–step for the third scenario
includes the formulas
Eθ(k)

q
X

!
o
xmis
xmis
i
i , yi

=

i=p+1

Eθ(k)

q
X

o
(k)
E(xmis
)
i |yi = y, θ

q
X

!
o
yio xmis
i , yi

=

Eθ(k)

!
o
x2i (mis) xmis
i , yi

yio

=

p
X

E(x2i (mis) |yio = y, θ(k) )

(3.25)

i=p+1

i=p+1

Eθ(k)

q
X
i=p+1

i=p+1
q
X

(3.24)

i=p+1

q
X
i=p+1

!
o
yi2 (o) xmis
i , yi

=

q
X

yi2 (o)

i=p+1

32

q
X

Eθ(k)

q
X

!
o mis o
xmis
i yi xi , yi

=

o
(k)
yio · E(xmis
).
i |yi = y, θ

i=p+1

i=p+1

o
(k)
E(xmis
) and E(x2i (mis) |yio = y, θ(k) ) in equations 3.24 and 3.25 can be
i |yi = y, θ

computed for each missing observation using the conditional distribution of BVN
distribution. The conditional mean and variance provided by Chacko and Mathew
[2020] equation 2.6 are incorporated to obtain the expectations of xmis
conditioned
i
on yio .

(k)
(k) σx
(k)
(k)
mis o
E(xi |yi = y, θ ) = µx + ρ
(y o − µ(k)
y )
(k) i
σy

(3.26)

o
(k)
V ar(xmis
) = σx(k) 2(1 − ρ(k)2 )
i |yi = y, θ

(3.27)

conditioned on yio and current parameter estimates θ(k)
The second moment of xmis
i
can be obtained using equations 3.26 and 3.27 as
o
(k) 2
o
(k)
E(x2i (mis) |yio = y, θ(k) ) = (E(xmis
)) + V ar(xmis
).
i |yi = y, θ
i |yi = y, θ

Computing the Total Expectation for Missing Data

The computations in E–step in equations 3.2 - 3.6 are obtained by summing up the
expectations obtained in the 3 scenarios. Thus,
!
p
n
m
X
X
X
(k)
o o
mis mis
o
s1 = Eθ(k)
xi xi , yi , xi , yi
=
xi +
xoi
i=1

i=1
q

i=m+1

X

+

(3.28)

o
(k)
E(xmis
)
i |yi = y, θ

i=p+1

(k)
s2 = Eθ(k)

n
X

!
mis
yi xoi , yio , xmis
i , yi

=

i=1

m
X

p
X

yio +

i=1
q

+

E(yimis |xoi = x, θ(k) )

i=m+1

X

(3.29)

yio

i=p+1

s21 (k) = Eθ(k)

n
X

!
mis
x2i xoi , yio , xmis
i , yi

=

i=1

m
X

x2i (o) +

i=1
p

+

X

p
X

x2i (o)

i=m+1

(3.30)

E(x2i (mis) |yio = y, θ(k) )

i=m+1

33

s22 (k) = Eθ(k)

n
X

!
mis
yi2 xoi , yio , xmis
i , yi

=

i=1

m
X

yi2 (o) +

E(yi2 (mis) |xoi = x, θ(k) )

i=m+1

i=1
q

+

p
X

X

yi2 (o)

i=p+1

(3.31)
(k)

s12 = Eθ(k)

n
X

!
xi yi xoi , yio

=

i=1

m
X

p

xoi · yio +

i=1
q

+

X

X

xoi · E(yimis |xoi = x, θ(k) )

i=m+1

(3.32)

o
(k)
yio · E(xmis
).
i |yi = y, θ

i=p+1

3.2.2

M–step

M–step is the next step of the algorithm and the sufficient statistics calculated in
the E–step equations 3.28–3.32 are maximised. The parameters are then updated
accordingly. The formulas for MLEs of complete data are obtained from formulas
2.8–2.11 in section 2.7.
The E–step and M–step of the EM algorithm for right censored, left censored,
and interval censored observations are presented in the Appendix A.

3.3

Performance Metrics to Evaluate the Regression Estimates

The performance of the regression estimates in the four simulation scenarios are evaluated using several widely used performance metrics. These metrics include bias,
empirical variance, asymptotic variance, Mean Squared Error (MSE), Root Mean
Squared Error (RMSE), Coverage Probability (CP), and Relative Root Mean Squared
Error (RRMSE) computed based on asymptotic values. Each of these metrics provide
insights about the accuracy and reliability of the parameter estimates for the three
methods under consideration.
Bias: measures the difference between the expected (or average) value of the estimates

34

and the true value of the parameter being estimated. It indicates the systematic error
in the model. Mathematically, the bias of βˆ0 can be expressed as
Bias(βˆ0 ) = E(βˆ0 ) − β0 .
The empirical estimate of E(βˆ0 ) is obtained by taking the average of the βˆ0 obtained
from R total number of simulated datasets and it is written as
R
1X
β̂01 + β̂02 + . . . + β̂0R
ˆ
=
β̂0j .
E(β0 ) =
R
R j=1

The βˆ0 is considered unbiased if E(βˆ0 ) = β0 . If this condition is not met, the estimator
has a positive or negative bias, implying that it tends to overestimate or underestimate
the true parameter on average. Therefore, a low bias is preferable as it signifies that
the model’s predictions, on average, are closer to the true parameter value.
Empirical Variance: is a statistical measure used to quantify the spread of the
estimates around their average value. It provides valuable information about the
variability or uncertainty of the estimates within the simulation, indicating how much
individual estimates deviate from the average of the estimates. It can be written
mathematically as
R

Var(βˆ0 ) =

1X
(β̂0j − E(βˆ0 ))2 .
R j=1

The low empirical variances indicate more precise and reliable estimates with minimum variability, while high empirical variance suggests greater uncertainty and variability among the estimates, leading to less precise and less reliable conclusions.
Asymptotic Variance: describes the approximation of the variance for large samples (n → ∞) when the central limit theorem is employed. The asymptotic variance
of intercept (βˆ0 ) and slope (βˆ1 ) estimates can be obtained using


R 
X
x̄2j
1
1
2
Var(βˆ0 ) =
σ
+ Pn
2
R j=1 j n
i=1 (xij − x̄j )

(3.33)


R 
σj2
1X
ˆ
Pn
Var(β1 ) =
2
R j=1
i=1 (xij − x̄j )

(3.34)

where n is the number of observations in each dataset, xij is the ith observation of x
in j th dataset, x̄j is average of x observations in j th dataset, and σj2 is the variance of
35

1
error term for j th dataset. σj2 = n−2

Pn

i=1 (yij − ŷij )

2

where yij and ŷij are ith observed

and predicted y in the j th dataset. However, the formulas 3.33 and 3.34 cannot
be directly used to calculate the asymptotic variance of βˆ0 and βˆ1 for the proposed
methodology due to incomplete observations present in the data. The variance of
2
error term for j th dataset of the proposed method (σj,proposed
) is expanded and re–

written by substituting the ŷij by βˆ0 + βˆ1 xij as
n
1 X
(yij − ŷij )2
n − 2 i=1
n
2
1 X
=
yij − (βˆ0 + βˆ1 xij )
n − 2 i=1
n

2
2
1 X 2
=
yij + βˆ0 + βˆ1 x2ij − 2βˆ0 yij − 2βˆ1 xij yij + 2βˆ0 βˆ1 xij
n − 2 i=1
" n
n
n
n
X
X
X
2
2X 2
1
2
ˆ
ˆ
ˆ
=
β0 + β1
xij − 2β0
yij
y +
n − 2 i=1 ij i=1
i=1
i=1
#
n
n
X
X
−2βˆ1
xij yij + 2βˆ0 βˆ1
xij .

2
σj,proposed
=

i=1

(3.35)

i=1

The sufficient statistics in the equation 3.35 are substituted with the expected values
of the sufficient statistics to obtain the asymptotic variances of the estimates in the
proposed methodology as
" n
n
n
X
X
2
2X
1
2
2
2
ˆ
ˆ
ˆ
σj,proposed =
E(yij ) + nβ0 + β1
E(xij ) − 2β0
E(yij )
n − 2 i=1
i=1
i=1
#
n
n
X
X
−2β1
E(xij yij ) + 2βˆ0 βˆ1
E(xij ) .
i=1

Similarly, x̄j and

i=1

Pn

2
i=1 (xij − x̄j ) are expanded and re–written by replacing the suf-

ficient statistics using the expected value of the sufficient statistics and obtained the
final formula as
Var(βˆ0 ) =

R
1X





Pn

i=1 E(xij )



2

2
n
σj,proposed
1 + P

Pn
2
n
2
R j=1
n
i=1 [E(xij )]
E(x
)
−
ij
i=1
n

Var(βˆ1 ) =

R
1X

"

2
σj,proposed

#

.
P
R j=1 Pn E(x2 ) − ni=1 [E(xij )]2
ij
i=1
n

The asymptotic variance of parameters for the BVN distribution can be derived by
computing the inverse of the FIM. First, compute the score function U (θ), which is
36

the gradient of the log-likelihood function with respect to θ, given by U (θ) = ∂ℓ(θ)
.
∂θ
Next, compute the Hessian matrix H(θ) by taking the second-order partial derivatives
2

∂ ℓ(θ)
of the log-likelihood function, H(θ) = ∂θ∂θ
T . Then, obtain the FIM by taking the

negative expected value of the Hessian matrix, I(θ) = −E[H(θ)]. Finally, obtain the
asymptotic variance of the parameters by inverting the FIM; the diagonals of I(θ)−1
provide the asymptotic variance of each parameter. The upper diagonal elements of
the FIM for the BVN distribution can be expressed considering its symmetry [Amini
and Ahmadi, 2013] as



1
σx2 (1−ρ2 )



ρ
− σx σy (1−ρ
2)
 2


∂ ℓ(θ)

I(θ) = −E
= n
0

∂θ∂θT


0

0

ρ
− σx σy (1−ρ
2)

0

0

1
σy2 (1−ρ2 )

0

0

0

2−ρ2
σx2 (1−ρ2 )

−ρ2
σx σy (1−ρ2 )

0

−ρ2
σx σy (1−ρ2 )

2−ρ2
σy2 (1−ρ2 )

0

−ρ
σx (1−ρ2 )

−ρ
σy (1−ρ2 )

0








−ρ
.
σx (1−ρ2 ) 

−ρ

σy (1−ρ2 ) 
0

1+ρ2
(1−ρ2 )2

Mean Squared Error (MSE): MSE of an estimator βˆ0 measures the mean squared
difference between the estimator βˆ0 and true parameter β0 . It is a metric used to
assess the accuracy of estimates. The MSE of an estimator can be obtained by taking
the summation of the variance of the estimator and the square of its bias. A lower
MSE indicates a better fit, while a higher MSE indicates a poorer fit.
MSE(βˆ0 ) = E(βˆ0 − β0 ) = Var(βˆ0 ) + Bias2 (βˆ0 )
Root Mean Squared Error (RMSE): RMSE of an estimator is the square root
of the MSE of the estimator and can be defined as
q
ˆ
RM SE(β0 ) = M SE(βˆ0 ).
Confidence Interval (CI): CI is a statistical measure that is used to make inferences
about a population. The narrower the CI, the greater the reliability of the estimator.
The formulas for calculating the CI are expressed as
CI = point estimator ± critical value ×

√
variance of the estimator.

The critical value is a value corresponding to a confidence level from the distribution.
The z–value is used for standard normal distribution. As an example, the lower and
37

upper bound of a CI of βˆ0 for 95% CI can be written as


q
q
ˆ
ˆ
ˆ
ˆ
CI = β0 − 1.96 × V ar(β0 ), β0 + 1.96 × V ar(β0 ) .
Coverage Probability (CP): CP is defined as the probability that a CI will contain
the true value of the parameter of interest. It measures the effectiveness of the CI in
capturing the true parameter value over repeated sampling. Typically, we aim for this
probability to match the nominal confidence level chosen for constructing the interval
of 95% CI.
Relative Root Mean Squared Error (RRMSE): measures the performance of
one model relative to the other model. RRMSE of model A to that of model B can
be defined as
RRM SE(β̂A , β̂B ) =

RM SEβ̂A
RM SEβ̂B

.

If the RRMSE is greater than 1, it implies the superior performance of β̂ of model A
compared to model B. If the RRMSE is closer to 1, it implies a similar performance
between the two estimates.
Akaike Information Criterion (AIC): AIC is a statistical method used to determine the best model from multiple models fitted to a given dataset. It selects the
model that best explains the variance in the dependent variable while using fewer
parameters, thereby avoiding overfitting by penalizing complexity. The likelihood
function for linear regression model under the assumption that the errors are normally distributed with mean 0 and variance σ 2 can be expressed as 2.1. The corresponding log-likelihood function is expressed in 2.2. Given the log likelihood function
ℓ(β0 , β1 , σ 2 ), the AIC can be computed using the formula
AIC = 2k − 2ℓ(β0 , β1 , σ 2 )
where n is the number of observations in the data and k is the number of parameters in
the model. However, when dealing with incomplete data (such as missing or censored
observations), the AIC needs to be adjusted to account for the effective sample size.
For methods that handle incomplete data, such as the naive method which only uses
complete observations, the AIC should be scaled. This scaling is done by multiplying
the AIC by the ratio of the total number of observations n to the number of complete
38

observations m. This adjustment corrects for the reduced effective sample size and
ensures a fair comparison with models using all available data. The adjusted AIC
formula for a subset of data can thus be written as
AICadjusted = 2k − 2ℓ(β0 , β1 , σ 2 ) ×

n
.
m

This adjustment is important because it ensures the actual amount of data used in
the estimation process, thereby providing a more accurate measure of model quality.

39

Chapter 4
Simulation Studies
This chapter covers the design of the simulation study, presents the extensive simulation results, and discusses the outcomes for each type of incompleteness.

4.1

Simulation Design

Let (x, y) be the BVN random variables with means µx and µy , standard deviations
σx and σy , and correlation ρ between x and y variables, respectively. The model is

 
  
x
µx
σx2
ρσx σy
 .
  ∼ N   , 
(4.1)
2
ρσx σy
σy
y
µy
We generated R = 500 simulated datasets from BVN model given in 4.1, by setting
µx = 3, µy = −1, σx = 10, σy = 5 for different correlations ρ = 0.2, 0.4, 0.6, 0.8, and
sample sizes n = 50, 100, and 500. Then, we introduced the four different types of
incomplete data separately to R = 500 simulated datasets considering three total
incomplete proportions p = 0.1, 0.3, and 0.5, which is equally split between x and
y. The possible combinations of datasets for different parameters are shown in Table
4.1 and a total of 12 data scenarios were evaluated for each sample size, resulting
in 36 total combinations across all three sample sizes. More precisely; for the given
parameter values (µx = 3, µy = −1, σx = 10, σy = 5) and for sample size n = 50,
ρ = 0.2, we generated a total of 3 datasets using proportion p = 0.1, 0.3, and 0.5,
40

respectively. And each set of data is simulated R = 500 times. Similarly, for each set
of correlation values and proportion values, we generated a total of 12 datasets for
n = 50, 12 datasets for n = 100, and 12 datasets for n = 500. The simulation is done
using R software and the rmvnorm function from the mvtnorm package in R [R Core
Team, 2022] is used to generate the data.
Sample Size (n)

Correlation (ρ)

Proportion (p)

50

0.2

0.1, 0.3, 0.5

0.4

0.1, 0.3, 0.5

0.6

0.1, 0.3, 0.5

0.8

0.1, 0.3, 0.5

0.2

0.1, 0.3, 0.5

0.4

0.1, 0.3, 0.5

0.6

0.1, 0.3, 0.5

0.8

0.1, 0.3, 0.5

0.2

0.1, 0.3, 0.5

0.4

0.1, 0.3, 0.5

0.6

0.1, 0.3, 0.5

0.8

0.1, 0.3, 0.5

100

500

Table 4.1: Simulation details for different sample sizes and correlations for a single
proportion of x and y.
Missingness is introduced to the datasets by considering a missing proportion p.
For a dataset with n observations, the missing proportion is evenly split between x
and y, resulting in nmissing = 0.5p × n, where nmissing is the number of missing values
for each variable. A random set of nmissing indices is generated from a sequence of 1
to n to determine which observations will be missing.
The left censored observations in x and y are introduced by determining the
quantiles corresponding to 0.5p. This process creates thresholds to identify the values
lc
below these percentiles. Specifically, the threshold clc
x for x, where P (x ≤ cx ) =
−1
0.5p, is calculated using the inverse CDF of x, i.e., clc
x = Φ (0.5p). Values below

this threshold are replaced with clc
x to indicate they are left censored. Similarly, left
41

censored observations are introduced for y using a threshold clc
y calculated in the same
manner. The quantile function in R is used to obtain the threshold values.
Right censored data is introduced in a similar way. The right censored observations in x and y are determined using the quantile method. The threshold crc
x for x,
rc
−1
where P (x ≥ crc
x ) = 0.5p, is calculated using the inverse CDF of x; cx = Φ (1−0.5p).

Values above this threshold are replaced with crc
x to indicate they are right censored.
Similarly, right censored observations are introduced for y using a threshold of crc
y
calculated in the same manner.
The process for introducing interval censored data in x and y follows a similar
methodology. Initially, random probabilities are generated to determine the points at
which the lower bound of interval censoring occurs in x and y. These probabilities are
constrained such that their sum, along with the desired censoring percentage for x and
y, each denoted as 0.5p, does not exceed 1. The lower bounds for interval censored
observations are then calculated using quantiles based on these random probabilities.
Subsequently, upper bounds for interval censoring are computed using quantiles based
on the sum of random probabilities and the desired censoring percentages of 0.5p for
both x and y.
The proposed model is compared with a naive model and a benchmark model.
First, a SLR model is fitted to the complete dataset using the CLS method, which
is the benchmark model. Then, the LS estimates of a naive model are obtained by
fitting a SLR model for a subset of complete data, considering only the fully observed
data. The parameter estimates of the EM based model, CLS model, and naive model
are compared to assess the performance of the proposed approach. In addition, a
comprehensive study is conducted to determine the influence and behaviour of sample size, incomplete data proportion, and correlation on the estimation process. The
parameter estimates of the BVN distribution µx , µy , σx , σy , ρ are estimated and presented in Appendix B.

42

4.2

Simulation Results

This section presents the simulation results for four different types of incomplete data:
missing, right censoring, left censoring, and interval censoring.

4.2.1

Results in the Presence of Missing Observations

The true intercept β0 and slope β1 are computed using the true parameters of the
BVN model that is used to generate the data, using the formulas
β0 = µy − β1 µx and β1 = ρ

σy
.
σx

Figure 4.1 illustrate the boxplots of parameter estimates for a SLR model with n =
100, ρ = 0.2 for the CLS models, the naive models, and the proposed models. The
averages of the estimates from all three models are approximately centered around
the true values of the intercept (-1.3) and slope (0.1), as indicated by the horizontal
red dashed lines in these figures. As the proportion of missing data increases, the
variability of the estimates also increases for all models. The boxplots displaying
parameter estimates for the BVN distribution for this combination are included in
Appendix A.

Figure 4.1: Boxplots for estimates of intercept (βˆ0 ) and slope (βˆ1 ), n = 100 and
ρ = 0.2 over different missing proportions of 0.1, 0.3, and 0.5
43

The asymptotic variances of the intercept (βˆ0 ) and slope (βˆ1 ) estimates for the
CLS, naive, and proposed models are depicted in Figure 4.2 (The red line is always
positioned under the blue line whenever it is obscured). These variances are evaluated
across varying missing proportions and for the three different sample sizes. The
results indicate that the proposed estimates consistently exhibit lower and more stable
asymptotic variances compared to the naive estimates; particularly as the proportion
of missing data increases. For smaller sample sizes n = 50 and n = 100, the naive
approach shows a significant increase in variance for the intercept and slope estimates,
at higher missing levels. In contrast, the CLS and proposed estimates maintain low
variances at these sample sizes. As the sample size increases to n = 500, the variances
of all three models decrease, with the proposed estimates consistently showing the
lowest variances, as expected. These findings demonstrate the superior stability and
reliability of the proposed methodology in regression modeling while handling missing
data, particularly in smaller sample scenarios.

Figure 4.2: Line charts for asymptotic variance of intercept (βˆ0 ) and slope (βˆ1 ) when
ρ = 0.2 over different missing proportions of 0.1, 0.3, and 0.5 and sample sizes 50,
100, and 500

Table 4.2 and 4.3 provides computations of bias, RMSE, 95% CP, and RRMSE
for βˆ0 and βˆ1 across different sample sizes and missing proportions. The intercept estimates exhibit significantly higher bias and RMSE in naive models compared to the
proposed models, irrespective of sample size and missing proportion. Moreover, both

44

estimates show high CI coverage rates ranging from 90% to 97% for the intercept.
However, the naive estimates demonstrate a higher CP, attributed to its higher variance. The relative RMSE values for the CLS estimates and the proposed estimates lie
around 1 or less. The relative RMSE between the naive estimates and the proposed
estimates exceeds 1, indicating the superior performance of the proposed models over
the naive models.
Table 4.2: Bias, asymptotic RMSE, 95% CP, and relative RMSE of intercept (βˆ0 ) for
ρ = 0.2 over sample sizes 50, 100, and 500 and missing proportions 0.1, 0.3 and 0.5

n

50

100

500

p

Bias

RMSE

CP

RRMSE

CLS

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

0.002

-0.019

-0.005

0.738

0.771

0.740

0.97

0.96

0.96

1.00

1.04

0.3

0.002

-0.013

-0.011

0.738

0.887

0.732

0.97

0.94

0.94

1.01

1.21

0.5

0.002

-0.019

-0.023

0.738

0.982

0.717

0.97

0.95

0.93

1.03

1.37

0.1

0.014

0.022

0.015

0.517

0.544

0.517

0.97

0.96

0.96

1.00

1.05

0.3

0.014

0.021

0.014

0.517

0.609

0.513

0.97

0.97

0.95

1.01

1.19

0.5

0.014

0.034

0.027

0.517

0.692

0.504

0.97

0.95

0.89

1.02

1.37

0.1

-0.002

0.002

-0.001

0.229

0.241

0.229

0.94

0.96

0.94

1.00

1.05

0.3

-0.002

0.001

0.000

0.229

0.269

0.227

0.94

0.95

0.91

1.01

1.19

0.5

-0.002 -0.009

-0.006

0.229

0.306

0.224

0.94

0.95

0.92

1.02

1.37

Table 4.3: Bias, asymptotic RMSE, 95% CP, and relative RMSE of slope (βˆ1 ) for
ρ = 0.2 over sample sizes 50, 100, and 500 and missing proportions 0.1, 0.3 and 0.5

n

50

100

500

p

Bias

RMSE

CP

RRMSE

CLS

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

0.000

0.002

0.002

0.072

0.076

0.074

0.94

0.95

0.95

0.98

1.03

0.3

0.000

0.003

0.003

0.072

0.088

0.078

0.94

0.95

0.91

0.93

1.13

0.5

0.000

0.002

0.002

0.072

0.098

0.079

0.94

0.94

0.88

0.91

1.23

0.1

-0.001 -0.003

-0.003

0.050

0.053

0.051

0.94

0.95

0.94

0.98

1.04

0.3

-0.001 -0.002

-0.001

0.050

0.059

0.054

0.94

0.94

0.91

0.93

1.10

0.5

-0.001 -0.004

-0.003

0.050

0.068

0.056

0.94

0.95

0.88

0.90

1.22

0.1

0.000

0.001

0.000

0.022

0.022

0.022

0.96

0.96

0.95

1.00

1.00

0.3

0.000

0.001

0.000

0.022

0.027

0.022

0.96

0.94

0.91

1.00

1.18

0.5

0.000

0.000

0.000

0.022

0.030

0.025

0.96

0.93

0.90

0.91

1.22

45

The bias of the slope estimate is approximately similar in both naive and proposed
models. In contrast, the RMSE of naive estimates is higher compared to the proposed
estimates due to its higher variance. Similar to the coverage probabilities obtained
for intercept, the 95% CP of naive slope estimates is higher than the slope estimates
of proposed models due to the higher variance. However, the RRMSE of naive and
proposed estimates indicates values greater than 1, which shows that the proposed
estimates outperform the naive estimates.

4.2.2

Results in the Presence of Right Censored Observations

Figure 4.3 illustrate the boxplots of parameter estimates for SLR models with n =
100, ρ = 0.2 for the CLS models, the naive models, and the proposed models in the
presence of right censored observations in data. The proposed models show consistent estimates for intercept with less variability around the true value of βˆ0 = −1.3
(indicated by the red dashed line). In contrast, naive models show a significant deviation from the actual value, especially with higher proportions of right censored
data. This indicates a high level of bias and variability, and as the proportion of right
censored data increases, the variability of the estimates also increases for all three
models. However, this increase in variability is more pronounced for the naive models
compared to the proposed models.
The slope (βˆ1 ) estimates obtained from the naive models lie closer to the actual
value than the intercept estimate, but gradually deviate from the actual value with
increased right censoring proportion. In addition, the variability of the slope estimates
obtained from the naive approach is increasing significantly with extreme outliers for
heavy censoring. The boxplots displaying parameter estimates for the BVN distribution for this combination are included in Appendix B.

46

Figure 4.3: Boxplots for estimates of intercept (βˆ0 ) and slope (βˆ1 ), when n = 100 and
ρ = 0.2 over different right censoring proportions of 0.1, 0.3, and 0.5

The asymptotic variances of the intercept (βˆ0 ) and slope (βˆ1 ) estimates for the
CLS, naive, and proposed models are depicted in Figure 4.4, in the presence of right
censored data. These variances are evaluated across varying proportions of right
censored data and for the three different sample sizes. The findings demonstrate that
our proposed methodology consistently yields lower and more consistent asymptotic
variances for estimating the slope compared to the naive approach, especially as the
proportion of right censored data increases. While the naive models exhibit relatively
closer variances for the intercept when censoring is light (0.1 and 0.3), they show
higher variance under small sample size (50,100) and heavy censoring (0.5).

47

Figure 4.4: Line charts for asymptotic variance of intercept (βˆ0 ) and slope (βˆ1 ) when
ρ = 0.2 over different right censoring proportions of 0.1, 0.3, and 0.5 and sample sizes
50, 100, and 500

Tables 4.4 and 4.5 provide computations of bias, RMSE, 95% CP, and relative
RMSE for βˆ0 and βˆ1 across different sample sizes and right censoring proportions. The
intercept estimates exhibit a significantly higher magnitude of bias and RMSE in naive
models compared to the proposed models, irrespective of sample size and proportion
of right censoring. Moreover, the CI coverage rate of naive estimates has significantly
reduced compared to proposed estimates. The proposed models demonstrate a higher
CP ranging from 93% to 97% for the intercept. The relative RMSE of CLS over
proposed estimates indicates the better performance of proposed estimates, and the
relative RMSE of naive over proposed estimates is greater than 1. This indicates that
the proposed approach outperforms the naive approach.

48

Table 4.4: Bias, asymptotic RMSE, 95% CP, and relative RMSE of intercept (βˆ0 ) for
ρ = 0.2 over sample sizes 50, 100, and 500 and right censoring proportions of 0.1, 0.3,
and 0.5

n

50

100

500

p

Bias

RMSE

CP

RRMSE

CLS

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

0.002

-0.533

0.021

0.738

0.924

0.746

0.97

0.91

0.97

0.99

1.24

0.3

0.002

-1.261

0.030

0.738

1.482

0.747

0.97

0.64

0.96

0.99

1.98

0.5

0.002

-1.967

0.035

0.738

2.125

0.757

0.97

0.29

0.95

0.97

2.81

0.1

0.014

-0.444

0.017

0.517

0.691

0.518

0.97

0.89

0.97

1.00

1.33

0.3

0.014

-1.191

0.018

0.517

1.308

0.517

0.97

0.40

0.97

1.00

2.53

0.5

0.014

-1.871

0.009

0.517

1.950

0.522

0.97

0.08

0.97

0.99

3.74

0.1

-0.002 -0.467

-0.001

0.229

0.522

0.229

0.94

0.48

0.94

1.00

2.28

0.3

-0.002 -1.220

-0.003

0.229

1.243

0.229

0.94

0.00

0.94

1.00

5.44

0.5

-0.002 -1.912

-0.013

0.229

1.927

0.231

0.94

0.00

0.93

0.99

8.34

Table 4.5: Bias, asymptotic RMSE, 95% CP, and relative RMSE of slope (βˆ1 ) for
ρ = 0.2 over sample sizes 50, 100, and 500 and right censoring proportions 0.1, 0.3,
and 0.5

n

50

100

500

p

Bias

RMSE

CP

RRMSE

CLS

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

0.000

-0.021

-0.000

0.072

0.077

0.073

0.94

0.93

0.93

0.99

1.05

0.3

0.000

-0.031

-0.003

0.072

0.093

0.074

0.94

0.91

0.94

0.97

1.26

0.5

0.000

-0.039

-0.006

0.072

0.108

0.076

0.94

0.92

0.95

0.95

1.43

0.1

-0.001 -0.018

-0.001

0.050

0.056

0.050

0.94

0.93

0.93

1.00

1.12

0.3

-0.001 -0.029

-0.003

0.050

0.067

0.051

0.94

0.91

0.93

0.98

1.30

0.5

-0.001 -0.035

-0.006

0.050

0.076

0.052

0.94

0.91

0.95

0.96

1.46

0.1

0.000

-0.017

-0.001

0.022

0.028

0.022

0.96

0.90

0.95

1.00

1.25

0.3

0.000

-0.032

-0.003

0.022

0.041

0.023

0.96

0.78

0.96

1.00

1.84

0.5

0.000

-0.041

-0.005

0.022

0.051

0.023

0.96

0.72

0.95

0.97

2.22

49

4.2.3

Results in the Presence of Left Censored Observations

Figure 4.5 illustrate the boxplots of parameter estimates for SLR models with n =
100, ρ = 0.2 using the CLS models, the naive models, and the proposed models in the
presence of left censored observations in data. The proposed models show consistent
estimates for intercept with less variability around the actual value of βˆ0 = −1.3
(indicated by the red dashed line). In contrast, naive models show a significant
deviation from the actual value, especially with higher proportions of left censored
data, indicating high bias and variability.
The slope (βˆ1 ) estimates obtained from the naive models lie closer to the actual
value compared to the intercept estimate, but gradually deviate from the actual value
with increased left censoring proportion. In addition, the variability of the slope estimates obtained from the naive models increases significantly with extreme outliers
for heavy censoring. The boxplots displaying parameter estimates for the BVN distribution for this combination are included in Appendix C.

Figure 4.5: Boxplots for estimates of intercept (βˆ0 ) and slope (βˆ1 ), n = 100 and
ρ = 0.2 over different left censoring proportions of 0.1, 0.3, and 0.5

The asymptotic variances of the intercept (βˆ0 ) and slope (βˆ1 ) estimates for the
CLS, Naive, and proposed methods are depicted in Figure 4.6, in the presence of
left censored data. These variances were evaluated across varying proportions of left

50

censored data and the three sample sizes. The findings demonstrate that our proposed
models yield lower and more consistent asymptotic variances in estimating slope and
intercept compared to the naive models, especially as the proportion of left censored
data increases. While the naive estimates exhibit relatively closer variances when the
sample size is high (500), they show higher variance under small sample sizes (50,100)
and heavy censoring (0.3, 0.5).

Figure 4.6: Line charts for asymptotic variance of intercept (βˆ0 ) and slope (βˆ1 ) when
ρ = 0.2 over different left censoring proportions of 0.1, 0.3 and 0.5 and sample sizes
50, 100, and 500

Tables 4.6 and 4.7 provide computations of bias, RMSE, 95% CP, and relative
RMSE for βˆ0 and βˆ1 across different sample sizes and left censoring proportions.
The intercept estimates exhibit significantly higher bias and RMSE in naive models
compared to the proposed models, irrespective of sample size and proportion of left
censoring. Moreover, the CI coverage rate of naive estimates has significantly reduced
over the left censoring proportion compared to proposed estimates. The proposed
models demonstrate a higher CP ranging from 93% to 97% for intercept and slope.
The relative RMSE of CLS over proposed estimates indicates the better performance
of proposed estimates, and the relative RMSE of naive over proposed estimates is
greater than 1. This indicates that the proposed estimates outperform the naive
estimates.

51

Table 4.6: Bias, asymptotic RMSE, 95% CP, and relative RMSE of intercept (βˆ0 ) for
ρ = 0.2 over sample sizes 50, 100, and 500 and left censoring proportions of 0.1, 0.3,
and 0.5

n

50

100

500

p

Bias

RMSE

CP

RRMSE

CLS

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

0.002

0.836

-0.008 0.738

1.106

0.743

0.97

0.79

0.97

0.99

1.49

0.3

0.002

1.480

0.012

0.738

1.717

0.749

0.97

0.60

0.96

0.99

2.29

0.5

0.002

2.209

0.030

0.738

2.448

0.754

0.97

0.43

0.96

0.98

3.25

0.1

0.014

0.575

0.018

0.517

0.777

0.517

0.97

0.80

0.97

1.00

1.50

0.3

0.014

1.409

0.043

0.517

1.530

0.522

0.97

0.33

0.96

0.99

2.93

0.5

0.014

2.155

0.065

0.517

2.270

0.528

0.97

0.14

0.95

0.98

4.30

0.1

-0.002

0.560

-0.000 0.229

0.606

0.229

0.94

0.32

0.95

1.00

2.65

0.3

-0.002

1.411

0.017

0.229

1.436

0.232

0.94

0.00

0.94

0.99

6.20

0.5

-0.002

2.167

0.047

0.229

2.189

0.236

0.94

0.00

0.93

0.97

9.27

Table 4.7: Bias, asymptotic RMSE, 95% CP, and relative RMSE of slope (βˆ1 ) for
ρ = 0.2 over sample sizes 50, 100, and 500 and left censoring proportions of 0.1, 0.3,
and 0.5

n

p

Bias

RMSE

CP

RRMSE

CLS

Naive

PM

CLS

Naive

PM

CLS Naive

PM

CLS/PM

Naive/PM

0.1

0.000

-0.024

-0.002

0.0721

0.077

0.072

0.94

0.92

0.93

1.00

1.07

0.3

0.000

-0.032

-0.004

0.072

0.092

0.074

0.94

0.91

0.94

0.98

1.25

0.5

0.000

-0.039

-0.007

0.072

0.107

0.075

0.94

0.91

0.95

0.96

1.42

100 0.1

-0.001

-0.019

-0.002

0.050

0.056

0.050

0.94

0.92

0.94

1.00

1.12

0.3

-0.001

-0.031

-0.005

0.050

0.067

0.051

0.94

0.91

0.93

0.98

1.31

0.5

-0.001

-0.039

-0.008

0.050

0.077

0.053

0.94

0.90

0.93

0.95

1.46

500 0.1

0.000

-0.017

-0.000

0.022

0.028

0.022

0.96

0.89

0.96

1.00

1.24

0.3

0.000

-0.032

-0.003

0.022

0.042

0.023

0.96

0.75

0.95

1.00

1.85

0.5

0.000

-0.042

-0.006

0.022

0.050

0.023

0.96

0.69

0.95

0.97

2.18

50

52

4.2.4

Results in the Presence of Interval Censored Observations

Figure 4.7 illustrate the boxplots of parameter estimates for SLR models with n = 100,
ρ = 0.2 of the CLS models, the naive models, and the proposed models in the presence
of interval censored observations in data. The naive and proposed models show the
average of the estimates centered around the actual value of βˆ0 = −1.3 (indicated by
the red dashed line). However, the naive estimates exhibit a high variability around
the actual value, especially with a higher proportion of interval censoring. In contrast,
the proposed estimates demonstrate consistent and less variability across all scenarios.
The slope (βˆ1 ) estimates obtained from the naive models are closer to the actual value
but show a slight deviation when the censoring proportion is 0.5. Additionally, the
variability of the slope estimates from the naive models increases significantly with
heavy censoring compared to the proposed models. The boxplots displaying parameter
estimates for the BVN distribution for this combination are included in Appendix D.

Figure 4.7: Boxplots for estimates of intercept (βˆ0 ) and slope (βˆ1 ), when n = 100 and
ρ = 0.2 over different interval censoring proportions of 0.1, 0.3, and 0.5

The asymptotic variances of the intercept (βˆ0 ) and slope (βˆ1 ) estimates for the
CLS, naive, and proposed models are depicted in Figure 4.8, in the presence of interval
censored data. These variances were evaluated across varying proportions of interval
censored data and for the three sample sizes. The findings demonstrate that our
53

proposed models yield lower and more consistent asymptotic variances for estimating
the parameters than the naive models, especially as the proportion of interval censored
data increases. While the naive estimates exhibit relatively closer variances when
censoring is light (0.1 and 0.3), they show higher variance under small sample size
(50, 100) and heavy censoring (0.5).

Figure 4.8: Line charts for asymptotic variance of intercept (βˆ0 ) and slope (βˆ1 ) when
ρ = 0.2 over different interval censoring proportions of 0.1, 0.3, and 0.5 and sample
sizes 50, 100, and 500

Tables 4.8 and 4.9 provide computations of bias, RMSE, 95% CP, and relative
RMSE for (βˆ0 ) and (βˆ1 ) across different sample sizes and interval censoring proportions. The estimates exhibit higher bias and RMSE when employing the naive
method than the proposed method, irrespective of sample size and proportion of interval censoring. Moreover, the CI coverage rate of the naive method has reduced over
the censoring proportion compared to the proposed method. The proposed method
demonstrates a higher CP ranging from 93% to 97% for intercept and slope. The relative RMSE of CLS over PM indicates a similar performance between the proposed
method and the CLS method, and the relative RMSE of Naive over PM is greater
than 1, which indicates that the proposed method outperforms the naive method.

54

Table 4.8: Bias, asymptotic RMSE, 95% CP, and relative RMSE of intercept (βˆ0 ) for
ρ = 0.2 over sample sizes 50, 100, and 500 and interval censoring proportions of 0.1,
0.3, and 0.5

n

50

100

500

p

Bias

RMSE

CP

RRMSE

CLS

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

0.002

-0.019

0.002

0.738

0.781

0.738

0.97

0.95

0.97

1.00

1.06

0.3

0.002

-0.066

-0.002

0.738

0.900

0.737

0.97

0.90

0.96

1.00

1.22

0.5

0.002

-0.130

-0.004

0.738

1.065

0.737

0.97

0.85

0.96

1.00

1.44

0.1

0.014

0.020

0.014

0.517

0.546

0.517

0.97

0.95

0.97

1.00

1.06

0.3

0.014

-0.000

0.017

0.517

0.626

0.517

0.97

0.82

0.97

1.00

1.21

0.5

0.014

0.027

0.013

0.517

0.743

0.517

0.97

0.70

0.97

1.00

1.44

0.1

-0.002 -0.009

-0.002

0.229

0.242

0.229

0.94

0.86

0.94

1.00

1.06

0.3

-0.002 -0.030

0.001

0.229

0.280

0.229

0.94

0.58

0.94

1.00

1.22

0.5

-0.002 -0.060

-0.001

0.229

0.334

0.229

0.94

0.40

0.94

1.00

1.46

Table 4.9: Bias, asymptotic RMSE, 95% CP, and relative RMSE of slope (βˆ1 ) for
ρ = 0.2 over sample sizes 50, 100, and 500 and interval censoring proportions of 0.1,
0.3, and 0.5

n

50

100

500

p

Bias

RMSE

CP

RRMSE

CLS

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

0.000

0.003

0.000

0.072

0.076

0.072

0.94

0.93

0.93

1.00

1.05

0.3

0.000

0.007

0.000

0.072

0.086

0.072

0.94

0.93

0.93

1.00

1.19

0.5

0.000

0.016

0.001

0.072

0.099

0.073

0.94

0.93

0.93

0.99

1.36

0.1

-0.001 -0.001

-0.001

0.050

0.053

0.050

0.94

0.94

0.94

1.00

1.06

0.3

-0.001

0.002

-0.001

0.050

0.059

0.050

0.94

0.94

0.93

1.00

1.18

0.5

-0.001

0.010

-0.001

0.050

0.068

0.050

0.94

0.93

0.94

1.00

1.36

0.1

0.000

0.001

0.000

0.022

0.022

0.022

0.96

0.95

0.96

1.00

1.00

0.3

0.000

0.006

0.000

0.022

0.027

0.022

0.96

0.92

0.96

1.00

1.21

0.5

0.000

0.013

0.000

0.022

0.033

0.022

0.96

0.87

0.95

1.00

1.46

55

4.2.5

Summary of the Simulation Results

Extensive simulation studies reveal that the proposed approach outperforms the naive
approach, by providing less biased estimates, reduced standard errors, higher coverage
probability, and RRMSE values greater than 1, regardless of correlation, sample size,
or the proportion of the incomplete data.

56

Chapter 5
Data Applications
This chapter presents the results obtained from applying the proposed approach to
two real data applications: the first for missing observations and the second for leftcensored observations.

5.1

Determining the Effect of Solar Radiation on
Ozone Concentration–An Application to the
Presence of Missing Observations

Ozone is a significant atmospheric air pollutant with dual effects on environmental
health, air quality, and climate dynamics. Ozone acts as a protective layer in the
stratosphere by absorbing the most harmful ultraviolet (UV) radiation released by
the sun, specifically UV–B and UV–C rays. This shielding effect is significant in safeguarding living organisms from various serious health issues, including skin cancers,
DNA damage, cataracts, and other related health problems. However, Ozone acts
as a harmful air pollutant in the troposphere, the lowest layer of the atmosphere.
Ground–level Ozone, primarily formed through solar radiation, presents significant
health hazards. It impacts respiratory health, leading to asthma, diminished lung
capacity, and increased respiratory infections.

57

Numerous researchers have delved into the relationship between Ozone concentration and solar radiation, as evidenced by studies conducted by Jasaitis et al. [2016],
Belan and Sklyadneva [1999], Ghalib et al. [2020], and Rivas and Rojas [2018]. Jasaitis et al. [2016] observed a positive linear correlation between solar radiation and
Ozone concentration in the atmospheric ground layer. Although they noted that the
relationship was not exceptionally strong, they emphasized its statistical significance,
which had a significance level of less than 0.5.
The air quality dataset in the MICE package for R is utilized to assess the regression models fitted via the proposed method and a naive approach. This dataset
comprises 154 daily air quality measurements collected over 154 consecutive days from
May to September 1973 in New York City. The dataset integrates Ozone data from
the New York State Department of Conservation with meteorological data from the
National Weather Service.
The dataset consists of 42 observations, with 37 observations missing in Ozone
concentration and 7 observations missing in solar radiation, which constitutes a 26.5%
missing percentage. The Little’s MCAR test was performed using the naniar package
in R to assess whether the missing data follows the MCAR assumption. The test
yielded a p–value of 0.0014, which is less than the significance level of 0.05, suggesting
the missing data is not MCAR. We assumed the data are MAR, indicating the missing
data is related with observed data. The scatter plot in Figure 5.1 illustrates the
relationship between Ozone concentration (measured in parts per billion) and solar
radiation (measured in Langleys), providing a visual representation of how these two
variables of the data interact.

58

Figure 5.1: Scatter plot of Ozone concentration (ppb) over Solar radiation (Lang)

Both the normal Q–Q plot and histogram are presented in Figures 5.2 and 5.3
to assess whether the distribution of Ozone concentration and solar radiation follows
a normal distribution as it is one of our primary assumptions. It is significant that
the Ozone concentrations follow a non–normal distribution, whereas solar radiation
approximately follows a normal distribution. Due to the non–normality in Ozone
concentrations, a log transformation is applied to approximate normality better. The
histogram and normal Q–Q plot for log–transformed Ozone concentration is illustrated
in figure 5.4.
SLR models are fitted using two distinct approaches: the naive approach, which
excludes observations with missing values, and the EM–based proposed approach,
which imputes missing values to obtain parameter estimates. The effectiveness of the
proposed estimates is assessed by comparing them to the naive estimates.

59

Figure 5.2: Histogram and normal Q–Q plot for solar radiation

Figure 5.3: Histogram and normal Q–Q plot for Ozone concentration

60

Figure 5.4: Histogram and normal Q–Q plot for log–transformed Ozone concentration

Tables 5.1 display parameter estimates for a SLR model obtained through both
the naive (naive model) and proposed (proposed model) approaches. Although the
intercept and slope estimates are approximately similar between the two models,
notable differences emerge in the standard errors of these estimates. Both the standard
errors of the slope and intercept of the proposed model are lower than those of the
naive model. Consequently, the CIs obtained from the proposed model are narrower,
indicating greater certainty in the estimated parameters. For instance, the ratio
0.6608
0.5692

= 1.16 indicates that the intercept interval of the naive model is 1.16 times

larger than that of the proposed model, representing a 16% increase. Similarly, the
0.0032
= 1.14 indicates that the slope interval of the naive model is 1.14 times
ratio 0.0028

larger than that of the proposed model, representing a 14% increase. The slightly
lower AIC in the proposed model (352.4097) compared to naive model (352.4452)
implies a good overall fit of the data. This suggests that the proposed model offers
more precise estimates, enhancing confidence in the model’s predictions and parameter
interpretations.

61

Table 5.1: Comparison of model coefficients, standard errors, Lower Confidence Limit
(LCL) and Upper Confidence Limit (UCL), and interval width of naive model and
proposed model
Model

Parameter

Naive model

Proposed model

5.2

Estimate

SE

95% CI
LCL

UCL

Interval width

β0

2.6152

0.1667

2.2849

2.9456

0.6608

β1

0.0043

0.0008

0.0027

0.0059

0.0032

β0

2.6173

0.1452

2.3327

2.9019

0.5692

β1

0.0043

0.0007

0.0030

0.0057

0.0028

Exploring the Relationship Between Blood and
Feather Lead Concentrations in Black–Crowned
Night Herons (Nycticorax nycticorax)–An Application to the Presence of Left Censored Observations

Recent research has demonstrated an increase in the interest of applying noninvasive
techniques to monitor environmental contaminant exposure and its subsequent effects
on wildlife populations. This focus is particularly relevant for avian species, where
heavy metals such as Lead (Pb), Mercury (Hg), Selenium (Se) and Cadmium (Cd)
accumulate in feathers and tissues, causing significant health problems [Golden et al.,
2003]. Feathers act as airborne particle collectors, providing a valuable method for
measuring a bird’s exposure to heavy metals. By analyzing the concentrations of
metals in feathers, researchers can gain important insights into the level of heavy
metal pollution in the birds’ surrounding environment ([Ellenberg et al., 1985]).
Juvenile birds typically stay in the same area where they were born until they are
mature enough to migrate. The metal concentrations in their feathers can provide
a better picture of the environmental contamination at a specific location. Feathers
62

are advantageous as a monitoring tool because they are easy to collect and store,
can be sampled from the same bird over multiple years, and cause minimal harm
to the bird. This makes feathers an excellent choice for noninvasive monitoring of
heavy metal exposure in avian species. Blood serves as a frequently utilized tissue for
monitoring lead exposure in birds. When a bird consumes lead–contaminated food
or water, the lead swiftly enters its bloodstream. Monitoring lead levels in the blood
clearly indicates recent exposure to lead sources. Feathers grow gradually over time
and incorporate substances like lead in the bloodstream. As feathers develop, they
accumulate lead circulating in the bloodstream during their formation. Thus, feather
lead concentration can indicate recent and historical exposure to lead. Understanding the relationship between lead concentrations in these two tissues can improve the
efficacy of monitoring programs and offer valuable insights into the dynamics of lead
exposure in herons. Numerous studies have investigated lead contamination in birds,
mainly focusing on the relationship between lead concentrations in blood, feathers
and other organs such as liver, kidney, and bone. These studies have generally found
a positive correlation, suggesting that feathers can serve as reliable indicators of lead
exposure in blood. Studies conducted by Golden et al. [2003], Dauwe et al. [2002],
Pain et al. [2019] have investigated the association between lead concentrations in
blood and feathers of different bird species. Golden et al. [2003] observed a positive
correlation between the lead concentrations of feathers and the blood of herons (Nycticorax nycticorax).
The association between lead concentrations in feathers and blood is investigated
using the publicly available PbHerons dataset from the NADA2 package in R. This
dataset originates from a study conducted by Golden et al. [2003], which aimed to
determine the relationship between lead concentrations in feathers and various other
organs, including the liver, kidney, bone, brain, and blood. This analysis focuses on
the lead concentrations in feathers and blood to explore the practical application of the
proposed methodology. Herons, such as the Black-crowned Night Heron (Nycticorax
nycticorax), are top predators in aquatic ecosystems and are particularly susceptible
to the bioaccumulation of lead from contaminated water bodies and prey.
The dataset comprises 27 observations. Among these, 15 observations have left-

63

Figure 5.5: Scatter plot of lead concentration in blood and feather of Herons
censored lead concentration values in blood at 0.06 and 0.07 µg/g, accounting for
55.5% of the dataset. Additionally, there are 2 left censored values for lead concentration in feathers at 0.02 µg/g, representing 7.4% of the total observations. It is
important to note that the two left censored feather lead concentration observations
are also left-censored for blood lead concentrations. The scatter plot showing the
relationship between blood and feather lead concentrations is presented in Figure 5.5.
Table 5.2 presents the parameter estimates for three different models: the naive
model, which ignored censored observations; the alternate model, which considered
complete observations by taking the threshold values in x and y as observed values;
and the EM–based proposed model. A comparison of the intercept and slope coefficients among the three models reveals notable differences. In particular, the standard
errors for the slope and intercept are highest in the naive model, followed by the
alternate model, and lowest in the proposed model.
According to the CIs for the intercept and slope estimates of the naive model, zero
lies within the respective intervals. Therefore, we fail to reject the null hypothesis,
indicating that there is no enough evidence to conclude that βˆ0 and βˆ1 are significantly

64

different from zero at the 95% confidence level. However, zero does not lie within the
respective CIs for the alternate and proposed models. This implies the rejection of the
null hypothesis. Therefore, there is sufficient evidence to conclude that the intercept
and slope estimates of the alternate and proposed models are significantly different
from zero at the 95% confidence level. In addition, the proposed model has narrow
CIs for both β0 and β1 compared to the alternate model. This suggests the better
fit and greater statistical significance of the proposed model for these parameters.
The AIC is slightly lower in the proposed model (91.5285), followed by the alternate
model (91.9588), and highest in the naive model (92.0023). The lower AIC in the
proposed model indicates a better balance between model complexity and goodness
of fit. These findings collectively indicate that the proposed model outperforms both
the naive and alternate models across multiple evaluation criteria. It demonstrates
superior explanatory power, better model fit, and higher predictive accuracy, making
it a more robust choice for analyzing and predicting lead concentration in feathers
using lead concentration in blood compared to the other models evaluated.
Table 5.2: Comparison of model coefficients, standard errors, Lower Confidence Limit
(LCL) and Upper Confidence Limit (UCL), and interval width of naive model, alternate model, and proposed model
Model

Naive model

Alternate model

Proposed model

Parameter

Estimate

SE

95% CI
LCL

UCL

Interval width

β0

-0.5848

0.5337

-1.7739

0.6042

2.3781

β1

6.8052

4.7582

-3.7968

17.4071

21.2039

β0

-1.4455

0.3061

-2.0760

-0.8150

1.2610

β1

-0.1558

4.0153

2.9473

19.4866

16.5393

β0

-0.9809

0.2352

-1.4419

-0.5199

0.9220

β1

8.8309

2.6283

3.6794

13.9824

10.3031

The SLR model expressed as y = −0.9809 + 8.8309x, where y represents the lead
concentration in feather, and x denotes the lead concentration in blood, offers valuable insights into the relationship between these two variables. This equation suggests
that for every unit increase in the lead concentration in blood (x), there is an esti65

mated increase of 8.8309 µg/g in the lead concentration in feathers (y). The intercept
of -0.9809 indicates the estimated lead concentration in feathers when the lead concentration in blood is zero, however it is impossible that for the lead concentration
in blood to be exactly zero due to environmental exposures. To ensure the model
is correctly specified, a further investigation into data accuracy and identification of
outliers is necessary.

5.3

Summary of the Data Applications

Based on the two real data applications, the proposed models provided better model
fitting compared to the naive models in terms of standard errors and widths of the
CIs. This implies the efficiency and superior consistency of the proposed approach
relative to the existing naive approach.

66

Chapter 6
Discussion
In this thesis, we explored the regression parameters in the presence of left censored,
right censored, interval censored, and missing observations based on the EM algorithm. We reviewed previous studies on handling incomplete data across different
distributions, identified existing research gaps, and justified the importance of our
study. We presented the theoretical assumptions and properties and explained how
the EM algorithm is adapted in regression modeling to address various types of censored data and missing values. We conducted simulation studies to evaluate the
performance of the proposed EM–based methodology under various scenarios of censoring and missing data. We presented the design of the simulations, the parameters
used, and the simulation results. Our analysis compared the performance of a naive
approach with the proposed methodology, evaluating key metrics such as asymptotic
variance, asymptotic RMSE, CP, and RRMSE. We applied the proposed methodology
to real–world datasets.
The findings revealed that the proposed method consistently provided more accurate and less biased estimates for regression parameters across all types of censored
and missing data scenarios. This method demonstrated superior consistency and efficiency, which is crucial for reliable inference in practical applications. The asymptotic
variance of the estimates was significantly lower with the proposed method, indicating
greater precision in the parameter estimates. This reduction in variance was observed
consistently across interval censored, left censored, right censored, and missing ob67

servations, highlighting the robustness of the proposed approach. Furthermore, the
proposed approach yielded lower asymptotic RMSE values than the naive approach,
suggesting improved predictive performance and reliability. The reduction in RMSE
was particularly notable in scenarios with high censoring or missing data levels, where
the naive approach often needed help to provide accurate estimates.
Regarding CP, the proposed method achieved consistently higher values, reflecting its effectiveness in providing accurate interval estimates and capturing true uncertainty in the parameter estimates. Additionally, the proposed method exhibited
significantly lower RRMSE values across all types of censored and missing data scenarios, indicating its superior reliability and robustness in dealing with complex data
structures and providing more accurate and stable estimates. These findings have
important implications for statistical modeling in the presence of censored and missing data. The proposed method’s ability to outperform the naive method in terms of
parameter accuracy, precision, and reliability makes it a valuable tool for researchers
and practitioners in various fields, including medical research, environmental studies,
and social sciences.

6.1

Limitations of the Study

Despite the promising simulation and real–data results of this study, it’s crucial to
identify the potential limitations. The high computing power and time can make it
impractical or difficult to use the algorithm effectively in specific real-world situations
where such extensive resources might not be readily available. Moreover, the performance of the algorithm might vary significantly depending on the initial parameters.
Inappropriate initial parameters can cause the algorithm to converge to a local maximum rather than the global maximum. Therefore, selecting the initial parameters
might require running the algorithm multiple times with different initial points, which
demands additional computational time and effort. Another limitation is the unexplored performance of the EM algorithm in high–dimensional settings, which could
restrict its applicability in cases involving large numbers of predictors.

68

6.2

Future Work

Future research should focus on several key areas to address these limitations. Comparing the proposed method with other imputation techniques such as mean imputation and regression imputation would allow to benchmark the performance of the proposed approach with existing techniques. Additionally, applying multiple imputation
to the proposed approach could be valuable. This would involve generating multiple
models from the same dataset by using different initial parameters during the initialization step of the EM algorithm. Additionally, developing robust versions of the EM
algorithm that are less sensitive to violations of model assumptions would increase its
applicability. Extending the proposed methodology to handle high–dimensional data
would also represent a significant advancement.
Moreover, investigations into the placement of censoring thresholds and the width
of intervals for interval censoring should be conducted. Understanding how close these
intervals are to the mean can provide insights into the accuracy and reliability of the
estimates. Applying the proposed methodology to a broader range of real–world applications, including diverse types of data and various fields of study, would actively
engage the community in validating and refining the approach. Finally, creating userfriendly software packages that implement the proposed methodology for regression
modeling would facilitate adoption by practitioners and researchers, making the techniques developed in this study more accessible and widely used.

69

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75

Appendices

Appendix A
EM Algorithm for Different
Censoring Types

A.1

EM Algorithm for Right Censored Observations

This section explains the computation of the E–step and the M–step when the data
contain right censored observations.

A.1.1

E–Step

The E–step of the EM algorithm requires computing the expected complete data
log–likelihood conditioned on the observed data and the current parameter estimates.
Incorporating the censoring mechanism for right censored data is also crucial in this
computation. Let (x, y) be the two BVN random variables, θ be the parameter vecrc
tor, x be right censored at crc
x , and y be right censored at cy . The right censored

observations are assumed to be present in the x and y of the dataset according to 4
scenarios, where “o” denotes observed data and “rc” denotes right censored data as
shown in the Table 3.3.

77

Scenario

Index

x

y

1

1, . . . , m

xoi

yio

2

m + 1, . . . , p

xoi

yirc

3

p + 1, . . . , q

xrc
i

yio

4

q + 1, . . . , n

xrc
i

yirc

Table A.1: Scenarios with observed and right-censored data for x and y

Scenario 1 : xi observed, yi observed
The E–step of the EM algorithm for right censored data, when observations are observed for both xi and yi , is the same as scenario 1 of the missing observations.
Therefore, it can be computed using the equations given in 3.15–3.19.

Scenario 2 : xi observed, yi right censored
The E–step involves computing the sufficient statistics expressed in equations 3.2–3.6
when xi is observed while yi is right censored in a set of observations. The expected
value of xi , when it is observed, is xi itself. However, the expected values of yirc need
to be calculated conditioned on the value of xi , censoring mechanism of yirc < crc
y ,
where crc
y is the threshold and current parameter estimates. The computation of the
E–step is expressed as
p
X

Eθ(k)

!
xoi xoi , yio

=

i=m+1
p
X

Eθ(k)

=

i=m+1

Eθ(k)

p
X

m
X

m
X

!
x2i (o) xoi , yio

Eθ(k)

i=1

(A.1)

yi2 (rc) xoi , yirc

=

p
X

x2i (o)

p
X

(k)
E(yi2 (rc) |xoi = x, yirc > crc
)
y ,θ

(A.2)

i=m+1

!
xoi yirc xoi , yirc

=

i=m+1

!

i=1
m
X

(k)
)
E(yirc |xoi = x, yirc > crc
y ,θ

i=m+1

i=1

Eθ(k)

xoi

i=m+1

!
yirc xoi , yirc

p
X

=

p
X

(k)
xoi · E(yirc |xoi = x, yirc > crc
).
y ,θ

i=m+1

78

(k)
(k)
E(yirc |xoi = x, yirc > crc
) and E(yi2 (rc) |xoi = x, yirc > crc
) in equations A.1 and
y ,θ
y ,θ

A.2 can be computed by considering the conditional distribution of BVN distribution
and right censored normal distribution. The conditional mean and variance when y
is conditioned on x can be obtained using
(k)

(k)

(k)
µy|x = E(yi |xoi = x, θ(k) ) = µ(k)
y +ρ

σy

(k)
σx

(xoi − µ(k)
x )

2(k)

σy|x = Var(yi | xoi = x, θ(k) ) = σy2(k) (1 − ρ2(k) )
(k)

where αy =

(crc
y −µy|x )
(k)
σy|x

(k)

(A.3)
(A.4)

(k)

. µy|x and σy|x are the mean and standard deviation of yirc

given xoi = x at the k th iteration, respectively. The conditional mean and variance
conditioned on both x and y can be obtained by incorporating A.3 and A.4 as


(k)
2(k)
(k)
(k)
E yirc | xoi = x, yirc > crc
,
µ
,
σ
= µy|x + σy|x
y
y|x
y|x

Var



(k)
2 (k)
yirc | xoi = x, yirc > crc
y , µy|x , σy|x



2 (k)
= σy|x


−

ϕ(αy )
1 − Φ(αy )


1 + αy

ϕ(αy )
1 − Φ(αy )

ϕ(αy )
1 − Φ(αy )
2 #


  
2
(k)
(k)
2 (k)
rc
o
rc
rc
2 (k)
E yi2 (rc) | xoi = x, yirc > crc
,
µ
,
σ
=
E
y
|
x
=
x,
y
>
c
,
µ
,
σ
y
y|x
i
i
i
y
y|x
y|x
y|x


(k)
2 (k)
,
σ
.
+ Var yirc | xoi = x, yirc > crc
,
µ
y|x
y
y|x

Scenario 3 : xi right censored, yi observed
This is similar to scenario 2, and sufficient statistics need to be computed when xi
is right censored while yi is observed in a set of observations. The expected values
of yi , when it is observed, is yi itself, and the right censored xi is replaced with the
o
rc
rc
expected values of xrc
i conditioned on the value of yi and xi > cx . The computation

of the E–step for the third scenario includes the formulas
!
q
q
X
X
o
rc
rc (k)
rc rc o
)
E(xrc
Eθ(k)
xi xi , yi =
i |yi = y, xi > cx , θ

(A.5)

i=p+1

i=p+1

Eθ(k)

q
X
i=p+1

!
o
yio xrc
i , yi

=

q
X

yio

i=p+1

79

q
X

Eθ(k)

!
o
x2i (rc) xrc
i , yi

=

q
X

rc (k)
E(x2i (rc) |yio = y, xrc
)
i > cx , θ

Eθ(k)

q
X

!
o
yi2 (o) xrc
i , yi

q
X

!
o rc o
xrc
i yi xi , yi

=

q
X

yi2 (o)

i=p+1

i=p+1

Eθ(k)

(A.6)

i=p+1

i=p+1

=

q
X

rc (k)
rc
o
).
yio · E(xrc
i |yi = y, xi > cx , θ

i=p+1

i=p+1

o
rc
rc (k)
rc (k)
E(xrc
) and E(x2i (rc) |yio = y, xrc
) in equations A.5 and
i |yi = y, xi > cx , θ
i > cx , θ

A.6 can be computed for each right censored observation, similar to scenario 2. The
conditional mean and variance can be obtained using
(k)
(k)
o
(k)
(k)
(k) σx
µx|y = E(xi |yi = y, θ ) = µx + ρ
(y o − µ(k)
y )
(k) i
σy

(A.7)

2 (k)
= V ar(xi |yio = y, θ(k) ) = σx(k) 2 (1 − ρ(k)2 )
σx|y

(A.8)

(k)

where αx =

(crc
x −µx|y )
(k)

σx|y )
th

yio = y at the k

(k)

(k)

. µx|y and σx|y are the mean and standard deviation of xrc
i given

iteration, respectively. The conditional mean and variance given

both x and y can be obtained by incorporating A.7 and A.8.
(k)

(k)

(k)

o
rc
rc
2 (k)
E(xrc
) = µx|y + σx|y
i |yi = y, xi > cx , µx|y , σx|y

ϕ(αx )
1 − Φ(αx )




(k)
rc
o
rc
rc
2 (k)
2 (k)
Var xi | yi = y, xi > cx , µx|y , σx|y
= σx|y
1 + αx

−

(k)
rc
2 (k)
E(x2i (rc) |yio = y, xrc
)=
i > cx , µx|y , σx|y



ϕ(αx )
1 − Φ(αx )

ϕ(αx )
1 − Φ(αx )
2 #

2

(k)
o
rc
rc
2 (k)
E(xrc
)
i |yi = y, xi > cx , µx|y , σx|y
(k)

o
rc
rc
2 (k)
+ V ar(xrc
)
i |yi = y, xi > cx , µx|y , σx|y

Scenario 4 : xi right censored, yi right censored
The expected values of xi and yi cannot be calculated directly due to the right censored
observations in both variables. The expected value of xi 6rc and expected value of yirc
rc
rc
rc
rc
rc
needed to be calculated conditioned on xrc
i > cx and yi > cy in which cx and cy

80

are thresholds of x and y respectively. The computation of the E–step includes the
formulas
n
X

Eθ(k)

!
rc rc
xrc
i xi , yi

i=q+1

Eθ(k)

n
X

Eθ(k)

!
rc
yirc xrc
i , yi

n
X

Eθ(k)

rc rc
rc (k)
E(yirc |xrc
)
i > cx , yi > cy , θ

i=q+1

!
rc
x2i (rc) xrc
i , yi

n
X

=

rc rc
rc (k)
E(x2i (rc) |xrc
)
i > cx , yi > cy , θ

i=q+1

!
rc
yi2 (rc) xrc
i , yi

n
X

=

rc rc
rc (k)
E(yi2 (rc) |xrc
)
i > cx , yi > cy , θ

i=q+1

i=q+1

n
X

n
X

=

i=q+1

Eθ(k)

rc
rc rc
rc (k)
E(xrc
)
i |xi > cx , yi > cy , θ

i=q+1

i=q+1
n
X

n
X

=

!
rc rc rc
xrc
i yi xi , yi

=

i=q+1

n
X

rc
rc rc
rc (k)
E xrc
i | xi > cx , yi > cy , θ




i=q+1



rc rc
rc (k)
· E yirc | xrc
.
i > cx , yi > cy , θ
The first moments of the BVN data with both variables being right censored can be
obtained using the integral formulas
R∞R∞
rc
rc rc
rc
E(xrc
i |xi > cx , yi > cy ) =

R∞R∞

xfx,y (x, y) dy dx
crc
crc
x
y
R
R
=
∞
∞
rc
rc rc
P (xrc
f (x, y) dy dx
i > cx , yi > cy )
crc crc x,y
crc
x

crc
y

xfx,y (x, y) dy dx

x

R∞R∞
rc rc
rc
E(yirc |xrc
i > cx , yi > cy ) =

yfx,y (x, y) dy dx
crc
y
=
rc rc
rc
P (xrc
i > cx , yi > cy )

y

R∞R∞

crc
x

yfx,y (x, y) dy dx
crc crc
R x∞ R y∞
.
f
(x,
y)
dy
dx
x,y
rc
rc
c
c
x

y

Similarly, the second moments of xi and yi can be computed using
R∞R∞ 2
R∞R∞ 2
x
f
(x,
y)
dy
dx
x fx,y (x, y) dy dx
x,y
rc
rc
cx cy
crc
crc
x
y
rc rc
rc
R
R
=
E(x2i (rc) |xrc
∞
∞
i > cx , yi > cy ) =
rc rc
rc
P (xrc
f (x, y) dy dx
i > cx , yi > cy )
crc crc x,y
x

R∞R∞
rc rc
rc
E(yi2 (rc) |xrc
i > cx , yi > cy ) =

y 2 fx,y (x, y) dy dx
crc
y
=
rc rc
rc
P (xrc
i > cx , yi > cy )

crc
x

y

R∞R∞

y 2 fx,y (x, y) dy dx
crc
crc
x
y
R∞R∞
fx,y (x, y) dy dx
crc
crc
x
y

where fx,y (x, y) is the joint PDFof x and y. The adaptIntegrate function in cubature
package of R software [R Core Team, 2022] is used to numerically evaluate the integrals
due to its complexity to obtain a closed–form solution.

81

Computing the Total Expectation for Right Censored Data

The total expectations in eq: 3.2 and 3.6 are obtained by summing up the expected
values obtained from all 4 scenarios.
n
X

(k)

s1 = Eθ(k)

!
rc
xi xoi , yio , xrc
i , yi

m
X

=

xoi +

X

+

o
rc
rc (k)
E(xrc
)
i |yi = y, xi > cx , θ

i=p+1
n
X

+

xoi

i=m+1

i=1
q

i=1

p
X

(A.9)

rc
rc rc
rc (k)
E(xrc
)
i |xi > cx , yi > cy , θ

i=q+1

(k)
s2 = Eθ(k)

n
X

!
rc
yi xoi , yio , xrc
i , yi

=

m
X

yio

i=1
p

i=1

X

+

(k)
E(yirc | xoi = x, yirc > crc
)
y ,θ

(A.10)

i=m+1
q

X

+
+

yio

i=p+1
n
X

rc rc
rc (k)
E(yirc | xrc
)
i > cx , yi > cy , θ

i=q+1

s21 (k) = Eθ(k)

n
X

!
rc
x2i xoi , yio , xrc
i , yi

m
X

=

i=1

i=1
q

X

+

i=p+1
n
X

+

p
X

x2i (o) +

x2i (o)

i=m+1
rc (k)
E(x2i (rc) |yio = y, xrc
)
i > cx , θ

rc rc
rc (k)
E(x2i (rc) |xrc
)
i > cx , yi > cy , θ

i=q+1

(A.11)
s22 (k) = Eθ(k)

n
X

!
rc
yi2 xoi , yio , xrc
i , yi

=

m
X

yi2 (o) +

i=1
q

i=1

+
+

X
i=p+1
n
X

p
X

(k)
)
E(yi2 (rc) |xoi = x, yirc > crc
y ,θ

i=m+1

yi2 (o)
rc rc
rc (k)
E(yi2 (rc) |xrc
)
i > cx , yi > cy , θ

i=q+1

(A.12)

82

(k)
s12 = Eθ(k)

n
X

!
rc
xi yi xoi , yio , xrc
i , yi

=

i=1

m
X

xoi · yio

i=1
p

+

X

(k)
xoi · E(yirc |xoi = x, yirc > crc
)
y ,θ

i=m+1
q

+
+

X
i=p+1
n
X

o
rc
rc (k)
yio · E(xrc
)
i |yi = y, xi > cx , θ

rc (k)
rc rc
rc
)
E(xrc
i |xi > cx , yi > cy , θ

i=q+1
rc rc
rc (k)
· E(yirc |xrc
)
i > cx , yi > cy , θ

(A.13)

A.1.2

M–step

M–step is the next step of the algorithm and the sufficient statistics calculated in
the E–step equations A.9–A.13 are maximised. The parameters are then updated
accordingly. The formulas for MLEs of complete data are obtained from formulas
2.8–2.11 in section 2.7.

A.2

EM Algorithm for Left Censored Observations

This section explains the computation of the E–step and the M–step when the data
contain left censored observations.

A.2.1

E–step

The E–step of the EM algorithm requires computing the expected complete data
log–likelihood conditioned on the observed data and the current parameter estimates.
Incorporating the censoring mechanism for left censored data is also crucial in this
computation. Let (x, y) be the two BVN random variables, θ be the parameter vector,
lc
clc
x be the threshold of x, and cy be the threshold of y. The left censored data is

assumed to be present in the dataset, and “o” denotes the observed data and “lc”
83

denotes the left censored data as shown in Table A.2.
Scenario

Index

x

y

1

1, · · · , m

xoi

yio

2

m + 1, · · · , p

xoi

yilc

3

p + 1, · · · , q

xlc
i

yio

4

q + 1, · · · , n

xlc
i

yilc

Table A.2: Scenarios with observed and left censored data for x and y

Scenario 1 : xi observed, yi observed
The E–step of the EM algorithm for left censored data when observations are observed
for both xi and yi are same as the scenario 1 of the missing observations. Therefore,
it can be computed using the equations given in 3.15–3.19.

Scenario 2 : xi observed, yi left censored
The E–step involves computing the sufficient statistics expressed in equations 3.2–
3.6 when xi is observed while yi is left censored at clc
y in a set of observations. The
expected values of xi , when it is observed, is xi itself. However, the expected values of
yilc must be calculated conditioned on the value of xoi , censoring mechanism of yilc < clc
y,
and current parameter estimates. The computation of the E–step is expressed as
!
p
p
X
X
o o lc
xoi
Eθ(k)
xi xi , yi =
i=m+1

i=m+1

Eθ(k)

p
X

!
yilc xoi , yilc

=

i=m+1

p
X

m
X

!
x2i (o) xoi , yilc

i=1

Eθ(k)

i=1

(A.14)

i=m+1

Eθ(k)
m
X

(k)
)
E(yilc |xoi = x, yilc < clc
y ,θ

!
yi2 (lc) xoi , yilc

=

=

p
X

x2i (o)

i=m+1
p
X

(k)
)
E(yi2 (lc) |xoi = x, yilc < clc
y ,θ

(A.15)

i=m+1

84

Eθ(k)

m
X

p
X

!
xoi yilc xoi , yilc

=

(k)
xoi · E(yilc |xoi = x, yilc < clc
).
y ,θ

i=m+1

i=1

(k)
(k)
E(yilc |xoi = x, yilc < clc
) and E(yi2 (lc) |xoi = x, yilc < clc
) in equations A.14
y ,θ
y ,θ

and A.15 can be computed for each left censored observation using the conditional
distribution of BVN distribution and left censored normal distribution. Suppose γy =
(k)

(clc
y −µy|x )
(k)
σy|x

(k)

(k)

, the conditional mean µy|x and variance σy|x at k th iteration can be obtained

using the equations A.3 and A.4.
(k)

(k) ϕ(γy )

(k)

2 (k)
E(yilc |xoi = x, yilc < clc
) = µy|x − σy|x
y , µy|x , σy|x

Φ(γy )
"

2 #
ϕ(γy )
ϕ(γy )
(k)
2 (k)
2 (k)
lc
lc
lc o
1 − γy
V ar(yi |xi = x, yi < cy , µy|x , σy|x ) = σy|x
−
Φ(γy )
Φ(γy )
2

(k)
(k)
2 (k)
lc o
lc
lc
2 (k)
,
σ
)
E(y
|x
=
x,
y
<
c
,
µ
,
σ
)
=
E(yi2 (lc) |xoi = x, yilc < clc
,
µ
y|x
i
i
i
y
y|x
y
y|x
y|x
(k)

2 (k)
)
+ V ar(yilc |xoi = x, yilc < clc
y , µy|x , σy|x

Scenario 3 : xi left censored, yi observed
This is similar to scenario 2 and the sufficient statistics need to compute when xi is
left censored at clc
x while yi is observed in a set of observations. The expected values
of yi , when it is observed, is yi itself and the left censored xi is replaced with the
o
lc
lc
expected values of xlc
i conditioned on the value of yi and xi < cx . The computation

of the E–step includes the formulas
!
q
q
X
X
lc lc o
o
lc
lc (k)
Eθ(k)
xi xi , yi =
E(xlc
)
i |yi = y, xi < cx , θ
i=p+1
q
X

Eθ(k)

!
o
yio xlc
i , yi

=

q
X

!
o
x2i (lc) xlc
i , yi

=

i=p+1
q
X

q
X

lc (k)
E(x2i (lc) |yio = y, xlc
)
i < cx , θ

!
o
yi2 (o) xlc
i , yi

i=p+1
q

X
i=p+1

yio

(A.17)

i=p+1

Eθ(k)

Eθ(k)

q
X
i=p+1

i=p+1

Eθ(k)

(A.16)

i=p+1

q
X

yi2 (o)

i=p+1
q

!
o lc o
xlc
i yi xi , yi

=

=

X

o
lc
lc (k)
yio · E(xlc
).
i |yi = y, xi < cx , θ

i=p+1

85

o
lc
lc (k)
lc (k)
E(xlc
) and E(x2i (lc) |yio = y, xlc
) in equations A.16 and
i |yi = y, xi < cx , θ
i < cx , θ
(k)

A.17 can be computed similar to scenario 2. Suppose γx =

(clc
x −µx|y )
σx|y (k)

(k)

and µx|y and

(k)

σx|y are the mean and standard deviation of xi , given yio = y at the k th iteration,
respectively. It can be calculated using the equations A.3 and A.4.
(k)

(k) ϕ(γx )

(k)

o
lc
lc
2 (k)
E(xlc
) = µx|y − σx|y
i |yi = y, xi < cx , µx|y , σx|y

"
(k)

2 (k)
2 (k)
lc
lc
o
1 − γx
) = σx|y
V ar(xlc
i |yi = y, xi < cx , µx|y , σx|y

Φ(γx )

ϕ(γx )
−
Φ(γx )



ϕ(γx )
Φ(γx )

2 #

Scenario 4 : xi left censored, yi left censored
The expected values of xi and yi cannot be calculated directly due to the left censored
lc
observations in both variables. The expected value of xlc
i and yi need to be computed
lc
lc
lc
lc
lc
conditionally on xlc
i < cx and yi < cy , where cx and cy are thresholds of x and y,

respectively. The computation of the E–step includes the formulas
!
n
n
X
X
lc lc lc
lc
lc lc
lc (k)
Eθ(k)
xi xi , yi =
E(xlc
)
i |xi < cx , yi < cy , θ
i=q+1

Eθ(k)

n
X

i=q+1

!
lc
yilc xlc
i , yi

=

i=q+1

Eθ(k)

n
X

n
X

lc
x2i (lc) xlc
i , yi

=

Eθ(k)

n
X

lc lc
lc (k)
E(x2i (lc) |xlc
)
i < cx , yi < cy , θ

i=q+1

!
lc
yi2 (lc) xlc
i , yi

=

n
X

lc lc
lc (k)
)
E(yi2 (lc) |xlc
i < cx , yi < cy , θ

i=q+1

i=q+1
n
X

lc lc
lc (k)
E(yilc |xlc
)
i < cx , yi < cy , θ

i=q+1

!

i=q+1

Eθ(k)

n
X

!
lc lc lc
xlc
i yi xi , yi

=

i=q+1

n
X

lc
lc lc
lc (k)
E(xlc
)
i | xi < cx , yi < cy , θ

i=q+1
lc lc
lc (k)
· E(yilc | xlc
).
i < cx , yi < cy , θ

The first moments of the BVN data with both variables being left censored can be
obtained using the integral formulas
R clcx R clcy
lc
lc lc
lc
E(xlc
i |xi < cx , yi < cy ) =

xfx,y (x, y) dy dx
−∞
=
lc
lc
lc
P (xi < clc
x , yi < cy )

−∞

R clcx R clcy
−∞

−∞

xfx,y (x, y) dy dx

R clcx R clcy
−∞

−∞

fx,y (x, y) dy dx
86

R clcx R clcy
lc
lc lc
E(yilc |xlc
i < cx , yi < cy ) =

yfx,y (x, y) dy dx
−∞ −∞
lc lc
lc
P (xlc
i < cx , yi < cy )

R clcx R clcy
−∞
= R−∞
clc R clc
x

yfx,y (x, y) dy dx

.

y

f (x, y) dy dx
−∞ −∞ x,y

lc
Similarly, the second moments of xlc
i and yi can be computed using the equations

R clcx R clcy
lc
lc lc
E(x2i (lc) |xlc
i < cx , yi < cy ) =

x2 fx,y (x, y) dy dx
−∞
=
lc lc
lc
P (xlc
i < cx , yi < cy )

−∞

R clcx R clcy
lc lc
lc
E(yi2 (lc) |xlc
i < cx , yi < cy ) =

y 2 fx,y (x, y) dy dx
−∞
=
lc lc
lc
P (xlc
i < cx , yi < cy )

−∞

R clcx R clcy

x2 fx,y (x, y) dy dx
−∞ −∞
R clcx R clcy
f (x, y) dy dx
−∞ −∞ x,y

R clcx R clcy

y 2 fx,y (x, y) dy dx
−∞ −∞
R clcx R clcy
f (x, y) dy dx
−∞ −∞ x,y

where fx,y (x, y) is the joint PDFof x and y. The adaptIntegrate function of cubature
package in R software [R Core Team, 2022] is used to numerically evaluate the integrals
due to its complexity to obtain a closed–form solution.

Computing the Total Expectation for Left Censored Data

The total expectations in eq: 3.3 and 3.6 are obtained by summing up the expected
values obtained from all 4 scenarios.
!
p
n
m
X
X
X
(k)
o o
lc lc
o
s1 = Eθ(k)
xi xi , yi , xi , yi =
xi +
xoi
i=1

i=1

+
+

i=m+1

q
X
i=p+1
n
X

o
lc
lc (k)
E xlc
i | yi = y, xi < cx , θ




lc
lc lc
lc (k)
E xlc
i | xi < cx , yi < cy , θ




(A.18)

i=q+1

(k)
s2 = Eθ(k)

n
X
i=1

!
lc
yi xoi , yio , xlc
i , yi

=

m
X

yio

i=1

+

p
X

(k)
E yilc | xoi = x, yilc < clc
y ,θ




i=m+1
q

+
+

X
i=p+1
n
X

yio
lc lc
lc (k)
E yilc | xlc
i < cx , yi < cy , θ




(A.19)

i=q+1

87

s21 (k) = Eθ(k)

n
X

!
lc
x2i xoi , yio , xlc
i , yi

=

i=1

m
X

x2i (o)

+

p
X

x2i (o)

i=m+1

i=1
q

X

+
+

lc (k)
E x2i (lc) | yio = y, xlc
i < cx , θ

i=p+1
n
X




lc lc
lc (k)
E x2i (lc) | xlc
i < cx , yi < cy , θ




i=q+1

(A.20)

s22 (k) = Eθ(k)

n
X

!
lc
yi2 xoi , yio , xlc
i , yi

=

i=1

m
X

yi2 (o)

i=1
p
X

+

(k)
E yi2 (lc) | xoi = x, yilc < clc
y ,θ




i=m+1
q

X

+
+

yi2 (o)

i=p+1
n
X

lc lc
lc (k)
E yi2 (lc) | xlc
i < cx , yi < cy , θ




i=q+1

(A.21)

(k)
s12 = Eθ(k)

n
X

!
lc
xi yi xoi , yio , xlc
i , yi

i=1

=

m
X

xoi · yio

i=1

+

p
X

(k)
xoi · E yilc | xoi = x, yilc < clc
y ,θ




i=m+1
q

+
+

X
i=p+1
n
X

o
lc
lc (k)
yio · E xlc
i | yi = y, xi < cx , θ



lc
lc lc
lc (k)
E xlc
i | xi < cx , yi < cy , θ







i=q+1
lc lc
lc (k)
·E yilc | xlc
i < cx , yi < cy , θ



(A.22)

A.2.2

M–step

M–step is the next step of the algorithm and the sufficient statistics calculated in
the E–step equations A.18–A.22 are maximised. The parameters are then updated
88

accordingly. The formulas for MLEs of complete data are obtained from formulas
2.8–2.11 in section 2.7.

A.3

EM Algorithm for Interval Censored Observations

This section explains the computation of the E–step and the M–step when the data
contain interval censored observations.

A.3.1

E–Step

The E–step of the EM algorithm requires computing the expected complete data
log–likelihood conditioned on the observed data and the current parameter estimates.
Incorporating the censoring mechanism for interval censored data is also crucial in
this computation. Let (x, y) be the two BVN random variables, θ be the parameter
ic
ic
vector, and [lxic , uic
x ] and [ly , uy ] are the lower and upper limits of the interval for

variables x and y, respectively. The interval censored data is assumed to be present
in the dataset according to 4 scenarios, “o” denotes observed data and “ic” denotes
interval censored data as shown in Table A.3.
Scenario

Index

x

y

1

1 ···m

xoi

yio

2

m + 1 ···p

xoi

yiic

3

p + 1 ···q

xic
i

yio

4

q + 1 ···n

xic
i

yiic

Table A.3: Scenarios with observed and interval censored data for x and y

89

Scenario 1 : xi observed, yi observed
The E–step of the EM algorithm when observations are observed for both xi and yi
are same as the scenario 1 of the missing observations. Therefore, it can be computed
using the equations given in 3.15–3.19.

Scenario 2 : xi observed, yi interval censored
The E–step involves computing the sufficient statistics expressed in equations 3.2–3.6
when xi is observed while yi is interval censored at [lyic , uic
y ] in a set of observations.
The expected values of xi can be directly obtained from the observed x. However, the
expected values of yiic need to be calculated conditioned on the value of xoi , censoring
mechanism of lyic < yiic < uic
y , and current parameter estimates. The computation of
the E–step is expressed as
p
X

Eθ(k)

!
xoi xoi , yiic

=

i=m+1
p
X

Eθ(k)

yiic xoi , yiic

=

i=m+1

(k)
E(yiic |xoi = x, lyic < yiic < uic
)
y ,θ

m
X

!
x2i (o) xoi , yiic

=

i=1

Eθ(k)

yi2 (ic) xoi , yiic

p
X

x2i (o)

i=m+1
p

!

X

=

i=1

Eθ(k)

(A.23)

i=m+1

Eθ(k)
m
X

xoi

i=m+1

p
X

!

p
X

(k)
E(yi2 (ic) |xoi = x, lyic < yiic < uic
)
y ,θ

(A.24)

i=m+1

m
X

!
xoi yiic xoi , yiic

=

i=1

p
X

(k)
xoi · E(yiic |xoi = x, lyic < yiic < uic
).
y ,θ

i=m+1

(k)
(k)
E(yiic |xoi = x, lyic < yiic < uic
) and E(yi2 (ic) |xoi = x, lyic < yiic < uic
) in equations
y ,θ
y ,θ

A.23 and A.24 can be computed for interval censored observations using the conditional distribution of BVN distribution and interval censored normal distribution.
(k)

(k)

Suppose ηy =

(lyic −µy|x )
(k)
σy|x

and δy =

(uic
y −µy|x )
(k)
σy|x

(k)

(k)

and µy|x and σy|x are the mean and standard

deviation of yi given xi = x at the k th iteration, respectively. It can be calculated
from the equations A.3 and A.4.
(k)

(k)

(k) ϕ(δy ) − ϕ(ηy )

2 (k)
E(yiic |xoi = x, lyic < yiic < uic
) = µy|x − σy|x
y , µy|x , σy|x

Φ(δy )Φ(ηy )
90



δy ϕ(δy ) − ηy ϕ(ηy )
Φ(δy ) − Φ(ηy )
2 

ϕ(δy ) − ϕ(ηy )
−
Φ(δy ) − Φ(ηy )

(k)
2 (k)
yiic | xoi = x, lyic < yiic < uic
y , µy|x , σy|x



2 (k)
= σy|x

(k)
2 (k)
E(yi2 (ic) |xoi = x, lyic < yiic < uic
)=
y , µy|x , σy|x



(k)
2 (k)
E(yiic |xoi = x, lyic < yiic < uic
)
y , µy|x , σy|x

Var



1−

2

(k)

2 (k)
+ V ar(yiic |xoi = x, lyic < yiic < uic
)
y , µy|x , σy|x

Scenario 3 : xi interval censored, yi observed
This is similar to scenario 2 and the sufficient statistics need to be computed when
xi is interval censored at [lxic , uic
x ] while yi is observed in a set of observations. The
expected values of yi can be directly obtained from the observed values in y and the
interval censored xi is replaced with the expected values of xic
i conditioned on the
ic
value of yio and lxic < xic
i < ux . The computation of the E–step for the third scenario

includes the formulas
q
X

Eθ(k)

!
ic o
xic
i xi , yi

=

i=p+1

q
X

o
ic
ic
ic (k)
E(xic
)
i |yi = y, lx < xi < ux , θ

q
X

Eθ(k)

!
o
yio xic
i , yi

=

i=p+1
q
X

Eθ(k)

!
o
x2i (ic) xic
i , yi

=

i=p+1

q
X

yio

i=p+1

p
X

ic (k)
E(x2i (ic) |yio = y, lxic < xic
)
i < ux , θ

(A.26)

i=m+1
q
X

Eθ(k)

!
o
yi2 (o) xic
i , yi

i=p+1

Eθ(k)

(A.25)

i=p+1

q
X

yi2 (o)

i=p+1
q
X

!
o ic o
xic
i yi xi , yi

=

q
X

=

ic
ic
ic (k)
o
).
yio · E(xic
i |yi = y, lx < xi < ux , θ

i=p+1

i=p+1

o
ic
ic
ic (k)
ic (k)
E(xic
) and E(x2i (ic) |yio = y, lxic < xic
) in equations
i |yi = y, lx < xi < ux , θ
i < ux , θ

A.25 and A.26 can be computed for interval censored observations using the conditional distribution of BVN distribution and interval censored normal distribution.
(k)

(k)

Suppose ηx =

ic −µ
(lx
)
x|y
(k)
σx|y )

, δx =

(uic
x −µx|y )
(k)
σx|y )

(k)

(k)

and µx|y and σx|y are the mean and standard

ic
th
deviation of xi given yio = y and lxic < xic
iteration, respectively.
i < ux at the k
(k)

(k)

(k) ϕ(δx ) − ϕ(ηx )

ic
ic
2 (k)
o
ic
) = µx|y − σx|y
E(xic
i |yi = y, lx < xi < ux , µx|y , σx|y

Φ(δx )Φ(ηx )
91



δx ϕ(δx ) − ηx ϕ(ηx )
Φ(δx ) − Φ(ηx )
2 

ϕ(δx ) − ϕ(ηx )
−
Φ(δx ) − Φ(ηx )

 h 
i2
(ic)2
(k)
(k)
ic
2 (k)
ic o
ic
ic
ic
2 (k)
E xi |yio = y, lxic < xic
<
u
,
µ
,
σ
=
E
x
|y
=
y,
l
<
x
<
u
,
µ
,
σ
i
x
x|y
i
i
x
i
x
x|y
x|y
x|y


(k)
o
ic
ic
ic
2 (k)
+ Var xic
|y
=
y,
l
<
x
<
u
,
µ
,
σ
i
i
x
i
x
x|y
x|y
(k)
o
ic
ic
ic
2 (k)
2 (k)
Var(xic
) = σx|y
i |yi = y, lx < xi < ux , µx|y , σx|y

1−

Scenario 4 : xi interval censored, yi interval censored
The expected values of xi and yi cannot be calculated directly due to the interval
censored observations in both variables. The expected value of xi and expected value
ic
ic
of yi need to be computed conditioned on lxic < xi < uic
x and ly < yi < uy in which
ic
ic
[lxic , uic
x ] and [ly , uy ] are lower and upper limits of the intervals of x and y, respectively.

The computation of the E–step includes the formulas:
!
n
n
X
X
ic ic
ic
ic
ic ic
ic
ic (k)
Eθ(k)
xic
x
,
y
=
E(xic
)
i
i
i
i |lx < xi < ux , ly < yi < uy , θ
i=q+1

Eθ(k)

i=q+1

!

n
X

ic
yiic xic
i , yi

=

i=q+1

Eθ(k)

n
X

Eθ(k)

ic
x2i (ic) xic
i , yi

=

n
X

ic ic
ic
ic (k)
E(x2i (ic) |lxic < xic
)
i < ux , ly < yi < uy , θ

i=q+1

!
ic
yi2 (ic) xic
i , yi

=

i=q+1

Eθ(k)

ic ic
ic
ic (k)
E(yiic |lxic < xic
)
i < ux , ly < yi < uy , θ

i=q+1

!

i=q+1
n
X

n
X

n
X
i=q+1

n
X

ic ic
ic
ic (k)
E(yi2 (ic) |lxic < xic
)
i < ux , ly < yi < uy , θ

i=q+1

!
ic ic ic
xic
i yi xi , yi

=

n
X

ic
ic
ic ic
ic
ic (k)
E(xic
i |lx < xi < ux , ly < yi < uy , θ

i=q+1
ic ic
ic
ic (k)
× E(yiic |lxic < xic
)
i < ux , ly < yi < uy , θ

The first moments of the BVN data with both variables being interval censored
can be obtained using the integral formulas
R uic
R uic
y
x

ic
ic
ic ic
ic
ic
E(xic
i |lx < xi < ux , ly < yi < uy ) =

xfx,y (x, y) dy dx
lyic
ic
ic ic
ic
P (lxic < xic
i < ux , ly < yi < uy )
ic
lx

R uic
R uic
y
x

xfx,y (x, y) dy dx
lic
lic
= Rxuic Ryuic
y
x
fx,y (x, y) dy dx
lic
lic
x

y

92

R uic
R uic
y
x

ic
ic
ic ic
E(yiic |lxic < xic
i < ux , ly < yi < uy ) =

yfx,y (x, y) dy dx
lyic
ic
ic ic
ic
P (lxic < xic
i < ux , ly < yi < uy )
ic
lx

R uic
R uic
y
x

yfx,y (x, y) dy dx
lic
lic
= Rxuic Ryuic
y
x
fx,y (x, y) dy dx
lic
lic
y

x

Similarly, the second moments of xi and yi can be computed using the equations
R uic
R uic
y
x

ic ic
ic
ic
E(x2i ic |lxic < xic
i < ux , ly < yi < uy ) =

x2 fx,y (x, y) dy dx
lyic
ic
ic ic
ic
P (lxic < xic
i < ux , ly < yi < uy )
ic
lx

R uic
R uic
y
x
lyic

ic
lx

x2 fx,y (x, y) dy dx

= R uic R uic
y
x
lyic

ic
lx

fx,y (x, y) dy dx

R uic
R uic
y
x

ic ic
ic
ic
E(yi2 ic |lxic < xic
i < ux , ly < yi < uy ) =

y 2 fx,y (x, y) dy dx
lyic
ic
ic ic
ic
P (lxic < xic
i < ux , ly < yi < uy )
ic
lx

R uic
R uic
y
x
ic
lx

lyic

y 2 fx,y (x, y) dy dx

lyic

fx,y (x, y) dy dx

= R uic R uic
y
x
ic
lx

where fx,y (x, y) is the joint PDFof x and y. The adaptIntegrate function of cubature
package in R software [R Core Team, 2022] is used to numerically evaluate the integrals
due to its complexity to obtain a closed–form solution.

Computing the Total Expectation for Interval Censored Data

The total expectations in equations 3.2 and 3.6 are obtained by summing up the
expected values obtained from all 4 scenarios.
!
p
m
n
X
X
X
(k)
ic ic
o
o o
s1 = Eθ(k)
xi +
xoi
xi xi , yi , xi , yi =
i=1
q

i=1

+
+

X
i=p+1
n
X

i=m+1
ic (k)
o
ic
ic
)
E(xic
i |yi = y, lx < xi < ux , θ

ic
ic
ic ic
ic
ic (k)
E(xic
)
i |lx < xi < ux , ly < yi < uy , θ

i=q+1

(A.27)

93

(k)
s2 = Eθ(k)

n
X

!
ic
yi xoi , yio , xic
i , yi

=

i=1

m
X

p
X

yio +

i=m+1

i=1
q

X

+

yio

i=p+1
n
X

+

(k)
E(yiic |xoi = x, lyic < yiic < uic
)
y ,θ

ic ic
ic
ic (k)
E(yi |lxic < xic
)
i < ux , ly < yi < uy , θ

i=q+1

(A.28)
s21 (k) = Eθ(k)

n
X

!
ic
x2i (ic) xoi , yio , xic
i , yi

=

m
X

p

x2i (ic) +

i=1
q

i=1

X

+
+

i=p+1
n
X

X

x2i (ic)

i=m+1
ic (k)
)
E(x2i (ic) |yio = y, lxic < xic
i < ux , θ

ic ic
ic
ic (k)
E(x2i (ic) |lxic < xic
)
i < ux , ly < yi < uy , θ

i=q+1

(A.29)
s22 (k) = Eθ(k)

n
X

!
ic
yi2 xoi , yio , xic
i , yi

=

i=1

m
X

yi2 (o)

i=1
p

+

X

(k)
E(yi2 (ic) |xoi = x, lyic < yiic < uic
)
y ,θ

i=m+1
q

+
+

X
i=p+1
n
X

yi2 (o)
ic ic
ic
ic (k)
E(yi2 (ic) |lxic < xic
)
i < ux , ly < yi < uy , θ

i=q+1

(A.30)
(k)

s12 = Eθ(k)

n
X

!
ic
xi yi xoi , yio , xic
i , yi

=

i=1

m
X

xi · yi +

i=1
q

+
+

X
i=p+1
n
X

p
X

(k)
xoi · E(yiic |xoi = x, lyic < yiic < uic
)
y ,θ

i=m+1
o
ic
ic
ic (k)
yio · E(xic
)
i |yi = y, lx < xi < ux , θ



ic
ic
ic ic
ic
ic (k)
E(xic
)
i |lx < xi < ux , ly < yi < uy , θ

i=q+1
ic ic
ic
ic (k)
× E(yiic |lxic < xic
)]
i < ux , ly < yi < uy , θ
(A.31)

94

A.3.2

M–step

M–step is the next step of the algorithm and the sufficient statistics calculated in
the E–step equations A.27–A.31 are maximised. The parameters are then updated
accordingly. The formulas for MLEs of complete data are obtained from formulas
2.8–2.11 in section 2.7.

95

Appendix B
Additional Simulation Results

B.1

Results in the Presence of Missing observations in the Data

Figure B.1: Boxplots for estimates of mean of x (µˆx ) and mean of y (µˆy ), when
n = 100, ρ = 0.2 over different missing proportions of 0.1, 0.3, and 0.5

96

Figure B.2: Boxplots for estimates of standard deviation of x (σˆx ) and standard
deviation of y (σˆy ), when n = 100, ρ = 0.2 over different missing proportions of 0.1,
0.3, and 0.5

Figure B.3: Boxplots for estimates of correlation between x and y (ρ̂), when n =
100, ρ = 0.2 over different missing proportions of 0.1, 0.3, and 0.5

97

Table B.1: Empirical variance and asymptotic variance of intercept (βˆ0 ) when ρ = 0.2
over different missing proportions of 0.1, 0.3, and 0.5 and sample sizes 50, 100, and
500
n

50

100

500

p

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.460

0.515

0.481

0.545

0.594

0.547

0.3

0.460

0.697

0.592

0.545

0.786

0.536

0.5

0.460

0.848

0.630

0.545

0.963

0.514

0.1

0.215

0.239

0.231

0.267

0.296

0.267

0.3

0.215

0.305

0.263

0.267

0.370

0.263

0.5

0.215

0.444

0.345

0.267

0.478

0.253

0.1

0.052

0.056

0.055

0.052

0.058

0.053

0.3

0.052

0.075

0.065

0.052

0.073

0.052

0.5

0.052

0.087

0.067

0.052

0.093

0.050

Table B.2: Empirical variance and asymptotic variance of slope (βˆ1 ) when ρ = 0.2
over different missing proportions of 0.1, 0.3, and 0.5 and sample sizes 50, 100, and
500
n

50

100

500

p

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.006

0.006

0.006

0.005

0.006

0.005

0.3

0.006

0.008

0.007

0.005

0.008

0.006

0.5

0.006

0.009

0.009

0.005

0.010

0.006

0.1

0.003

0.003

0.003

0.003

0.003

0.003

0.3

0.003

0.003

0.003

0.003

0.004

0.003

0.5

0.003

0.004

0.004

0.003

0.005

0.003

0.1

0.001

0.001

0.001

0.001

0.001

0.001

0.3

0.001

0.001

0.001

0.001

0.001

0.001

0.5

0.001

0.001

0.001

0.001

0.001

0.001

98

Table B.3: Empirical variance and asymptotic variance of mean of x (µˆx ) when ρ = 0.2
over different missing proportions of 0.1, 0.3, and 0.5 and sample sizes 50, 100, and
500
n

p

50

100

500

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

1.955

2.211

2.067

1.984

2.151

1.938

0.3

1.955

2.757

2.296

1.984

2.803

1.931

0.5

1.955

3.598

2.581

1.984

3.423

1.933

0.1

0.958

1.065

0.964

0.997

1.103

0.987

0.3

0.958

1.323

1.137

0.997

1.382

0.984

0.5

0.958

1.734

1.255

0.997

1.763

0.985

0.1

0.206

0.226

0.212

0.200

0.222

0.200

0.3

0.206

0.278

0.245

0.200

0.277

0.200

0.5

0.206

0.361

0.276

0.200

0.356

0.199

Table B.4: Bias, RMSE, 95% CP, and relative RMSE of mean of x (µˆx ) for ρ = 0.2
over sample sizes 50, 100, and 500 and missing proportions of 0.1, 0.3, and 0.5

n

50

100

500

p

Bias

RMSE

CP

RRMSE

CLS

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

0.099

0.121

0.116 1.412

1.472

1.397

0.95

0.93

0.93

1.01

1.05

0.3

0.099

0.135

0.120 1.412

1.680

1.395

0.95

0.95

0.90

1.01

1.20

0.5

0.099

0.097

0.148 1.412

1.853

1.398

0.95

0.95

0.89

1.01

1.33

0.1

0.064

0.052

0.069 1.001

1.051

0.996

0.96

0.95

0.94

1.00

1.06

0.3

0.064

0.064

0.051 1.001

1.177

0.993

0.96

0.96

0.93

1.01

1.19

0.5

0.064

0.043

0.047 1.001

1.328

0.994

0.96

0.94

0.92

1.01

1.34

0.1

0.027

0.011

0.020 0.448

0.471

0.447

0.94

0.94

0.94

1.00

1.05

0.3

0.027

0.001

0.012 0.448

0.526

0.447

0.94

0.93

0.92

1.00

1.18

0.5

0.027

0.010

0.002 0.448

0.596

0.447

0.94

0.95

0.89

1.00

1.34

99

Table B.5: Empirical variance and asymptotic variance of mean of y (µˆy ) when ρ = 0.2
over different missing proportions of 0.1, 0.3, and 0.5 and sample sizes 50, 100, and
500
n

p

50

100

500

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.444

0.497

0.469

0.505

0.549

0.496

0.3

0.444

0.668

0.548

0.505

0.720

0.494

0.5

0.444

0.815

0.579

0.505

0.874

0.489

0.1

0.216

0.236

0.229

0.251

0.277

0.248

0.3

0.216

0.296

0.249

0.251

0.346

0.248

0.5

0.216

0.436

0.320

0.251

0.444

0.248

0.1

0.051

0.054

0.052

0.050

0.055

0.050

0.3

0.051

0.073

0.061

0.050

0.069

0.050

0.5

0.051

0.088

0.064

0.050

0.089

0.050

Table B.6: Bias, RMSE, 95% CP, and relative RMSE of mean of y (µˆy ) for ρ = 0.2
over sample sizes 50, 100, and 500 and interval censoring proportions of 0.1, 0.3, and
0.5

n

p

Bias
CLS

50

100

500

RMSE

CP

RRMSE

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

0.005 -0.007

0.008

0.711

0.741

0.704

0.95

0.96

0.95

1.01

1.05

0.3

0.005

0.003

0.006

0.711

0.849

0.703

0.95

0.95

0.93

1.01

1.21

0.5

0.005

0.000

-0.006

0.711

0.935

0.699

0.95

0.95

0.91

1.02

1.34

0.1

0.015

0.017

0.012

0.501

0.527

0.498

0.97

0.97

0.97

1.01

1.06

0.3

0.015

0.019

0.013

0.501

0.589

0.498

0.97

0.95

0.95

1.01

1.18

0.5

0.015

0.025

0.019

0.501

0.667

0.498

0.97

0.96

0.92

1.01

1.34

0.1

0.001

0.004

0.002

0.223

0.235

0.223

0.93

0.95

0.94

1.00

1.05

0.3

0.001

0.002

0.002

0.223

0.263

0.223

0.93

0.94

0.91

1.00

1.18

0.5

0.001 -0.007

-0.005

0.223

0.298

0.223

0.93

0.95

0.92

1.00

1.34

100

Table B.7: Empirical variance and asymptotic variance of standard deviation of x
(σˆx ) when ρ = 0.2 over different missing proportions of 0.1, 0.3, and 0.5 and sample
sizes 50, 100, and 500
n

p

50

100

500

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

1.104

1.164

1.122

1.023

1.112

0.971

0.3

1.104

1.585

1.228

1.023

1.466

1.002

0.5

1.104

1.971

1.410

1.023

1.809

1.052

0.1

0.551

0.597

0.556

0.506

0.561

0.495

0.3

0.551

0.810

0.634

0.506

0.706

0.508

0.5

0.551

0.876

0.723

0.506

0.906

0.540

0.1

0.102

0.113

0.106

0.100

0.111

0.100

0.3

0.102

0.145

0.120

0.100

0.139

0.103

0.5

0.102

0.183

0.138

0.100

0.179

0.109

Table B.8: Bias, RMSE, 95% CP, and relative RMSE of standard deviation of x (σˆx )
for ρ = 0.2 over sample sizes 50, 100, and 500 and missing proportions of 0.1, 0.3, and
0.5

n

p

Bias
CLS

50

100

500

RMSE

CP

RRMSE

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

-0.096 -0.102

-0.214

1.016

1.060

1.009

0.94

0.94

0.93

1.01

1.05

0.3

-0.096 -0.135

-0.239

1.016

1.218

1.029

0.94

0.93

0.89

0.99

1.18

0.5

-0.096 -0.150

-0.249

1.016

1.353

1.056

0.94

0.91

0.88

0.96

1.28

0.1

-0.043 -0.056

-0.095

0.713

0.751

0.710

0.94

0.94

0.92

1.00

1.06

0.3

-0.043 -0.050

-0.115

0.713

0.842

0.722

0.94

0.94

0.91

0.99

1.17

0.5

-0.043 -0.078

-0.119

0.713

0.955

0.745

0.94

0.95

0.89

0.96

1.28

0.1

-0.008 -0.008

-0.017

0.317

0.333

0.317

0.95

0.95

0.95

1.00

1.05

0.3

-0.008 -0.004

-0.015

0.317

0.373

0.321

0.95

0.95

0.94

0.99

1.16

0.5

-0.008 -0.008

-0.028

0.317

0.423

0.332

0.95

0.95

0.92

0.96

1.28

101

Table B.9: Empirical variance and asymptotic variance of standard deviation of y
(σˆy ) when ρ = 0.2 over different missing proportions of 0.1, 0.3, and 0.5 and sample
sizes 50, 100, and 500
n

p

50

100

500

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.257

0.284

0.265

0.261

0.284

0.249

0.3

0.257

0.378

0.300

0.261

0.376

0.257

0.5

0.257

0.454

0.331

0.261

0.462

0.268

0.1

0.126

0.136

0.128

0.127

0.141

0.125

0.3

0.126

0.177

0.148

0.127

0.177

0.128

0.5

0.126

0.222

0.168

0.127

0.228

0.137

0.1

0.025

0.027

0.026

0.025

0.028

0.025

0.3

0.025

0.033

0.028

0.025

0.035

0.026

0.5

0.025

0.042

0.034

0.025

0.045

0.027

Table B.10: Bias, RMSE, 95% CP, and relative RMSE of standard deviation of y (σˆy )
for ρ = 0.2 over sample sizes 50, 100, and 500 and missing proportions of 0.1, 0.3, and
0.5

n

50

100

500

p

Bias

RMSE

CP

RRMSE

CLS

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

0.000

0.002

-0.048

0.511

0.533

0.501

0.94

0.95

0.93

1.02

1.06

0.3

0.000

0.002

-0.060

0.511

0.614

0.511

0.94

0.94

0.91

1.00

1.20

0.5

0.000

-0.019

-0.092

0.511

0.680

0.526

0.94

0.93

0.88

0.97

1.29

0.1

-0.005 -0.012

-0.031

0.357

0.376

0.354

0.94

0.94

0.94

1.01

1.06

0.3

-0.005 -0.017

-0.036

0.357

0.421

0.360

0.94

0.93

0.93

0.99

1.17

0.5

-0.005 -0.024

-0.041

0.357

0.478

0.372

0.94

0.95

0.91

0.96

1.29

0.1

-0.008 -0.002

-0.010

0.158

0.167

0.158

0.95

0.96

0.95

1.00

1.05

0.3

-0.008 -0.007

-0.019

0.158

0.186

0.161

0.95

0.96

0.93

0.98

1.15

0.5

-0.008 -0.007

-0.013

0.158

0.211

0.166

0.95

0.96

0.93

0.95

1.27

102

Table B.11: Empirical variance and asymptotic variance of correlation (ρ̂) when ρ =
0.2 over different missing proportions of 0.1, 0.3, and 0.5 and sample sizes 50, 100,
and 500
n

p

50

100

500

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.020

0.020

0.021

0.019

0.020

0.018

0.3

0.020

0.027

0.027

0.019

0.026

0.018

0.5

0.020

0.033

0.033

0.019

0.032

0.020

0.1

0.010

0.010

0.010

0.009

0.010

0.009

0.3

0.010

0.013

0.013

0.009

0.013

0.009

0.5

0.010

0.016

0.016

0.009

0.017

0.010

0.1

0.002

0.002

0.002

0.002

0.002

0.002

0.3

0.002

0.003

0.003

0.002

0.003

0.002

0.5

0.002

0.004

0.004

0.002

0.003

0.002

Table B.12: Bias, RMSE, 95% CP, and relative RMSE of correlation(ρ̂) for ρ = 0.2
over sample sizes 50, 100, and 500 and missing proportions of 0.1, 0.3, and 0.5

n

p

Bias
CLS

50

100

500

RMSE

CP

RRMSE

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

-0.005 -0.003

-0.003

0.137

0.142

0.133

0.93

0.94

0.92

1.02

1.07

0.3

-0.005 -0.002

-0.001

0.137

0.163

0.136

0.93

0.93

0.87

1.01

1.20

0.5

-0.005 -0.002

-0.002

0.137

0.180

0.140

0.93

0.93

0.84

0.98

1.29

0.1

-0.004 -0.008

-0.007

0.097

0.102

0.096

0.93

0.94

0.92

1.01

1.06

0.3

-0.004 -0.006

-0.005

0.097

0.114

0.097

0.93

0.94

0.89

0.99

1.17

0.5

-0.004 -0.011

-0.009

0.097

0.129

0.101

0.93

0.95

0.87

0.95

1.28

0.1

0.000

0.001

0.001

0.042

0.045

0.042

0.95

0.95

0.94

1.00

1.05

0.3

0.000

0.001

0.001

0.042

0.050

0.044

0.95

0.94

0.91

0.97

1.15

0.5

0.000

0.000

0.000

0.042

0.057

0.046

0.95

0.94

0.86

0.93

1.25

103

Table B.13: Empirical variance and asymptotic variance of intercept (βˆ0 ) when ρ = 0.4
over different missing proportions of 0.1, 0.3, and 0.5 and sample sizes 50, 100, and
500
n

p

50

100

500

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.400

0.452

0.425

0.476

0.519

0.478

0.3

0.400

0.610

0.530

0.476

0.687

0.468

0.5

0.400

0.742

0.569

0.476

0.841

0.450

0.1

0.189

0.211

0.204

0.234

0.259

0.234

0.3

0.189

0.269

0.236

0.234

0.324

0.230

0.5

0.189

0.389

0.313

0.234

0.418

0.222

0.1

0.045

0.049

0.048

0.046

0.051

0.046

0.3

0.045

0.065

0.057

0.046

0.063

0.045

0.5

0.045

0.075

0.059

0.046

0.082

0.044

Table B.14: Bias, RMSE, 95% CP, and relative RMSE of intercept (βˆ0 ) when ρ = 0.4
over sample sizes 50, 100, and 500 and missing proportions of 0.1, 0.3, and 0.5

n

p

Bias
CLS

50

100

500

RMSE

CP

RRMSE

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

-0.003 -0.023

-0.013

0.690

0.721

0.691

0.96

0.96

0.96

1.00

1.04

0.3

-0.003 -0.020

-0.019

0.690

0.829

0.684

0.96

0.95

0.93

1.01

1.21

0.5

-0.003 -0.024

-0.030

0.690

0.918

0.671

0.96

0.95

0.91

1.03

1.37

0.1

0.010

0.018

0.010

0.483

0.509

0.484

0.97

0.96

0.96

1.00

1.05

0.3

0.010

0.017

0.009

0.483

0.569

0.480

0.97

0.97

0.94

1.01

1.19

0.5

0.010

0.029

0.020

0.483

0.647

0.472

0.97

0.95

0.89

1.02

1.37

0.1

-0.003

0.001

-0.001

0.214

0.226

0.214

0.95

0.95

0.95

1.00

1.05

0.3

-0.003

0.001

0.002

0.214

0.252

0.212

0.95

0.95

0.91

1.01

1.19

0.5

-0.003 -0.009

-0.004

0.214

0.286

0.209

0.95

0.95

0.91

1.02

1.37

104

Table B.15: Empirical variance and asymptotic variance of slope (βˆ1 ) when ρ = 0.4
over different missing proportions of 0.1, 0.3, and 0.5 and sample sizes 50, 100, and
500 in the presence of missing data
n

p

50

100

500

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.005

0.005

0.005

0.005

0.005

0.005

0.3

0.005

0.007

0.007

0.005

0.007

0.005

0.5

0.005

0.008

0.008

0.005

0.008

0.005

0.1

0.002

0.002

0.002

0.002

0.002

0.002

0.3

0.002

0.003

0.003

0.002

0.003

0.003

0.5

0.002

0.004

0.004

0.002

0.004

0.003

0.1

0.000

0.001

0.001

0.000

0.001

0.000

0.3

0.000

0.001

0.001

0.000

0.001

0.001

0.5

0.000

0.001

0.001

0.000

0.001

0.001

Table B.16: Bias, RMSE, 95% CP, and relative RMSE of slope (βˆ1 ) when ρ = 0.4
over sample sizes 50, 100, and 500 and missing proportions of 0.1, 0.3, and 0.5

n

50

100

500

p

Bias

RMSE

CP

RRMSE

CLS

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

0.001

0.002

0.002

0.068

0.071

0.069

0.93

0.94

0.94

0.99

1.03

0.3

0.001

0.004

0.004

0.068

0.082

0.072

0.93

0.94

0.91

0.94

1.13

0.5

0.001

0.003

0.003

0.068

0.092

0.074

0.93

0.94

0.89

0.92

1.25

0.1

-0.001 -0.002

-0.002

0.047

0.049

0.048

0.94

0.96

0.95

0.98

1.02

0.3

-0.001 -0.001

-0.000

0.047

0.056

0.050

0.94

0.95

0.92

0.94

1.11

0.5

-0.001 -0.003

-0.002

0.047

0.063

0.051

0.94

0.95

0.89

0.92

1.24

0.1

0.000

0.001

0.000

0.020

0.022

0.020

0.96

0.96

0.95

1.00

1.12

0.3

0.000

0.000

0.000

0.020

0.025

0.022

0.96

0.94

0.92

0.89

1.09

0.5

0.000

0.000

0.000

0.020

0.028

0.022

0.96

0.93

0.90

0.89

1.26

105

Table B.17: Empirical variance and asymptotic variance of mean of x (µˆx ) when
ρ = 0.4 over different missing proportions of 0.1, 0.3, and 0.5 and sample sizes 50,
100, and 500
n

p

50

100

500

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

1.955

2.205

2.058

1.984

2.151

1.939

0.3

1.955

2.752

2.271

1.984

2.805

1.932

0.5

1.955

3.600

2.549

1.984

3.426

1.934

0.1

0.948

1.054

0.952

0.997

1.102

0.987

0.3

0.948

1.311

1.102

0.997

1.381

0.984

0.5

0.948

1.728

1.211

0.997

1.760

0.985

0.1

0.207

0.227

0.212

0.200

0.222

0.200

0.3

0.207

0.280

0.242

0.200

0.277

0.200

0.5

0.207

0.365

0.270

0.200

0.356

0.200

Table B.18: Bias, RMSE, 95% CP, and relative RMSE of mean of x (µˆx ) when ρ = 0.4
over sample sizes 50, 100, and 500 and missing proportions of 0.1, 0.3, and 0.5

n

50

100

500

p

Bias

RMSE

CP

RRMSE

CLS

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

0.099

0.118

0.117 1.412

1.472

1.397

0.94

0.93

0.93

1.01

1.05

0.3

0.099

0.133

0.119 1.412

1.680

1.395

0.94

0.95

0.91

1.01

1.20

0.5

0.099

0.095

0.144 1.412

1.853

1.398

0.94

0.94

0.90

1.01

1.33

0.1

0.065

0.053

0.068 1.000

1.051

0.996

0.96

0.96

0.95

1.00

1.06

0.3

0.065

0.066

0.050 1.000

1.177

0.993

0.96

0.96

0.93

1.01

1.19

0.5

0.065

0.046

0.045 1.000

1.328

0.994

0.96

0.94

0.92

1.01

1.34

0.1

0.027

0.011

0.020 0.448

0.471

0.447

0.94

0.94

0.93

1.00

1.05

0.3

0.027

0.002

0.012 0.448

0.526

0.447

0.94

0.94

0.92

1.00

1.18

0.5

0.027

0.009

0.004 0.448

0.596

0.447

0.94

0.94

0.91

1.00

1.33

106

Table B.19: Empirical variance and asymptotic variance of mean of y (µˆy ) when
ρ = 0.4 over different missing proportions of 0.1, 0.3, and 0.5 and sample sizes 50,
100, and 500
n

p

50

100

500

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.448

0.499

0.470

0.504

0.548

0.495

0.3

0.448

0.668

0.541

0.504

0.719

0.494

0.5

0.448

0.823

0.574

0.504

0.874

0.489

0.1

0.214

0.233

0.225

0.251

0.277

0.248

0.3

0.214

0.294

0.245

0.251

0.346

0.248

0.5

0.214

0.433

0.308

0.251

0.443

0.248

0.1

0.051

0.055

0.053

0.050

0.055

0.050

0.3

0.051

0.073

0.060

0.050

0.069

0.050

0.5

0.051

0.090

0.064

0.050

0.089

0.050

Table B.20: Bias, RMSE, 95% CP, and relative RMSE of mean of y (µˆy ) when ρ = 0.4
over sample sizes 50, 100, and 500 and missing proportions of 0.1, 0.3, and 0.5

n

50

100

500

p

Bias

RMSE

CP

RRMSE

CLS

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

0.012

0.002

0.014

0.710

0.741

0.704

0.95

0.96

0.95

1.01

1.05

0.3

0.012

0.013

0.014

0.710

0.848

0.703

0.95

0.95

0.93

1.01

1.21

0.5

0.012

0.007

0.005

0.710

0.935

0.699

0.95

0.96

0.93

1.02

1.34

0.1

0.019

0.020

0.017

0.501

0.526

0.498

0.97

0.97

0.97

1.01

1.06

0.3

0.019

0.023

0.017

0.501

0.589

0.498

0.97

0.96

0.95

1.01

1.18

0.5

0.019

0.027

0.021

0.501

0.666

0.498

0.97

0.95

0.92

1.01

1.34

0.1

0.003

0.005

0.004

0.223

0.235

0.223

0.93

0.95

0.94

1.00

1.05

0.3

0.003

0.002

0.004

0.223

0.263

0.223

0.93

0.95

0.92

1.00

1.18

0.5

0.003 -0.006

-0.003

0.223

0.298

0.223

0.93

0.95

0.92

1.00

1.33

107

Table B.21: Empirical variance and asymptotic variance of standard deviation of x
(σˆx ) when ρ = 0.4 over different missing proportions of 0.1, 0.3, and 0.5 and sample
sizes 50, 100, and 500
n

p

50

100

500

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

1.096

1.155

1.115

1.025

1.115

0.972

0.3

1.096

1.570

1.217

1.025

1.471

1.004

0.5

1.096

1.968

1.402

1.025

1.817

1.057

0.1

0.546

0.593

0.554

0.507

0.561

0.495

0.3

0.546

0.804

0.625

0.507

0.706

0.508

0.5

0.546

0.868

0.715

0.507

0.906

0.543

0.1

0.103

0.114

0.107

0.100

0.111

0.100

0.3

0.103

0.146

0.122

0.100

0.139

0.103

0.5

0.103

0.183

0.139

0.100

0.179

0.110

Table B.22: Bias, RMSE, 95% CP, and relative RMSE of standard deviation of x (σˆx )
when ρ = 0.4 over sample sizes 50, 100, and 500 and missing proportions of 0.1, 0.3,
and 0.5

n

p

Bias
CLS

50

100

500

RMSE

CP

RRMSE

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

-0.096 -0.100

-0.212

1.017

1.061

1.008

0.94

0.94

0.92

1.01

1.05

0.3

-0.096 -0.131

-0.236

1.017

1.220

1.030

0.94

0.93

0.89

0.99

1.18

0.5

-0.096 -0.146

-0.247

1.017

1.356

1.058

0.94

0.93

0.89

0.96

1.28

0.1

-0.044 -0.059

-0.096

0.713

0.751

0.710

0.93

0.93

0.92

1.00

1.06

0.3

-0.044 -0.053

-0.115

0.713

0.842

0.722

0.93

0.94

0.91

0.99

1.17

0.5

-0.044 -0.084

-0.119

0.713

0.956

0.747

0.93

0.94

0.90

0.96

1.28

0.1

-0.008 -0.008

-0.017

0.317

0.334

0.317

0.95

0.95

0.94

1.00

1.05

0.3

-0.008 -0.003

-0.015

0.317

0.373

0.322

0.95

0.95

0.93

0.99

1.16

0.5

-0.008 -0.008

-0.027

0.317

0.423

0.333

0.95

0.95

0.91

0.95

1.27

108

Table B.23: Empirical variance and asymptotic variance of standard deviation of y
(σˆy ) when ρ = 0.4 over different missing proportions of 0.1, 0.3, and 0.5 and sample
sizes 50, 100, and 500
n

p

50

100

500

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.264

0.291

0.272

0.261

0.284

0.248

0.3

0.264

0.383

0.310

0.261

0.377

0.257

0.5

0.264

0.462

0.338

0.261

0.464

0.269

0.1

0.127

0.135

0.129

0.127

0.141

0.124

0.3

0.127

0.179

0.150

0.127

0.177

0.128

0.5

0.127

0.220

0.165

0.127

0.228

0.137

0.1

0.024

0.027

0.026

0.025

0.028

0.025

0.3

0.024

0.032

0.028

0.025

0.035

0.026

0.5

0.024

0.042

0.034

0.025

0.045

0.028

Table B.24: Bias, RMSE, 95% CP, and relative RMSE of standard deviation of y (σˆy )
when ρ = 0.4 over sample sizes 50, 100, and 500 and missing percentage of 10%, 30%,
and 50%

n

p

Bias
CLS

50

100

500

RMSE

CP

RRMSE

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

-0.004 -0.001

-0.052

0.511

0.533

0.501

0.93

0.95

0.91

1.02

1.06

0.3

-0.004 -0.001

-0.063

0.511

0.614

0.511

0.93

0.93

0.90

1.00

1.20

0.5

-0.004 -0.020

-0.094

0.511

0.681

0.527

0.93

0.93

0.89

0.97

1.29

0.1

-0.008 -0.017

-0.035

0.357

0.376

0.354

0.94

0.94

0.93

1.01

1.06

0.3

-0.008 -0.020

-0.038

0.357

0.421

0.360

0.94

0.94

0.92

0.99

1.17

0.5

-0.008 -0.031

-0.042

0.357

0.478

0.373

0.94

0.94

0.92

0.96

1.28

0.1

-0.007 -0.001

-0.009

0.158

0.167

0.158

0.96

0.96

0.95

1.00

1.05

0.3

-0.007 -0.006

-0.017

0.158

0.186

0.161

0.96

0.96

0.94

0.98

1.16

0.5

-0.007 -0.006

-0.013

0.158

0.211

0.167

0.96

0.97

0.93

0.95

1.27

109

Table B.25: Empirical variance and asymptotic variance of correlation (ρ̂) when ρ =
0.4 over different missing proportions of 0.1, 0.3, and 0.5 and sample sizes 50, 100,
and 500
n

p

50

100

500

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.015

0.016

0.016

0.015

0.016

0.014

0.3

0.015

0.021

0.021

0.015

0.021

0.014

0.5

0.015

0.026

0.025

0.015

0.026

0.015

0.1

0.007

0.008

0.008

0.007

0.008

0.007

0.3

0.007

0.010

0.010

0.007

0.010

0.007

0.5

0.007

0.012

0.012

0.007

0.013

0.008

0.1

0.002

0.002

0.002

0.001

0.002

0.001

0.3

0.002

0.002

0.002

0.001

0.002

0.002

0.5

0.002

0.003

0.003

0.001

0.003

0.002

Table B.26: Bias, RMSE, 95% CP, and relative RMSE of correlation (ρ̂) when ρ = 0.4
over sample sizes 50, 100, and 500 and missing proportions of 0.1, 0.3, and 0.5

n

p

Bias
CLS

50

100

500

RMSE

CP

RRMSE

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

-0.007 -0.005

-0.005

0.121

0.126

0.118

0.93

0.93

0.92

1.03

1.07

0.3

-0.007 -0.005

-0.003

0.121

0.145

0.120

0.93

0.93

0.87

1.01

1.21

0.5

-0.007 -0.005

-0.003

0.121

0.160

0.124

0.93

0.93

0.86

0.98

1.29

0.1

-0.005 -0.008

-0.006

0.085

0.090

0.085

0.93

0.95

0.93

1.01

1.06

0.3

-0.005 -0.006

-0.005

0.085

0.100

0.086

0.93

0.94

0.90

0.99

1.17

0.5

-0.005 -0.011

-0.007

0.085

0.115

0.090

0.93

0.94

0.88

0.95

1.28

0.1

0.000

0.000

0.000

0.037

0.040

0.037

0.95

0.95

0.94

1.00

1.07

0.3

0.000

0.001

0.001

0.037

0.045

0.039

0.95

0.94

0.90

0.97

1.16

0.5

0.000

-0.000

-0.000

0.037

0.050

0.040

0.95

0.94

0.86

0.94

1.25

110

Table B.27: Empirical variance and asymptotic variance of intercept (βˆ0 ) when ρ = 0.6
over different missing proportions of 0.1, 0.3, and 0.5 and sample sizes 50, 100, and
500
n

p

50

100

500

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.304

0.346

0.330

0.363

0.395

0.363

0.3

0.304

0.466

0.420

0.363

0.522

0.356

0.5

0.304

0.568

0.461

0.363

0.640

0.343

0.1

0.145

0.162

0.157

0.178

0.197

0.178

0.3

0.145

0.207

0.186

0.178

0.247

0.175

0.5

0.145

0.298

0.251

0.178

0.319

0.170

0.1

0.034

0.037

0.036

0.035

0.039

0.035

0.3

0.034

0.049

0.044

0.035

0.048

0.034

0.5

0.034

0.057

0.047

0.035

0.062

0.033

Table B.28: Bias, RMSE, 95% CP, and relative RMSE of intercept (βˆ0 ) when ρ = 0.6
over sample sizes 50, 100, and 500 and missing proportions of 0.1, 0.3, and 0.5

n

p

Bias
CLS

50

100

500

RMSE

CP

RRMSE

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

-0.008 -0.026

-0.019

0.602

0.629

0.603

0.97

0.96

0.95

1.00

1.04

0.3

-0.008 -0.023

-0.024

0.602

0.723

0.597

0.97

0.94

0.92

1.01

1.21

0.5

-0.008 -0.026

-0.034

0.602

0.800

0.587

0.97

0.95

0.90

1.03

1.36

0.1

0.006

0.012

0.006

0.422

0.445

0.422

0.97

0.96

0.96

1.00

1.05

0.3

0.006

0.012

0.005

0.422

0.497

0.419

0.97

0.96

0.93

1.01

1.19

0.5

0.006

0.023

0.013

0.422

0.565

0.412

0.97

0.96

0.88

1.02

1.37

0.1

-0.003

0.000

-0.000

0.187

0.197

0.187

0.94

0.96

0.95

1.00

1.05

0.3

-0.003

0.001

0.003

0.187

0.220

0.186

0.94

0.95

0.91

1.01

1.18

0.5

-0.003 -0.008

-0.003

0.187

0.250

0.183

0.94

0.95

0.90

1.02

1.36

111

Table B.29: Empirical variance and asymptotic variance of slope (βˆ1 ) when ρ = 0.6
over different missing proportions of 0.1, 0.3, and 0.5 and sample sizes 50, 100, and
500
n

p

50

100

500

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.004

0.004

0.004

0.004

0.004

0.004

0.3

0.004

0.005

0.005

0.004

0.005

0.004

0.5

0.004

0.006

0.006

0.004

0.006

0.004

0.1

0.002

0.002

0.002

0.002

0.002

0.002

0.3

0.002

0.003

0.002

0.002

0.002

0.002

0.5

0.002

0.003

0.003

0.002

0.003

0.002

0.1

0.000

0.000

0.000

0.000

0.000

0.000

0.3

0.000

0.001

0.000

0.000

0.000

0.000

0.5

0.000

0.001

0.001

0.000

0.001

0.000

Table B.30: Bias, RMSE, 95% CP, and relative RMSE of slope (βˆ1 ) when ρ = 0.6
over sample sizes 50, 100, and 500 and missing proportions of 0.1, 0.3, and 0.5

n

50

100

500

p

Bias

RMSE

CP

RRMSE

CLS

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

0.001

0.002

0.003

0.059

0.062

0.060

0.94

0.94

0.93

0.99

1.03

0.3

0.001

0.004

0.005

0.059

0.072

0.062

0.94

0.94

0.91

0.96

1.16

0.5

0.001

0.004

0.004

0.059

0.080

0.063

0.94

0.94

0.88

0.93

1.25

0.1

-0.000 -0.002

-0.001

0.041

0.044

0.041

0.93

0.95

0.94

1.00

1.06

0.3

-0.000 -0.001

0.001

0.041

0.048

0.042

0.93

0.94

0.91

0.97

1.13

0.5

-0.000 -0.002

-0.000

0.041

0.055

0.044

0.93

0.95

0.91

0.95

1.26

0.1

0.000

0.000

0.000

0.017

0.020

0.017

0.96

0.96

0.95

1.00

1.16

0.3

0.000

0.000

-0.000

0.017

0.020

0.020

0.96

0.95

0.92

0.87

1.00

0.5

0.000

0.000

0.000

0.017

0.025

0.020

0.96

0.94

0.90

0.87

1.23

112

Table B.31: Empirical variance and asymptotic variance of mean of x (µˆx ) when
ρ = 0.6 over different missing proportions of 0.1, 0.3, and 0.5 and sample sizes 50,
100, and 500
n

p

50

100

500

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

1.954

2.196

2.040

1.984

2.153

1.940

0.3

1.954

2.746

2.218

1.984

2.807

1.934

0.5

1.954

3.598

2.470

1.984

3.430

1.936

0.1

0.937

1.041

0.938

0.996

1.101

0.987

0.3

0.937

1.298

1.056

0.996

1.380

0.984

0.5

0.937

1.723

1.149

0.996

1.758

0.987

0.1

0.208

0.227

0.211

0.200

0.222

0.200

0.3

0.208

0.282

0.237

0.200

0.277

0.200

0.5

0.208

0.370

0.260

0.200

0.356

0.200

Table B.32: Bias, RMSE, 95% CP, and relative RMSE of mean of x (µˆx ) when ρ = 0.6
over sample sizes 50, 100, and 500 and missing proportions of 0.1, 0.3, and 0.5

n

50

100

500

p

Bias

RMSE

CP

RRMSE

CLS

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

0.097

0.114

0.116 1.412

1.472

1.398

0.94

0.93

0.93

1.01

1.05

0.3

0.097

0.130

0.115 1.412

1.681

1.395

0.94

0.96

0.92

1.01

1.20

0.5

0.097

0.093

0.136 1.412

1.854

1.398

0.94

0.94

0.91

1.01

1.33

0.1

0.066

0.055

0.066 1.000

1.051

0.996

0.95

0.95

0.94

1.00

1.06

0.3

0.066

0.067

0.049 1.000

1.177

0.993

0.95

0.95

0.93

1.01

1.18

0.5

0.066

0.049

0.042 1.000

1.327

0.994

0.95

0.94

0.93

1.01

1.33

0.1

0.026

0.012

0.019 0.448

0.471

0.447

0.94

0.93

0.93

1.00

1.05

0.3

0.026

0.002

0.013 0.448

0.526

0.447

0.94

0.93

0.93

1.00

1.18

0.5

0.026

0.008

0.007 0.448

0.596

0.447

0.94

0.94

0.91

1.00

1.33

113

Table B.33: Empirical variance and asymptotic variance of mean of y (µˆy ) when
ρ = 0.6 over different missing proportions of 0.1, 0.3, and 0.5 and sample sizes 50,
100, and 500
n

p

50

100

500

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.454

0.503

0.472

0.503

0.547

0.494

0.3

0.454

0.668

0.529

0.503

0.717

0.493

0.5

0.454

0.835

0.560

0.503

0.873

0.489

0.1

0.212

0.232

0.221

0.250

0.276

0.247

0.3

0.212

0.293

0.239

0.250

0.345

0.248

0.5

0.212

0.430

0.289

0.250

0.441

0.248

0.1

0.052

0.056

0.053

0.050

0.056

0.050

0.3

0.052

0.074

0.059

0.050

0.069

0.050

0.5

0.052

0.093

0.063

0.050

0.089

0.050

Table B.34: Bias, RMSE, 95% CP, and relative RMSE of mean of y (µˆy ) when ρ = 0.6
over sample sizes 50, 100, and 500 and missing proportions of 0.1, 0.3, and 0.5

n

50

100

500

p

Bias

RMSE

CP

RRMSE

CLS

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

0.019

0.012

0.021

0.710

0.740

0.703

0.96

0.96

0.95

1.01

1.05

0.3

0.019

0.023

0.021

0.710

0.847

0.703

0.96

0.95

0.94

1.01

1.21

0.5

0.019

0.015

0.016

0.710

0.934

0.700

0.96

0.95

0.92

1.01

1.34

0.1

0.023

0.023

0.022

0.501

0.526

0.498

0.97

0.97

0.97

1.01

1.06

0.3

0.023

0.027

0.020

0.501

0.588

0.498

0.97

0.97

0.96

1.01

1.18

0.5

0.023

0.029

0.023

0.501

0.665

0.498

0.97

0.95

0.94

1.00

1.33

0.1

0.005

0.005

0.006

0.223

0.236

0.224

0.93

0.95

0.94

1.00

1.05

0.3

0.005

0.002

0.006

0.223

0.263

0.223

0.93

0.95

0.92

1.00

1.18

0.5

0.005 -0.005

-0.001

0.223

0.298

0.223

0.93

0.95

0.91

1.00

1.33

114

Table B.35: Empirical variance and asymptotic variance of standard deviation of x
(σˆx ) when ρ = 0.6 over different missing proportions of 0.1, 0.3, and 0.5 and sample
sizes 50, 100, and 500
n

p

50

100

500

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

1.088

1.146

1.106

1.030

1.121

0.973

0.3

1.088

1.552

1.197

1.030

1.482

1.008

0.5

1.088

1.958

1.376

1.030

1.834

1.066

0.1

0.540

0.587

0.550

0.507

0.562

0.495

0.3

0.540

0.795

0.615

0.507

0.708

0.510

0.5

0.540

0.860

0.700

0.507

0.909

0.549

0.1

0.104

0.116

0.108

0.100

0.111

0.100

0.3

0.104

0.147

0.122

0.100

0.139

0.104

0.5

0.104

0.184

0.138

0.100

0.179

0.111

Table B.36: Bias, RMSE, 95% CP, and relative RMSE of standard deviation of x (σˆx )
when ρ = 0.6 over sample sizes 50, 100, and 500 and missing proportions of 0.1, 0.3,
and 0.5

n

p

Bias
CLS

50

100

500

RMSE

CP

RRMSE

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

-0.095 -0.097

-0.208

1.019

1.063

1.008

0.94

0.93

0.91

1.01

1.05

0.3

-0.095 -0.126

-0.231

1.019

1.224

1.030

0.94

0.92

0.90

0.99

1.19

0.5

-0.095 -0.141

-0.244

1.019

1.362

1.061

0.94

0.93

0.89

0.96

1.28

0.1

-0.045 -0.063

-0.095

0.714

0.752

0.710

0.93

0.93

0.93

1.01

1.06

0.3

-0.045 -0.055

-0.111

0.714

0.843

0.723

0.93

0.93

0.91

0.99

1.17

0.5

-0.045 -0.089

-0.113

0.714

0.957

0.749

0.93

0.94

0.91

0.95

1.28

0.1

-0.008 -0.007

-0.018

0.317

0.334

0.317

0.94

0.95

0.94

1.00

1.05

0.3

-0.008 -0.003

-0.016

0.317

0.373

0.322

0.94

0.95

0.93

0.98

1.16

0.5

-0.008 -0.008

-0.026

0.317

0.423

0.334

0.94

0.94

0.92

0.95

1.27

115

Table B.37: Empirical variance and asymptotic variance of standard deviation of y
(σˆy ) when ρ = 0.6 over different missing proportions of 0.1, 0.3, and 0.5 and sample
sizes 50, 100, and 500
n

p

50

100

500

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.271

0.296

0.278

0.261

0.285

0.248

0.3

0.271

0.385

0.314

0.261

0.379

0.258

0.5

0.271

0.467

0.338

0.261

0.467

0.271

0.1

0.127

0.134

0.129

0.127

0.141

0.124

0.3

0.127

0.181

0.151

0.127

0.177

0.129

0.5

0.127

0.218

0.160

0.127

0.228

0.138

0.1

0.024

0.027

0.025

0.025

0.028

0.025

0.3

0.024

0.033

0.028

0.025

0.035

0.026

0.5

0.024

0.042

0.033

0.025

0.045

0.028

Table B.38: Bias, RMSE, 95% CP, and relative RMSE of standard deviation of y (σˆy )
when ρ = 0.6 over sample sizes 50, 100, and 500 and missing proportions of 0.1, 0.3,
and 0.5

n

p

Bias
CLS

50

100

500

RMSE

CP

RRMSE

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

-0.011 -0.007

-0.059

0.511

0.534

0.501

0.93

0.94

0.93

1.02

1.07

0.3

-0.011 -0.007

-0.067

0.511

0.615

0.512

0.93

0.94

0.91

1.00

1.20

0.5

-0.011 -0.024

-0.095

0.511

0.684

0.529

0.93

0.93

0.89

0.97

1.29

0.1

-0.011 -0.023

-0.039

0.357

0.376

0.355

0.94

0.94

0.94

1.01

1.06

0.3

-0.011 -0.024

-0.040

0.357

0.422

0.361

0.94

0.93

0.92

0.99

1.17

0.5

-0.011 -0.038

-0.044

0.357

0.479

0.375

0.94

0.95

0.93

0.95

1.28

0.1

-0.007 -0.001

-0.009

0.158

0.167

0.158

0.96

0.96

0.95

1.00

1.05

0.3

-0.007 -0.004

-0.016

0.158

0.187

0.161

0.96

0.96

0.94

0.98

1.16

0.5

-0.007 -0.005

-0.012

0.158

0.212

0.167

0.96

0.96

0.94

0.95

1.27

116

Table B.39: Empirical variance and asymptotic variance of correlation (ρ̂) when ρ =
0.6 over different missing proportions of 0.1, 0.3, and 0.5 and sample sizes 50, 100,
and 500
n

p

50

100

500

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.009

0.009

0.009

0.009

0.010

0.008

0.3

0.009

0.012

0.012

0.009

0.013

0.009

0.5

0.009

0.015

0.014

0.009

0.016

0.009

0.1

0.004

0.005

0.005

0.004

0.005

0.004

0.3

0.004

0.006

0.005

0.004

0.006

0.004

0.5

0.004

0.007

0.007

0.004

0.008

0.005

0.1

0.001

0.001

0.001

0.001

0.001

0.001

0.3

0.001

0.001

0.001

0.001

0.001

0.001

0.5

0.001

0.002

0.002

0.001

0.002

0.001

Table B.40: Bias, RMSE, 95% CP, and relative RMSE of correlation (ρ̂) when ρ = 0.6
over sample sizes 50, 100, and 500 and missing proportions of 0.1, 0.3, and 0.5

n

p

Bias
CLS

50

100

500

RMSE

CP

RRMSE

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

-0.007 -0.006

-0.006

0.094

0.098

0.092

0.92

0.94

0.91

1.03

1.07

0.3

-0.007 -0.006

-0.003

0.094

0.113

0.093

0.92

0.94

0.87

1.01

1.21

0.5

-0.007 -0.007

-0.002

0.094

0.126

0.097

0.92

0.93

0.88

0.98

1.30

0.1

-0.004 -0.007

-0.005

0.066

0.070

0.065

0.94

0.95

0.94

1.01

1.07

0.3

-0.004 -0.006

-0.004

0.066

0.078

0.066

0.94

0.95

0.90

1.00

1.18

0.5

-0.004 -0.009

-0.005

0.066

0.089

0.069

0.94

0.94

0.88

0.95

1.29

0.1

-0.000

0.000

0.000

0.028

0.030

0.028

0.95

0.95

0.93

1.00

1.06

0.3

-0.000

0.000

0.000

0.028

0.033

0.030

0.95

0.94

0.91

0.94

1.11

0.5

-0.000 -0.000

-0.000

0.028

0.039

0.030

0.95

0.93

0.87

0.94

1.29

117

Table B.41: Empirical variance and asymptotic variance of intercept (βˆ0 ) when ρ = 0.8
over different missing proportions of 0.1, 0.3, and 0.5 and sample sizes 50, 100, and
500
n

p

50

100

500

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.170

0.196

0.191

0.204

0.222

0.204

0.3

0.170

0.263

0.249

0.204

0.293

0.200

0.5

0.170

0.323

0.286

0.204

0.359

0.195

0.1

0.083

0.092

0.091

0.100

0.111

0.100

0.3

0.083

0.118

0.110

0.100

0.139

0.099

0.5

0.083

0.168

0.152

0.100

0.180

0.097

0.1

0.019

0.021

0.020

0.020

0.022

0.020

0.3

0.019

0.027

0.026

0.020

0.027

0.019

0.5

0.019

0.031

0.028

0.020

0.035

0.019

Table B.42: Bias, RMSE, 95% CP, and relative RMSE of intercept (βˆ0 ) when ρ = 0.8
over sample sizes 50, 100, and 500 and missing proportions of 0.1, 0.3, and 0.5

n

p

Bias
CLS

50

100

500

RMSE

CP

RRMSE

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

-0.010 -0.023

-0.021

0.452

0.472

0.452

0.96

0.96

0.95

1.00

1.04

0.3

-0.010 -0.022

-0.024

0.452

0.542

0.448

0.96

0.95

0.92

1.01

1.21

0.5

-0.010 -0.023

-0.032

0.452

0.599

0.442

0.96

0.95

0.88

1.02

1.35

0.1

0.001

0.007

0.003

0.316

0.333

0.316

0.97

0.96

0.96

1.00

1.05

0.3

0.001

0.007

0.002

0.316

0.373

0.314

0.97

0.96

0.93

1.01

1.19

0.5

0.001

0.015

0.007

0.316

0.424

0.311

0.97

0.96

0.88

1.02

1.36

0.1

-0.003

0.000

-0.000

0.140

0.148

0.140

0.95

0.95

0.95

1.00

1.05

0.3

-0.003

0.001

0.002

0.140

0.165

0.139

0.95

0.95

0.92

1.01

1.18

0.5

-0.003 -0.006

-0.003

0.140

0.187

0.138

0.95

0.96

0.89

1.02

1.35

118

Table B.43: Empirical variance and asymptotic variance of slope (βˆ1 ) when ρ = 0.8
over different missing proportions of 0.1, 0.3, and 0.5 and sample sizes 50, 100, and
500
n

p

50

100

500

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.002

0.002

0.002

0.002

0.002

0.002

0.3

0.002

0.003

0.003

0.002

0.003

0.002

0.5

0.002

0.004

0.003

0.002

0.004

0.002

0.1

0.001

0.001

0.001

0.001

0.001

0.001

0.3

0.001

0.001

0.001

0.001

0.001

0.001

0.5

0.001

0.002

0.002

0.001

0.002

0.001

0.1

0.000

0.000

0.000

0.000

0.000

0.000

0.3

0.000

0.000

0.000

0.000

0.000

0.000

0.5

0.000

0.000

0.000

0.000

0.000

0.000

Table B.44: Bias, RMSE, 95% CP, and relative RMSE of slope (βˆ1 ) when ρ = 0.8
over sample sizes 50, 100, and 500 and missing proportions of 0.1, 0.3, and 0.5

n

50

100

500

p

Bias

RMSE

CP

RRMSE

CLS

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

0.001

0.002

0.002

0.044

0.046

0.045

0.94

0.94

0.94

0.97

1.02

0.3

0.001

0.004

0.004

0.044

0.053

0.046

0.94

0.93

0.89

0.95

1.15

0.5

0.001

0.003

0.004

0.044

0.059

0.046

0.94

0.93

0.87

0.95

1.29

0.1

0.000

-0.001

-0.000

0.030

0.032

0.032

0.93

0.94

0.93

0.95

1.00

0.3

0.000

-0.000

0.001

0.030

0.036

0.032

0.93

0.94

0.92

0.95

1.14

0.5

0.000

-0.001

0.001

0.030

0.041

0.032

0.93

0.95

0.89

0.95

1.31

0.1

-0.000

0.000

0.000

0.014

0.014

0.014

0.96

0.96

0.95

1.00

1.00

0.3

-0.000

0.000

-0.000

0.014

0.017

0.014

0.96

0.95

0.92

1.00

1.23

0.5

-0.000

0.000

0.000

0.014

0.017

0.014

0.96

0.94

0.90

1.00

1.23

119

Table B.45: Empirical variance and asymptotic variance of mean of x (µˆx ) when
ρ = 0.8 over different missing proportions of 0.1, 0.3, and 0.5 and sample sizes 50,
100, and 500
n

p

50

100

500

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

1.950

2.183

2.009

1.984

2.151

1.941

0.3

1.950

2.738

2.134

1.984

2.807

1.933

0.5

1.950

3.591

2.331

1.984

3.419

1.936

0.1

0.924

1.025

0.922

0.996

1.100

0.987

0.3

0.924

1.281

0.998

0.996

1.380

0.986

0.5

0.924

1.717

1.069

0.996

1.756

0.990

0.1

0.209

0.228

0.211

0.200

0.222

0.200

0.3

0.209

0.285

0.229

0.200

0.277

0.200

0.5

0.209

0.375

0.246

0.200

0.356

0.200

Table B.46: Bias, RMSE, 95% CP, and relative RMSE of mean of x (µˆx ) when ρ = 0.8
over sample sizes 50, 100, and 500 and missing proportions of 0.1, 0.3, and 0.5

n

50

100

500

p

Bias

RMSE

CP

RRMSE

CLS

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

0.095

0.109

0.111 1.412

1.471

1.398

0.95

0.94

0.93

1.01

1.05

0.3

0.095

0.126

0.108 1.412

1.680

1.395

0.95

0.96

0.92

1.01

1.20

0.5

0.095

0.090

0.122 1.412

1.851

1.397

0.95

0.94

0.92

1.01

1.33

0.1

0.066

0.056

0.064 1.000

1.050

0.996

0.96

0.95

0.94

1.00

1.06

0.3

0.066

0.068

0.049 1.000

1.177

0.994

0.96

0.95

0.94

1.01

1.18

0.5

0.066

0.052

0.041 1.000

1.326

0.996

0.96

0.94

0.94

1.00

1.33

0.1

0.026

0.012

0.019 0.448

0.471

0.447

0.93

0.94

0.92

1.00

1.05

0.3

0.026

0.002

0.016 0.448

0.526

0.447

0.93

0.94

0.93

1.00

1.18

0.5

0.026

0.006

0.012 0.448

0.596

0.447

0.93

0.94

0.92

1.00

1.33

120

Table B.47: Empirical variance and asymptotic variance of mean of y (µˆy ) when
ρ = 0.8 over different missing proportions of 0.1, 0.3, and 0.5 and sample sizes 50,
100, and 500
n

p

50

100

500

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.463

0.511

0.475

0.501

0.545

0.492

0.3

0.463

0.670

0.512

0.501

0.714

0.492

0.5

0.463

0.853

0.538

0.501

0.865

0.488

0.1

0.213

0.233

0.218

0.250

0.275

0.247

0.3

0.213

0.295

0.232

0.250

0.345

0.247

0.5

0.213

0.427

0.266

0.250

0.440

0.248

0.1

0.052

0.056

0.053

0.050

0.056

0.050

0.3

0.052

0.074

0.057

0.050

0.069

0.050

0.5

0.052

0.095

0.060

0.050

0.089

0.050

Table B.48: Bias, RMSE, 95% CP, and relative RMSE of mean of y (µˆy ) when ρ = 0.8
over sample sizes 50, 100, and 500 and missing proportions of 0.1, 0.3, and 0.5

n

50

100

500

p

Bias

RMSE

CP

RRMSE

CLS

Naive

PM

CLS

Naive

PM

CLS Naive

PM

CLS/PM

Naive/PM

0.1

0.028

0.024

0.029

0.708

0.738

0.702

0.95

0.96

0.95

1.01

1.05

0.3

0.028

0.035

0.030

0.708

0.846

0.702

0.95

0.95

0.94

1.01

1.21

0.5

0.028

0.024

0.028

0.708

0.930

0.699

0.95

0.96

0.93

1.01

1.33

0.1

0.027

0.026

0.026

0.500

0.525

0.498

0.96

0.97

0.96

1.01

1.06

0.3

0.027

0.030

0.023

0.500

0.588

0.498

0.96

0.96

0.95

1.01

1.18

0.5

0.027

0.030

0.024

0.500

0.664

0.499

0.96

0.95

0.95

1.00

1.33

0.1

0.007

0.006

0.008

0.224

0.236

0.224

0.94

0.94

0.94

1.00

1.05

0.3

0.007

0.002

0.008

0.224

0.263

0.223

0.94

0.95

0.92

1.00

1.18

0.5

0.007

-0.003

0.003

0.224

0.298

0.224

0.94

0.94

0.91

1.00

1.33

121

Table B.49: Empirical variance and asymptotic variance of standard deviation of x
(σˆx ) when ρ = 0.8 over different missing proportions of 0.1, 0.3, and 0.5 and sample
sizes 50, 100, and 500
n

p

50

100

500

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

1.082

1.139

1.091

1.039

1.131

0.973

0.3

1.082

1.531

1.155

1.039

1.501

1.007

0.5

1.082

1.942

1.296

1.039

1.857

1.062

0.1

0.533

0.579

0.542

0.510

0.564

0.495

0.3

0.533

0.783

0.594

0.510

0.712

0.511

0.5

0.533

0.850

0.665

0.510

0.915

0.551

0.1

0.105

0.117

0.108

0.100

0.111

0.100

0.3

0.105

0.147

0.119

0.100

0.139

0.104

0.5

0.105

0.184

0.131

0.100

0.179

0.111

Table B.50: Bias, RMSE, 95% CP, and relative RMSE of standard deviation of x (σˆx )
when ρ = 0.8 over sample sizes 50, 100, and 500 and missing proportions of 0.1, 0.3,
and 0.5

n

p

Bias
CLS

50

100

500

RMSE

CP

RRMSE

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

-0.092 -0.092

-0.202

1.023

1.068

1.007

0.93

0.93

0.90

1.02

1.06

0.3

-0.092 -0.117

-0.222

1.023

1.231

1.028

0.93

0.94

0.90

1.00

1.20

0.5

-0.092 -0.133

-0.235

1.023

1.369

1.057

0.93

0.92

0.89

0.97

1.30

0.1

-0.046 -0.066

-0.093

0.715

0.754

0.710

0.94

0.93

0.93

1.01

1.06

0.3

-0.046 -0.057

-0.103

0.715

0.846

0.722

0.94

0.94

0.92

0.99

1.17

0.5

-0.046 -0.095

-0.101

0.715

0.961

0.749

0.94

0.95

0.92

0.96

1.28

0.1

-0.008 -0.006

-0.019

0.317

0.334

0.317

0.94

0.95

0.94

1.00

1.05

0.3

-0.008 -0.002

-0.018

0.317

0.373

0.322

0.94

0.95

0.93

0.98

1.16

0.5

-0.008 -0.007

-0.024

0.317

0.423

0.334

0.94

0.94

0.93

0.95

1.27

122

Table B.51: Empirical variance and asymptotic variance of standard deviation of y
(σˆy ) when ρ = 0.8 over different missing proportions of 0.1, 0.3, and 0.5 and sample
sizes 50, 100, and 500
n

p

50

100

500

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.274

0.296

0.279

0.262

0.286

0.247

0.3

0.274

0.382

0.305

0.262

0.382

0.256

0.5

0.274

0.468

0.320

0.262

0.470

0.269

0.1

0.127

0.134

0.127

0.128

0.141

0.124

0.3

0.127

0.182

0.145

0.128

0.178

0.128

0.5

0.127

0.213

0.152

0.128

0.229

0.138

0.1

0.025

0.028

0.025

0.025

0.028

0.025

0.3

0.025

0.034

0.028

0.025

0.035

0.026

0.5

0.025

0.043

0.031

0.025

0.045

0.028

Table B.52: Bias, RMSE, 95% CP, and relative RMSE of standard deviation of y (σˆy )
when ρ = 0.8 over sample sizes 50, 100, and 500 and missing proportions of 0.1, 0.3,
and 0.5

n

p

Bias
CLS

50

100

500

RMSE

CP

RRMSE

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

-0.021 -0.016

-0.069

0.513

0.535

0.501

0.93

0.93

0.92

1.02

1.07

0.3

-0.021 -0.018

-0.073

0.513

0.618

0.512

0.93

0.94

0.90

1.00

1.21

0.5

-0.021 -0.032

-0.093

0.513

0.686

0.527

0.93

0.92

0.90

0.97

1.30

0.1

-0.016 -0.029

-0.042

0.358

0.377

0.355

0.94

0.94

0.93

1.01

1.06

0.3

-0.016 -0.027

-0.043

0.358

0.423

0.361

0.94

0.92

0.92

0.99

1.17

0.5

-0.016 -0.046

-0.044

0.358

0.481

0.375

0.94

0.95

0.94

0.95

1.28

0.1

-0.006 -0.001

-0.008

0.159

0.167

0.158

0.96

0.95

0.95

1.00

1.06

0.3

-0.006 -0.003

-0.013

0.159

0.187

0.162

0.96

0.95

0.95

0.98

1.16

0.5

-0.006 -0.005

-0.011

0.159

0.212

0.167

0.96

0.95

0.93

0.95

1.26

123

Table B.53: Empirical variance and asymptotic variance of correlation (ρ̂) when ρ =
0.8 over different missing proportions of 0.1, 0.3, and 0.5 and sample sizes 50, 100,
and 500
n

p

50

100

500

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.003

0.003

0.003

0.003

0.003

0.003

0.3

0.003

0.004

0.004

0.003

0.004

0.003

0.5

0.003

0.005

0.004

0.003

0.006

0.003

0.1

0.001

0.002

0.002

0.001

0.002

0.001

0.3

0.001

0.002

0.002

0.001

0.002

0.001

0.5

0.001

0.002

0.002

0.001

0.003

0.002

0.1

0.000

0.000

0.000

0.000

0.000

0.000

0.3

0.000

0.000

0.000

0.000

0.000

0.000

0.5

0.000

0.001

0.000

0.000

0.001

0.000

Table B.54: Bias, RMSE, 95% CP, and relative RMSE of correlation (ρ̂) when ρ = 0.8
over sample sizes 50, 100, and 500 and missing proportions of 0.1, 0.3, and 0.5

n

p

Bias
CLS

50

100

500

RMSE

CP

RRMSE

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

-0.005 -0.004

-0.004

0.055

0.057

0.053

0.94

0.94

0.93

1.04

1.07

0.3

-0.005 -0.005

-0.002

0.055

0.067

0.054

0.94

0.94

0.90

1.02

1.23

0.5

-0.005 -0.005

-0.001

0.055

0.075

0.055

0.94

0.95

0.89

1.00

1.37

0.1

-0.003 -0.004

-0.003

0.038

0.040

0.038

0.94

0.94

0.93

1.00

1.07

0.3

-0.003 -0.004

-0.002

0.038

0.045

0.038

0.94

0.94

0.91

1.00

1.20

0.5

-0.003 -0.006

-0.002

0.038

0.051

0.039

0.94

0.95

0.88

0.97

1.32

0.1

-0.000 -0.000

-0.000

0.017

0.017

0.017

0.94

0.94

0.93

1.00

1.00

0.3

-0.000

0.000

0.000

0.017

0.020

0.017

0.94

0.94

0.90

1.00

1.16

0.5

-0.000 -0.000

0.000

0.017

0.022

0.017

0.94

0.93

0.88

1.00

1.29

124

B.2

Results in the Presence of Right Censored Observations in the Data

Figure B.4: Boxplots for estimators of mean of x (µˆx ) and mean of y (µˆy ), when
n = 100, ρ = 0.2 over different right censoring proportions of 0.1, 0.3, and 0.5

Figure B.5: Boxplots for estimators of standard deviation of x (σˆx ) and standard
deviation if y (σˆy ), when n = 100, ρ = 0.2 over different right censoring proportions
of 0.1, 0.3, and 0.5

125

Figure B.6: Boxplots for estimators of correlation between x and y (ρ̂), when n =
100, ρ = 0.2 over different right censoring proportions of 0.1, 0.3, and 0.5

Table B.55: Empirical variance and asymptotic variance of intercept(βˆ0 ) when ρ = 0.2
over different missing proportions of 0.1, 0.3, and 0.5 and sample sizes 50, 100, and
500 in the presence of right censored data
n

50

100

500

p

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.460

0.470

0.458

0.545

0.570

0.556

0.3

0.460

0.536

0.464

0.545

0.606

0.557

0.5

0.460

0.672

0.505

0.545

0.643

0.572

0.1

0.215

0.218

0.217

0.267

0.280

0.268

0.3

0.215

0.254

0.223

0.267

0.292

0.267

0.5

0.215

0.313

0.234

0.267

0.303

0.272

0.1

0.052

0.051

0.052

0.052

0.054

0.052

0.3

0.052

0.057

0.053

0.052

0.057

0.052

0.5

0.052

0.068

0.057

0.052

0.059

0.053

126

Table B.56: Empirical variance and asymptotic variance of slope(βˆ1 ) when ρ = 0.2
over different missing proportions of 0.1, 0.3, and 0.5 and sample sizes 50, 100, and
500 in the presence of right censored data
n

50

100

500

p

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.006

0.006

0.006

0.005

0.006

0.005

0.3

0.006

0.008

0.006

0.005

0.008

0.006

0.5

0.006

0.011

0.006

0.005

0.010

0.006

0.1

0.003

0.003

0.003

0.003

0.003

0.003

0.3

0.003

0.004

0.003

0.003

0.004

0.003

0.5

0.003

0.005

0.003

0.003

0.005

0.003

0.1

0.001

0.001

0.001

0.001

0.001

0.001

0.3

0.001

0.001

0.001

0.001

0.001

0.001

0.5

0.001

0.001

0.001

0.001

0.001

0.001

Table B.57: Empirical variance and asymptotic variance of mean of x (µˆx ) when
ρ = 0.2 over different right censoring proportions of 0.1, 0.3, and 0.5 and sample sizes
50, 100, and 500
n

50

100

500

p

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

1.955

2.124

1.965

1.984

2.078

1.971

0.3

1.955

2.211

2.031

1.984

2.128

2.001

0.5

1.955

2.488

2.140

1.984

2.278

1.996

0.1

0.958

1.000

0.967

0.997

0.988

0.986

0.3

0.958

1.078

0.987

0.997

1.055

0.992

0.5

0.958

1.237

1.050

0.997

1.124

0.991

0.1

0.206

0.213

0.204

0.200

0.197

0.199

0.3

0.206

0.238

0.210

0.200

0.212

0.202

0.5

0.206

0.282

0.222

0.200

0.226

0.201

127

Table B.58: Bias, RMSE, 95% CP, and relative RMSE of mean of x (µˆx ) when ρ = 0.2
over sample sizes 50, 100, and 500 and right censoring proportions of 0.1, 0.3, and 0.5

n

p

Bias
CLS

50

100

500

RMSE

CP

RRMSE

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

0.099 -0.980

0.131

1.412

1.743

1.410

0.95

0.90

0.94

1.00

1.24

0.3

0.099 -3.029

0.130

1.412

3.362

1.420

0.95

0.46

0.93

0.99

2.37

0.5

0.099 -4.562

0.092

1.412

4.805

1.416

0.95

0.16

0.92

1.00

3.39

0.1

0.064 -1.149

0.061

1.001

1.519

0.995

0.96

0.78

0.96

1.01

1.53

0.3

0.064 -2.927

0.022

1.001

3.102

0.996

0.96

0.19

0.94

1.00

3.11

0.5

0.064 -4.485

-0.020

1.001

4.609

0.996

0.96

0.01

0.93

1.00

4.63

0.1

0.027 -1.209

0.025

0.448

1.288

0.447

0.94

0.22

0.94

1.00

2.88

0.3

0.027 -3.022

0.001

0.448

3.057

0.449

0.94

0.00

0.94

1.00

6.81

0.5

0.027 -4.605

-0.043

0.448

4.629

0.450

0.94

0.00

0.93

0.99

10.28

Table B.59: Empirical variance and asymptotic variance of mean of y (µˆy ) when
ρ = 0.2 over different right censoring proportions of 0.1, 0.3, and 0.5 and sample sizes
50, 100, and 500
n

50

100

500

p

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.444

0.465

0.444

0.505

0.522

0.504

0.3

0.444

0.542

0.450

0.505

0.551

0.513

0.5

0.444

0.629

0.488

0.505

0.595

0.513

0.1

0.216

0.228

0.218

0.251

0.272

0.249

0.3

0.216

0.256

0.227

0.251

0.270

0.252

0.5

0.216

0.291

0.238

0.251

0.290

0.251

0.1

0.051

0.051

0.051

0.050

0.054

0.050

0.3

0.051

0.059

0.052

0.050

0.053

0.050

0.5

0.051

0.068

0.056

0.050

0.057

0.050

128

Table B.60: Bias, RMSE, 95% CP, and relative RMSE of mean of y (µˆy ) when ρ = 0.2
over sample sizes 50, 100, and 500 and right censoring proportions of 0.1, 0.3, and 0.5

n

p

Bias
CLS

50

100

500

RMSE

CP

RRMSE

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

0.005 -0.834

0.028

0.711

1.103

0.711

0.95

0.81

0.96

1.00

1.55

0.3

0.005 -1.568

0.026

0.711

1.735

0.716

0.95

0.44

0.96

0.99

2.42

0.5

0.005 -2.362

0.013

0.711

2.485

0.716

0.95

0.11

0.94

0.99

3.47

0.1

0.015 -0.596

0.016

0.501

0.792

0.499

0.97

0.80

0.96

1.00

1.59

0.3

0.015 -1.491

0.007

0.501

1.579

0.502

0.97

0.16

0.96

1.00

3.15

0.5

0.015 -2.276

-0.016

0.501

2.339

0.502

0.97

0.01

0.95

1.00

4.66

0.1

0.001 -0.619

-0.000

0.223

0.661

0.223

0.93

0.23

0.94

1.00

2.97

0.3

0.001 -1.523

-0.011

0.223

1.540

0.225

0.93

0.00

0.94

0.99

6.85

0.5

0.001 -2.307

-0.034

0.223

2.319

0.227

0.93

0.00

0.93

0.99

10.23

Table B.61: Empirical variance and asymptotic variance of standard deviation of x
(σˆx ) when ρ = 0.2 over different right censoring proportions of 0.1, 0.3, and 0.5 and
sample sizes 50, 100, and 500
n

50

100

500

p

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

1.104

1.072

1.144

1.023

1.076

0.986

0.3

1.104

1.014

1.304

1.023

1.111

1.000

0.5

1.104

1.162

1.506

1.023

1.204

0.998

0.1

0.551

0.523

0.587

0.506

0.502

0.493

0.3

0.551

0.545

0.661

0.506

0.539

0.496

0.5

0.551

0.600

0.759

0.506

0.577

0.495

0.1

0.102

0.100

0.110

0.100

0.099

0.100

0.3

0.102

0.108

0.133

0.100

0.107

0.101

0.5

0.102

0.123

0.152

0.100

0.113

0.101

129

Table B.62: Bias, RMSE, 95% CP, and relative RMSE of standard deviation of x (σˆx )
when ρ = 0.2 over sample sizes 50, 100, and 500 and right censoring proportions of
0.1, 0.3, and 0.5

n

p

Bias
CLS

50

100

500

RMSE

CP

RRMSE

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

-0.096 -0.906

-0.129

1.016

1.377

1.001

0.94

0.80

0.93

1.02

1.38

0.3

-0.096 -2.086

-0.163

1.016

2.337

1.013

0.94

0.47

0.90

1.00

2.31

0.5

-0.096 -2.745

-0.234

1.016

2.956

1.026

0.94

0.32

0.86

0.99

2.88

0.1

-0.043 -1.014

-0.101

0.713

1.237

0.709

0.94

0.66

0.91

1.00

1.74

0.3

-0.043 -1.998

-0.170

0.713

2.129

0.725

0.94

0.26

0.89

0.98

2.94

0.5

-0.043 -2.650

-0.231

0.713

2.757

0.741

0.94

0.10

0.86

0.96

3.72

0.1

-0.008 -0.995

-0.023

0.317

1.044

0.316

0.95

0.14

0.94

1.00

3.30

0.3

-0.008 -1.948

-0.066

0.317

1.976

0.324

0.95

0.00

0.91

0.98

6.09

0.5

-0.008 -2.599

-0.129

0.317

2.621

0.342

0.95

0.00

0.86

0.93

7.66

Table B.63: Empirical variance and asymptotic variance of standard deviation of y
(σˆy ) when ρ = 0.2 over different right censoring proportions of 0.1, 0.3, and 0.5 and
sample sizes 50, 100, and 500
n

50

100

500

p

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.257

0.235

0.280

0.261

0.270

0.252

0.3

0.257

0.254

0.318

0.261

0.288

0.256

0.5

0.257

0.291

0.373

0.261

0.315

0.256

0.1

0.126

0.113

0.132

0.127

0.138

0.124

0.3

0.126

0.117

0.147

0.127

0.138

0.126

0.5

0.126

0.143

0.163

0.127

0.149

0.126

0.1

0.025

0.022

0.026

0.025

0.027

0.025

0.3

0.025

0.023

0.028

0.025

0.027

0.025

0.5

0.025

0.027

0.034

0.025

0.028

0.025

130

Table B.64: Bias, RMSE, 95% CP, and relative RMSE of standard deviation of y (σˆy )
when ρ = 0.2 over sample sizes 50, 100, and 500 and right censoring proportions of
0.1, 0.3, and 0.5

n

50

100

500

p

Bias

RMSE

CP

RRMSE

CLS

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

0.000

-0.635

-0.006

0.511

0.821

0.502

0.94

0.73

0.93

1.02

1.63

0.3

0.000

-0.973

-0.020

0.511

1.111

0.507

0.94

0.52

0.90

1.01

2.19

0.5

0.000

-1.291

-0.048

0.511

1.408

0.509

0.94

0.39

0.88

1.00

2.77

0.1

-0.005 -0.481

-0.026

0.357

0.608

0.354

0.94

0.73

0.93

1.01

1.72

0.3

-0.005 -0.951

-0.044

0.357

1.021

0.358

0.94

0.29

0.91

1.00

2.86

0.5

-0.005 -1.270

-0.077

0.357

1.327

0.363

0.94

0.14

0.89

0.98

3.66

0.1

-0.008 -0.502

-0.015

0.158

0.528

0.159

0.95

0.14

0.96

1.00

3.33

0.3

-0.008 -0.974

-0.034

0.158

0.987

0.162

0.95

0.00

0.92

0.98

6.09

0.5

-0.008 -1.295

-0.067

0.158

1.306

0.172

0.95

0.00

0.88

0.92

7.59

Table B.65: Empirical variance and asymptotic variance of correlation (ρ̂) when ρ =
0.2 over different right censoring proportions of 0.1, 0.3, and 0.5 and sample sizes 50,
100, and 500
n

50

100

500

p

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.020

0.022

0.020

0.019

0.022

0.018

0.3

0.020

0.030

0.020

0.019

0.027

0.018

0.5

0.020

0.039

0.020

0.019

0.034

0.018

0.1

0.010

0.011

0.010

0.009

0.011

0.009

0.3

0.010

0.013

0.010

0.009

0.013

0.009

0.5

0.010

0.018

0.010

0.009

0.017

0.009

0.1

0.002

0.002

0.002

0.002

0.002

0.002

0.3

0.002

0.003

0.002

0.002

0.003

0.002

0.5

0.002

0.004

0.002

0.002

0.003

0.002

131

Table B.66: Bias, RMSE, 95% CP, and relative RMSE of correlation y (ρ̂) when
ρ = 0.2 over sample sizes 50, 100, and 500 and right censoring proportions of 0.1, 0.3,
and 0.5

n

p

Bias
CLS

50

100

500

RMSE

CP

RRMSE

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

-0.005 -0.040

-0.006

0.137

0.152

0.134

0.93

0.93

0.91

1.02

1.14

0.3

-0.005 -0.069

-0.011

0.137

0.178

0.134

0.93

0.92

0.92

1.02

1.32

0.5

-0.005 -0.086

-0.018

0.137

0.204

0.135

0.93

0.90

0.92

1.01

1.51

0.1

-0.004 -0.038

-0.005

0.097

0.109

0.096

0.93

0.92

0.93

1.01

1.14

0.3

-0.004 -0.061

-0.009

0.097

0.130

0.096

0.93

0.92

0.93

1.01

1.35

0.5

-0.004 -0.074

-0.015

0.097

0.149

0.097

0.93

0.90

0.93

0.99

1.53

0.1

0.000

-0.034

-0.001

0.042

0.057

0.042

0.95

0.89

0.95

1.00

1.35

0.3

0.000

-0.064

-0.005

0.042

0.082

0.043

0.95

0.79

0.96

0.99

1.91

0.5

0.000

-0.083

-0.011

0.042

0.101

0.045

0.95

0.71

0.95

0.95

2.25

Table B.67: Empirical variance and asymptotic variance of intercept (βˆ0 ) when ρ = 0.4
over different right censoring proportions of 0.1, 0.3, and 0.5 and sample sizes 50, 100,
and 500
n

50

100

500

p

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.400

0.401

0.400

0.476

0.508

0.486

0.3

0.400

0.460

0.401

0.476

0.509

0.493

0.5

0.400

0.593

0.437

0.476

0.594

0.495

0.1

0.189

0.195

0.191

0.234

0.249

0.234

0.3

0.189

0.222

0.194

0.234

0.245

0.236

0.5

0.189

0.277

0.204

0.234

0.280

0.236

0.1

0.045

0.044

0.045

0.046

0.049

0.046

0.3

0.045

0.050

0.046

0.046

0.048

0.046

0.5

0.045

0.058

0.049

0.046

0.055

0.046

132

Table B.68: Bias, RMSE, 95% CP, and relative RMSE of intercept (βˆ0 ) when ρ = 0.4
over sample sizes 50, 100, and 500 and right censoring proportions of 0.1, 0.3, and 0.5

n

p

Bias
CLS

50

100

500

RMSE

CP

RRMSE

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

-0.003 -0.418

0.014

0.690

0.827

0.697

0.96

0.93

0.97

0.99

1.19

0.3

-0.003 -1.037

0.016

0.690

1.259

0.703

0.96

0.70

0.96

0.98

1.79

0.5

-0.003 -1.640

0.020

0.690

1.812

0.704

0.96

0.42

0.95

0.98

2.57

0.1

0.010

-0.346

0.013

0.483

0.608

0.484

0.97

0.91

0.97

1.00

1.26

0.3

0.010

-0.971

0.012

0.483

1.090

0.486

0.97

0.50

0.96

0.99

2.24

0.5

0.010

-1.565

0.006

0.483

1.652

0.486

0.97

0.15

0.97

1.00

3.40

0.1

-0.003 -0.363

-0.002

0.214

0.425

0.214

0.95

0.65

0.95

1.00

1.98

0.3

-0.003 -0.995

-0.003

0.214

1.018

0.215

0.95

0.00

0.94

1.00

4.73

0.5

-0.003 -1.602

-0.014

0.214

1.619

0.215

0.95

0.00

0.93

1.00

7.52

Table B.69: Empirical variance and asymptotic variance of slope (βˆ1 ) when ρ = 0.4
over different right censoring proportions of 0.1, 0.3, and 0.5 and sample sizes 50, 100,
and 500
n

50

100

500

p

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.005

0.006

0.005

0.005

0.006

0.005

0.3

0.005

0.008

0.005

0.005

0.007

0.005

0.5

0.005

0.011

0.006

0.005

0.009

0.005

0.1

0.002

0.003

0.002

0.002

0.003

0.002

0.3

0.002

0.004

0.002

0.002

0.003

0.002

0.5

0.002

0.005

0.003

0.002

0.004

0.002

0.1

0.000

0.000

0.000

0.000

0.001

0.000

0.3

0.000

0.001

0.000

0.000

0.001

0.000

0.5

0.000

0.001

0.000

0.000

0.001

0.001

133

Table B.70: Bias, RMSE, 95% CP, and relative RMSE of slope (βˆ1 ) when ρ = 0.4
over sample sizes 50, 100, and 500 and right censoring proportions of 0.1, 0.3, and 0.5

n

50

100

500

p

Bias

RMSE

CP

RRMSE

CLS

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

0.001

-0.030

0.000

0.068

0.080

0.069

0.93

0.90

0.92

0.99

1.17

0.3

0.001

-0.053

-0.005

0.068

0.099

0.070

0.93

0.87

0.92

0.98

1.43

0.5

0.001

-0.069

-0.010

0.068

0.118

0.072

0.93

0.84

0.92

0.94

1.64

0.1

-0.001 -0.028

-0.002

0.047

0.058

0.047

0.94

0.90

0.93

1.00

1.23

0.3

-0.001 -0.051

-0.005

0.047

0.076

0.048

0.94

0.83

0.94

0.97

1.58

0.5

-0.001 -0.065

-0.010

0.047

0.092

0.050

0.94

0.81

0.92

0.94

1.83

0.1

0.000

-0.028

-0.001

0.020

0.036

0.020

0.96

0.78

0.96

1.00

1.79

0.3

0.000

-0.054

-0.005

0.020

0.059

0.021

0.96

0.43

0.95

0.97

2.86

0.5

0.000

-0.071

-0.010

0.020

0.076

0.025

0.96

0.27

0.92

0.82

3.10

Table B.71: Empirical variance and asymptotic variance of mean of x (µˆx ) when
ρ = 0.4 over different right censoring proportions of 0.1, 0.3, and 0.5 and sample sizes
50, 100, and 500
n

50

100

500

p

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

1.955

2.084

1.982

1.984

2.110

1.959

0.3

1.955

2.209

2.044

1.984

2.081

1.976

0.5

1.955

2.414

2.125

1.984

2.194

1.947

0.1

0.948

0.985

0.956

0.997

1.064

0.982

0.3

0.948

1.076

0.983

0.997

1.036

0.980

0.5

0.948

1.167

1.025

0.997

1.088

0.967

0.1

0.207

0.211

0.205

0.200

0.213

0.199

0.3

0.207

0.231

0.210

0.200

0.207

0.199

0.5

0.207

0.271

0.220

0.200

0.218

0.196

134

Table B.72: Bias, RMSE, 95% CP, and relative RMSE of mean of x (µˆx ) when ρ = 0.4
over sample sizes 50, 100, and 500 and right censoring proportions of 0.1, 0.3, and 0.5

n

p

Bias
CLS

50

100

500

RMSE

CP

RRMSE

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

0.099 -1.420

0.119

1.412

2.032

1.405

0.94

0.84

0.94

1.01

1.45

0.3

0.099 -3.256

0.097

1.412

3.562

1.409

0.94

0.39

0.94

1.00

2.53

0.5

0.099 -4.876

0.018

1.412

5.096

1.395

0.94

0.10

0.93

1.01

3.65

0.1

0.065 -1.268

0.054

1.000

1.635

0.992

0.96

0.77

0.96

1.01

1.65

0.3

0.065 -3.160

-0.008

1.000

3.320

0.990

0.96

0.12

0.95

1.01

3.35

0.5

0.065 -4.777

-0.091

1.000

4.890

0.988

0.96

0.01

0.93

1.01

4.95

0.1

0.027 -1.326

0.017

0.448

1.404

0.446

0.94

0.17

0.94

1.00

3.15

0.3

0.027 -3.246

-0.033

0.448

3.278

0.447

0.94

0.00

0.94

1.00

7.33

0.5

0.027 -4.884

-0.117

0.448

4.907

0.458

0.94

0.00

0.92

0.98

10.71

Table B.73: Empirical variance and asymptotic variance of mean of y (µˆy ) when
ρ = 0.4 over different right censoring proportions of 0.1, 0.3, and 0.5 and sample sizes
50, 100, and 500
n

50

100

500

p

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.448

0.468

0.448

0.504

0.541

0.502

0.3

0.448

0.532

0.458

0.504

0.536

0.504

0.5

0.448

0.619

0.492

0.504

0.572

0.500

0.1

0.214

0.224

0.216

0.251

0.269

0.248

0.3

0.214

0.245

0.225

0.251

0.264

0.248

0.5

0.214

0.276

0.236

0.251

0.279

0.245

0.1

0.051

0.051

0.051

0.050

0.053

0.050

0.3

0.051

0.059

0.053

0.050

0.052

0.050

0.5

0.051

0.066

0.056

0.050

0.055

0.049

135

Table B.74: Bias, RMSE, 95% CP, and relative RMSE of mean of y (µˆy ) when ρ = 0.4
over sample sizes 50, 100, and 500 and right censoring proportions of 0.1, 0.3, and 0.5

n

p

Bias
CLS

50

100

500

RMSE

CP

RRMSE

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

0.012 -0.754

0.031

0.710

1.053

0.709

0.95

0.84

0.95

1.00

1.49

0.3

0.012 -1.678

0.011

0.710

1.831

0.710

0.95

0.36

0.95

1.00

2.58

0.5

0.012 -2.489

-0.017

0.710

2.602

0.707

0.95

0.08

0.93

1.00

3.68

0.1

0.019 -0.651

0.017

0.501

0.832

0.498

0.97

0.76

0.96

1.01

1.67

0.3

0.019 -1.598

-0.009

0.501

1.678

0.498

0.97

0.11

0.95

1.01

3.37

0.5

0.019 -2.413

-0.047

0.501

2.471

0.498

0.97

0.01

0.95

1.01

4.97

0.1

0.003 -0.675

-0.002

0.223

0.714

0.223

0.93

0.15

0.93

1.00

3.20

0.3

0.003 -1.632

-0.025

0.223

1.647

0.225

0.93

0.00

0.93

1.00

7.34

0.5

0.003 -2.447

-0.067

0.223

2.458

0.232

0.93

0.00

0.91

0.97

10.62

Table B.75: Empirical variance and asymptotic variance of standard deviation of x
(σˆx ) when ρ = 0.4 over different right censoring proportions of 0.1, 0.3, and 0.5 and
sample sizes 50, 100, and 500
n

50

100

500

p

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

1.096

0.984

1.154

1.025

1.094

0.980

0.3

1.096

0.997

1.271

1.025

1.087

0.988

0.5

1.096

1.092

1.425

1.025

1.159

0.974

0.1

0.546

0.506

0.575

0.507

0.541

0.491

0.3

0.546

0.520

0.637

0.507

0.529

0.490

0.5

0.546

0.561

0.714

0.507

0.559

0.484

0.1

0.103

0.100

0.110

0.100

0.107

0.099

0.3

0.103

0.105

0.131

0.100

0.104

0.100

0.5

0.103

0.117

0.147

0.100

0.110

0.098

136

Table B.76: Bias, RMSE, 95% CP, and relative RMSE of standard deviation of x (σˆx )
when ρ = 0.4 over sample sizes 50, 100, and 500 and right censoring proportions of
0.1, 0.3, and 0.5

n

p

Bias
CLS

50

100

500

RMSE

CP

RRMSE

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

-0.096 -1.198

-0.160

1.017

1.590

1.003

0.94

0.73

0.91

1.01

1.59

0.3

-0.096 -2.090

-0.222

1.017

2.336

1.019

0.94

0.46

0.88

1.00

2.29

0.5

-0.096 -2.725

-0.352

1.017

2.929

1.048

0.94

0.33

0.84

0.97

2.80

0.1

-0.044 -1.051

-0.120

0.713

1.283

0.711

0.93

0.67

0.91

1.00

1.80

0.3

-0.044 -1.993

-0.230

0.713

2.121

0.737

0.93

0.26

0.88

0.97

2.88

0.5

-0.044 -2.626

-0.347

0.713

2.730

0.777

0.93

0.09

0.84

0.92

3.51

0.1

-0.008 -1.029

-0.041

0.317

1.080

0.318

0.95

0.13

0.93

1.00

3.40

0.3

-0.008 -1.963

-0.128

0.317

1.989

0.340

0.95

0.00

0.88

0.93

5.84

0.5

-0.008 -2.583

-0.247

0.317

2.604

0.399

0.95

0.00

0.80

0.79

6.53

Table B.77: Empirical variance and asymptotic variance of standard deviation of y
(σˆy ) when ρ = 0.4 over different right censoring proportions of 0.1, 0.3, and 0.5 and
sample sizes 50, 100, and 500
n

50

100

500

p

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.264

0.246

0.275

0.261

0.280

0.251

0.3

0.264

0.252

0.319

0.261

0.280

0.252

0.5

0.264

0.278

0.349

0.261

0.302

0.250

0.1

0.127

0.110

0.129

0.127

0.137

0.124

0.3

0.127

0.115

0.145

0.127

0.135

0.124

0.5

0.127

0.132

0.157

0.127

0.143

0.123

0.1

0.024

0.022

0.026

0.025

0.027

0.025

0.3

0.024

0.023

0.029

0.025

0.026

0.025

0.5

0.024

0.026

0.033

0.025

0.027

0.025

137

Table B.78: Bias, RMSE, 95% CP, and relative RMSE of standard deviation of y (σˆy )
when ρ = 0.4 over sample sizes 50, 100, and 500 and right censoring proportions of
0.1, 0.3, and 0.5

n

p

Bias
CLS

50

100

500

RMSE

CP

RRMSE

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

-0.004 -0.544

-0.018

0.511

0.759

0.501

0.93

0.77

0.92

1.02

1.51

0.3

-0.004 -0.984

-0.063

0.511

1.118

0.506

0.93

0.52

0.89

1.01

2.21

0.5

-0.004 -1.285

-0.111

0.511

1.398

0.512

0.93

0.37

0.87

1.00

2.73

0.1

-0.008 -0.501

-0.038

0.357

0.622

0.354

0.94

0.72

0.93

1.01

1.76

0.3

-0.008 -0.959

-0.083

0.357

1.027

0.362

0.94

0.28

0.91

0.99

2.84

0.5

-0.008 -1.263

-0.136

0.357

1.319

0.376

0.94

0.13

0.87

0.95

3.51

0.1

-0.007 -0.517

-0.024

0.158

0.543

0.159

0.96

0.11

0.96

0.99

3.41

0.3

-0.007 -0.979

-0.063

0.158

0.993

0.170

0.96

0.00

0.89

0.93

5.84

0.5

-0.007 -1.286

-0.123

0.158

1.297

0.199

0.96

0.00

0.84

0.80

6.51

Table B.79: Empirical variance and asymptotic variance of correlation (ρ̂) when ρ =
0.4 over different right censoring proportions of 0.1, 0.3, and 0.5 and sample sizes 50,
100, and 500
n

50

100

500

p

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.015

0.020

0.015

0.015

0.018

0.014

0.3

0.015

0.026

0.016

0.015

0.023

0.014

0.5

0.015

0.035

0.016

0.015

0.030

0.015

0.1

0.007

0.009

0.007

0.007

0.009

0.007

0.3

0.007

0.012

0.008

0.007

0.011

0.007

0.5

0.007

0.016

0.008

0.007

0.014

0.007

0.1

0.002

0.002

0.002

0.001

0.002

0.001

0.3

0.002

0.002

0.002

0.001

0.002

0.001

0.5

0.002

0.003

0.002

0.001

0.003

0.002

138

Table B.80: Bias, RMSE, 95% CP, and relative RMSE of correlation (ρ̂) when ρ = 0.4
over sample sizes 50, 100, and 500 and right censoring proportions of 0.1, 0.3, and 0.5

n

p

Bias
CLS

50

100

500

RMSE

CP

RRMSE

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

-0.007 -0.070

-0.010

0.121

0.152

0.119

0.93

0.91

0.92

1.02

1.28

0.3

-0.007 -0.117

-0.019

0.121

0.192

0.121

0.93

0.90

0.93

1.00

1.59

0.5

-0.007 -0.151

-0.030

0.121

0.230

0.124

0.93

0.86

0.91

0.98

1.85

0.1

-0.005 -0.061

-0.007

0.085

0.111

0.085

0.93

0.91

0.93

1.00

1.32

0.3

-0.005 -0.108

-0.015

0.085

0.152

0.086

0.93

0.85

0.94

0.99

1.76

0.5

-0.005 -0.136

-0.026

0.085

0.182

0.089

0.93

0.80

0.92

0.95

2.04

0.1

0.000

-0.056

-0.002

0.037

0.069

0.038

0.95

0.75

0.95

1.00

1.85

0.3

0.000

-0.108

-0.010

0.037

0.118

0.039

0.95

0.38

0.95

0.96

3.03

0.5

0.000

-0.142

-0.020

0.037

0.152

0.044

0.95

0.23

0.92

0.85

3.46

Table B.81: Empirical variance and asymptotic variance of intercept (βˆ0 ) when ρ = 0.6
over different right censoring proportions of 0.1, 0.3, and 0.5 and sample sizes 50, 100,
and 500
n

50

100

500

p

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.304

0.315

0.303

0.363

0.351

0.370

0.3

0.304

0.350

0.305

0.363

0.359

0.377

0.5

0.304

0.449

0.327

0.363

0.418

0.380

0.1

0.145

0.151

0.146

0.178

0.172

0.178

0.3

0.145

0.167

0.152

0.178

0.173

0.180

0.5

0.145

0.209

0.154

0.178

0.198

0.181

0.1

0.034

0.034

0.034

0.035

0.033

0.035

0.3

0.034

0.038

0.035

0.035

0.034

0.035

0.5

0.034

0.044

0.035

0.035

0.039

0.035

139

Table B.82: Bias, RMSE, 95% CP, and relative RMSE of intercept (βˆ0 ) when ρ = 0.6
over sample sizes 50, 100, and 500 and right censoring proportions of 0.1, 0.3, and 0.5

n

p

Bias
CLS

50

100

500

RMSE

CP

RRMSE

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

-0.008 -0.280

0.005

0.602

0.655

0.608

0.97

0.93

0.96

0.99

1.08

0.3

-0.008 -0.741

0.012

0.602

0.953

0.614

0.97

0.77

0.96

0.98

1.55

0.5

-0.008 -1.199

0.012

0.602

1.362

0.617

0.97

0.54

0.96

0.98

2.21

0.1

0.006

-0.226

0.009

0.422

0.472

0.422

0.97

0.92

0.97

1.00

1.12

0.3

0.006

-0.680

0.011

0.422

0.797

0.425

0.97

0.61

0.96

0.99

1.88

0.5

0.006

-1.148

0.011

0.422

1.231

0.425

0.97

0.27

0.96

0.99

2.90

0.1

-0.003 -0.238

-0.001

0.187

0.300

0.187

0.94

0.75

0.95

1.00

1.61

0.3

-0.003 -0.701

-0.001

0.187

0.724

0.188

0.94

0.05

0.95

0.99

3.85

0.5

-0.003 -1.175

-0.007

0.187

1.191

0.188

0.94

0.00

0.94

1.00

6.35

Table B.83: Empirical variance and asymptotic variance of slope (βˆ1 ) when ρ = 0.6
over different right censoring proportions of 0.1, 0.3, and 0.5 and sample sizes 50, 100,
and 500
n

50

100

500

p

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.004

0.005

0.004

0.004

0.004

0.004

0.3

0.004

0.007

0.004

0.004

0.006

0.004

0.5

0.004

0.009

0.005

0.004

0.008

0.004

0.1

0.002

0.002

0.002

0.002

0.002

0.002

0.3

0.002

0.003

0.002

0.002

0.003

0.002

0.5

0.002

0.004

0.002

0.002

0.003

0.002

0.1

0.000

0.000

0.000

0.000

0.000

0.000

0.3

0.000

0.001

0.000

0.000

0.001

0.000

0.5

0.000

0.001

0.000

0.000

0.001

0.000

140

Table B.84: Bias, RMSE, 95% CP, and relative RMSE of slope (βˆ1 ) when ρ = 0.6
over sample sizes 50, 100, and 500 and right censoring proportions of 0.1, 0.3, and 0.5

n

50

100

500

p

Bias

RMSE

CP

RRMSE

CLS

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

0.001

-0.031

0.000

0.059

0.073

0.060

0.94

0.89

0.92

0.99

1.22

0.3

0.001

-0.059

-0.006

0.059

0.096

0.061

0.94

0.85

0.92

0.97

1.57

0.5

0.001

-0.078

-0.013

0.059

0.117

0.065

0.94

0.80

0.90

0.92

1.81

0.1

-0.000 -0.030

-0.002

0.041

0.054

0.041

0.93

0.87

0.93

1.00

1.31

0.3

-0.000 -0.058

-0.006

0.041

0.078

0.043

0.93

0.78

0.92

0.96

1.82

0.5

-0.000 -0.077

-0.013

0.041

0.096

0.046

0.93

0.71

0.91

0.91

2.12

0.1

0.000

-0.030

-0.002

0.017

0.036

0.017

0.96

0.67

0.95

0.99

2.07

0.3

0.000

-0.059

-0.007

0.017

0.063

0.019

0.96

0.27

0.92

0.94

3.43

0.5

0.000

-0.081

-0.013

0.017

0.084

0.024

0.96

0.14

0.88

0.72

3.50

Table B.85: Empirical variance and asymptotic variance of mean of x (µˆx ) when
ρ = 0.6 over different right censoring proportions of 0.1, 0.3, and 0.5 and sample sizes
50, 100, and 500
n

50

100

500

p

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

1.954

2.087

1.979

1.984

1.802

1.947

0.3

1.954

2.222

2.053

1.984

1.744

1.943

0.5

1.954

2.345

2.083

1.984

1.772

1.900

0.1

0.937

0.985

0.945

0.996

0.911

0.976

0.3

0.937

1.055

0.974

0.996

0.869

0.966

0.5

0.937

1.146

0.999

0.996

0.880

0.946

0.1

0.208

0.214

0.207

0.200

0.182

0.198

0.3

0.208

0.233

0.213

0.200

0.174

0.196

0.5

0.208

0.261

0.216

0.200

0.176

0.192

141

Table B.86: Bias, RMSE, 95% CP, and relative RMSE of mean of x (µˆx ) when ρ = 0.6
over sample sizes 50, 100, and 500 and right censoring proportions of 0.1, 0.3, and 0.5

n

p

Bias
CLS

50

100

500

RMSE

CP

RRMSE

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

0.097 -1.517

0.103

1.412

2.026

1.399

0.94

0.79

0.93

1.01

1.45

0.3

0.097 -3.400

0.049

1.412

3.648

1.395

0.94

0.29

0.93

1.01

2.62

0.5

0.097 -5.032

-0.042

1.412

5.205

1.379

0.94

0.05

0.93

1.02

3.77

0.1

0.066 -1.342

0.043

1.000

1.647

0.989

0.95

0.70

0.96

1.01

1.67

0.3

0.066 -3.296

-0.044

1.000

3.425

0.984

0.95

0.08

0.95

1.02

3.48

0.5

0.066 -4.951

-0.158

1.000

5.039

0.985

0.95

0.00

0.93

1.02

5.11

0.1

0.026 -1.398

0.005

0.448

1.462

0.444

0.94

0.10

0.94

1.01

3.29

0.3

0.026 -3.363

-0.071

0.448

3.389

0.449

0.94

0.00

0.93

1.00

7.55

0.5

0.026 -5.031

-0.188

0.448

5.049

0.477

0.94

0.00

0.90

0.94

10.59

Table B.87: Empirical variance and asymptotic variance of mean of y (µˆy ) when
ρ = 0.6 over different right censoring proportions of 0.1, 0.3, and 0.5 and sample sizes
50, 100, and 500
n

50

100

500

p

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.454

0.482

0.456

0.503

0.462

0.497

0.3

0.454

0.533

0.471

0.503

0.447

0.495

0.5

0.454

0.608

0.491

0.503

0.460

0.486

0.1

0.212

0.221

0.213

0.250

0.230

0.246

0.3

0.212

0.236

0.227

0.250

0.221

0.244

0.5

0.212

0.262

0.231

0.250

0.225

0.239

0.1

0.052

0.051

0.052

0.050

0.046

0.049

0.3

0.052

0.058

0.054

0.050

0.044

0.049

0.5

0.052

0.064

0.055

0.050

0.044

0.048

142

Table B.88: Bias, RMSE, 95% CP, and relative RMSE of mean of y (µˆy ) when ρ = 0.6
over sample sizes 50, 100, and 500 and right censoring proportions of 0.1, 0.3, and 0.5

n

p

Bias
CLS

50

100

500

RMSE

CP

RRMSE

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

0.019 -0.784

0.030

0.710

1.038

0.706

0.96

0.80

0.96

1.01

1.47

0.3

0.019 -1.740

-0.000

0.710

1.864

0.704

0.96

0.28

0.95

1.01

2.65

0.5

0.019 -2.553

-0.047

0.710

2.641

0.699

0.96

0.05

0.94

1.02

3.78

0.1

0.023 -0.680

0.015

0.501

0.832

0.496

0.97

0.71

0.96

1.01

1.68

0.3

0.023 -1.654

-0.023

0.501

1.720

0.495

0.97

0.07

0.95

1.01

3.47

0.5

0.023 -2.488

-0.079

0.501

2.533

0.495

0.97

0.00

0.95

1.01

5.11

0.1

0.005 -0.706

-0.005

0.223

0.737

0.222

0.93

0.09

0.93

1.00

3.32

0.3

0.005 -1.689

-0.041

0.223

1.702

0.225

0.93

0.00

0.93

0.99

7.55

0.5

0.005 -2.521

-0.101

0.223

2.530

0.241

0.93

0.00

0.90

0.93

10.48

Table B.89: Empirical variance and asymptotic variance of standard deviation of x
(σˆx ) when ρ = 0.6 over different right censoring proportions of 0.1, 0.3, and 0.5 and
sample sizes 50, 100, and 500
n

50

100

500

p

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

1.088

0.943

1.125

1.030

0.938

0.973

0.3

1.088

0.944

1.245

1.030

0.914

0.972

0.5

1.088

1.006

1.348

1.030

0.937

0.950

0.1

0.540

0.495

0.564

0.507

0.465

0.488

0.3

0.540

0.491

0.614

0.507

0.444

0.483

0.5

0.540

0.522

0.679

0.507

0.452

0.473

0.1

0.104

0.099

0.112

0.100

0.092

0.099

0.3

0.104

0.101

0.130

0.100

0.088

0.098

0.5

0.104

0.110

0.145

0.100

0.088

0.096

143

Table B.90: Bias, RMSE, 95% CP, and relative RMSE of standard deviation of x (σˆx )
when ρ = 0.6 over sample sizes 50, 100, and 500 and right censoring proportions of
0.1, 0.3, and 0.5

n

p

Bias
CLS

50

100

500

RMSE

CP

RRMSE

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

-0.095 -1.270

-0.191

1.019

1.597

1.005

0.94

0.68

0.91

1.01

1.59

0.3

-0.095 -2.163

-0.304

1.019

2.364

1.032

0.94

0.40

0.87

0.99

2.29

0.5

-0.095 -2.773

-0.467

1.019

2.937

1.081

0.94

0.24

0.83

0.94

2.72

0.1

-0.045 -1.114

-0.149

0.714

1.306

0.714

0.93

0.61

0.91

1.00

1.83

0.3

-0.045 -2.067

-0.298

0.714

2.172

0.756

0.93

0.18

0.87

0.94

2.87

0.5

-0.045 -2.682

-0.454

0.714

2.765

0.824

0.93

0.06

0.82

0.87

3.36

0.1

-0.008 -1.087

-0.067

0.317

1.129

0.321

0.94

0.07

0.93

0.99

3.51

0.3

-0.008 -2.030

-0.198

0.317

2.051

0.370

0.94

0.00

0.85

0.86

5.54

0.5

-0.008 -2.645

-0.357

0.317

2.662

0.473

0.94

0.00

0.73

0.67

5.63

Table B.91: Empirical variance and asymptotic variance of standard deviation of y
(σˆx ) when ρ = 0.6 over different right censoring proportions of 0.1, 0.3, and 0.5 and
sample sizes 50, 100, and 500
n

50

100

500

p

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.271

0.244

0.277

0.261

0.241

0.249

0.3

0.271

0.243

0.319

0.261

0.234

0.248

0.5

0.271

0.263

0.342

0.261

0.243

0.243

0.1

0.127

0.108

0.127

0.127

0.117

0.123

0.3

0.127

0.109

0.144

0.127

0.113

0.122

0.5

0.127

0.122

0.155

0.127

0.116

0.120

0.1

0.024

0.022

0.025

0.025

0.023

0.025

0.3

0.024

0.023

0.029

0.025

0.022

0.025

0.5

0.024

0.023

0.033

0.025

0.022

0.024

144

Table B.92: Bias, RMSE, 95% CP, and relative RMSE of standard deviation of y (σˆy )
when ρ = 0.6 over sample sizes 50, 100, and 500 and right censoring proportions of
0.1, 0.3, and 0.5

n

p

Bias
CLS

50

100

500

RMSE

CP

RRMSE

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

-0.011 -0.579

-0.043

0.511

0.759

0.500

0.93

0.73

0.91

1.02

1.52

0.3

-0.011 -1.032

-0.106

0.511

1.140

0.509

0.93

0.43

0.88

1.00

2.24

0.5

-0.011 -1.319

-0.180

0.511

1.408

0.525

0.93

0.29

0.86

0.97

2.69

0.1

-0.011 -0.532

-0.055

0.357

0.633

0.355

0.94

0.64

0.92

1.01

1.78

0.3

-0.011 -1.000

-0.120

0.357

1.055

0.370

0.94

0.20

0.89

0.97

2.85

0.5

-0.011 -1.298

-0.196

0.357

1.342

0.398

0.94

0.07

0.84

0.90

3.38

0.1

-0.007 -0.543

-0.034

0.158

0.564

0.161

0.96

0.07

0.94

0.98

3.51

0.3

-0.007 -1.014

-0.098

0.158

1.025

0.184

0.96

0.00

0.85

0.86

5.56

0.5

-0.007 -1.318

-0.179

0.158

1.326

0.236

0.96

0.00

0.74

0.67

5.61

Table B.93: Empirical variance and asymptotic variance of correlation (ρ̂) when ρ =
0.6 over different right censoring proportions of 0.1, 0.3, and 0.5 and sample sizes 50,
100, and 500
n

50

100

500

p

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.009

0.013

0.009

0.009

0.012

0.009

0.3

0.009

0.018

0.010

0.009

0.017

0.009

0.5

0.009

0.026

0.010

0.009

0.023

0.009

0.1

0.004

0.006

0.004

0.004

0.006

0.004

0.3

0.004

0.008

0.005

0.004

0.008

0.004

0.5

0.004

0.012

0.005

0.004

0.011

0.005

0.1

0.001

0.001

0.001

0.001

0.001

0.001

0.3

0.001

0.002

0.001

0.001

0.002

0.001

0.5

0.001

0.002

0.001

0.001

0.002

0.001

145

Table B.94: Bias, RMSE, 95% CP, and relative RMSE of correlation (ρ̂) when ρ = 0.6
over sample sizes 50, 100, and 500 and right censoring proportions of 0.1, 0.3, and 0.5

n

p

Bias
CLS

50

100

500

RMSE

CP

RRMSE

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

-0.007 -0.075

-0.012

0.094

0.133

0.093

0.92

0.92

0.93

1.01

1.44

0.3

-0.007 -0.132

-0.024

0.094

0.185

0.097

0.92

0.89

0.92

0.97

1.90

0.5

-0.007 -0.173

-0.040

0.094

0.229

0.105

0.92

0.84

0.95

0.90

2.18

0.1

-0.004 -0.066

-0.008

0.066

0.100

0.065

0.94

0.89

0.94

1.01

1.54

0.3

-0.004 -0.123

-0.019

0.066

0.152

0.069

0.94

0.77

0.93

0.95

2.21

0.5

-0.004 -0.163

-0.033

0.066

0.193

0.076

0.94

0.69

0.93

0.87

2.55

0.1

-0.000 -0.060

-0.004

0.028

0.069

0.029

0.95

0.58

0.95

0.99

2.41

0.3

-0.000 -0.119

-0.014

0.028

0.125

0.033

0.95

0.12

0.93

0.86

3.80

0.5

-0.000 -0.162

-0.027

0.028

0.169

0.041

0.95

0.05

0.88

0.70

4.16

Table B.95: Empirical variance and asymptotic variance of intercept (βˆ0 ) when ρ = 0.8
over different right censoring proportions of 0.1, 0.3, and 0.5 and sample sizes 50, 100,
and 500
n

50

100

500

p

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.170

0.172

0.170

0.204

0.206

0.208

0.3

0.170

0.197

0.174

0.204

0.219

0.218

0.5

0.170

0.260

0.188

0.204

0.258

0.231

0.1

0.083

0.086

0.083

0.100

0.101

0.101

0.3

0.083

0.095

0.086

0.100

0.105

0.105

0.5

0.083

0.116

0.091

0.100

0.122

0.109

0.1

0.019

0.019

0.019

0.020

0.020

0.020

0.3

0.019

0.021

0.020

0.020

0.020

0.020

0.5

0.019

0.024

0.020

0.020

0.024

0.021

146

Table B.96: Bias, RMSE, 95% CP, and relative RMSE of intercept (βˆ0 ) when ρ = 0.8
over sample sizes 50, 100, and 500 and right censoring proportions of 0.1, 0.3, and 0.5

n

p

Bias
CLS

50

100

500

RMSE

CP

RRMSE

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

-0.010 -0.136

-0.001

0.452

0.474

0.457

0.96

0.95

0.97

0.99

1.04

0.3

-0.010 -0.380

0.012

0.452

0.603

0.467

0.96

0.89

0.97

0.97

1.29

0.5

-0.010 -0.650

0.032

0.452

0.825

0.482

0.96

0.75

0.97

0.94

1.71

0.1

0.001

-0.103

0.006

0.316

0.334

0.317

0.97

0.96

0.97

1.00

1.05

0.3

0.001

-0.339

0.017

0.316

0.469

0.324

0.97

0.83

0.96

0.98

1.45

0.5

0.001

-0.607

0.027

0.316

0.701

0.332

0.97

0.59

0.97

0.95

2.11

0.1

-0.003 -0.109

0.000

0.140

0.178

0.140

0.95

0.88

0.95

1.00

1.27

0.3

-0.003 -0.353

0.008

0.140

0.381

0.143

0.95

0.32

0.94

0.98

2.66

0.5

-0.003 -0.626

0.020

0.140

0.644

0.148

0.95

0.02

0.94

0.95

4.36

Table B.97: Empirical variance and asymptotic variance of slope (βˆ1 ) when ρ = 0.8
over different right censoring proportions of 0.1, 0.3, and 0.5 and sample sizes 50, 100,
and 500
n

50

100

500

p

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.002

0.003

0.002

0.002

0.003

0.002

0.3

0.002

0.004

0.003

0.002

0.004

0.002

0.5

0.002

0.006

0.003

0.002

0.005

0.002

0.1

0.001

0.001

0.001

0.001

0.001

0.001

0.3

0.001

0.002

0.001

0.001

0.002

0.001

0.5

0.001

0.002

0.002

0.001

0.002

0.001

0.1

0.000

0.000

0.000

0.000

0.000

0.000

0.3

0.000

0.000

0.000

0.000

0.000

0.000

0.5

0.000

0.000

0.000

0.000

0.000

0.000

147

Table B.98: Bias, RMSE, 95% CP, and relative RMSE of slope (βˆ1 ) when ρ = 0.8
over sample sizes 50, 100, and 500 and right censoring proportions of 0.1, 0.3, and 0.5

n

50

100

500

p

Bias

RMSE

CP

RRMSE

CLS

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

0.001

-0.022

-0.000

0.044

0.056

0.045

0.94

0.91

0.92

0.98

1.26

0.3

0.001

-0.045

-0.006

0.044

0.075

0.047

0.94

0.84

0.90

0.92

1.59

0.5

0.001

-0.062

-0.017

0.044

0.093

0.052

0.94

0.82

0.89

0.84

1.78

0.1

0.000

-0.021

-0.002

0.030

0.041

0.032

0.93

0.88

0.93

0.95

1.28

0.3

0.000

-0.042

-0.007

0.030

0.059

0.032

0.93

0.79

0.91

0.93

1.82

0.5

0.000

-0.060

-0.016

0.030

0.076

0.037

0.93

0.72

0.88

0.81

2.07

0.1

-0.000 -0.021

-0.002

0.014

0.025

0.014

0.96

0.74

0.94

0.99

1.77

0.3

-0.000 -0.044

-0.008

0.014

0.047

0.016

0.96

0.32

0.91

0.88

2.94

0.5

-0.000 -0.061

-0.016

0.014

0.065

0.022

0.96

0.13

0.77

0.65

2.98

Table B.99: Empirical variance and asymptotic variance of mean of x (µˆx ) when
ρ = 0.8 over different right censoring proportions of 0.1, 0.3, and 0.5 and sample sizes
50, 100, and 500
n

50

100

500

p

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

1.950

2.087

1.974

1.985

1.760

1.935

0.3

1.950

2.169

2.040

1.985

1.648

1.915

0.5

1.950

2.223

2.058

1.985

1.623

1.879

0.1

0.924

0.952

0.930

0.996

0.891

0.970

0.3

0.924

1.000

0.956

0.996

0.823

0.955

0.5

0.924

1.047

0.976

0.996

0.807

0.936

0.1

0.209

0.216

0.209

0.200

0.178

0.197

0.3

0.209

0.235

0.214

0.200

0.165

0.194

0.5

0.209

0.250

0.219

0.200

0.161

0.190

148

Table B.100: Bias, RMSE, 95% CP, and relative RMSE of mean of x (µˆx ) when
ρ = 0.8 over sample sizes 50, 100, and 500 and right censoring proportions of 0.1, 0.3,
and 0.5

n

p

Bias
CLS

50

100

500

RMSE

CP

RRMSE

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

0.095 -1.512

0.086

1.412

2.011

1.394

0.95

0.77

0.94

1.01

1.44

0.3

0.095 -3.392

0.000

1.412

3.627

1.384

0.95

0.26

0.94

1.02

2.62

0.5

0.095 -5.019

-0.060

1.412

5.178

1.372

0.95

0.04

0.93

1.03

3.77

0.1

0.066 -1.346

0.029

1.000

1.644

0.985

0.96

0.69

0.96

1.02

1.67

0.3

0.066 -3.285

-0.078

1.000

3.408

0.980

0.96

0.06

0.95

1.02

3.48

0.5

0.066 -4.923

-0.192

1.000

5.005

0.987

0.96

0.00

0.93

1.01

5.07

0.1

0.026 -1.391

-0.008

0.448

1.453

0.443

0.93

0.10

0.94

1.01

3.28

0.3

0.026 -3.340

-0.105

0.448

3.364

0.453

0.93

0.00

0.93

0.99

7.43

0.5

0.026 -5.002

-0.233

0.448

5.018

0.494

0.93

0.00

0.88

0.91

10.17

Table B.101: Empirical variance and asymptotic variance of mean of y (µˆy ) when
ρ = 0.8 over different right censoring proportions of 0.1, 0.3, and 0.5 and sample sizes
50, 100, and 500
n

50

100

500

p

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.463

0.483

0.467

0.501

0.447

0.491

0.3

0.463

0.521

0.483

0.501

0.419

0.486

0.5

0.463

0.569

0.497

0.501

0.416

0.475

0.1

0.213

0.218

0.213

0.250

0.224

0.244

0.3

0.213

0.227

0.226

0.250

0.208

0.241

0.5

0.213

0.245

0.239

0.250

0.205

0.235

0.1

0.052

0.053

0.052

0.050

0.045

0.049

0.3

0.052

0.058

0.054

0.050

0.041

0.049

0.5

0.052

0.061

0.055

0.050

0.040

0.047

149

Table B.102: Bias, RMSE, 95% CP, and relative RMSE of mean of y (µˆx ) when
ρ = 0.8 over sample sizes 50, 100, and 500 and right censoring proportions of 0.1, 0.3,
and 0.5

n

p

Bias
CLS

50

100

500

RMSE

CP

RRMSE

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

0.028 -0.775

0.028

0.709

1.024

0.701

0.95

0.79

0.94

1.01

1.46

0.3

0.028 -1.722

-0.013

0.709

1.840

0.698

0.95

0.26

0.94

1.02

2.64

0.5

0.028 -2.535

-0.046

0.709

2.615

0.691

0.95

0.04

0.93

1.03

3.79

0.1

0.027 -0.677

0.011

0.500

0.826

0.494

0.96

0.70

0.96

1.01

1.67

0.3

0.027 -1.643

-0.037

0.500

1.705

0.492

0.96

0.06

0.96

1.02

3.47

0.5

0.027 -2.464

-0.098

0.500

2.505

0.495

0.96

0.00

0.94

1.01

5.06

0.1

0.007 -0.699

-0.009

0.224

0.731

0.222

0.94

0.10

0.94

1.01

3.29

0.3

0.007 -1.675

-0.056

0.224

1.687

0.227

0.94

0.00

0.93

0.98

7.42

0.5

0.007 -2.504

-0.119

0.224

2.512

0.248

0.94

0.00

0.89

0.90

10.13

Table B.103: Empirical variance and asymptotic variance of standard deviation of x
(σˆx ) when ρ = 0.8 over different right censoring proportions of 0.1, 0.3, and 0.5 and
sample sizes 50, 100, and 500
n

50

100

500

p

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

1.082

0.940

1.102

1.040

0.924

0.967

0.3

1.082

0.901

1.241

1.040

0.870

0.957

0.5

1.082

0.943

1.396

1.040

0.864

0.940

0.1

0.533

0.478

0.549

0.510

0.456

0.485

0.3

0.533

0.463

0.606

0.510

0.423

0.478

0.5

0.533

0.481

0.683

0.510

0.416

0.468

0.1

0.105

0.099

0.112

0.100

0.090

0.098

0.3

0.105

0.099

0.129

0.100

0.083

0.097

0.5

0.105

0.102

0.143

0.100

0.081

0.095

150

Table B.104: Bias, RMSE, 95% CP, and relative RMSE of standard deviation of x
(σˆx ) when ρ = 0.8 over sample sizes 50, 100, and 500 and right censoring proportions
of 0.1, 0.3, and 0.5

n

p

Bias
CLS

50

100

500

RMSE

CP

RRMSE

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

-0.092 -1.323

-0.220

1.024

1.635

1.008

0.93

0.66

0.90

1.02

1.62

0.3

-0.092 -2.252

-0.377

1.024

2.438

1.048

0.93

0.37

0.85

0.98

2.33

0.5

-0.092 -2.884

-0.522

1.024

3.030

1.101

0.93

0.19

0.82

0.93

2.75

0.1

-0.046 -1.173

-0.181

0.715

1.354

0.719

0.94

0.57

0.91

0.99

1.88

0.3

-0.046 -2.161

-0.356

0.715

2.256

0.777

0.94

0.14

0.86

0.92

2.90

0.5

-0.046 -2.796

-0.503

0.715

2.869

0.849

0.94

0.03

0.80

0.84

3.38

0.1

-0.008 -1.141

-0.095

0.317

1.180

0.327

0.94

0.06

0.93

0.97

3.60

0.3

-0.008 -2.126

-0.253

0.317

2.145

0.401

0.94

0.00

0.82

0.79

5.35

0.5

-0.008 -2.765

-0.416

0.317

2.780

0.517

0.94

0.00

0.67

0.61

5.37

Table B.105: Empirical variance and asymptotic variance of standard deviation of y
(σˆy ) when ρ = 0.8 over different right censoring proportions of 0.1, 0.3, and 0.5 and
sample sizes 50, 100, and 500
n

50

100

500

p

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.274

0.244

0.280

0.263

0.235

0.245

0.3

0.274

0.236

0.324

0.263

0.221

0.243

0.5

0.274

0.248

0.350

0.263

0.221

0.238

0.1

0.127

0.111

0.126

0.128

0.115

0.122

0.3

0.127

0.107

0.145

0.128

0.107

0.120

0.5

0.127

0.113

0.158

0.128

0.106

0.118

0.1

0.025

0.023

0.026

0.025

0.022

0.025

0.3

0.025

0.023

0.030

0.025

0.021

0.024

0.5

0.025

0.023

0.036

0.025

0.020

0.024

151

Table B.106: Bias, RMSE, 95% CP, and relative RMSE of standard deviation of y
(σˆy ) when ρ = 0.8 over sample sizes 50, 100, and 500 and right censoring proportions
of 0.1, 0.3, and 0.5

n

p

Bias
CLS

50

100

500

RMSE

CP

RRMSE

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

-0.021 -0.628

-0.075

0.513

0.793

0.501

0.93

0.71

0.90

1.02

1.58

0.3

-0.021 -1.094

-0.151

0.513

1.191

0.516

0.93

0.39

0.85

0.99

2.31

0.5

-0.021 -1.399

-0.234

0.513

1.476

0.541

0.93

0.22

0.82

0.95

2.73

0.1

-0.016 -0.571

-0.078

0.358

0.664

0.358

0.94

0.58

0.92

1.00

1.86

0.3

-0.016 -1.056

-0.156

0.358

1.105

0.380

0.94

0.14

0.87

0.94

2.91

0.5

-0.016 -1.367

-0.237

0.358

1.405

0.417

0.94

0.04

0.82

0.86

3.37

0.1

-0.006 -0.570

-0.048

0.159

0.590

0.164

0.96

0.04

0.94

0.97

3.60

0.3

-0.006 -1.061

-0.125

0.159

1.071

0.200

0.96

0.00

0.84

0.79

5.36

0.5

-0.006 -1.379

-0.205

0.159

1.386

0.256

0.96

0.00

0.70

0.62

5.41

Table B.107: Empirical variance and asymptotic variance of correlation (ρ̂) when
ρ = 0.8 over different right censoring proportions of 0.1, 0.3, and 0.5 and sample sizes
50, 100, and 500
n

50

100

500

p

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.003

0.005

0.003

0.003

0.005

0.003

0.3

0.003

0.008

0.003

0.003

0.007

0.003

0.5

0.003

0.012

0.004

0.003

0.011

0.004

0.1

0.001

0.002

0.002

0.001

0.002

0.001

0.3

0.001

0.004

0.002

0.001

0.003

0.002

0.5

0.001

0.005

0.002

0.001

0.005

0.002

0.1

0.000

0.000

0.000

0.000

0.000

0.000

0.3

0.000

0.001

0.000

0.000

0.001

0.000

0.5

0.000

0.001

0.000

0.000

0.001

0.000

152

Table B.108: Bias, RMSE, 95% CP, and relative RMSE of correlation (ρ̂) when ρ = 0.8
over sample sizes 50, 100, and 500 and right censoring proportions of 0.1, 0.3, and 0.5

n

p

Bias
CLS

50

100

500

B.3

RMSE

CP

RRMSE

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

-0.005 -0.054

-0.010

0.055

0.087

0.055

0.94

0.95

0.93

1.00

1.59

0.3

-0.005 -0.102

-0.024

0.055

0.133

0.061

0.94

0.90

0.96

0.90

2.17

0.5

-0.005 -0.139

-0.045

0.055

0.174

0.076

0.94

0.86

0.96

0.72

2.28

0.1

-0.003 -0.047

-0.007

0.038

0.065

0.038

0.94

0.90

0.95

0.98

1.72

0.3

-0.003 -0.092

-0.020

0.038

0.108

0.045

0.94

0.71

0.95

0.84

2.43

0.5

-0.003 -0.129

-0.038

0.038

0.147

0.057

0.94

0.57

0.93

0.66

2.59

0.1

-0.000 -0.042

-0.004

0.017

0.047

0.018

0.94

0.45

0.95

0.97

2.63

0.3

-0.000 -0.088

-0.016

0.017

0.092

0.023

0.94

0.02

0.88

0.74

3.91

0.5

-0.000 -0.124

-0.034

0.017

0.128

0.038

0.94

0.00

0.57

0.45

3.35

Results in the Presence of Left Censored Observations in the Data

Figure B.7: Boxplots for estimators of mean of x (µˆx ) and mean of y (µˆy ), when
n = 100, ρ = 0.2 over different left censoring proportions of 0.1, 0.3, and 0.5

153

Figure B.8: Boxplots for estimators of standard deviation of x (σˆx ) and standard
deviation of y (σˆy ), when n = 100, ρ = 0.2 over different left censoring proportions of
0.1, 0.3, and 0.5

Figure B.9: Boxplots for estimators of correlation between x and y (ρ̂), when n =
100, ρ = 0.2 over different left censoring proportions of 0.1, 0.3, and 0.5

154

Table B.109: Empirical variance and asymptotic variance of intercept (βˆ0 ) when ρ =
0.2 over different left censoring proportions of 0.1, 0.3, and 0.5 and sample sizes 50,
100, and 500
n

50

100

500

p

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.460

0.525

0.457

0.545

0.563

0.552

0.3

0.460

0.807

0.483

0.545

0.760

0.561

0.5

0.460

1.204

0.530

0.545

1.113

0.567

0.1

0.215

0.252

0.221

0.267

0.273

0.267

0.3

0.215

0.366

0.232

0.267

0.356

0.271

0.5

0.215

0.536

0.252

0.267

0.509

0.274

0.1

0.052

0.059

0.053

0.052

0.053

0.053

0.3

0.052

0.082

0.055

0.052

0.070

0.053

0.5

0.052

0.109

0.060

0.052

0.099

0.054

Table B.110: Empirical variance and asymptotic variance of slope (βˆ1 ) when ρ = 0.2
over different left censoring proportions of 0.1, 0.3, and 0.5 and sample sizes 50, 100,
and 500
n

50

100

500

p

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.006

0.006

0.006

0.005

0.005

0.005

0.3

0.006

0.008

0.005

0.005

0.008

0.005

0.5

0.006

0.011

0.006

0.005

0.010

0.006

0.1

0.003

0.003

0.003

0.003

0.003

0.003

0.3

0.003

0.004

0.003

0.003

0.004

0.003

0.5

0.003

0.004

0.003

0.003

0.004

0.003

0.1

0.001

0.001

0.001

0.001

0.001

0.001

0.3

0.001

0.001

0.001

0.001

0.001

0.001

0.5

0.001

0.001

0.001

0.001

0.001

0.001

155

Table B.111: Empirical variance and asymptotic variance of mean of x (µˆx ) when
ρ = 0.2 over different left censoring proportions of 0.1, 0.3, and 0.5 and sample sizes
50, 100, and 500
n

p

50

100

500

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

1.955

1.989

1.972

1.984

1.909

1.978

0.3

1.955

2.375

2.046

1.984

1.845

2.017

0.5

1.955

2.686

2.172

1.984

1.999

2.024

0.1

0.958

0.990

0.971

0.997

0.905

0.990

0.3

0.958

1.172

1.016

0.997

0.913

1.003

0.5

0.958

1.355

1.092

0.997

0.986

1.004

0.1

0.206

0.207

0.207

0.200

0.180

0.200

0.3

0.206

0.235

0.216

0.200

0.181

0.202

0.5

0.206

0.275

0.225

0.200

0.195

0.202

Table B.112: Bias, RMSE, 95% CP, and relative RMSE of mean of x (µˆx ) when
ρ = 0.2 over sample sizes 50, 100, and 500 and left censoring proportions of 0.1, 0.3,
and 0.5

n

50

100

500

p

Bias

RMSE

CP

RRMSE

CLS

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

0.099

1.193

0.060 1.412

1.825

1.408

0.95

0.84

0.94

1.00

1.30

0.3

0.099

3.225

0.052 1.412

3.500

1.421

0.95

0.37

0.94

0.99

2.46

0.5

0.099

4.787

0.067 1.412

4.991

1.424

0.95

0.10

0.93

0.99

3.50

0.1

0.064

1.278

0.059 1.001

1.593

0.997

0.96

0.72

0.95

1.00

1.60

0.3

0.064

3.072

0.079 1.001

3.217

1.005

0.96

0.11

0.95

1.00

3.20

0.5

0.064

4.625

0.113 1.001

4.730

1.008

0.96

0.00

0.93

0.99

4.69

0.1

0.027

1.264

0.027 0.448

1.333

0.448

0.94

0.17

0.95

1.00

2.98

0.3

0.027

3.074

0.043 0.448

3.103

0.452

0.94

0.00

0.94

0.99

6.87

0.5

0.027

4.647

0.084 0.448

4.668

0.457

0.94

0.00

0.93

0.98

10.21

156

Table B.113: Empirical variance and asymptotic variance of mean of y (µˆy ) when
ρ = 0.2 over different left censoring proportions of 0.1, 0.3, and 0.5 and sample sizes
50, 100, and 500
n

p

50

100

500

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.444

0.471

0.441

0.505

0.432

0.503

0.3

0.444

0.525

0.456

0.505

0.461

0.507

0.5

0.444

0.616

0.483

0.505

0.497

0.507

0.1

0.216

0.238

0.221

0.251

0.226

0.248

0.3

0.216

0.272

0.229

0.251

0.227

0.250

0.5

0.216

0.329

0.246

0.251

0.243

0.250

0.1

0.051

0.054

0.051

0.050

0.045

0.050

0.3

0.051

0.059

0.053

0.050

0.045

0.050

0.5

0.051

0.068

0.058

0.050

0.048

0.050

Table B.114: Bias, RMSE, 95% CP, and relative RMSE of mean of y (µˆy ) when
ρ = 0.2 over sample sizes 50, 100, and 500 and left censoring proportions of 0.1, 0.3,
and 0.5

n

50

100

500

p

Bias

RMSE

CP

RRMSE

CLS

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

0.005

0.850

-0.014

0.711

1.074

0.709

0.95

0.73

0.96

1.00

1.51

0.3

0.005

1.598

-0.001

0.711

1.736

0.712

0.95

0.35

0.96

1.00

2.44

0.5

0.005

2.376

0.015

0.711

2.479

0.712

0.95

0.10

0.95

1.00

3.48

0.1

0.015

0.621

0.017

0.501

0.782

0.498

0.97

0.74

0.96

1.01

1.57

0.3

0.015

1.523

0.035

0.501

1.595

0.501

0.97

0.12

0.97

1.00

3.18

0.5

0.015

2.320

0.053

0.501

2.371

0.503

0.97

0.01

0.95

1.00

4.71

0.1

0.001

0.617

0.002

0.223

0.652

0.223

0.93

0.17

0.93

1.00

2.92

0.3

0.001

1.524

0.014

0.223

1.539

0.225

0.93

0.00

0.93

0.99

6.84

0.5

0.001

2.315

0.038

0.223

2.325

0.227

0.93

0.00

0.91

0.98

10.24

157

Table B.115: Empirical variance and asymptotic variance of standard deviation of x
(σˆx ) when ρ = 0.2 over different left censoring proportions of 0.1, 0.3, and 0.5 and
sample sizes 50, 100, and 500
n

p

50

100

500

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

1.104

1.071

1.117

1.023

0.989

0.989

0.3

1.104

1.108

1.307

1.023

0.964

1.008

0.5

1.104

1.375

1.549

1.023

1.057

1.012

0.1

0.551

0.509

0.586

0.506

0.460

0.495

0.3

0.551

0.539

0.648

0.506

0.466

0.502

0.5

0.551

0.607

0.768

0.506

0.506

0.502

0.1

0.102

0.099

0.111

0.100

0.090

0.100

0.3

0.102

0.102

0.129

0.100

0.091

0.101

0.5

0.102

0.115

0.149

0.100

0.098

0.101

Table B.116: Bias, RMSE, 95% CP, and relative RMSE of standard deviation of x
(σˆx ) when ρ = 0.2 over sample sizes 50, 100, and 500 and left censoring proportions
of 0.1, 0.3, and 0.5

n

p

Bias
CLS

50

100

500

RMSE

CP

RRMSE

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

-0.096 -0.865

-0.112

1.016

1.318

1.001

0.94

0.80

0.93

1.02

1.32

0.3

-0.096 -2.022

-0.123

1.016

2.247

1.012

0.94

0.44

0.91

1.00

2.22

0.5

-0.096 -2.677

-0.166

1.016

2.868

1.020

0.94

0.31

0.88

1.00

2.81

0.1

-0.043 -0.976

-0.082

0.713

1.188

0.708

0.94

0.66

0.92

1.01

1.68

0.3

-0.043 -1.932

-0.116

0.713

2.049

0.718

0.94

0.23

0.91

0.99

2.86

0.5

-0.043 -2.581

-0.166

0.713

2.677

0.728

0.94

0.09

0.88

0.98

3.68

0.1

-0.008 -0.991

-0.018

0.317

1.035

0.316

0.95

0.11

0.93

1.00

3.27

0.3

-0.008 -1.929

-0.046

0.317

1.952

0.321

0.95

0.00

0.91

0.99

6.07

0.5

-0.008 -2.569

-0.105

0.317

2.588

0.335

0.95

0.00

0.88

0.95

7.73

158

Table B.117: Empirical variance and asymptotic variance of standard deviation of y
(σˆx ) when ρ = 0.2 over different left censoring proportions of 0.1, 0.3, and 0.5 and
sample sizes 50, 100, and 500
n

p

50

100

500

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.257

0.258

0.302

0.261

0.224

0.251

0.3

0.257

0.272

0.333

0.261

0.241

0.254

0.5

0.257

0.315

0.394

0.261

0.263

0.254

0.1

0.126

0.125

0.139

0.127

0.115

0.124

0.3

0.126

0.138

0.156

0.127

0.116

0.125

0.5

0.126

0.164

0.193

0.127

0.125

0.125

0.1

0.025

0.024

0.027

0.025

0.023

0.025

0.3

0.025

0.026

0.031

0.025

0.023

0.025

0.5

0.025

0.030

0.038

0.025

0.024

0.025

Table B.118: Bias, RMSE, 95% CP, and relative RMSE of standard deviation of y
(σˆx ) when ρ = 0.2 over sample sizes 50, 100, and 500 and left censoring proportions
of 0.1, 0.3, and 0.5

n

50

100

500

p

Bias

RMSE

CP

RRMSE

CLS

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

0.000

-0.656

-0.016

0.511

0.809

0.502

0.94

0.65

0.93

1.02

1.61

0.3

0.000

-1.012

-0.047

0.511

1.124

0.506

0.94

0.44

0.89

1.01

2.22

0.5

0.000

-1.346

-0.079

0.511

1.440

0.510

0.94

0.29

0.86

1.00

2.83

0.1

-0.005 -0.491

-0.033

0.357

0.597

0.354

0.94

0.65

0.93

1.01

1.69

0.3

-0.005 -0.981

-0.065

0.357

1.039

0.359

0.94

0.24

0.91

0.99

2.89

0.5

-0.005 -1.319

-0.091

0.357

1.366

0.365

0.94

0.09

0.86

0.98

3.74

0.1

-0.008 -0.501

-0.014

0.158

0.523

0.158

0.95

0.10

0.95

1.00

3.30

0.3

-0.008 -0.976

-0.035

0.158

0.987

0.163

0.95

0.00

0.91

0.97

6.07

0.5

-0.008 -1.300

-0.071

0.158

1.309

0.174

0.95

0.00

0.85

0.91

7.54

159

Table B.119: Empirical variance and asymptotic variance of correlation (ρ̂) when
ρ = 0.2 over different left censoring proportions of 0.1, 0.3, and 0.5 and sample sizes
50, 100, and 500
n

p

50

100

500

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.020

0.023

0.020

0.019

0.022

0.018

0.3

0.020

0.031

0.019

0.019

0.027

0.018

0.5

0.020

0.039

0.020

0.019

0.034

0.018

0.1

0.010

0.011

0.010

0.009

0.011

0.009

0.3

0.010

0.013

0.010

0.009

0.013

0.009

0.5

0.010

0.017

0.010

0.009

0.017

0.009

0.1

0.002

0.002

0.002

0.002

0.002

0.002

0.3

0.002

0.003

0.002

0.002

0.003

0.002

0.5

0.002

0.003

0.002

0.002

0.003

0.002

Table B.120: Bias, RMSE, 95% CP, and relative RMSE of correlation (ρ̂) when ρ = 0.2
over sample sizes 50, 100, and 500 and left censoring proportions of 0.1, 0.3, and 0.5

n

p

Bias
CLS

50

100

500

RMSE

CP

RRMSE

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

-0.005 -0.043

-0.009

0.137

0.153

0.134

0.93

0.92

0.91

1.02

1.14

0.3

-0.005 -0.064

-0.013

0.137

0.176

0.135

0.93

0.90

0.91

1.01

1.31

0.5

-0.005 -0.080

-0.017

0.137

0.202

0.135

0.93

0.90

0.92

1.01

1.49

0.1

-0.004 -0.039

-0.006

0.097

0.110

0.096

0.93

0.93

0.93

1.01

1.15

0.3

-0.004 -0.063

-0.011

0.097

0.131

0.096

0.93

0.91

0.93

1.00

1.37

0.5

-0.004 -0.077

-0.017

0.097

0.151

0.097

0.93

0.91

0.92

0.99

1.55

0.1

0.000

-0.033

-0.001

0.042

0.056

0.042

0.95

0.90

0.95

1.00

1.33

0.3

0.000

-0.064

-0.005

0.042

0.082

0.043

0.95

0.77

0.95

0.99

1.92

0.5

0.000

-0.083

-0.011

0.042

0.101

0.045

0.95

0.69

0.94

0.95

2.25

160

Table B.121: Empirical variance and asymptotic variance of intercept (βˆ0 ) when ρ =
0.4 over different left censoring proportions of 0.1, 0.3, and 0.5 and sample sizes 50,
100, and 500
n

p

50

100

500

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.400

0.486

0.402

0.476

0.524

0.482

0.3

0.400

0.733

0.434

0.476

0.709

0.489

0.5

0.400

1.202

0.504

0.476

1.043

0.498

0.1

0.189

0.221

0.197

0.234

0.248

0.233

0.3

0.189

0.328

0.210

0.234

0.334

0.237

0.5

0.189

0.499

0.237

0.234

0.482

0.239

0.1

0.045

0.051

0.046

0.046

0.049

0.046

0.3

0.045

0.074

0.049

0.046

0.065

0.047

0.5

0.045

0.097

0.056

0.046

0.094

0.047

Table B.122: Bias, RMSE, 95% CP, and relative RMSE of intercept (βˆ0 ) when ρ = 0.8
over sample sizes 50, 100, and 500 and right censoring proportions of 0.1, 0.3, and 0.5

n

50

100

500

p

Bias

RMSE

CP

RRMSE

CLS

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

-0.003

0.611

-0.001 0.690

0.947

0.694

0.96

0.88

0.97

0.99

1.37

0.3

-0.003

1.373

0.031

0.690

1.610

0.700

0.96

0.62

0.97

0.99

2.30

0.5

-0.003

2.073

0.054

0.690

2.311

0.708

0.96

0.47

0.94

0.98

3.27

0.1

0.010

0.544

0.019

0.483

0.738

0.484

0.97

0.81

0.97

1.00

1.53

0.3

0.010

1.318

0.053

0.483

1.439

0.490

0.97

0.35

0.96

0.99

2.94

0.5

0.010

2.027

0.093

0.483

2.143

0.498

0.97

0.17

0.93

0.97

4.30

0.1

-0.003

0.526

0.003

0.214

0.570

0.214

0.95

0.32

0.94

1.00

2.66

0.3

-0.003

1.316

0.032

0.214

1.340

0.218

0.95

0.00

0.94

0.98

6.14

0.5

-0.003

2.036

0.077

0.214

2.059

0.230

0.95

0.00

0.91

0.93

8.97

161

Table B.123: Empirical variance and asymptotic variance of slope (βˆ1 ) when ρ = 0.4
over different left censoring proportions of 0.1, 0.3, and 0.5 and sample sizes 50, 100,
and 500
n

p

50

100

500

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.005

0.006

0.005

0.005

0.005

0.005

0.3

0.005

0.008

0.005

0.005

0.007

0.005

0.5

0.005

0.011

0.005

0.005

0.009

0.005

0.1

0.002

0.003

0.002

0.002

0.003

0.002

0.3

0.002

0.003

0.003

0.002

0.003

0.002

0.5

0.002

0.004

0.003

0.002

0.004

0.002

0.1

0.000

0.001

0.000

0.000

0.001

0.000

0.3

0.000

0.001

0.000

0.000

0.001

0.000

0.5

0.000

0.001

0.001

0.000

0.001

0.001

Table B.124: Bias, RMSE, 95% CP, and relative RMSE of slope (βˆ1 ) when ρ = 0.4
over sample sizes 50, 100, and 500 and left censoring proportions of 0.1, 0.3, and 0.5

n

50

100

500

p

Bias

RMSE

CP

RRMSE

CLS

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

0.001

-0.032

-0.002

0.068

0.080

0.068

0.93

0.91

0.93

1.00

1.18

0.3

0.001

-0.054

-0.007

0.068

0.098

0.069

0.93

0.86

0.94

0.98

1.43

0.5

0.001

-0.069

-0.011

0.068

0.117

0.072

0.93

0.84

0.93

0.95

1.63

0.1

-0.001 -0.031

-0.002

0.047

0.059

0.047

0.94

0.88

0.93

1.00

1.25

0.3

-0.001 -0.054

-0.007

0.047

0.078

0.049

0.94

0.83

0.92

0.97

1.61

0.5

-0.001 -0.070

-0.012

0.047

0.094

0.051

0.94

0.77

0.92

0.93

1.87

0.1

0.000

-0.028

-0.001

0.020

0.036

0.020

0.96

0.77

0.96

1.00

1.80

0.3

0.000

-0.054

-0.005

0.020

0.059

0.021

0.96

0.43

0.95

0.97

2.85

0.5

0.000

-0.071

-0.011

0.020

0.077

0.025

0.96

0.27

0.90

0.81

3.10

162

Table B.125: Empirical variance and asymptotic variance of mean of x (µˆx ) when
ρ = 0.4 over different left censoring proportions of 0.1, 0.3, and 0.5 and sample sizes
50, 100, and 500
n

p

50

100

500

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

1.955

2.026

1.984

1.984

1.777

1.972

0.3

1.955

2.319

2.039

1.984

1.807

1.991

0.5

1.955

2.647

2.153

1.984

1.935

1.975

0.1

0.948

0.981

0.958

0.997

0.894

0.986

0.3

0.948

1.125

1.007

0.997

0.893

0.991

0.5

0.948

1.290

1.072

0.997

0.953

0.981

0.1

0.207

0.207

0.209

0.200

0.178

0.199

0.3

0.207

0.229

0.215

0.200

0.177

0.200

0.5

0.207

0.264

0.226

0.200

0.189

0.197

Table B.126: Bias, RMSE, 95% CP, and relative RMSE of mean of x (µˆx ) when
ρ = 0.4 over sample sizes 50, 100, and 500 and left censoring proportions of 0.1, 0.3,
and 0.5

n

50

100

500

p

Bias

RMSE

CP

RRMSE

CLS

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

0.099

1.626

0.063 1.412

2.102

1.406

0.94

0.75

0.94

1.00

1.50

0.3

0.099

3.455

0.084 1.412

3.708

1.414

0.94

0.30

0.94

1.00

2.62

0.5

0.099

5.068

0.143 1.412

5.256

1.413

0.94

0.06

0.93

1.00

3.72

0.1

0.065

1.404

0.067 1.000

1.693

0.995

0.96

0.69

0.95

1.01

1.70

0.3

0.065

3.303

0.111 1.000

3.436

1.002

0.96

0.08

0.94

1.00

3.43

0.5

0.065

4.929

0.185 1.000

5.025

1.007

0.96

0.00

0.91

0.99

4.99

0.1

0.027

1.387

0.034 0.448

1.449

0.447

0.94

0.12

0.95

1.00

3.24

0.3

0.027

3.293

0.077 0.448

3.320

0.454

0.94

0.00

0.94

0.99

7.32

0.5

0.027

4.931

0.160 0.448

4.950

0.472

0.94

0.00

0.92

0.95

10.49

163

Table B.127: Empirical variance and asymptotic variance of mean of y (µˆy ) when
ρ = 0.4 over different left censoring proportions of 0.1, 0.3, and 0.5 and sample sizes
50, 100, and 500
n

p

50

100

500

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

1.955

2.026

1.984

1.984

1.777

1.972

0.3

1.955

2.319

2.039

1.984

1.807

1.991

0.5

1.955

2.647

2.153

1.984

1.935

1.975

0.1

0.948

0.981

0.958

0.997

0.894

0.986

0.3

0.948

1.125

1.007

0.997

0.893

0.991

0.5

0.948

1.290

1.072

0.997

0.953

0.981

0.1

0.207

0.207

0.209

0.200

0.178

0.199

0.3

0.207

0.229

0.215

0.200

0.177

0.200

0.5

0.207

0.264

0.226

0.200

0.189

0.197

Table B.128: Bias, RMSE, 95% CP, and relative RMSE of mean of y (µˆy ) when
ρ = 0.4 over sample sizes 50, 100, and 500 and left censoring proportions of 0.1, 0.3,
and 0.5

n

50

100

500

p

Bias

RMSE

CP

RRMSE

CLS

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

0.012

0.784

0.001 0.710

1.031

0.706

0.95

0.80

0.96

1.01

1.46

0.3

0.012

1.709

0.025 0.710

1.836

0.707

0.95

0.29

0.95

1.00

2.60

0.5

0.012

2.519

0.052 0.710

2.612

0.707

0.95

0.05

0.94

1.01

3.70

0.1

0.019

0.691

0.024 0.501

0.837

0.498

0.97

0.68

0.96

1.01

1.68

0.3

0.019

1.641

0.053 0.501

1.707

0.500

0.97

0.08

0.97

1.00

3.41

0.5

0.019

2.454

0.094 0.501

2.502

0.503

0.97

0.00

0.95

1.00

4.98

0.1

0.003

0.680

0.007 0.223

0.712

0.223

0.93

0.11

0.93

1.00

3.19

0.3

0.003

1.638

0.032 0.223

1.651

0.225

0.93

0.00

0.93

0.99

7.33

0.5

0.003

2.460

0.076 0.223

2.469

0.234

0.93

0.00

0.90

0.95

10.55

164

Table B.129: Empirical variance and asymptotic variance of standard deviation of x
(σˆx ) when ρ = 0.4 over different left censoring proportions of 0.1, 0.3, and 0.5 and
sample sizes 50, 100, and 500
n

p

50

100

500

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

1.096

1.036

1.138

1.025

0.921

0.986

0.3

1.096

1.083

1.265

1.025

0.944

0.996

0.5

1.096

1.268

1.482

1.025

1.022

0.988

0.1

0.546

0.498

0.581

0.507

0.455

0.493

0.3

0.546

0.512

0.629

0.507

0.456

0.496

0.5

0.546

0.569

0.743

0.507

0.489

0.490

0.1

0.103

0.096

0.111

0.100

0.089

0.099

0.3

0.103

0.098

0.128

0.100

0.089

0.100

0.5

0.103

0.104

0.140

0.100

0.095

0.099

Table B.130: Bias, RMSE, 95% CP, and relative RMSE of standard deviation of x
(σˆx ) when ρ = 0.4 over sample sizes 50, 100, and 500 and left censoring proportions
of 0.1, 0.3, and 0.5

n

p

Bias
CLS

50

100

500

RMSE

CP

RRMSE

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

-0.096 -1.162

-0.128

1.017

1.507

1.001

0.94

0.70

0.92

1.02

1.51

0.3

-0.096 -2.016

-0.184

1.017

2.238

1.015

0.94

0.43

0.92

1.00

2.21

0.5

-0.096 -2.636

-0.284

1.017

2.824

1.034

0.94

0.31

0.88

0.98

2.73

0.1

-0.044 -1.014

-0.100

0.713

1.218

0.709

0.93

0.66

0.92

1.01

1.72

0.3

-0.044 -1.944

-0.175

0.713

2.058

0.725

0.93

0.21

0.90

0.98

2.84

0.5

-0.044 -2.553

-0.281

0.713

2.647

0.754

0.93

0.08

0.86

0.95

3.51

0.1

-0.008 -1.022

-0.035

0.317

1.065

0.317

0.95

0.09

0.93

1.00

3.36

0.3

-0.008 -1.940

-0.109

0.317

1.963

0.334

0.95

0.00

0.90

0.95

5.87

0.5

-0.008 -2.552

-0.226

0.317

2.571

0.387

0.95

0.00

0.84

0.82

6.64

165

Table B.131: Empirical variance and asymptotic variance of standard deviation of y
(σˆy ) when ρ = 0.4 over different left censoring proportions of 0.1, 0.3, and 0.5 and
sample sizes 50, 100, and 500
n

p

50

100

500

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.264

0.262

0.299

0.261

0.233

0.249

0.3

0.264

0.271

0.338

0.261

0.235

0.250

0.5

0.264

0.315

0.395

0.261

0.254

0.248

0.1

0.127

0.125

0.138

0.127

0.113

0.124

0.3

0.127

0.132

0.157

0.127

0.113

0.124

0.5

0.127

0.157

0.184

0.127

0.121

0.122

0.1

0.024

0.022

0.025

0.025

0.022

0.025

0.3

0.024

0.025

0.030

0.025

0.022

0.025

0.5

0.024

0.028

0.036

0.025

0.024

0.025

Table B.132: Bias, RMSE, 95% CP, and relative RMSE of standard deviation of y
(σˆy ) when ρ = 0.4 over sample sizes 50, 100, and 500 and left censoring proportions
of 0.1, 0.3, and 0.5

n

p

Bias
CLS

50

100

500

RMSE

CP

RRMSE

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

-0.004 -0.557

-0.036

0.511

0.737

0.501

0.93

0.73

0.91

1.02

1.47

0.3

-0.004 -1.013

-0.087

0.511

1.124

0.507

0.93

0.43

0.87

1.01

2.22

0.5

-0.004 -1.326

-0.131

0.511

1.419

0.515

0.93

0.29

0.85

0.99

2.75

0.1

-0.008 -0.514

-0.044

0.357

0.615

0.354

0.94

0.63

0.92

1.01

1.74

0.3

-0.008 -0.986

-0.095

0.357

1.042

0.364

0.94

0.22

0.89

0.98

2.86

0.5

-0.008 -1.304

-0.153

0.357

1.349

0.381

0.94

0.08

0.85

0.94

3.54

0.1

-0.007 -0.516

-0.021

0.158

0.537

0.159

0.96

0.07

0.95

1.00

3.38

0.3

-0.007 -0.982

-0.065

0.158

0.993

0.171

0.96

0.00

0.90

0.93

5.82

0.5

-0.007 -1.290

-0.128

0.158

1.299

0.202

0.96

0.00

0.82

0.78

6.43

166

Table B.133: Empirical variance and asymptotic variance of correlation (ρ̂) when
ρ = 0.4 over different left censoring proportions of 0.1, 0.3, and 0.5 and sample sizes
50, 100, and 500
n

p

50

100

500

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.015

0.018

0.015

0.015

0.018

0.014

0.3

0.015

0.028

0.015

0.015

0.023

0.014

0.5

0.015

0.039

0.016

0.015

0.030

0.015

0.1

0.007

0.010

0.008

0.007

0.009

0.007

0.3

0.007

0.012

0.008

0.007

0.011

0.007

0.5

0.007

0.016

0.008

0.007

0.015

0.007

0.1

0.002

0.002

0.002

0.001

0.002

0.001

0.3

0.002

0.002

0.002

0.001

0.002

0.001

0.5

0.002

0.003

0.002

0.001

0.003

0.002

Table B.134: Bias, RMSE, 95% CP, and relative RMSE of correlation (ρ̂) when ρ = 0.4
over sample sizes 50, 100, and 500 and left censoring proportions of 0.1, 0.3, and 0.5

n

p

Bias
CLS

50

100

500

RMSE

CP

RRMSE

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

-0.007 -0.070

-0.011

0.121

0.152

0.119

0.93

0.93

0.93

1.02

1.27

0.3

-0.007 -0.111

-0.019

0.121

0.189

0.121

0.93

0.88

0.93

1.00

1.56

0.5

-0.007 -0.142

-0.029

0.121

0.223

0.124

0.93

0.86

0.93

0.98

1.80

0.1

-0.005 -0.062

-0.008

0.085

0.112

0.085

0.93

0.90

0.93

1.00

1.33

0.3

-0.005 -0.109

-0.017

0.085

0.152

0.087

0.93

0.86

0.93

0.98

1.76

0.5

-0.005 -0.140

-0.026

0.085

0.185

0.089

0.93

0.80

0.93

0.95

2.07

0.1

0.000

-0.056

-0.002

0.037

0.070

0.038

0.95

0.74

0.95

1.00

1.86

0.3

0.000

-0.107

-0.010

0.037

0.117

0.039

0.95

0.40

0.94

0.96

3.01

0.5

0.000

-0.142

-0.020

0.037

0.151

0.044

0.95

0.24

0.93

0.86

3.46

167

Table B.135: Empirical variance and asymptotic variance of intercept (βˆ0 ) when ρ =
0.6 over different left censoring proportions of 0.1, 0.3, and 0.5 and sample sizes 50,
100, and 500
n

p

50

100

500

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.304

0.371

0.313

0.363

0.416

0.367

0.3

0.304

0.593

0.350

0.363

0.588

0.375

0.5

0.304

0.928

0.429

0.363

0.874

0.385

0.1

0.145

0.175

0.155

0.178

0.196

0.178

0.3

0.145

0.258

0.168

0.178

0.275

0.182

0.5

0.145

0.371

0.202

0.178

0.402

0.184

0.1

0.034

0.040

0.035

0.035

0.039

0.035

0.3

0.034

0.058

0.040

0.035

0.054

0.036

0.5

0.034

0.080

0.045

0.035

0.079

0.036

Table B.136: Bias, RMSE, 95% CP, and relative RMSE of intercept (βˆ0 ) when ρ = 0.6
over sample sizes 50, 100, and 500 and right censoring proportions of 0.1, 0.3, and 0.5

n

50

100

500

p

Bias

RMSE

CP

RRMSE

CLS

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

-0.008

0.482

0.002

0.602

0.806

0.606

0.97

0.90

0.96

0.99

1.33

0.3

-0.008

1.128

0.039

0.602

1.363

0.614

0.97

0.68

0.96

0.98

2.22

0.5

-0.008

1.714

0.074

0.602

1.952

0.625

0.97

0.54

0.92

0.96

3.12

0.1

0.006

0.435

0.019

0.422

0.621

0.422

0.97

0.84

0.96

1.00

1.47

0.3

0.006

1.066

0.055

0.422

1.188

0.430

0.97

0.46

0.95

0.98

2.77

0.5

0.006

1.665

0.107

0.422

1.782

0.443

0.97

0.25

0.91

0.95

4.03

0.1

-0.003

0.418

0.007

0.187

0.462

0.187

0.94

0.42

0.94

1.00

2.47

0.3

-0.003

1.057

0.042

0.187

1.082

0.193

0.94

0.01

0.93

0.97

5.60

0.5

-0.003

1.665

0.086

0.187

1.689

0.209

0.94

0.00

0.90

0.90

8.09

168

Table B.137: Empirical variance and asymptotic variance of slope (βˆ1 ) when ρ = 0.6
over different left censoring proportions of 0.1, 0.3, and 0.5 and sample sizes 50, 100,
and 500
n

p

50

100

500

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.004

0.005

0.004

0.004

0.004

0.004

0.3

0.004

0.007

0.004

0.004

0.006

0.004

0.5

0.004

0.010

0.005

0.004

0.007

0.004

0.1

0.002

0.002

0.002

0.002

0.002

0.002

0.3

0.002

0.003

0.002

0.002

0.003

0.002

0.5

0.002

0.004

0.002

0.002

0.003

0.002

0.1

0.000

0.000

0.000

0.000

0.000

0.000

0.3

0.000

0.001

0.000

0.000

0.001

0.000

0.5

0.000

0.001

0.000

0.000

0.001

0.000

Table B.138: Bias, RMSE, 95% CP, and relative RMSE of slope (βˆ1 ) when ρ = 0.6
over sample sizes 50, 100, and 500 and right censoring proportions of 0.1, 0.3, and 0.5

n

50

100

500

p

Bias

RMSE

CP

RRMSE

CLS

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

0.001

-0.033

-0.002

0.059

0.074

0.059

0.94

0.91

0.94

1.00

1.24

0.3

0.001

-0.062

-0.008

0.059

0.097

0.061

0.94

0.83

0.92

0.98

1.61

0.5

0.001

-0.081

-0.015

0.059

0.118

0.064

0.94

0.78

0.91

0.92

1.83

0.1

-0.000 -0.032

-0.003

0.041

0.055

0.041

0.93

0.87

0.92

1.00

1.33

0.3

-0.000 -0.061

-0.009

0.041

0.079

0.042

0.93

0.78

0.92

0.98

1.88

0.5

-0.000 -0.081

-0.016

0.041

0.100

0.045

0.93

0.69

0.90

0.91

2.20

0.1

0.000

-0.030

-0.002

0.017

0.036

0.017

0.96

0.67

0.95

0.99

2.09

0.3

0.000

-0.060

-0.007

0.017

0.064

0.019

0.96

0.25

0.93

0.92

3.39

0.5

0.000

-0.081

-0.014

0.017

0.085

0.024

0.96

0.10

0.86

0.71

3.48

169

Table B.139: Empirical variance and asymptotic variance of mean of x (µˆx ) when
ρ = 0.6 over different left censoring proportions of 0.1, 0.3, and 0.5 and sample sizes
50, 100, and 500
n

p

50

100

500

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

1.954

2.042

1.985

1.984

1.740

1.959

0.3

1.954

2.270

2.034

1.984

1.732

1.959

0.5

1.954

2.492

2.110

1.984

1.821

1.924

0.1

0.937

0.953

0.947

0.996

0.876

0.981

0.3

0.937

1.123

1.009

0.996

0.857

0.977

0.5

0.937

1.184

1.038

0.996

0.897

0.960

0.1

0.208

0.208

0.211

0.200

0.174

0.198

0.3

0.208

0.229

0.216

0.200

0.170

0.197

0.5

0.208

0.251

0.225

0.200

0.177

0.193

Table B.140: Bias, RMSE, 95% CP, and relative RMSE of mean of x (µˆx ) when
ρ = 0.6 over sample sizes 50, 100, and 500 and right censoring proportions of 0.1, 0.3,
and 0.5

n

50

100

500

p

Bias

RMSE

CP

RRMSE

CLS

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

0.097

1.699

0.076 1.401

2.151

1.402

0.94

0.73

0.94

1.01

1.53

0.3

0.097

3.587

0.127 1.401

3.821

1.405

0.94

0.24

0.94

1.00

2.72

0.5

0.097

5.231

0.213 1.401

5.402

1.404

0.94

0.05

0.93

1.01

3.85

0.1

0.066

1.467

0.079 0.970

1.740

0.993

0.95

0.66

0.95

1.01

1.75

0.3

0.066

3.418

0.148 0.970

3.541

1.000

0.95

0.06

0.93

1.00

3.54

0.5

0.066

5.080

0.251 0.970

5.167

1.011

0.95

0.00

0.91

0.99

5.11

0.1

0.026

1.453

0.044 0.457

1.512

0.447

0.94

0.08

0.93

1.00

3.38

0.3

0.026

3.407

0.114 0.457

3.432

0.458

0.94

0.00

0.93

0.98

7.49

0.5

0.026

5.083

0.230 0.457

5.100

0.495

0.94

0.00

0.90

0.90

10.30

170

Table B.141: Empirical variance and asymptotic variance of mean of y (µˆy ) when
ρ = 0.6 over different left censoring proportions of 0.1, 0.3, and 0.5 and sample sizes
50, 100, and 500
n

p

50

100

3*500

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.454

0.469

0.452

0.503

0.438

0.495

0.3

0.454

0.527

0.467

0.503

0.431

0.491

0.5

0.454

0.583

0.483

0.503

0.454

0.483

0.1

0.212

0.223

0.215

0.250

0.218

0.245

0.3

0.212

0.257

0.219

0.250

0.213

0.243

0.5

0.212

0.287

0.235

0.250

0.221

0.238

0.1

0.052

0.054

0.052

0.050

0.043

0.049

0.3

0.052

0.060

0.054

0.050

0.042

0.049

0.5

0.052

0.065

0.057

0.050

0.044

0.048

Table B.142: Bias, RMSE, 95% CP, and relative RMSE of mean of y (µˆy ) when
ρ = 0.6 over sample sizes 50, 100, and 500 and right censoring proportions of 0.1, 0.3,
and 0.5

n

50

100

500

p

Bias

RMSE

CP

RRMSE

CLS

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

0.019

0.832

0.015 0.710

1.063

0.703

0.96

0.77

0.96

1.01

1.51

0.3

0.019

1.784

0.050 0.710

1.901

0.703

0.96

0.23

0.95

1.01

2.71

0.5

0.019

2.608

0.094 0.710

2.694

0.701

0.96

0.04

0.94

1.01

3.84

0.1

0.023

0.733

0.034 0.501

0.869

0.496

0.97

0.64

0.97

1.01

1.75

0.3

0.023

1.703

0.075 0.501

1.764

0.499

0.97

0.06

0.96

1.00

3.54

0.5

0.023

2.534

0.135 0.501

2.578

0.506

0.97

0.00

0.95

0.99

5.09

0.1

0.005

0.719

0.015 0.223

0.749

0.223

0.93

0.09

0.93

1.00

3.36

0.3

0.005

1.699

0.054 0.223

1.711

0.228

0.93

0.00

0.93

0.98

7.51

0.5

0.005

2.535

0.111 0.223

2.543

0.245

0.93

0.00

0.90

0.91

10.36

171

Table B.143: Empirical variance and asymptotic variance of standard deviation of x
(µˆx ) when ρ = 0.6 over different left censoring proportions of 0.1, 0.3, and 0.5 and
sample sizes 50, 100, and 500
n

p

50

100

500

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

1.088

1.008

1.127

1.030

0.906

0.980

0.3

1.088

1.076

1.234

1.030

0.908

0.980

0.5

1.088

1.195

1.421

1.030

0.963

0.962

0.1

0.540

0.503

0.565

0.507

0.447

0.490

0.3

0.540

0.493

0.621

0.507

0.438

0.489

0.5

0.540

0.529

0.722

0.507

0.461

0.480

0.1

0.104

0.094

0.111

0.100

0.087

0.099

0.3

0.104

0.093

0.124

0.100

0.085

0.099

0.5

0.104

0.096

0.137

0.100

0.089

0.096

Table B.144: Bias, RMSE, 95% CP, and relative RMSE of standard deviation of x
(µˆx ) when ρ = 0.6 over sample sizes 50, 100, and 500 and right censoring proportions
of 0.1, 0.3, and 0.5

n

p

Bias
CLS

50

100

500

RMSE

CP

RRMSE

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

-0.095 -1.219

-0.159

1.019

1.546

1.003

0.94

0.68

0.91

1.02

1.54

0.3

-0.095 -2.086

-0.263

1.019

2.294

1.024

0.94

0.40

0.89

1.00

2.24

0.5

-0.095 -2.690

-0.409

1.019

2.864

1.063

0.94

0.27

0.84

0.96

2.69

0.1

-0.045 -1.074

-0.126

0.714

1.265

0.712

0.93

0.61

0.91

1.00

1.78

0.3

-0.045 -2.006

-0.244

0.714

2.112

0.740

0.93

0.17

0.88

0.96

2.85

0.5

-0.045 -2.613

-0.385

0.714

2.699

0.792

0.93

0.06

0.85

0.90

3.41

0.1

-0.008 -1.079

-0.058

0.317

1.119

0.320

0.94

0.07

0.93

0.99

3.50

0.3

-0.008 -2.007

-0.179

0.317

2.029

0.361

0.94

0.00

0.88

0.88

5.62

0.5

-0.008 -2.616

-0.336

0.317

2.633

0.457

0.94

0.00

0.77

0.69

5.76

172

Table B.145: Empirical variance and asymptotic variance of standard deviation of y
(µˆy ) when ρ = 0.6 over different left censoring proportions of 0.1, 0.3, and 0.5 and
sample sizes 50, 100, and 500
n

p

50

100

500

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.271

0.262

0.302

0.261

0.228

0.247

0.3

0.271

0.270

0.337

0.261

0.226

0.246

0.5

0.271

0.301

0.390

0.261

0.240

0.242

0.1

0.127

0.123

0.137

0.127

0.111

0.123

0.3

0.127

0.129

0.152

0.127

0.109

0.122

0.5

0.127

0.142

0.178

0.127

0.114

0.119

0.1

0.024

0.021

0.025

0.025

0.022

0.025

0.3

0.024

0.023

0.029

0.025

0.021

0.025

0.5

0.024

0.025

0.034

0.025

0.022

0.024

Table B.146: Bias, RMSE, 95% CP, and relative RMSE of standard deviation of y
(µˆy ) when ρ = 0.6 over sample sizes 50, 100, and 500 and right censoring proportions
of 0.1, 0.3, and 0.5

n

p

Bias
CLS

50

100

500

RMSE

CP

RRMSE

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

-0.011 -0.593

-0.058

0.511

0.762

0.501

0.93

0.72

0.91

1.02

1.52

0.3

-0.011 -1.050

-0.127

0.511

1.153

0.512

0.93

0.39

0.87

1.00

2.25

0.5

-0.011 -1.351

-0.197

0.511

1.437

0.530

0.93

0.26

0.83

0.97

2.71

0.1

-0.011 -0.547

-0.062

0.357

0.641

0.356

0.94

0.59

0.92

1.00

1.80

0.3

-0.011 -1.018

-0.132

0.357

1.070

0.373

0.94

0.18

0.88

0.96

2.87

0.5

-0.011 -1.331

-0.214

0.357

1.373

0.406

0.94

0.07

0.83

0.88

3.38

0.1

-0.007 -0.546

-0.034

0.158

0.565

0.161

0.96

0.04

0.94

0.98

3.52

0.3

-0.007 -1.014

-0.102

0.158

1.025

0.187

0.96

0.00

0.86

0.85

5.49

0.5

-0.007 -1.320

-0.179

0.158

1.328

0.237

0.96

0.00

0.74

0.67

5.61

173

Table B.147: Empirical variance and asymptotic variance of correlation (ρ̂) when
ρ = 0.6 over different left censoring proportions of 0.1, 0.3, and 0.5 and sample sizes
50, 100, and 500
n

p

50

100

500

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.009

0.013

0.009

0.009

0.012

0.009

0.3

0.009

0.020

0.009

0.009

0.017

0.009

0.5

0.009

0.030

0.010

0.009

0.022

0.009

0.1

0.004

0.006

0.004

0.004

0.006

0.004

0.3

0.004

0.009

0.005

0.004

0.008

0.004

0.5

0.004

0.012

0.005

0.004

0.011

0.005

0.1

0.001

0.001

0.001

0.001

0.001

0.001

0.3

0.001

0.002

0.001

0.001

0.002

0.001

0.5

0.001

0.002

0.001

0.001

0.002

0.001

ˆ when
Table B.148: Bias, RMSE, 95% CP, and relative RMSE of correlation (rho)
ρ = 0.6 over sample sizes 50, 100, and 500 and right censoring proportions of 0.1, 0.3,
and 0.5

n

p

Bias
CLS

50

100

500

RMSE

CP

RRMSE

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

-0.007 -0.074

-0.012

0.094

0.133

0.094

0.92

0.92

0.93

1.01

1.42

0.3

-0.007 -0.129

-0.023

0.094

0.183

0.097

0.92

0.89

0.93

0.97

1.88

0.5

-0.007 -0.169

-0.037

0.094

0.225

0.103

0.92

0.83

0.95

0.91

2.18

0.1

-0.004 -0.065

-0.008

0.066

0.100

0.065

0.94

0.89

0.94

1.01

1.53

0.3

-0.004 -0.122

-0.019

0.066

0.151

0.069

0.94

0.79

0.93

0.95

2.20

0.5

-0.004 -0.162

-0.033

0.066

0.192

0.075

0.94

0.68

0.93

0.87

2.55

0.1

-0.000 -0.060

-0.004

0.028

0.069

0.029

0.95

0.57

0.95

0.99

2.42

0.3

-0.000 -0.118

-0.014

0.028

0.125

0.033

0.95

0.12

0.93

0.86

3.79

0.5

-0.000 -0.162

-0.027

0.028

0.168

0.040

0.95

0.04

0.89

0.70

4.16

174

Table B.149: Empirical variance and asymptotic variance of intercept(βˆ0 ) when ρ =
0.8 over different left censoring proportions of 0.1, 0.3, and 0.5 and sample sizes 50,
100, and 500
n

p

50

100

500

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.170

0.211

0.183

0.204

0.246

0.208

0.3

0.170

0.351

0.228

0.204

0.362

0.220

0.5

0.170

0.584

0.283

0.204

0.549

0.235

0.1

0.083

0.104

0.092

0.100

0.116

0.101

0.3

0.083

0.149

0.111

0.100

0.168

0.106

0.5

0.083

0.228

0.138

0.100

0.253

0.112

0.1

0.019

0.022

0.020

0.020

0.023

0.020

0.3

0.019

0.033

0.025

0.020

0.033

0.021

0.5

0.019

0.045

0.029

0.020

0.049

0.022

ˆ 0 ) when
Table B.150: Bias, RMSE, 95% CP, and relative RMSE of intercept (beta
ρ = 0.8 over sample sizes 50, 100, and 500 and left censoring proportions of 0.1, 0.3,
and 0.5

n

50

100

500

p

Bias

RMSE

CP

RRMSE

CLS

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

-0.010

0.273

0.004

0.452

0.566

0.456

0.96

0.93

0.96

0.99

1.24

0.3

-0.010

0.659

0.030

0.452

0.892

0.470

0.96

0.80

0.94

0.96

1.90

0.5

-0.010

1.053

0.069

0.452

1.288

0.490

0.96

0.69

0.91

0.92

2.63

0.1

0.001

0.248

0.015

0.316

0.421

0.318

0.97

0.87

0.96

1.00

1.33

0.3

0.001

0.624

0.045

0.316

0.747

0.328

0.97

0.67

0.94

0.96

2.28

0.5

0.001

1.017

0.097

0.316

1.135

0.349

0.97

0.44

0.88

0.91

3.25

0.1

-0.003

0.236

0.007

0.140

0.280

0.141

0.95

0.64

0.94

1.00

1.99

0.3

-0.003

0.615

0.039

0.140

0.642

0.149

0.95

0.07

0.90

0.94

4.30

0.5

-0.003

1.005

0.081

0.140

1.029

0.169

0.95

0.00

0.88

0.83

6.09

175

Table B.151: Empirical variance and asymptotic variance of slope (βˆ1 ) when ρ = 0.8
over different left censoring proportions of 0.1, 0.3, and 0.5 and sample sizes 50, 100,
and 500
n

p

50

100

500

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.002

0.003

0.002

0.002

0.003

0.002

0.3

0.002

0.004

0.003

0.002

0.004

0.002

0.5

0.002

0.006

0.003

0.002

0.005

0.002

0.1

0.001

0.001

0.001

0.001

0.001

0.001

0.3

0.001

0.002

0.001

0.001

0.002

0.001

0.5

0.001

0.003

0.002

0.001

0.002

0.001

0.1

0.000

0.000

0.000

0.000

0.000

0.000

0.3

0.000

0.000

0.000

0.000

0.000

0.000

0.5

0.000

0.000

0.000

0.000

0.000

0.000

ˆ 0 ) when ρ = 0.8
Table B.152: Bias, RMSE, 95% CP, and relative RMSE of slope (beta
over sample sizes 50, 100, and 500 and left censoring proportions of 0.1, 0.3, and 0.5

n

50

100

500

p

Bias

RMSE

CP

RRMSE

CLS

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

0.001

-0.024

-0.002

0.044

0.056

0.045

0.94

0.90

0.93

0.97

1.25

0.3

0.001

-0.047

-0.009

0.044

0.075

0.047

0.94

0.85

0.92

0.94

1.62

0.5

0.001

-0.065

-0.019

0.044

0.094

0.053

0.94

0.80

0.90

0.83

1.79

0.1

0.000

-0.023

-0.003

0.030

0.041

0.032

0.93

0.88

0.93

0.95

1.31

0.3

0.000

-0.045

-0.009

0.030

0.060

0.033

0.93

0.79

0.90

0.91

1.84

0.5

0.000

-0.064

-0.019

0.030

0.079

0.038

0.93

0.71

0.85

0.79

2.07

0.1

-0.000 -0.022

-0.002

0.014

0.026

0.014

0.96

0.71

0.94

0.99

1.80

0.3

-0.000 -0.044

-0.008

0.014

0.048

0.016

0.96

0.27

0.89

0.86

2.90

0.5

-0.000 -0.063

-0.017

0.014

0.066

0.022

0.96

0.11

0.77

0.63

2.94

176

Table B.153: Empirical variance and asymptotic variance of mean of x (µˆx ) when
ρ = 0.8 over different left censoring proportions of 0.1, 0.3, and 0.5 and sample sizes
50, 100, and 500
n

p

50

100

500

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

1.950

2.056

1.980

1.985

1.695

1.947

0.3

1.950

2.172

2.027

1.985

1.631

1.932

0.5

1.950

2.348

2.050

1.985

1.667

1.897

0.1

0.924

0.954

0.935

0.996

0.856

0.974

0.3

0.924

1.083

0.987

0.996

0.811

0.964

0.5

0.924

1.139

1.028

0.996

0.823

0.947

0.1

0.209

0.208

0.212

0.200

0.170

0.197

0.3

0.209

0.225

0.214

0.200

0.161

0.195

0.5

0.209

0.243

0.220

0.200

0.163

0.190

Table B.154: Bias, RMSE, 95% CP, and relative RMSE of mean of x (mu
ˆ x ) when
ρ = 0.8 over sample sizes 50, 100, and 500 and left censoring proportions of 0.1, 0.3,
and 0.5

n

50

100

500

p

Bias

RMSE

CP

RRMSE

CLS

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

0.095

1.695

0.089 1.412

2.137

1.398

0.95

0.72

0.94

1.01

1.53

0.3

0.095

3.582

0.169 1.412

3.803

1.400

0.95

0.23

0.94

1.01

2.72

0.5

0.095

5.201

0.179 1.412

5.359

1.389

0.95

0.05

0.93

1.02

3.86

0.1

0.066

1.473

0.094 1.000

1.739

0.991

0.96

0.64

0.95

1.01

1.75

0.3

0.066

3.405

0.187 1.000

3.522

1.000

0.96

0.05

0.93

1.00

3.52

0.5

0.066

5.054

0.295 1.000

5.135

1.017

0.96

0.00

0.92

0.98

5.05

0.1

0.026

1.441

0.055 0.448

1.499

0.447

0.93

0.08

0.93

1.00

3.35

0.3

0.026

3.392

0.149 0.448

3.415

0.466

0.93

0.00

0.92

0.96

7.33

0.5

0.026

5.053

0.274 0.448

5.069

0.515

0.93

0.00

0.89

0.87

9.84

177

Table B.155: Empirical variance and asymptotic variance of mean of y (µˆy ) when
ρ = 0.8 over different left censoring proportions of 0.1, 0.3, and 0.5 and sample sizes
50, 100, and 500
n

p

50

100

500

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.463

0.482

0.464

0.501

0.426

0.490

0.3

0.463

0.518

0.475

0.501

0.408

0.485

0.5

0.463

0.567

0.478

0.501

0.415

0.474

0.1

0.213

0.220

0.217

0.250

0.213

0.243

0.3

0.213

0.245

0.221

0.250

0.202

0.240

0.5

0.213

0.269

0.234

0.250

0.204

0.235

0.1

0.052

0.053

0.053

0.050

0.043

0.049

0.3

0.052

0.058

0.055

0.050

0.040

0.049

0.5

0.052

0.062

0.056

0.050

0.041

0.047

Table B.156: Bias, RMSE, 95% CP, and relative RMSE of mean of y (µˆy ) when
ρ = 0.8 over sample sizes 50, 100, and 500 and left censoring proportions of 0.1, 0.3,
and 0.5

n

50

100

500

p

Bias

RMSE

CP

RRMSE

CLS

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

0.028

0.836

0.031 0.709

1.061

0.700

0.95

0.73

0.95

1.01

1.51

0.3

0.028

1.781

0.072 0.709

1.892

0.700

0.95

0.22

0.94

1.01

2.70

0.5

0.028

2.598

0.083 0.709

2.676

0.694

0.95

0.04

0.93

1.02

3.86

0.1

0.027

0.737

0.045 0.500

0.869

0.495

0.96

0.64

0.96

1.01

1.75

0.3

0.027

1.698

0.094 0.500

1.756

0.499

0.96

0.04

0.95

1.00

3.52

0.5

0.027

2.525

0.156 0.500

2.565

0.509

0.96

0.00

0.93

0.98

5.04

0.1

0.007

0.717

0.023 0.224

0.746

0.223

0.94

0.09

0.92

1.00

3.35

0.3

0.007

1.690

0.073 0.224

1.702

0.232

0.94

0.00

0.92

0.96

7.34

0.5

0.007

2.524

0.135 0.224

2.532

0.256

0.94

0.00

0.89

0.87

9.89

178

Table B.157: Empirical variance and asymptotic variance of standard deviation of x
(σˆx ) when ρ = 0.8 over different left censoring proportions of 0.1, 0.3, and 0.5 and
sample sizes 50, 100, and 500
n

p

50

100

500

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

1.082

0.991

1.124

1.040

0.889

0.974

0.3

1.082

1.031

1.245

1.040

0.861

0.966

0.5

1.082

1.166

1.462

1.040

0.888

0.949

0.1

0.533

0.491

0.550

0.510

0.438

0.487

0.3

0.533

0.473

0.623

0.510

0.416

0.482

0.5

0.533

0.512

0.707

0.510

0.424

0.474

0.1

0.105

0.094

0.110

0.100

0.086

0.098

0.3

0.105

0.089

0.127

0.100

0.081

0.097

0.5

0.105

0.091

0.140

0.100

0.082

0.095

Table B.158: Bias, RMSE, 95% CP, and relative RMSE ofstandard deviation of x
(σˆx ) when ρ = 0.8 over sample sizes 50, 100, and 500 and left censoring proportions
of 0.1, 0.3, and 0.5

n

p

Bias
CLS

50

100

500

RMSE

CP

RRMSE

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

-0.092 -1.280

-0.190

1.024

1.590

1.005

0.93

0.65

0.90

1.02

1.58

0.3

-0.092 -2.194

-0.333

1.024

2.382

1.038

0.93

0.36

0.88

0.99

2.30

0.5

-0.092 -2.804

-0.480

1.024

2.959

1.086

0.93

0.21

0.83

0.94

2.73

0.1

-0.046 -1.139

-0.160

0.715

1.317

0.716

0.94

0.58

0.91

1.00

1.84

0.3

-0.046 -2.107

-0.310

0.715

2.204

0.760

0.94

0.13

0.86

0.94

2.90

0.5

-0.046 -2.733

-0.447

0.715

2.810

0.821

0.94

0.03

0.84

0.87

3.42

0.1

-0.008 -1.130

-0.086

0.317

1.168

0.325

0.94

0.04

0.93

0.97

3.59

0.3

-0.008 -2.108

-0.239

0.317

2.127

0.393

0.94

0.00

0.84

0.81

5.41

0.5

-0.008 -2.739

-0.397

0.317

2.754

0.503

0.94

0.00

0.71

0.63

5.48

179

Table B.159: Empirical variance and asymptotic variance of standard deviation of y
(σˆy ) when ρ = 0.8 over different left censoring proportions of 0.1, 0.3, and 0.5 and
sample sizes 50, 100, and 500
n

p

50

100

500

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.274

0.265

0.301

0.263

0.223

0.245

0.3

0.274

0.270

0.341

0.263

0.215

0.243

0.5

0.274

0.293

0.381

0.263

0.221

0.237

0.1

0.127

0.124

0.137

0.128

0.109

0.122

0.3

0.127

0.122

0.154

0.128

0.103

0.120

0.5

0.127

0.133

0.182

0.128

0.105

0.117

0.1

0.025

0.022

0.025

0.025

0.021

0.025

0.3

0.025

0.021

0.029

0.025

0.020

0.024

0.5

0.025

0.022

0.033

0.025

0.020

0.024

Table B.160: Bias, RMSE, 95% CP, and relative RMSE of standard deviation of y
(σˆy ) when ρ = 0.8 over sample sizes 50, 100, and 500 and left censoring proportions
of 0.1, 0.3, and 0.5

n

p

Bias
CLS

50

100

500

RMSE

CP

RRMSE

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

-0.021 -0.633

-0.083

0.513

0.790

0.502

0.93

0.68

0.89

1.02

1.57

0.3

-0.021 -1.098

-0.158

0.513

1.192

0.517

0.93

0.35

0.87

0.99

2.30

0.5

-0.021 -1.410

-0.241

0.513

1.487

0.544

0.93

0.23

0.82

0.94

2.74

0.1

-0.016 -0.578

-0.080

0.358

0.666

0.358

0.94

0.57

0.93

1.00

1.86

0.3

-0.016 -1.065

-0.161

0.358

1.113

0.382

0.94

0.13

0.86

0.94

2.91

0.5

-0.016 -1.387

-0.244

0.358

1.424

0.421

0.94

0.04

0.81

0.85

3.39

0.1

-0.006 -0.572

-0.046

0.159

0.590

0.164

0.96

0.02

0.92

0.97

3.61

0.3

-0.006 -1.060

-0.128

0.159

1.069

0.202

0.96

0.00

0.82

0.79

5.31

0.5

-0.006 -1.379

-0.206

0.159

1.386

0.257

0.96

0.00

0.71

0.62

5.39

180

Table B.161: Empirical variance and asymptotic variance of correlation (ρ̂) when
ρ = 0.8 over different left censoring proportions of 0.1, 0.3, and 0.5 and sample sizes
50, 100, and 500
n

p

50

100

500

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.003

0.005

0.003

0.003

0.005

0.003

0.3

0.003

0.009

0.003

0.003

0.007

0.003

0.5

0.003

0.015

0.003

0.003

0.011

0.004

0.1

0.001

0.002

0.001

0.001

0.002

0.001

0.3

0.001

0.004

0.002

0.001

0.003

0.002

0.5

0.001

0.006

0.002

0.001

0.005

0.002

0.1

0.000

0.001

0.000

0.000

0.000

0.000

0.3

0.000

0.001

0.000

0.000

0.001

0.000

0.5

0.000

0.001

0.000

0.000

0.001

0.000

Table B.162: Bias, RMSE, 95% CP, and relative RMSE of correlation (ρ̂) when ρ = 0.8
over sample sizes 50, 100, and 500 and left censoring proportions of 0.1, 0.3, and 0.5

n

p

Bias
CLS

50

100

500

RMSE

CP

RRMSE

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

-0.005 -0.053

-0.010

0.055

0.086

0.055

0.94

0.94

0.94

1.01

1.58

0.3

-0.005 -0.098

-0.023

0.055

0.130

0.061

0.94

0.88

0.97

0.90

2.13

0.5

-0.005 -0.135

-0.043

0.055

0.170

0.075

0.94

0.83

0.96

0.74

2.27

0.1

-0.003 -0.046

-0.007

0.038

0.065

0.038

0.94

0.89

0.95

0.98

1.70

0.3

-0.003 -0.091

-0.019

0.038

0.107

0.043

0.94

0.72

0.95

0.87

2.49

0.5

-0.003 -0.127

-0.037

0.038

0.145

0.057

0.94

0.60

0.91

0.66

2.56

0.1

-0.000 -0.042

-0.004

0.017

0.047

0.018

0.94

0.45

0.95

0.97

2.63

0.3

-0.000 -0.088

-0.016

0.017

0.091

0.023

0.94

0.04

0.89

0.74

3.89

0.5

-0.000 -0.124

-0.034

0.017

0.127

0.038

0.94

0.00

0.58

0.46

3.35

181

B.4

Results in the Presence of Interval Censored
Observations in the Data

Figure B.10: Boxplots for estimates of mean of x (µˆx ) and mean of y (µˆy ), when
n = 100, ρ = 0.2 over different interval censoring proportions of 0.1, 0.3, and 0.5

Figure B.11: Boxplots for estimates of standard deviation of x (σˆx ) and standard
deviation if y (σˆy ), when n = 100, ρ = 0.2 over different interval censoring proportions
of 0.1, 0.3, and 0.5

182

Figure B.12: Boxplots for estimates of correlation between x and y (ρ̂), when n =
100, ρ = 0.2 over different interval censoring proportions of 0.1, 0.3, and 0.5

Table B.163: Empirical variance and asymptotic variance of intercept (βˆ0 ) when ρ =
0.2 over different interval censoring proportions of 0.1, 0.3, and 0.5 and sample sizes
50, 100, and 500
n

50

100

500

p

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.460

0.490

0.460

0.545

0.627

0.545

0.3

0.460

0.630

0.458

0.545

0.832

0.545

0.5

0.460

0.949

0.463

0.545

1.086

0.544

0.1

0.215

0.239

0.215

0.267

0.307

0.267

0.3

0.215

0.342

0.216

0.267

0.407

0.267

0.5

0.215

0.583

0.217

0.267

0.533

0.267

0.1

0.052

0.061

0.052

0.052

0.060

0.052

0.3

0.052

0.121

0.052

0.052

0.120

0.052

0.5

0.052

0.289

0.053

0.052

0.210

0.052

183

Table B.164: Empirical variance and asymptotic variance of slope (βˆ1 ) when ρ = 0.2
over different interval censoring proportions of 0.1, 0.3, and 0.5 and sample sizes 50,
100, and 500
n

50

100

500

p

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.006

0.006

0.006

0.005

0.006

0.005

0.3

0.006

0.007

0.006

0.005

0.007

0.005

0.5

0.006

0.010

0.006

0.005

0.009

0.005

0.1

0.003

0.003

0.003

0.003

0.003

0.003

0.3

0.003

0.004

0.003

0.003

0.003

0.003

0.5

0.003

0.005

0.003

0.003

0.004

0.003

0.1

0.001

0.001

0.001

0.001

0.001

0.001

0.3

0.001

0.001

0.001

0.001

0.001

0.001

0.5

0.001

0.001

0.001

0.001

0.001

0.001

Table B.165: Empirical variance and asymptotic variance of mean of x (µˆx ) when
ρ = 0.2 over different interval censoring proportions of 0.1, 0.3, and 0.5 and sample
sizes 50, 100, and 500
n

50

100

500

p

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

1.955

2.172

1.956

1.984

2.307

1.944

0.3

1.955

2.918

1.969

1.984

3.139

1.943

0.5

1.955

4.371

2.011

1.984

4.216

1.939

0.1

0.958

1.067

0.958

0.997

1.158

0.987

0.3

0.958

1.545

0.959

0.997

1.585

0.987

0.5

0.958

2.591

0.969

0.997

2.163

0.986

0.1

0.206

0.232

0.206

0.200

0.232

0.200

0.3

0.206

0.523

0.208

0.200

0.476

0.200

0.5

0.206

1.459

0.206

0.200

0.859

0.199

184

Table B.166: Bias, RMSE, 95% CP, and relative RMSE of mean x (µˆx ) when ρ = 0.2
over sample sizes 50, 100, and 500 and interval censoring proportions of 0.1, 0.3, and
0.5

n

50

100

500

p

Bias

RMSE

CP

RRMSE

CLS

Naive

PM

CLS

Naive

PM

CLS Naive

PM

CLS/PM

Naive/PM

0.1

0.099

0.066

0.100

1.412

1.520

1.398

0.95

0.94

0.94

1.01

1.09

0.3

0.099

-0.307

0.100

1.412

1.798

1.397

0.95

0.94

0.94

1.01

1.29

0.5

0.099

-1.223

0.092

1.412

2.390

1.395

0.95

0.89

0.93

1.01

1.71

0.1

0.064

0.050

0.064

1.001

1.077

0.996

0.96

0.96

0.96

1.01

1.08

0.3

0.064

-0.329

0.064

1.001

1.301

0.995

0.96

0.94

0.95

1.01

1.31

0.5

0.064

-1.279

0.057

1.001

1.949

0.994

0.96

0.82

0.95

1.01

1.96

0.1

0.027

-0.018

0.027

0.448

0.482

0.448

0.94

0.94

0.94

1.00

1.08

0.3

0.027

-0.438

0.027

0.448

0.818

0.448

0.94

0.88

0.94

1.00

1.83

0.5

0.027

-1.459

0.023

0.448

1.729

0.447

0.94

0.60

0.94

1.00

3.87

Table B.167: Empirical variance and asymptotic variance of mean of y (µˆy ) when
ρ = 0.2 over different interval censoring proportions of 0.1, 0.3, and 0.5 and sample
sizes 50, 100, and 500
n

50

100

500

p

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.444

0.470

0.444

0.505

0.587

0.495

0.3

0.444

0.616

0.443

0.505

0.802

0.495

0.5

0.444

0.975

0.450

0.505

1.081

0.494

0.1

0.216

0.239

0.217

0.251

0.291

0.248

0.3

0.216

0.345

0.216

0.251

0.398

0.248

0.5

0.216

0.614

0.218

0.251

0.541

0.248

0.1

0.051

0.060

0.051

0.050

0.058

0.050

0.3

0.051

0.135

0.051

0.050

0.119

0.050

0.5

0.051

0.356

0.051

0.050

0.217

0.050

185

Table B.168: Bias, RMSE, 95% CP, and relative RMSE of mean y (µˆy ) when ρ = 0.2
over sample sizes 50, 100, and 500 and interval censoring proportions of 0.1, 0.3, and
0.5

n

p

Bias
CLS

50

100

500

RMSE

CP

RRMSE

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

0.005 -0.021

0.005

0.711

0.766

0.704

0.95

0.96

0.95

1.01

1.09

0.3

0.005 -0.211

0.005

0.711

0.920

0.704

0.95

0.96

0.95

1.01

1.31

0.5

0.005 -0.689

0.004

0.711

1.247

0.703

0.95

0.90

0.95

1.01

1.77

0.1

0.015 -0.002

0.015

0.501

0.539

0.499

0.97

0.97

0.97

1.01

1.08

0.3

0.015 -0.190

0.017

0.501

0.659

0.499

0.97

0.94

0.97

1.00

1.32

0.5

0.015 -0.675

0.015

0.501

0.998

0.498

0.97

0.79

0.97

1.01

2.00

0.1

0.001 -0.022

0.001

0.223

0.242

0.223

0.93

0.94

0.94

1.00

1.08

0.3

0.001 -0.216

0.001

0.223

0.407

0.223

0.93

0.87

0.93

1.00

1.82

0.5

0.001 -0.711

-0.000

0.223

0.850

0.223

0.93

0.62

0.93

1.00

3.81

Table B.169: Empirical variance and asymptotic variance of standard deviation of x
(σˆx ) when ρ = 0.2 over different interval censoring proportions of 0.1, 0.3, and 0.5
and sample sizes 50, 100, and 500
n

50

100

500

p

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

1.104

1.179

1.082

1.023

1.194

0.972

0.3

1.104

1.453

1.087

1.023

1.639

0.971

0.5

1.104

2.163

1.089

1.023

2.230

0.969

0.1

0.551

0.590

0.546

0.506

0.589

0.494

0.3

0.551

0.758

0.545

0.506

0.810

0.493

0.5

0.551

1.130

0.551

0.506

1.112

0.493

0.1

0.102

0.116

0.102

0.100

0.117

0.100

0.3

0.102

0.159

0.102

0.100

0.239

0.100

0.5

0.102

0.499

0.103

0.100

0.432

0.100

186

Table B.170: Bias, RMSE, 95% CP, and relative RMSE of standard deviation of x (σˆx )
when ρ = 0.2 over sample sizes 50, 100, and 500 and interval censoring proportions
of 0.1, 0.3, and 0.5

n

50

100

500

p

Bias

RMSE

CP

RRMSE

CLS

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

-0.096

0.150

-0.196 1.016

1.103

1.005

0.94

0.95

0.93

1.01

1.10

0.3

-0.096

0.610

-0.199 1.016

1.418

1.006

0.94

0.97

0.93

1.01

1.41

0.5

-0.096

0.891

-0.210 1.016

1.739

1.007

0.94

0.95

0.93

1.01

1.73

0.1

-0.043

0.198

-0.093 0.713

0.793

0.709

0.94

0.96

0.93

1.01

1.12

0.3

-0.043

0.689

-0.095 0.713

1.134

0.709

0.94

0.92

0.93

1.01

1.60

0.5

-0.043

1.033

-0.100 0.713

1.476

0.709

0.94

0.89

0.93

1.01

2.08

0.1

-0.008

0.233

-0.018 0.317

0.413

0.316

0.95

0.89

0.95

1.00

1.31

0.3

-0.008

0.712

-0.018 0.317

0.863

0.316

0.95

0.75

0.95

1.00

2.73

0.5

-0.008

0.996

-0.022 0.317

1.193

0.317

0.95

0.64

0.95

1.00

3.77

Table B.171: Empirical variance and asymptotic variance of standard deviation of y
(σˆy ) when ρ = 0.2 over different interval censoring proportions of 0.1, 0.3, and 0.5
and sample sizes 50, 100, and 500
n

50

100

500

p

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.257

0.288

0.252

0.261

0.304

0.248

0.3

0.257

0.340

0.252

0.261

0.419

0.247

0.5

0.257

0.489

0.253

0.261

0.572

0.247

0.1

0.126

0.143

0.125

0.127

0.148

0.124

0.3

0.126

0.181

0.125

0.127

0.203

0.124

0.5

0.126

0.295

0.124

0.127

0.278

0.124

0.1

0.025

0.027

0.025

0.025

0.029

0.025

0.3

0.025

0.038

0.025

0.025

0.060

0.025

0.5

0.025

0.113

0.025

0.025

0.109

0.025

187

Table B.172: Bias, RMSE, 95% CP, and relative RMSE of standard deviation of y (σˆy )
when ρ = 0.2 over sample sizes 50, 100, and 500 and interval censoring proportions
of 0.1, 0.3, and 0.5

n

50

100

500

p

Bias

RMSE

CP

RRMSE

CLS

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

0.000

0.120

-0.050 0.511

0.564

0.500

0.94

0.95

0.93

1.02

1.13

0.3

0.000

0.367

-0.051 0.511

0.744

0.500

0.94

0.96

0.93

1.02

1.49

0.5

0.000

0.523

-0.054 0.511

0.920

0.500

0.94

0.95

0.93

1.02

1.84

0.1

-0.005

0.111

-0.030 0.357

0.400

0.354

0.94

0.96

0.93

1.01

1.13

0.3

-0.005

0.356

-0.030 0.357

0.574

0.354

0.94

0.93

0.93

1.01

1.62

0.5

-0.005

0.518

-0.031 0.357

0.740

0.354

0.94

0.88

0.93

1.01

2.09

0.1

-0.008

0.118

-0.013 0.158

0.207

0.158

0.95

0.90

0.95

1.00

1.31

0.3

-0.008

0.363

-0.013 0.158

0.438

0.158

0.95

0.75

0.95

1.00

2.77

0.5

-0.008

0.522

-0.014 0.158

0.618

0.158

0.95

0.62

0.95

1.00

3.90

Table B.173: Empirical variance and asymptotic variance of correlation (ρ̂) when
ρ = 0.2 over different interval censoring proportions of 0.1, 0.3, and 0.5 and sample
sizes 50, 100, and 500
n

50

100

500

p

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.020

0.022

0.020

0.019

0.021

0.018

0.3

0.020

0.026

0.020

0.019

0.025

0.018

0.5

0.020

0.034

0.020

0.019

0.032

0.018

0.1

0.010

0.011

0.010

0.009

0.010

0.009

0.3

0.010

0.013

0.010

0.009

0.013

0.009

0.5

0.010

0.017

0.010

0.009

0.016

0.009

0.1

0.002

0.002

0.002

0.002

0.002

0.002

0.3

0.002

0.003

0.002

0.002

0.003

0.002

0.5

0.002

0.004

0.002

0.002

0.003

0.002

188

Table B.174: Bias, RMSE, 95% CP, and relative RMSE of correlation (ρ̂) when ρ = 0.2
over sample sizes 50, 100, and 500 and interval censoring proportions of 0.1, 0.3, and
0.5

n

50

100

500

p

Bias

RMSE

CP

RRMSE

CLS

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

-0.005

0.004

-0.005 0.137

0.143

0.134

0.93

0.93

0.92

1.02

1.07

0.3

-0.005

0.022

-0.005 0.137

0.160

0.134

0.93

0.91

0.91

1.02

1.20

0.5

-0.005

0.032

-0.006 0.137

0.181

0.134

0.93

0.90

0.92

1.02

1.35

0.1

-0.004

0.005

-0.004 0.097

0.101

0.096

0.93

0.93

0.93

1.01

1.06

0.3

-0.004

0.023

-0.005 0.097

0.114

0.096

0.93

0.93

0.93

1.01

1.19

0.5

-0.004

0.039

-0.005 0.097

0.131

0.096

0.93

0.91

0.93

1.01

1.37

0.1

0.000

0.009

0.000

0.042

0.046

0.042

0.95

0.94

0.95

1.00

1.08

0.3

0.000

0.028

0.000

0.042

0.057

0.042

0.95

0.91

0.95

1.00

1.35

0.5

0.000

0.041

-0.000 0.042

0.069

0.042

0.95

0.85

0.95

1.00

1.63

Table B.175: Empirical variance and asymptotic variance of intercept (βˆ0 ) when ρ =
0.4 over different interval censoring proportions of 0.1, 0.3, and 0.5 and sample sizes
50, 100, and 500
n

50

100

500

p

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.400

0.421

0.417

0.476

0.544

0.521

0.3

0.400

0.552

0.401

0.476

0.706

0.537

0.5

0.400

0.769

0.406

0.476

0.902

0.538

0.1

0.189

0.210

0.203

0.234

0.267

0.250

0.3

0.189

0.291

0.193

0.234

0.348

0.260

0.5

0.189

0.456

0.193

0.234

0.442

0.262

0.1

0.045

0.052

0.047

0.046

0.053

0.048

0.3

0.045

0.093

0.047

0.046

0.102

0.051

0.5

0.045

0.192

0.047

0.046

0.174

0.052

189

Table B.176: Bias, RMSE, 95% CP, and relative RMSE of intercept (βˆ0 ) when ρ = 0.4
over sample sizes 50, 100, and 500 and interval censoring proportions of 0.1, 0.3, and
0.5

n

p

Bias
CLS

50

100

500

RMSE

CP

RRMSE

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

-0.003 -0.050

0.012

0.690

0.739

0.722

1.00

0.97

0.97

0.96

1.02

0.3

-0.003 -0.233

0.037

0.690

0.872

0.734

1.00

0.96

0.98

0.94

1.19

0.5

-0.003 -0.550

0.044

0.690

1.097

0.735

1.00

0.90

0.97

0.94

1.49

0.1

0.010

-0.019

0.029

0.483

0.518

0.501

1.00

0.98

0.96

0.97

1.03

0.3

0.010

-0.201

0.051

0.483

0.624

0.513

1.00

0.95

0.98

0.94

1.22

0.5

0.010

-0.552

0.060

0.483

0.864

0.516

1.00

0.84

0.97

0.94

1.68

0.1

-0.003 -0.041

0.019

0.214

0.233

0.221

1.00

0.94

0.93

0.97

1.06

0.3

-0.003 -0.213

0.044

0.214

0.384

0.230

1.00

0.90

0.95

0.93

1.67

0.5

-0.003 -0.550

0.050

0.214

0.690

0.233

1.00

0.69

0.96

0.92

2.97

Table B.177: Empirical variance and asymptotic variance of slope (βˆ1 ) when ρ = 0.4
over different interval censoring proportions of 0.1, 0.3, and 0.5 and sample sizes 50,
100, and 500
n

50

100

500

p

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.005

0.005

0.006

0.005

0.005

0.038

0.3

0.005

0.006

0.005

0.005

0.006

0.005

0.5

0.005

0.008

0.005

0.005

0.008

0.005

0.1

0.002

0.002

0.003

0.002

0.002

0.002

0.3

0.002

0.003

0.002

0.002

0.003

0.002

0.5

0.002

0.004

0.002

0.002

0.004

0.003

0.1

0.000

0.000

0.001

0.000

0.001

0.000

0.3

0.000

0.001

0.001

0.000

0.001

0.001

0.5

0.000

0.001

0.000

0.000

0.001

0.001

190

Table B.178: Bias, RMSE, 95% CP, and relative RMSE of slope (βˆ1 ) when ρ = 0.4
over sample sizes 50, 100, and 500 and interval censoring proportions of 0.1, 0.3, and
0.5

n

50

100

500

p

Bias

RMSE

CP

RRMSE

CLS

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

0.001

0.009

-0.006 0.068

0.071

0.195

0.93

0.93

0.91

0.35

0.37

0.3

0.001

0.023

-0.013 0.068

0.081

0.073

0.93

0.93

0.95

0.93

1.11

0.5

0.001

0.035

-0.016 0.068

0.094

0.074

0.93

0.91

0.95

0.92

1.27

0.1

-0.001

0.007

-0.007 0.047

0.049

0.049

0.94

0.94

0.91

0.97

1.02

0.3

-0.001

0.021

-0.014 0.047

0.058

0.051

0.94

0.93

0.93

0.92

1.14

0.5

-0.001

0.034

-0.017 0.047

0.069

0.053

0.94

0.89

0.94

0.89

1.31

0.1

0.000

0.008

-0.007 0.020

0.024

0.021

0.96

0.95

0.88

0.94

1.11

0.3

0.000

0.025

-0.015 0.020

0.035

0.027

0.96

0.78

0.89

0.74

1.29

0.5

0.000

0.036

-0.017 0.020

0.045

0.028

0.96

0.67

0.89

0.71

1.59

Table B.179: Empirical variance and asymptotic variance of mean of x (µˆx ) when
ρ = 0.4 over different interval censoring proportions of 0.1, 0.3, and 0.5 and sample
sizes 50, 100, and 500
n

50

100

500

p

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

1.955

2.104

1.963

1.984

2.321

2.043

0.3

1.955

2.840

1.969

1.984

3.190

2.104

0.5

1.955

4.585

2.005

1.984

4.277

2.114

0.1

0.948

1.031

0.947

0.997

1.167

1.044

0.3

0.948

1.516

0.957

0.997

1.614

1.065

0.5

0.948

2.598

0.967

0.997

2.184

1.073

0.1

0.207

0.245

0.207

0.200

0.234

0.208

0.3

0.207

0.565

0.208

0.200

0.483

0.216

0.5

0.207

1.538

0.206

0.200

0.869

0.217

191

Table B.180: Bias, RMSE, 95% CP, and relative RMSE of mean of x (µˆx ) when
ρ = 0.4 over sample sizes 50, 100, and 500 and interval censoring proportions of 0.1,
0.3, and 0.5

n

50

100

500

p

Bias

RMSE

CP

RRMSE

CLS

Naive

PM

CLS

Naive

PM

CLS Naive

PM

CLS/PM

Naive/PM

0.1

0.099

0.092

0.106

1.412

1.526

1.433

0.94

0.95

0.94

0.99

1.06

0.3

0.099

-0.324

0.099

1.412

1.815

1.454

0.94

0.95

0.95

0.97

1.25

0.5

0.099

-1.405

0.096

1.412

2.500

1.457

0.94

0.87

0.95

0.97

1.72

0.1

0.065

0.019

0.065

1.000

1.081

1.024

0.96

0.96

0.96

0.98

1.06

0.3

0.065

-0.419

0.065

1.000

1.338

1.034

0.96

0.93

0.96

0.97

1.29

0.5

0.065

-1.525

0.061

1.000

2.123

1.037

0.96

0.78

0.96

0.96

2.05

0.1

0.027

-0.033

0.027

0.448

0.485

0.457

0.94

0.94

0.94

0.98

1.06

0.3

0.027

-0.505

0.028

0.448

0.860

0.466

0.94

0.86

0.95

0.96

1.85

0.5

0.027

-1.707

0.030

0.448

1.945

0.467

0.94

0.54

0.96

0.96

4.16

Table B.181: Empirical variance and asymptotic variance of mean of y (µˆy ) when
ρ = 0.4 over different interval censoring proportions of 0.1, 0.3, and 0.5 and sample
sizes 50, 100, and 500
n

50

100

500

p

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.448

0.464

0.444

0.504

0.588

0.531

0.3

0.448

0.623

0.447

0.504

0.807

0.540

0.5

0.448

0.971

0.456

0.504

1.086

0.539

0.1

0.214

0.232

0.214

0.251

0.293

0.261

0.3

0.214

0.344

0.214

0.251

0.404

0.269

0.5

0.214

0.631

0.215

0.251

0.546

0.270

0.1

0.051

0.060

0.051

0.050

0.058

0.052

0.3

0.051

0.143

0.051

0.050

0.121

0.054

0.5

0.051

0.386

0.052

0.050

0.218

0.054

192

Table B.182: Bias, RMSE, 95% CP, and relative RMSE of mean of y (µˆy ) when
ρ = 0.4 over sample sizes 50, 100, and 500 and interval censoring proportions of 0.1,
0.3, and 0.5

n

50

100

500

p

Bias

RMSE

CP

RRMSE

CLS

Naive

PM

CLS

Naive

PM

CLS Naive

PM

CLS/PM

Naive/PM

0.1

0.012

-0.013

0.012

0.710

0.767

0.729

0.95

0.97

0.96

0.97

1.05

0.3

0.012

-0.246

0.010

0.710

0.931

0.735

0.95

0.97

0.96

0.97

1.27

0.5

0.012

-0.784

0.009

0.710

1.304

0.734

0.95

0.87

0.96

0.97

1.78

0.1

0.019

0.003

0.019

0.501

0.541

0.511

0.97

0.97

0.98

0.98

1.06

0.3

0.019

-0.232

0.020

0.501

0.676

0.519

0.97

0.94

0.97

0.97

1.30

0.5

0.019

-0.801

0.019

0.501

1.090

0.520

0.97

0.75

0.97

0.96

2.10

0.1

0.003

-0.025

0.003

0.223

0.243

0.227

0.93

0.94

0.93

0.98

1.07

0.3

0.003

-0.251

0.004

0.223

0.429

0.232

0.93

0.83

0.94

0.96

1.85

0.5

0.003

-0.836

0.004

0.223

0.958

0.233

0.93

0.54

0.94

0.96

4.11

Table B.183: Empirical variance and asymptotic variance of standard deviation of x
(σˆx ) when ρ = 0.4 over different interval censoring proportions of 0.1, 0.3, and 0.5
and sample sizes 50, 100, and 500
n

50

100

500

p

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

1.096

1.196

1.670

1.025

1.204

1.021

0.3

1.096

1.477

1.301

1.025

1.672

1.052

0.5

1.096

2.124

1.192

1.025

2.273

1.057

0.1

0.546

0.599

1.303

0.507

0.594

0.522

0.3

0.546

0.748

0.731

0.507

0.826

0.532

0.5

0.546

1.161

0.640

0.507

1.125

0.536

0.1

0.103

0.117

0.304

0.100

0.117

0.104

0.3

0.103

0.170

0.205

0.100

0.243

0.108

0.5

0.103

0.527

0.138

0.100

0.437

0.109

193

Table B.184: Bias, RMSE, 95% CP, and relative RMSE of standard deviation of x (σˆx )
when ρ = 0.4 over sample sizes 50, 100, and 500 and interval censoring proportions
of 0.1, 0.3, and 0.5

n

50

100

500

p

Bias

RMSE

CP

RRMSE

CLS

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

-0.096

0.182

0.024

1.017

1.112

1.011

0.94

0.95

0.87

1.01

1.10

0.3

-0.096

0.717

0.193

1.017

1.478

1.044

0.94

0.95

0.93

0.97

1.42

0.5

-0.096

1.026

0.223

1.017

1.824

1.052

0.94

0.96

0.95

0.97

1.73

0.1

-0.044

0.241

0.151

0.713

0.808

0.738

0.93

0.95

0.84

0.97

1.09

0.3

-0.044

0.800

0.283

0.713

1.211

0.782

0.93

0.90

0.90

0.91

1.55

0.5

-0.044

1.129

0.325

0.713

1.549

0.801

0.93

0.85

0.93

0.89

1.93

0.1

-0.008

0.264

0.194

0.317

0.433

0.377

0.95

0.88

0.73

0.84

1.15

0.3

-0.008

0.800

0.391

0.317

0.940

0.511

0.95

0.67

0.70

0.62

1.84

0.5

-0.008

1.104

0.419

0.317

1.287

0.533

0.95

0.54

0.73

0.59

2.41

Table B.185: Empirical variance and asymptotic variance of standard deviation of y
(σˆy ) when ρ = 0.4 over different interval censoring proportions of 0.1, 0.3, and 0.5
and sample sizes 50, 100, and 500
n

50

100

500

p

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.264

0.297

0.457

0.261

0.305

0.266

0.3

0.264

0.355

0.319

0.261

0.423

0.270

0.5

0.264

0.506

0.280

0.261

0.577

0.270

0.1

0.127

0.141

0.238

0.127

0.149

0.131

0.3

0.127

0.179

0.163

0.127

0.207

0.135

0.5

0.127

0.309

0.138

0.127

0.281

0.135

0.1

0.024

0.027

0.066

0.025

0.029

0.026

0.3

0.024

0.039

0.046

0.025

0.061

0.027

0.5

0.024

0.118

0.032

0.025

0.110

0.027

194

Table B.186: Bias, RMSE, 95% CP, and relative RMSE of standard deviation of y (σˆy )
when ρ = 0.4 over sample sizes 50, 100, and 500 and interval censoring proportions
of 0.1, 0.3, and 0.5

n

50

100

500

p

Bias

RMSE

CP

RRMSE

CLS

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

-0.004

0.126

0.108

0.511

0.567

0.527

0.93

0.95

0.89

0.97

1.08

0.3

-0.004

0.392

0.165

0.511

0.759

0.545

0.93

0.95

0.94

0.94

1.39

0.5

-0.004

0.560

0.165

0.511

0.944

0.545

0.93

0.95

0.95

0.94

1.73

0.1

-0.008

0.132

0.087

0.357

0.408

0.372

0.94

0.96

0.85

0.96

1.10

0.3

-0.008

0.401

0.171

0.357

0.606

0.405

0.94

0.91

0.91

0.88

1.50

0.5

-0.008

0.565

0.182

0.357

0.775

0.410

0.94

0.84

0.94

0.87

1.89

0.1

-0.007

0.131

0.078

0.158

0.216

0.179

0.96

0.90

0.76

0.88

1.20

0.3

-0.007

0.401

0.184

0.158

0.470

0.246

0.96

0.69

0.73

0.64

1.91

0.5

-0.007

0.566

0.205

0.158

0.656

0.263

0.96

0.56

0.74

0.60

2.49

Table B.187: Empirical variance and asymptotic variance of correlation (ρ̂) when
ρ = 0.4 over different interval censoring proportions of 0.1, 0.3, and 0.5 and sample
sizes 50, 100, and 500
n

50

100

500

p

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.015

0.017

0.018

0.015

0.016

0.015

0.3

0.015

0.019

0.015

0.015

0.019

0.015

0.5

0.015

0.023

0.015

0.015

0.023

0.015

0.1

0.007

0.008

0.009

0.007

0.008

0.007

0.3

0.007

0.009

0.008

0.007

0.009

0.007

0.5

0.007

0.012

0.007

0.007

0.011

0.007

0.1

0.002

0.002

0.002

0.001

0.002

0.001

0.3

0.002

0.002

0.002

0.001

0.002

0.002

0.5

0.002

0.003

0.002

0.001

0.002

0.002

195

Table B.188: Bias, RMSE, 95% CP, and relative RMSE of correlation (ρ̂) when ρ = 0.4
over sample sizes 50, 100, and 500 and interval censoring proportions of 0.1, 0.3, and
0.5

n

50

100

500

p

Bias

RMSE

CP

RRMSE

CLS

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

-0.007

0.009

-0.031 0.121

0.126

0.124

0.93

0.91

0.89

0.97

1.01

0.3

-0.007

0.038

-0.037 0.121

0.142

0.127

0.93

0.89

0.93

0.96

1.12

0.5

-0.007

0.058

-0.040 0.121

0.162

0.128

0.93

0.87

0.93

0.94

1.26

0.1

-0.005

0.010

-0.019 0.085

0.089

0.087

0.93

0.94

0.91

0.98

1.02

0.3

-0.005

0.039

-0.032 0.085

0.103

0.092

0.93

0.91

0.93

0.92

1.12

0.5

-0.005

0.064

-0.037 0.085

0.123

0.094

0.93

0.84

0.94

0.91

1.31

0.1

0.000

0.016

-0.014 0.037

0.042

0.040

0.95

0.93

0.90

0.94

1.05

0.3

0.000

0.049

-0.030 0.037

0.065

0.049

0.95

0.73

0.87

0.77

1.33

0.5

0.000

0.070

-0.034 0.037

0.084

0.052

0.95

0.60

0.87

0.72

1.62

Table B.189: Empirical variance and asymptotic variance of intercept (βˆ0 ) when ρ =
0.6 over different interval censoring proportions of 0.1, 0.3, and 0.5 and sample sizes
50, 100, and 500
n

50

100

500

p

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.304

0.329

0.303

0.363

0.410

0.363

0.3

0.304

0.419

0.302

0.363

0.517

0.363

0.5

0.304

0.585

0.303

0.363

0.633

0.364

0.1

0.145

0.165

0.146

0.178

0.201

0.178

0.3

0.145

0.216

0.145

0.178

0.254

0.178

0.5

0.145

0.291

0.149

0.178

0.312

0.178

0.1

0.034

0.039

0.034

0.035

0.039

0.035

0.3

0.034

0.058

0.034

0.035

0.050

0.035

0.5

0.034

0.096

0.036

0.035

0.061

0.035

196

Table B.190: Bias, RMSE, 95% CP, and relative RMSE of intercept (βˆ0 ) when ρ = 0.6
over sample sizes 50, 100, and 500 and interval censoring proportions of 0.1, 0.3, and
0.5

n

p

Bias
CLS

50

100

500

RMSE

CP

RRMSE

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

-0.008 -0.040

-0.008

0.602

0.641

0.602

1.00

0.96

0.97

1.00

1.06

0.3

-0.008 -0.211

-0.007

0.602

0.749

0.603

1.00

0.96

0.96

1.00

1.24

0.5

-0.008 -0.451

-0.009

0.602

0.914

0.603

1.00

0.90

0.97

1.00

1.52

0.1

0.006

-0.027

0.006

0.422

0.449

0.422

1.00

0.97

0.97

1.00

1.07

0.3

0.006

-0.184

0.007

0.422

0.536

0.422

1.00

0.95

0.97

1.00

1.27

0.5

0.006

-0.432

0.009

0.422

0.706

0.422

1.00

0.89

0.97

1.00

1.67

0.1

-0.003 -0.041

-0.003

0.187

0.203

0.187

1.00

0.95

0.94

1.00

1.09

0.3

-0.003 -0.183

-0.004

0.187

0.288

0.187

1.00

0.83

0.94

1.00

1.54

0.5

-0.003 -0.436

-0.005

0.187

0.501

0.187

1.00

0.55

0.94

1.00

2.68

Table B.191: Empirical variance and asymptotic variance of slope (βˆ1 ) when ρ = 0.6
over different interval censoring proportions of 0.1, 0.3, and 0.5 and sample sizes 50,
100, and 500
n

50

100

500

p

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.004

0.004

0.004

0.004

0.004

0.004

0.3

0.004

0.005

0.004

0.004

0.004

0.004

0.5

0.004

0.006

0.004

0.004

0.005

0.004

0.1

0.002

0.002

0.002

0.002

0.002

0.002

0.3

0.002

0.002

0.002

0.002

0.002

0.002

0.5

0.002

0.003

0.002

0.002

0.003

0.002

0.1

0.000

0.000

0.000

0.000

0.000

0.000

0.3

0.000

0.000

0.000

0.000

0.000

0.000

0.5

0.000

0.001

0.000

0.000

0.001

0.000

197

Table B.192: Bias, RMSE, 95% CP, and relative RMSE of slope (βˆ1 ) when ρ = 0.6
over sample sizes 50, 100, and 500 and interval censoring proportions of 0.1, 0.3, and
0.5

n

50

100

500

p

Bias

RMSE

CP

RRMSE

CLS

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

0.001

0.010

0.001

0.059

0.062

0.059

0.94

0.93

0.93

1.00

1.04

0.3

0.001

0.028

0.001

0.059

0.071

0.059

0.94

0.90

0.93

1.00

1.20

0.5

0.001

0.041

-0.000 0.059

0.084

0.059

0.94

0.88

0.93

1.00

1.41

0.1

-0.000

0.009

-0.000 0.041

0.043

0.041

0.93

0.95

0.93

1.00

1.05

0.3

-0.000

0.025

-0.000 0.041

0.052

0.041

0.93

0.90

0.94

1.00

1.27

0.5

-0.000

0.038

-0.000 0.041

0.063

0.041

0.93

0.84

0.93

1.00

1.53

0.1

0.000

0.009

0.000

0.017

0.020

0.017

0.96

0.94

0.96

1.00

1.13

0.3

0.000

0.027

0.000

0.017

0.034

0.017

0.96

0.69

0.95

1.00

1.95

0.5

0.000

0.040

0.000

0.017

0.045

0.017

0.96

0.52

0.96

1.00

2.62

Table B.193: Empirical variance and asymptotic variance of mean of x (µˆx ) when
ρ = 0.6 over different interval censoring proportions of 0.1, 0.3, and 0.5 and sample
sizes 50, 100, and 500
n

50

100

500

p

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

1.954

2.026

1.954

1.984

2.336

1.944

0.3

1.954

2.702

1.966

1.984

3.255

1.943

0.5

1.954

4.234

1.992

1.984

4.381

1.939

0.1

0.937

0.984

0.936

0.996

1.174

0.986

0.3

0.937

1.434

0.946

0.996

1.643

0.986

0.5

0.937

2.634

0.952

0.996

2.218

0.985

0.1

0.208

0.242

0.208

0.200

0.236

0.200

0.3

0.208

0.612

0.209

0.200

0.329

0.200

0.5

0.208

1.749

0.208

0.200

0.444

0.199

198

Table B.194: Bias, RMSE, 95% CP, and relative RMSE of mean of x (µˆx ) when
ρ = 0.6 over sample sizes 50, 100, and 500 and interval censoring proportions of 0.1,
0.3, and 0.5

n

50

100

500

p

Bias

RMSE

CP

RRMSE

CLS

Naive

PM

CLS

Naive

PM

CLS Naive

PM

CLS/PM

Naive/PM

0.1

0.097

0.039

0.097

1.412

1.529

1.398

0.94

0.97

0.94

1.01

1.09

0.3

0.097

-0.421

0.097

1.412

1.853

1.397

0.94

0.95

0.94

1.01

1.33

0.5

0.097

-1.634

0.096

1.412

2.655

1.396

0.94

0.86

0.94

1.01

1.90

0.1

0.066

0.008

0.066

1.000

1.083

0.995

0.95

0.97

0.95

1.00

1.09

0.3

0.066

-0.462

0.064

1.000

1.363

0.995

0.95

0.96

0.95

1.01

1.37

0.5

0.066

-1.702

0.060

1.000

2.262

0.994

0.95

0.72

0.95

1.01

2.27

0.1

0.026

-0.042

0.026

0.448

0.487

0.447

0.94

0.94

0.94

1.00

1.09

0.3

0.026

-0.589

0.028

0.448

0.823

0.448

0.94

0.72

0.93

1.00

1.84

0.5

0.026

-1.917

0.026

0.448

2.030

0.447

0.94

0.35

0.94

1.00

4.54

Table B.195: Empirical variance and asymptotic variance of mean of y (µˆy ) when
ρ = 0.6 over different interval censoring proportions of 0.1, 0.3, and 0.5 and sample
sizes 50, 100, and 500
n

50

100

500

p

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.454

0.486

0.454

0.503

0.593

0.493

0.3

0.454

0.632

0.452

0.503

0.825

0.493

0.5

0.454

0.993

0.456

0.503

1.108

0.491

0.1

0.212

0.232

0.212

0.250

0.295

0.248

0.3

0.212

0.352

0.213

0.250

0.412

0.248

0.5

0.212

0.630

0.214

0.250

0.557

0.248

0.1

0.052

0.061

0.052

0.050

0.059

0.050

0.3

0.052

0.154

0.052

0.050

0.082

0.050

0.5

0.052

0.421

0.053

0.050

0.111

0.050

199

Table B.196: Bias, RMSE, 95% CP, and relative RMSE of mean of y (µˆy ) when
ρ = 0.6 over sample sizes 50, 100, and 500 and interval censoring proportions of 0.1,
0.3, and 0.5

n

50

100

500

p

Bias

RMSE

CP

RRMSE

CLS

Naive

PM

CLS

Naive

PM

CLS Naive

PM

CLS/PM

Naive/PM

0.1

0.019

-0.004

0.019

0.710

0.770

0.703

0.96

0.96

0.95

1.01

1.10

0.3

0.019

-0.273

0.019

0.710

0.949

0.702

0.96

0.97

0.95

1.01

1.35

0.5

0.019

-0.892

0.014

0.710

1.379

0.701

0.96

0.87

0.95

1.01

1.97

0.1

0.023

-0.000

0.023

0.501

0.543

0.498

0.97

0.96

0.97

1.01

1.09

0.3

0.023

-0.257

0.024

0.501

0.691

0.498

0.97

0.94

0.96

1.00

1.39

0.5

0.023

-0.882

0.026

0.501

1.155

0.498

0.97

0.72

0.96

1.00

2.32

0.1

0.005

-0.027

0.005

0.223

0.244

0.223

0.93

0.94

0.93

1.00

1.09

0.3

0.005

-0.293

0.005

0.223

0.410

0.223

0.93

0.71

0.93

1.00

1.84

0.5

0.005

-0.957

0.003

0.223

1.013

0.223

0.93

0.35

0.93

1.00

4.54

Table B.197: Empirical variance and asymptotic variance of standard deviation of x
(σˆx ) when ρ = 0.6 over different interval censoring proportions of 0.1, 0.3, and 0.5
and sample sizes 50, 100, and 500
n

50

100

500

p

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

1.088

1.208

1.067

1.030

1.218

0.972

0.3

1.088

1.551

1.069

1.030

1.718

0.971

0.5

1.088

2.277

1.077

1.030

2.352

0.970

0.1

0.540

0.592

0.535

0.507

0.599

0.493

0.3

0.540

0.737

0.535

0.507

0.844

0.493

0.5

0.540

1.160

0.541

0.507

1.148

0.493

0.1

0.104

0.117

0.104

0.100

0.118

0.100

0.3

0.104

0.171

0.104

0.100

0.166

0.100

0.5

0.104

0.577

0.105

0.100

0.223

0.100

200

Table B.198: Bias, RMSE, 95% CP, and relative RMSE of standard deviation of x (σˆx )
when ρ = 0.6 over sample sizes 50, 100, and 500 and interval censoring proportions
of 0.1, 0.3, and 0.5

n

50

100

500

p

Bias

RMSE

CP

RRMSE

CLS

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

-0.095

0.220

-0.194 1.019

1.126

1.005

0.94

0.96

0.93

1.01

1.12

0.3

-0.095

0.844

-0.199 1.019

1.559

1.005

0.94

0.96

0.93

1.01

1.55

0.5

-0.095

1.210

-0.209 1.019

1.954

1.007

0.94

0.94

0.93

1.01

1.94

0.1

-0.045

0.270

-0.095 0.714

0.820

0.709

0.93

0.95

0.92

1.01

1.16

0.3

-0.045

0.915

-0.097 0.714

1.296

0.709

0.93

0.89

0.92

1.01

1.83

0.5

-0.045

1.291

-0.101 0.714

1.678

0.709

0.93

0.83

0.92

1.01

2.37

0.1

-0.008

0.314

-0.017 0.317

0.466

0.316

0.94

0.86

0.94

1.00

1.47

0.3

-0.008

0.938

-0.017 0.317

1.022

0.316

0.94

0.35

0.94

1.00

3.23

0.5

-0.008

1.287

-0.019 0.317

1.371

0.316

0.94

0.26

0.94

1.00

4.33

Table B.199: Empirical variance and asymptotic variance of standard deviation of y
(σˆy ) when ρ = 0.6 over different interval censoring proportions of 0.1, 0.3, and 0.5
and sample sizes 50, 100, and 500
n

50

100

500

p

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.271

0.302

0.265

0.261

0.309

0.247

0.3

0.271

0.380

0.265

0.261

0.436

0.246

0.5

0.271

0.536

0.268

0.261

0.595

0.246

0.1

0.127

0.142

0.126

0.127

0.150

0.124

0.3

0.127

0.186

0.125

0.127

0.211

0.124

0.5

0.127

0.322

0.127

0.127

0.288

0.124

0.1

0.024

0.027

0.024

0.025

0.030

0.025

0.3

0.024

0.040

0.024

0.025

0.041

0.025

0.5

0.024

0.132

0.024

0.025

0.056

0.025

201

Table B.200: Bias, RMSE, 95% CP, and relative RMSE of standard deviation of y (σˆy )
when ρ = 0.6 over sample sizes 50, 100, and 500 and interval censoring proportions
of 0.1, 0.3, and 0.5

n

50

100

500

p

Bias

RMSE

CP

RRMSE

CLS

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

-0.011

0.148

-0.061 0.511

0.576

0.500

0.93

0.95

0.92

1.02

1.15

0.3

-0.011

0.461

-0.062 0.511

0.805

0.500

0.93

0.94

0.92

1.02

1.61

0.5

-0.011

0.640

-0.071 0.511

1.002

0.501

0.93

0.95

0.92

1.02

2.00

0.1

-0.011

0.148

-0.037 0.357

0.415

0.354

0.94

0.95

0.94

1.01

1.17

0.3

-0.011

0.463

-0.037 0.357

0.652

0.354

0.94

0.88

0.94

1.01

1.84

0.5

-0.011

0.655

-0.036 0.357

0.847

0.354

0.94

0.77

0.93

1.01

2.39

0.1

-0.007

0.152

-0.012 0.158

0.229

0.158

0.96

0.88

0.96

1.00

1.45

0.3

-0.007

0.468

-0.012 0.158

0.511

0.158

0.96

0.34

0.96

1.00

3.23

0.5

-0.007

0.650

-0.013 0.158

0.692

0.158

0.96

0.25

0.95

1.00

4.37

Table B.201: Empirical variance and asymptotic variance of correlation (ρ̂) when
ρ = 0.6 over different interval censoring proportions of 0.1, 0.3, and 0.5 and sample
sizes 50, 100, and 500
n

50

100

500

p

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.009

0.009

0.009

0.009

0.009

0.008

0.3

0.009

0.010

0.009

0.009

0.010

0.008

0.5

0.009

0.011

0.009

0.009

0.012

0.009

0.1

0.004

0.005

0.004

0.004

0.004

0.004

0.3

0.004

0.005

0.004

0.004

0.005

0.004

0.5

0.004

0.006

0.004

0.004

0.006

0.004

0.1

0.001

0.001

0.001

0.001

0.001

0.001

0.3

0.001

0.001

0.001

0.001

0.001

0.001

0.5

0.001

0.002

0.001

0.001

0.001

0.001

202

Table B.202: Bias, RMSE, 95% CP, and relative RMSE of correlation (ρ̂) when ρ = 0.6
over sample sizes 50, 100, and 500 and interval censoring proportions of 0.1, 0.3, and
0.5

n

50

100

500

p

Bias

RMSE

CP

RRMSE

CLS

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

-0.007

0.010

-0.007 0.094

0.097

0.092

0.92

0.91

0.92

1.02

1.05

0.3

-0.007

0.045

-0.008 0.094

0.110

0.092

0.92

0.84

0.91

1.02

1.20

0.5

-0.007

0.069

-0.010 0.094

0.129

0.093

0.92

0.78

0.92

1.02

1.39

0.1

-0.004

0.013

-0.004 0.066

0.068

0.065

0.94

0.92

0.94

1.01

1.04

0.3

-0.004

0.047

-0.004 0.066

0.084

0.065

0.94

0.83

0.94

1.01

1.29

0.5

-0.004

0.072

-0.005 0.066

0.103

0.065

0.94

0.72

0.94

1.01

1.59

0.1

-0.000

0.018

-0.000 0.028

0.035

0.028

0.95

0.89

0.95

1.00

1.24

0.3

-0.000

0.054

0.000

0.028

0.062

0.028

0.95

0.55

0.94

1.00

2.18

0.5

-0.000

0.077

0.000

0.028

0.084

0.028

0.95

0.34

0.95

1.00

2.95

Table B.203: Empirical variance and asymptotic variance of intercept (βˆ0 ) when ρ =
0.8 over different interval censoring proportions of 0.1, 0.3, and 0.5 and sample sizes
50, 100, and 500
n

50

100

500

p

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.170

0.188

0.136

0.204

0.226

0.211

0.3

0.170

0.229

0.171

0.204

0.273

0.222

0.5

0.170

0.304

0.178

0.204

0.322

0.230

0.1

0.083

0.090

0.086

0.100

0.111

0.105

0.3

0.083

0.116

0.086

0.100

0.135

0.107

0.5

0.083

0.147

0.090

0.100

0.160

0.109

0.1

0.019

0.022

0.020

0.020

0.022

0.020

0.3

0.019

0.030

0.020

0.020

0.026

0.021

0.5

0.019

0.043

0.020

0.020

0.031

0.021

203

Table B.204: Bias, RMSE, 95% CP, and relative RMSE of intercept (βˆ0 ) when ρ = 0.8
over sample sizes 50, 100, and 500 and interval censoring proportions of 0.1, 0.3, and
0.5

n

p

Bias
CLS

50

100

500

RMSE

CP

RRMSE

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

-0.010 -0.039

-0.002

0.452

0.477

0.459

1.00

0.96

0.99

0.98

1.04

0.3

-0.010 -0.123

0.031

0.452

0.537

0.473

1.00

0.96

0.97

0.96

1.14

0.5

-0.010 -0.261

0.037

0.452

0.624

0.481

1.00

0.93

0.97

0.94

1.30

0.1

0.001

-0.031

0.009

0.316

0.335

0.323

1.00

0.96

0.97

0.98

1.04

0.3

0.001

-0.110

0.024

0.316

0.383

0.327

1.00

0.96

0.97

0.97

1.17

0.5

0.001

-0.230

0.057

0.316

0.461

0.335

1.00

0.92

0.97

0.94

1.38

0.1

-0.003 -0.031

0.006

0.140

0.151

0.143

1.00

0.94

0.96

0.98

1.06

0.3

-0.003 -0.119

0.011

0.140

0.202

0.144

1.00

0.86

0.95

0.97

1.40

0.5

-0.003 -0.254

0.018

0.140

0.309

0.146

1.00

0.68

0.95

0.96

2.12

Table B.205: Empirical variance and asymptotic variance of slope (βˆ1 ) when ρ = 0.8
over different interval censoring proportions of 0.1, 0.3, and 0.5 and sample sizes 50,
100, and 500
n

50

100

500

p

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.002

0.002

0.002

0.002

0.002

0.002

0.3

0.002

0.002

0.002

0.002

0.002

0.002

0.5

0.002

0.003

0.003

0.002

0.003

0.002

0.1

0.001

0.001

0.001

0.001

0.001

0.001

0.3

0.001

0.001

0.001

0.001

0.001

0.001

0.5

0.001

0.002

0.001

0.001

0.001

0.001

0.1

0.000

0.000

0.000

0.000

0.000

0.000

0.3

0.000

0.000

0.000

0.000

0.000

0.000

0.5

0.000

0.000

0.000

0.000

0.000

0.000

204

Table B.206: Bias, RMSE, 95% CP, and relative RMSE of slope (βˆ1 ) when ρ = 0.8
over sample sizes 50, 100, and 500 and interval censoring proportions of 0.1, 0.3, and
0.5

n

50

100

500

p

Bias

RMSE

CP

RRMSE

CLS

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

0.001

0.009

-0.002 0.044

0.046

0.045

0.94

0.93

0.92

0.97

1.02

0.3

0.001

0.022

-0.010 0.044

0.052

0.048

0.94

0.92

0.93

0.91

1.08

0.5

0.001

0.029

-0.014 0.044

0.059

0.050

0.94

0.88

0.93

0.87

1.17

0.1

0.000

0.007

-0.002 0.030

0.032

0.032

0.93

0.94

0.90

0.95

1.02

0.3

0.000

0.019

-0.006 0.030

0.038

0.032

0.93

0.91

0.93

0.93

1.19

0.5

0.000

0.027

-0.009 0.030

0.044

0.033

0.93

0.84

0.92

0.91

1.33

0.1

-0.000

0.007

-0.003 0.014

0.016

0.014

0.96

0.94

0.92

0.98

1.08

0.3

-0.000

0.019

-0.005 0.014

0.024

0.015

0.96

0.73

0.93

0.95

1.58

0.5

-0.000

0.026

-0.005 0.014

0.030

0.015

0.96

0.55

0.93

0.94

1.97

Table B.207: Empirical variance and asymptotic variance of mean of x (µˆx ) when
ρ = 0.8 over different interval censoring proportions of 0.1, 0.3, and 0.5 and sample
sizes 50, 100, and 500
n

50

100

500

p

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

1.950

2.007

1.525

1.985

2.361

1.950

0.3

1.950

2.533

2.003

1.985

3.337

1.957

0.5

1.950

3.870

2.078

1.985

4.491

1.891

0.1

0.924

0.960

0.925

0.996

1.187

0.995

0.3

0.924

1.422

0.924

0.996

1.682

0.993

0.5

0.924

2.753

0.930

0.996

2.257

0.973

0.1

0.209

0.253

0.209

0.200

0.239

0.201

0.3

0.209

0.680

0.210

0.200

0.338

0.202

0.5

0.209

1.923

0.212

0.200

0.452

0.200

205

Table B.208: Bias, RMSE, 95% CP, and relative RMSE of mean of x (µˆx ) when
ρ = 0.8 over sample sizes 50, 100, and 500 and interval censoring proportions of 0.1,
0.3, and 0.5

n

50

100

500

p

Bias

RMSE

CP

RRMSE

CLS

Naive

PM

CLS

Naive

PM

CLS Naive

PM

CLS/PM

Naive/PM

0.1

0.095

0.042

0.094

1.412

1.537

1.400

0.95

0.97

0.98

1.01

1.10

0.3

0.095

-0.513

0.088

1.412

1.897

1.402

0.95

0.96

0.94

1.01

1.35

0.5

0.095

-1.813

0.224

1.412

2.789

1.393

0.95

0.85

0.93

1.01

2.00

0.1

0.066

0.017

0.069

1.000

1.090

1.000

0.96

0.98

0.96

1.00

1.09

0.3

0.066

-0.546

0.075

1.000

1.407

0.999

0.96

0.94

0.95

1.00

1.41

0.5

0.066

-1.937

0.195

1.000

2.452

1.005

0.96

0.70

0.96

1.00

2.44

0.1

0.026

-0.041

0.025

0.448

0.490

0.449

0.93

0.93

0.94

1.00

1.09

0.3

0.026

-0.630

0.025

0.448

0.857

0.450

0.93

0.69

0.94

1.00

1.91

0.5

0.026

-2.070

0.074

0.448

2.177

0.454

0.93

0.31

0.93

0.99

4.80

Table B.209: Empirical variance and asymptotic variance of mean of y (µˆy ) when
ρ = 0.8 over different interval censoring proportions of 0.1, 0.3, and 0.5 and sample
sizes 50, 100, and 500
n

50

100

500

p

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.463

0.470

0.349

0.501

0.597

0.492

0.3

0.463

0.596

0.473

0.501

0.846

0.491

0.5

0.463

0.965

0.484

0.501

1.134

0.480

0.1

0.213

0.224

0.213

0.250

0.298

0.250

0.3

0.213

0.345

0.215

0.250

0.422

0.249

0.5

0.213

0.664

0.221

0.250

0.567

0.245

0.1

0.052

0.062

0.052

0.050

0.060

0.050

0.3

0.052

0.170

0.052

0.050

0.084

0.050

0.5

0.052

0.477

0.051

0.050

0.113

0.050

206

Table B.210: Bias, RMSE, 95% CP, and relative RMSE of mean of y (µˆy ) when
ρ = 0.8 over sample sizes 50, 100, and 500 and interval censoring proportions of 0.1,
0.3, and 0.5

n

50

100

500

p

Bias

RMSE

CP

RRMSE

CLS

Naive

PM

CLS

Naive

PM

CLS Naive

PM

CLS/PM

Naive/PM

0.1

0.028

0.000

0.035

0.709

0.773

0.702

0.95

0.97

0.98

1.01

1.10

0.3

0.028

-0.277

0.033

0.709

0.961

0.701

0.95

0.97

0.94

1.01

1.37

0.5

0.028

-0.950

0.081

0.709

1.427

0.697

0.95

0.86

0.93

1.02

2.05

0.1

0.027

-0.005

0.027

0.500

0.546

0.501

0.96

0.97

0.96

1.00

1.09

0.3

0.027

-0.279

0.035

0.500

0.707

0.500

0.96

0.95

0.96

1.00

1.41

0.5

0.027

-0.963

0.106

0.500

1.223

0.506

0.96

0.69

0.95

0.99

2.41

0.1

0.007

-0.028

0.007

0.224

0.246

0.224

0.94

0.93

0.94

1.00

1.10

0.3

0.007

-0.325

0.008

0.224

0.436

0.225

0.94

0.69

0.94

1.00

1.94

0.5

0.007

-1.048

0.033

0.224

1.100

0.227

0.94

0.31

0.93

0.99

4.86

Table B.211: Empirical variance and asymptotic variance of standard deviation of x
(σˆx ) when ρ = 0.8 over different interval censoring proportions of 0.1, 0.3, and 0.5
and sample sizes 50, 100, and 500
n

50

100

500

p

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

1.082

1.188

1.030

1.040

1.244

0.975

0.3

1.082

1.485

1.078

1.040

1.784

0.978

0.5

1.082

2.142

1.039

1.040

2.448

0.946

0.1

0.533

0.596

0.612

0.510

0.609

0.498

0.3

0.533

0.768

0.506

0.510

0.869

0.496

0.5

0.533

1.284

0.441

0.510

1.177

0.486

0.1

0.105

0.118

0.113

0.100

0.120

0.101

0.3

0.105

0.177

0.106

0.100

0.170

0.101

0.5

0.105

0.658

0.106

0.100

0.228

0.100

207

Table B.212: Bias, RMSE, 95% CP, and relative RMSE of standard deviation of x (σˆx )
when ρ = 0.8 over sample sizes 50, 100, and 500 and interval censoring proportions
of 0.1, 0.3, and 0.5

n

50

100

500

p

Bias

RMSE

CP

RRMSE

CLS

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

-0.092

0.279

-0.120 1.024

1.150

0.995

0.93

0.96

0.92

1.03

1.16

0.3

-0.092

1.013

-0.164 1.024

1.676

1.003

0.93

0.95

0.92

1.02

1.67

0.5

-0.092

1.466

-0.329 1.024

2.144

1.027

0.93

0.93

0.91

1.00

2.09

0.1

-0.046

0.330

-0.056 0.715

0.847

0.708

0.94

0.94

0.92

1.01

1.20

0.3

-0.046

1.075

-0.062 0.715

1.423

0.707

0.94

0.85

0.94

1.01

2.01

0.5

-0.046

1.497

-0.160 0.715

1.848

0.716

0.94

0.76

0.95

1.00

2.58

0.1

-0.008

0.376

0.019

0.317

0.511

0.318

0.94

0.82

0.94

1.00

1.61

0.3

-0.008

1.120

0.038

0.317

1.193

0.320

0.94

0.20

0.95

0.99

3.73

0.5

-0.008

1.517

0.001

0.317

1.591

0.316

0.94

0.21

0.93

1.00

5.03

Table B.213: Empirical variance and asymptotic variance of standard deviation of y
(σˆy ) when ρ = 0.8 over different interval censoring proportions of 0.1, 0.3, and 0.5
and sample sizes 50, 100, and 500
n

50

100

500

p

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.274

0.306

0.196

0.263

0.315

0.246

0.3

0.274

0.376

0.259

0.263

0.453

0.245

0.5

0.274

0.544

0.272

0.263

0.618

0.240

0.1

0.127

0.142

0.138

0.128

0.153

0.125

0.3

0.127

0.185

0.121

0.128

0.218

0.125

0.5

0.127

0.322

0.114

0.128

0.296

0.123

0.1

0.025

0.027

0.027

0.025

0.030

0.025

0.3

0.025

0.041

0.025

0.025

0.042

0.025

0.5

0.025

0.157

0.023

0.025

0.057

0.025

208

Table B.214: Bias, RMSE, 95% CP, and relative RMSE of standard deviation of y (σˆy )
when ρ = 0.8 over sample sizes 50, 100, and 500 and interval censoring proportions
of 0.1, 0.3, and 0.5

n

50

100

500

p

Bias

RMSE

CP

RRMSE

CLS

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

-0.021

0.169

-0.041 0.513

0.586

0.498

0.93

0.95

0.95

1.03

1.18

0.3

-0.021

0.546

-0.073 0.513

0.866

0.501

0.93

0.94

0.92

1.02

1.73

0.5

-0.021

0.759

-0.130 0.513

1.093

0.507

0.93

0.91

0.89

1.01

2.16

0.1

-0.016

0.174

-0.012 0.358

0.428

0.354

0.94

0.96

0.92

1.01

1.21

0.3

-0.016

0.546

-0.021 0.358

0.719

0.354

0.94

0.85

0.94

1.01

2.03

0.5

-0.016

0.763

-0.060 0.358

0.937

0.355

0.94

0.71

0.94

1.01

2.64

0.1

-0.006

0.186

0.010

0.159

0.254

0.159

0.96

0.83

0.96

1.00

1.60

0.3

-0.006

0.558

0.019

0.159

0.594

0.160

0.96

0.21

0.97

0.99

3.72

0.5

-0.006

0.756

0.012

0.159

0.793

0.159

0.96

0.19

0.98

1.00

4.99

Table B.215: Empirical variance and asymptotic variance of correlation (ρ̂) when
ρ = 0.8 over different interval censoring proportions of 0.1, 0.3, and 0.5 and sample
sizes 50, 100, and 500
n

50

100

500

p

Empirical Variance

Asymptotic Variance

CLS

Naive

PM

CLS

Naive

PM

0.1

0.003

0.003

0.004

0.003

0.003

0.004

0.3

0.003

0.003

0.004

0.003

0.003

0.003

0.5

0.003

0.003

0.004

0.003

0.003

0.004

0.1

0.001

0.001

0.002

0.001

0.001

0.001

0.3

0.001

0.001

0.002

0.001

0.001

0.002

0.5

0.001

0.002

0.002

0.001

0.002

0.002

0.1

0.000

0.000

0.000

0.000

0.000

0.000

0.3

0.000

0.000

0.000

0.000

0.000

0.000

0.5

0.000

0.001

0.000

0.000

0.000

0.000

209

Table B.216: Bias, RMSE, 95% CP, and relative RMSE of correlation (ρ̂) when ρ = 0.8
over sample sizes 50, 100, and 500 and interval censoring proportions of 0.1, 0.3, and
0.5

n

50

100

500

p

Bias

RMSE

CP

RRMSE

CLS

Naive

PM

CLS

Naive

PM

CLS

Naive

PM

CLS/PM

Naive/PM

0.1

-0.005

0.009

-0.011 0.055

0.055

0.060

0.94

0.90

0.92

0.91

0.91

0.3

-0.005

0.034

-0.025 0.055

0.064

0.063

0.94

0.79

0.94

0.88

1.02

0.5

-0.005

0.049

-0.037 0.055

0.074

0.071

0.94

0.73

0.93

0.78

1.06

0.1

-0.003

0.010

-0.010 0.038

0.039

0.039

0.94

0.91

0.90

0.97

1.01

0.3

-0.003

0.035

-0.015 0.038

0.050

0.042

0.94

0.75

0.92

0.90

1.21

0.5

-0.003

0.049

-0.023 0.038

0.062

0.046

0.94

0.61

0.92

0.81

1.35

0.1

-0.000

0.013

-0.006 0.017

0.022

0.018

0.94

0.82

0.88

0.94

1.18

0.3

-0.000

0.038

-0.010 0.017

0.040

0.020

0.94

0.33

0.88

0.87

2.04

0.5

-0.000

0.052

-0.012 0.017

0.055

0.021

0.94

0.23

0.88

0.82

2.60

210