APPROVAL - Summitsummit.sfu.ca/system/files/iritems1/13564/etd6333_CCo.pdf · ing me direction in...

INVESTIGATING THE USE OF THE ACCELERATEDHAZARDS MODEL FOR

SURVIVAL ANALYSIS

by Caroll Anne Co

B.Sc., Simon Fraser University, 2007

PROJECT SUBMITTED IN PARTIAL FULFILLMENT

OF THE REQUIREMENTS FOR THE DEGREE OF

MASTER OF SCIENCE

in the

Department of Statistics & Actuarial Science

Faculty of Science

Caroll Anne Co 2010

SIMON FRASER UNIVERSITY Fall 2010

All rights reserved. However, in accordance with the Copyright Act of Canada, this work may

be reproduced, without authorization, under the conditions for “Fair Dealing.” Therefore, limited reproduction of this work for the

purposes of private study, research, criticism, review, and news reporting is likely to be in accordance with the law, particularly if cited appropriately.

APPROVAL

Name: Caroll Anne Co

Degree: Master of Science

Title of Thesis: Investigating the Use of the Accelerated Hazards Model for

Survival Analysis

Examining Committee: Dr. Derek Bingham (Chair)

Dr. Charmaine Dean, Senior Supervisor

Dr. Leilei Zeng, Supervisor

Dr. Joan Hu, External Examiner

Date Approved: December 9, 2010

ii

Last revision: Spring 09

Declaration of Partial Copyright Licence The author, whose copyright is declared on the title page of this work, has granted to Simon Fraser University the right to lend this thesis, project or extended essay to users of the Simon Fraser University Library, and to make partial or single copies only for such users or in response to a request from the library of any other university, or other educational institution, on its own behalf or for one of its users.

The author has further granted permission to Simon Fraser University to keep or make a digital copy for use in its circulating collection (currently available to the public at the “Institutional Repository” link of the SFU Library website <www.lib.sfu.ca> at: <http://ir.lib.sfu.ca/handle/1892/112>) and, without changing the content, to translate the thesis/project or extended essays, if technically possible, to any medium or format for the purpose of preservation of the digital work.

The author has further agreed that permission for multiple copying of this work for scholarly purposes may be granted by either the author or the Dean of Graduate Studies.

It is understood that copying or publication of this work for financial gain shall not be allowed without the author’s written permission.

Permission for public performance, or limited permission for private scholarly use, of any multimedia materials forming part of this work, may have been granted by the author. This information may be found on the separately catalogued multimedia material and in the signed Partial Copyright Licence.

While licensing SFU to permit the above uses, the author retains copyright in the thesis, project or extended essays, including the right to change the work for subsequent purposes, including editing and publishing the work in whole or in part, and licensing other parties, as the author may desire.

The original Partial Copyright Licence attesting to these terms, and signed by this author, may be found in the original bound copy of this work, retained in the Simon Fraser University Archive.

Simon Fraser University Library Burnaby, BC, Canada

Abstract

This project contrasts the Proportional Hazards, Accelerated Failure Time and Accelerated

Hazards (AH) models in the analysis of time to event data. The AH model handles data that

exhibit crossing of the survival and hazard curves, unlike the other two models considered.

The three models are illustrated on five contrasting data sets. A simulation study is con-

ducted to assess the small sample performance of the AH model by quantifying the mean

squared error of the predicted survivor curves under scenarios of crossing and non-crossing

survivor curves. The results show that the AH model can perform poorly under model

misspecification for models with a crossing hazard. Problems with variance estimation of

parameters in the AH model are observed for small sample sizes and a bootstrap approach

is offered as an alternate method of quantifying precision of estimates.

iii

Acknowledgments

This manuscript would not have gotten this far without the guidance and supervision from

Drs. Charmaine Dean and Leilei Zeng. I thank you both for your mindful insights in giv-

ing me direction in my research and course-work throughout the program. Many thanks

to Dr. Rachel Altman for providing me with helpful advice during the early stages of the

research project. I am grateful to Drs. Joan Hu and Derek Bingham for serving as the

external examiner and chair in my graduate committee. To my fellow graduate students in

the Department of Statistics & Actuarial Science, your friendships have truly made learn-

ing and research enjoyable. I have learned as much from the class lectures as I have from

the discussions with you on both academic and non-academic subjects. To my friends who

have kept me motivated in my academic pursuit, I would not be able to reach the finish line

without you.

I am indebted to my manager and co-workers from BC Mental Health & Addiction

Services for their understanding and encouragement in allowing me to pursue higher edu-

cation while working part-time; and to the staff at Cardiac Services BC for their patience

and support as I finish the last few steps in the program. I thank my family for their un-

wavering love and care throughout my education; and to PB, my best friend and confidant,

thank you for everything.

iv

Contents

Approval ii

Abstract iii

Acknowledgments iv

Contents v

List of Tables vii

List of Figures ix

1 Introduction 1

2 Models for Survival Analysis 42.1 The Proportional Hazards Model . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 The Accelerated Failure Time Model . . . . . . . . . . . . . . . . . . . . . 7

2.2.1 Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3 The Accelerated Hazards Model . . . . . . . . . . . . . . . . . . . . . . . 11

2.3.1 Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4 Comparison of the functional forms for the PH, AFT, and AH Models . . . 15

2.5 Small sample investigation of the performance of the AH estimator . . . . . 17

3 Data Analysis 333.1 Breast Cancer Clinical Trial . . . . . . . . . . . . . . . . . . . . . . . . . 33

v

CONTENTS vi

3.2 Coronary Artery Bypass Graft Surgery . . . . . . . . . . . . . . . . . . . . 39

3.3 Veteran Lung Cancer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.4 Kidney Catheter Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.5 Brain Tumor Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.6 Summary AH Parameter Interpretation . . . . . . . . . . . . . . . . . . . . 60

4 Exploring the fit of the PH and AH survivor curves 614.1 Case I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.2 Case II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5 Discussion 74

A Appendix 77

Bibliography 79

List of Tables

2.1 Parameter interpretation for PH, AFT and AH models . . . . . . . . . . . . 16

2.2 Summary of bias, variance, and variance ratios for the PH and AH models

fitted to a Weibull distribution for the 0% censoring case. Note that βPH =

βAH (k-1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3 Summary of bias, variance, and variance ratios for the PH and AH models

fitted to a Weibull distribution for the 27% censoring case. Note that βPH

= βAH (k-1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.4 Summary of bias, variance, and variance ratio for the PH and AH models

fitted to a Weibull distribution for the 53% censoring case. Note that βPH

= βAH (k-1). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.1 Estimated treatment effects in the analysis of the breast cancer data using

PH, AFT, and AH models. Values with * in the AH model represent boot-

strapped estimates. The p-values correspond to Wald tests of a hypothesis

of no treatment effect. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.2 Estimated gender effects on the coronary artery bypass graft data using

PH, AFT and AH models. Values with * in the AH model represent boot-

strapped estimates. The p-values compared to Wald tests of a hypothesis of

no treatment effect. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.3 Estimated treatment effects on the veteran lung cancer data using PH, AFT,

and AH models. Values with * in the AH model represent bootstrapped

estimates. The p-values correspond to Wald tests of a hypothesis of no

treatment effect. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

vii

LIST OF TABLES viii

3.4 Estimated treatment effects on the percutaneous catheter placement using

PH, AFT, and AH models. Values with * in the AH model represent boot-

strapped estimates. The p-values correspond to Wald tests of a hypothesis

of no treatment effect. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.5 Estimated treatment effects on the carmustine (BCNU) polymer disc data

using PH, AFT, and AH models. Values with * in the AH model repre-

sent bootstrapped estimates. The p-values correspond to Wald tests of a

hypothesis of no treatment effect. . . . . . . . . . . . . . . . . . . . . . . . 56

3.6 Summary of features of the five datasets considered in the chapter, and the

best model fitted in each dataset. The column headers reflect features seen

from the estimated Kaplan-Meier curves, without regard for significance of

these effects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

List of Figures

1.1 Hypothetical example of a scenario exhibiting crossing survivor curves,

with the control group representing individuals on an oral medication ther-

apy (black), and the treatment group representing individuals who were

surgically treated (red). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1 Hazard (top row) and survivor (bottom row) functions of PH, AFT, and

AH models. Non-crossing hazard functions for the AFT and AH models

are not shown. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2 Levelplot of variance ratios of the empirical and model-based accelerated

hazards model variance estimates, for treatment effects (te) 0,0.5, and 2;

shape parameter, k=0.5, 1.5 and 3; scale parameter, λ=0.25, 1, and 1.3;

sample sizes, n=100, 500, 1000, 5000, and 10,000. A negative ratio implies

an underestimation of the model-based variance denoted by a warm orange

color, while a positive ratio implies an overestimation of the model-based

variance denoted by a dark green color. . . . . . . . . . . . . . . . . . . . 25

2.3 Levelplot of variance ratios of empirical and model-based proportional haz-

ard model variance estimates, for treatment effects (te) 0,0.5, and 2; shape

parameter, k=0.5, 1.5 and 3; scale parameter, λ=0.25, 1, and 1.3; sample

sizes, n=100, 500, 1000, 5000, and 10,000. A negative ratio implies an

underestimation of the model-based variance denoted by a yellow color,

while a positive ratio implies an overestimation of the model-based vari-

ance denoted by green. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

ix

LIST OF FIGURES x

2.4 Dotplot of three variance estimates - empirical variance (blue), model-

based variance (pink), and non-parametric bootstrapped variance (green)

for data taken from a Weibull distribution with n=100, shape (k)=0.5, 1.5,

and 3, scale (λ)=0.25, 1, and 1.3, and treatment effect, βAH denoted (te) =

0, 0.5, and 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27





0, 0.5, and 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28





0, 0.5, and 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.7 Dotplot of two coverage probabilities using the model-based variance (blue),

and non-parametric bootstrapped variance (pink) for data taken from a

Weibull distribution with n=100, shape (k)=0.5, 1.5, and 3, scale (λ)=0.25,

1, and 1.3, and treatment effect, βAH denoted (te) = 0, 0.5, and 2. . . . . . . 30









3.1 Left: Kaplan-Meier survivor curves for the control group (black) and treat-

ment group (red). Right: Cumulative hazard curves for the control group

and treatment group. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.2 Smoothed hazard curves for the breast cancer data. . . . . . . . . . . . . . 36

LIST OF FIGURES xi

3.3 Top: Cumulative hazard curves for the PH, Weibull AFT, and AH models.

The fitted cumulative hazard curves are shown in solid lines vs the non-

parametric hazard curves in dashed lines. The baseline (control) hazard

curves are shown in black. Bottom: Kaplan-Meier survivor curves for the

control group (black dashed) and treatment group (red dashed), with the

fitted survivor curves for treatment group shown in red solid lines. . . . . . 37

3.4 Residual plot from the fit of the proportional hazards model to the breast

cancer data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.5 Left: Kaplan-Meier coronary artery bypass graft surgery 2-year survivor

curves for males (black) and females (red). Right: Two-year cumulative

hazard curve for coronary artery bypass graft surgeries. . . . . . . . . . . . 40

3.6 Smoothed hazard curves for males (black solid) and females (red dashed)

who underwent a coronary artery bypass graft surgery. . . . . . . . . . . . 42

3.7 Top: Cumulative hazard curves for the 2-year coronary artery bypass data

using the PH, AFT, and AH models. The non-parametric estimates are

shown in dashed lines, while the estimated cumulative hazard curves are

shown in solid lines for males (black) and females (red). Bottom: Kaplan-

Meier 2-year survivor curves for males (black dashed) and females (red

dashed), with the fitted survivor curves shown in solid lines. . . . . . . . . 43

3.8 Residual plot from the fit of the proportional hazards model to the coronary

artery bypass graft surgery data during the first two years post-surgery. . . . 44

3.9 Left: Kaplan-Meier survivor curves for the standard (black) and test (red)

chemotherapy groups. Right: Cumulative hazard curves for the two groups. 45

3.10 Smoothed hazard curves for the standard and test chemotherapy for the

treatment of lung cancer. . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.11 Top: Cumulative hazard curves for the veteran lung cancer data using the

PH, AFT, and AH models. The non-parametric estimates are shown in

dashed lines, while the estimated cumulative hazard curves are shown in

solid lines for standard treatment (black) and test treatment (red). Bottom:

Kaplan-Meier survivor curves for standard treatment (black dashed) and

test treatment (red dashed), with the fitted survivor curves shown in solid

lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

LIST OF FIGURES xii

3.12 Residual plot from the fit of the proportional hazards model to the veteran

lung cancer data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.13 Left: Kaplan-Meier survivor curves for surgical (black) and percutaneous

placement (red) of the catheter for patients undergoing kidney dialysis.

Right: Cumulative hazard curves for the two groups. . . . . . . . . . . . . 50

3.14 Smoothed hazard curves for surgically and percutaneously-placed catheters

for patients undergoing kidney dialysis. . . . . . . . . . . . . . . . . . . . 52

3.15 Top: Cumulative hazard curves for the kidney catheter placement data us-

ing the PH, AFT, and AH models. The non-parametric estimates are shown

in dashed lines, while the estimated cumulative hazard curves are shown

in solid lines for the surgically placement group (black) and the percuta-

neously placement group (red). Bottom: Kaplan-Meier survivor curves

for the surgically placement group (black dashed) and the percutaneously

placement group (red dashed), with the fitted survivor curves shown in solid

lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.16 Residual plot from the fit of the proportional hazards model to the kidney

catheter placement data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.17 Left: Kaplan-Meier survivor curves for the placebo (black) and BCNU

polymer (red) groups. Right: Cumulative hazard curves for the two groups. 55

3.18 Smoothed hazard curves for the placebo (black solid) and BCNU polymer

(red dashed) for the treatment of brain tumor. . . . . . . . . . . . . . . . . 57

3.19 Top: Cumulative hazard curves for the BCNU polymer disc data using

the PH, AFT, and AH models. The non-parametric estimates are shown

in dashed lines, while the estimated cumulative hazard curves are shown

in solid lines for the placebo group (black) and the BCNU polymer group

(red). Bottom: Kaplan-Meier survivor curves for the placebo group (black

dashed) and the BCNU polymer group (red dashed), with the fitted survivor

curves shown in solid lines. . . . . . . . . . . . . . . . . . . . . . . . . . . 58

3.20 Residual plot from the fit of the proportional hazards model to the brain

tumor data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

LIST OF FIGURES xiii

4.1 Hazard curves for the loglogistic distribution with shape parameters, a=0.5,

1, and 1.5 and scale parameter, s=1. . . . . . . . . . . . . . . . . . . . . . 62

4.2 Hazard (left) and survivor (right) curves for a loglogistic distribution with

shape=1.5 and scale=4 for a treatment effect size of -1. . . . . . . . . . . . 64

4.3 Comparison of the PH and AH fits for effect sizes of -0.5 (top), -1 (mid-

dle), and -1.5 (bottom) when the hazards do not start at the same point at

t=0. The dashed curves display true survivor functions corresponding to

the baseline and treatment groups. . . . . . . . . . . . . . . . . . . . . . . 65

4.4 Comparison of the fitted PH and AH curves with the true survivor curves

(shown with dashed lines) for three effect sizes (-0.5, -1, -1.5), with the

fitted baseline curves on the left panels, and the fitted treatment curves on

the right panels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

4.5 Boxplots of mean squared errors for the AH and PH models for β = -0.5,

-1, -1.5 in Case I, for a sample size of 100, for 1000 simulation runs. . . . . 67

4.6 Hazard (left) and survivor (right) curves for a loglogistic distribution with

shape parameters, a=4 and scale parameter, s=50 for a treatment effect size

of β=-1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

4.7 Comparison of the PH and AH fits for effect sizes of -0.5 (top), -1 (middle),

and -1.5 (bottom) when the hazards start at the same point at t=0. The

dashed curves display true survivor functions corresponding to the baseline

and treatment groups. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.8 Comparison of the fitted PH and AH curves with the true survivor curves

(shown with dashed lines) for three effect sizes (-0.5, -1, -1.5), with the

fitted baseline curves on the left panels, and the fitted treatment curves on

the right panels. Fixed censoring was done at t=100. . . . . . . . . . . . . 71

4.9 Boxplots of mean squared errors for the AH and PH models for β = -0.5,

-1, -1.5 in Case II, for a sample size of 100, for 1000 simulation runs. . . . 72

Chapter 1

Introduction

Survival analysis is concerned with the analysis of time to event data, such as time to

recurrence of a disease, or time to death, and determining the effects of treatments and

other factors on times to such events. In the modeling of survival time data, it is common

to use the hazard function and the survivor function to describe the distribution of lifetime

and the effect of the predictors or covariates on the cohorts’ risk and survival time. A hazard

function is defined as the instantaneous rate of failure (or having an event occur) at time t,

given survival up to time t. It is defined as:

h(t) = lim∆t→0+

P(t ≤ T < t +∆t|T ≥ t)∆t

=f (t)S(t)

(1.1)

where f (t) is the probability density function of the lifetime, and S(t) = P(T ≥ t) =1−F(t) is the survivor function, the probability of surviving beyond time t. In contrast,

F(t) is the probability of surviving up to time t. The hazard and survivor function are inter-

related by the following equation:

S(t) = e−∫ t

0 h(s)ds = e−H(t) (1.2)

where H(t) is defined as the cumulative hazard function.

The more commonly used models for survival analysis are the so-called Proportional

Hazards model (PH) and the Accelerated Failure Time model (AFT). Both of these models

assume an immediate treatment effect at the start of the study/trial. However, in random-

ized clinical trials, which compare a treatment with a placebo, it may be more reasonable to

1

CHAPTER 1. INTRODUCTION 2

assume that the risks of failure for the treatment and control (placebo) groups are equivalent

at the start of the trial and change as the trial proceeds. The Accelerated Hazards model

(AH), proposed by Chen & Wang (2000), holds this property. It is also able to handle data

that exhibit crossing of the survivor and hazard functions for treatment and control groups,

a feature that the PH and AFT models cannot accommodate. The model’s ability to capture

cross-overs of either or both of the hazard and survivor curves is often useful in practice.

An example of a situation where a crossing in the survivor curves may occur is in a ran-

domized controlled trial where one group receives oral medication, while another receives

a riskier treatment involving surgery. It may be that the operative mortality of individuals

undergoing surgery is initially high post-surgery but the rate of failure tapers off after a

certain time, indicating a cure. In the oral medication group however, mortality rate may

decline far more slowly at the start of the trial, since the treatment is less invasive, but may

continue to decline over time, with lower survivor rates exhibited after some period than

those for individuals undergoing the surgery. Figure 1.1 illustrates this hypothetical sce-

nario. This project contrasts the PH, AFT and AH models, develops methods for analysis

using the AH model, and considers the application of these models to several data sets.

In the proportional hazards model, the effect of the treatment is quantified on the haz-

ard scale. The effect of the treatment over a placebo, for example, is modeled in terms of

the hazard ratio of the placebo and treatment groups. This ratio can be thought of as the

relative risk, which quantifies how much more (less) risk the treatment group has over the

placebo group. In the accelerated failure time model, the effect of the treatment is quan-

tified by how fast (or slow) the treatment group ages (along the survivor curve) relative

to the control group. The treatment effect acts multiplicatively on time when calculating

the survivor function, and can be described as a survivor time ratio of the two groups. In

the accelerated hazards model, the treatment effect is quantified on the hazard scale and

describes how much faster (or slower) the risk progression is for the treatment group when

compared to the placebo group.

The precise formulation and inference for these three models are described and dis-

cussed in Chapter 2. A common feature of survival data is that observations may be cen-

sored; censoring occurs when individuals do not fail before the termination of the trial

CHAPTER 1. INTRODUCTION 3

0.2

0.4

0.6

0.8

1.0

Time in months

P(S

urvi

val)

0 1 2 3 4 5 6 7 8 9 10

Oral Medication Therapy

Surgery

Figure 1.1: Hypothetical example of a scenario exhibiting crossing survivor curves, with

the control group representing individuals on an oral medication therapy (black), and the

treatment group representing individuals who were surgically treated (red).

and what is recorded is the censoring time, or time at which failure had not yet occured.

For individuals who fail before the termination of the trial we have available their lifetime

or failure time. Understanding the censoring process is fundamental to correct statistical

inference; here, we assume that failure times are independent from censoring times and

develop inferential procedures based on this assumption in Chapter 2. Chapter 3 presents

data analyses of five datasets to illustrate the performance of the models, and we use these

examples to illustrate how parameter interpretation varies among the models under study.

Chapter 4 describes the results of a simulation study that considers the goodness of fit

of the proportional hazards and accelerated hazards models when the underlying hazard

curves for treatment and control groups cross. The goodness of fit is quantified by calculat-

ing the mean squared error of the predicted survivor curves for both the control group and

the treatment group. Some suggestions for future work and a discussion of the findings in

this project are presented in Chapter 5.

Chapter 2

Models for Survival Analysis

This chapter discusses inference for the proportional hazards, accelerated failure time, and

accelerated hazards models. We discuss semiparametric methods of inference for the pro-

portional hazards and the accelerated hazards models, and likelihood inference for the

Weibull model, a commonly used accelerated failure time model. We illustrate the vari-

ous functional forms these models exhibit, contrasting the shapes they may attain. We also

investigate small sample properties of estimators of treatment effects in these models, and

describe resampling approaches for improving the performance of the small sample vari-

ance estimator in the accelerated hazards model.

We first introduce some basic notation that is used throughout the Chapter. We denote

the observed event time, Xi = min(Ti,Ci), where Ti is the actual event time, and Ci is the

censoring time for the individual i, i = 1, ...,n. Throughout the analyses in this project, we

assume independence between Ti and Ci conditional on a given p×1 vector of covariates,

ZZZi = (Zi1, ...,Zip)′. We define the lifetime indicator as ∆i = I(Ti < Ci), where the function

I(.) takes a value of 1 if the condition is satisfied, and 0 otherwise. The observed data is

written as a triple, (Xi,∆i,ZZZi), for i = 1, ...n. The realization of (Xi,∆i,ZZZi) is denoted by

(xi,δi,zzzi).

4

CHAPTER 2. MODELS FOR SURVIVAL ANALYSIS 5

2.1 The Proportional Hazards Model

The proportional hazards model is among the most commonly used model in survival anal-

ysis, due to its simple conceptual framework, the fact that the model is semiparametric,

its excellent small sample performance, and the widespread availability of inferential tech-

niques for this model in statistical computing packages. In fact, this model, commonly

called the Cox model (Cox, 1972), is the standard model for survival analysis. This model

assumes hazard curves are proportional for individuals with different covariates.

The hazard function for an individual i with a covariate vector zzzi is formulated as

hPHi(t) = hPH0(t)g(zzzi;βββPH) (2.1)

where hPH0(t) is the so-called baseline hazard function, g(.), the relative risk function,

describes the effect of the covariates, ZZZi, on the baseline hazard, and βββPH is a p×1 vector

of regression parameters associated with covariates ZZZi. A typical form for g(.) is ezzz′iβββPH . In

this case, the survivor function for individual i is

SPHi(t) = e−∫ t

0 hPHi(s)ds

= [e−∫ t

0 hPH0(s)ds]ezzz′iβββPH

= [SPH0(t)]ezzz′iβββPH

where SPH0(t) is the survivor function corresponding to the baseline hazard; ie. the sur-

vivor function corresponding to an individual with ZZZi = 0. With semiparametric methods,

the model consists of a mixture of a non-parametric and parametric form. Here, the base-

line hazard function is non-parametric, while the relative risk function, ezzz′iβββPH , is paramet-

ric. Although the covariates in the formulation above are expressed as fixed over time,

extensions of the Cox model to accommodate time-dependent covariates are handled in a

straightforward manner with simple adjustments to inference for the basic framework.

2.1.1 Inference

The partial likelihood method was proposed by Cox (1972) for estimation of the regression

parameter βββPH . For simplicity, we assume there are no ties in the observed event times.


Techniques in Breslow (1974) can be used for tied event times. Let 0 < t1 < ... < tm denote

the m distinct ordered event times. The risk set Rl is defined as the set of individuals who

are still alive (ie. subjects who have not had the event) just prior to time tl . Let (l) denote

the subject with event at time tl , l = 1, ...,m. Under the proportional hazards model (2.1),

the conditional probability that individual (l) fails at tl , given the risk set (a summarized

history of the process) is,

h(tl;zzz(l))

∑ j∈Rlh(tl;zzz( j))

=hPH0(tl)e

zzz′(l)βββPH

∑ j∈RlhPH0(tl)e

zzz′( j)βββPH

=ezzz′(l)βββPH

∑ j∈Rlezzz′( j)βββPH

, (2.2)

where l = 1, ...,m.

The numerator on the right-hand side of (2.2) pertains to the hazard/risk that person (l)fails over a small interval. The denominator quantifies the combined hazards of everyone

who is at risk just prior to tl . The partial likelihood is formed by taking the product over all

m distinct failure points (or events) to give:

LPH(βββPH) =m

∏l=1

[ezzz′(l)βββPH


]. (2.3)

It can be shown that the baseline hazard function, hPH0(t), is uninformative for the estima-

tion of βββPH . An alternative way of writing the partial likelihood function (2.3) is

LPH(βββPH) =n

∏i=1

[ezzz′iβββPH

∑nj=1 I(x j ≥ xi)e

zzz′jβββPH

]δi

. (2.4)

The log (partial) likelihood of βββPH based on (2.3) is,

logLPH(βββPH) =m

∑l=1

zzz′(l)βββPH−m

∑l=1

[log( ∑

j∈Rl

ezzz′( j)βββPH )

](2.5)

Taking the first derivative of (2.5) with respect to the rth element of βββPH yields:

∂ logLPH

∂βPHr

=m

∑l=1

z(l)r−

∑ j∈Rlz( j)re

zzz′( j)βββPH


. (2.6)


The maximum likelihood estimate (MLE) of βββPH is easily obtained by using a Newton-

Raphson iterative procedure to solve UUU(βββPH) = 000, where UUU(βββPH) is the score vector with

elements ∂ logLPH∂βPHr

, r = 1, ..., p. This estimation procedure is shown to provide consistent and

asymptotically normally distributed estimates for βββPH . The asymptotic variance of the esti-

mate βββPH is the inverse of the information matrix, I(βββPH), where I(βββPH) = E(− ∂2 logLPH

∂βββPH∂βββ′PH

).

Usual maximum likelihood theory holds for Cox regression analysis. In particular, asymp-

totically it can be shown that:

• The likelihood ratio test statistic,−2log[L(βββPH)/L(βββPH)], has a χ2 distribution with

p degrees of freedom.

• The score function, UUU(βββPH), has a N(0, I(βββPH)) distribution.

• The Wald test statistic, (βββPH −βββPH)′I(βββPH)(βββPH −βββPH), has a χ2 distribution with

p degrees of freedom.

2.2 The Accelerated Failure Time Model

The accelerated failure time model is commonly used in parametric estimation, where spec-

ification of the probability distribution function is required. When the failure time, Ti,

arises from a log-linear family (that is, Yi = logTi corresponds to a linear model, with the

covariates ZZZi having an additive effect on the log of failure time, Ti) then the family is an

accelerated failure time model. This model is written as:

Yi = logTi = zzz′iβββAFT +σεi, εi =Yi− zzz′iβββAFT

σ(2.7)

where σ is the scale parameter, and εi is a random error term assumed to have a particular

density function, f (.). The parameter βββAFT describes the effect of the covariates, ZZZi, on

the log failure time, Yi. A positive value for βββAFT signifies a deceleration of failure (ie.

survival time is lengthened) for an increasing value of ZZZi. Similarly, a negative value for

βββAFT implies an acceleration in failure (ie. survival time is shortened) for an increasing

value of ZZZi. The distribution of failure time Ti depends on the distribution assumed for εi.


Exponentiating both sides of (2.7) yields:

Ti = ezzz′iβββAFT eσεi

= ezzz′iβββAFT T ∗i , (2.8)

where T ∗i = eσεi . Suppose T ∗i has a known hazard function, hAFT0(t), then the correspond-

ing hazard function for Ti is

hAFTi(t) = hAFT0(te−zzz′iβββAFT )e−zzz′iβββAFT , (2.9)

where hAFT0(t) is also referred to as the baseline hazard function of failure time Ti.

The survivor function of failure time Ti takes the form:

SAFTi(t) = SAFT0(tezzz′iβββAFT ) (2.10)

where SAFT0(t) is the survival function of T ∗i and also termed the baseline survivor function

of Ti. In this formulation, the model assumes that the covariate effect, βββAFT , acts multi-

plicatively on the time scale for the survivor function. It implies that the survivor curve

for an individual in a treatment group is a time-scale change of that for the control group

with all other covariates fixed, and survivor functions for both treatment and control groups

exhibit the same shape but with one group showing either a delay or an advancement of

failure time. Equation (2.9) shows the model formulation on the hazard scale. Note that

this is different from the PH model, where the parameter βββPH describes the multiplicative

effect of the covariates on the hazard.

The AFT model can handle crossing of hazard functions (but not survival functions),

and like the PH model, assumes an immediate treatment effect at t=0. Any distribution

that belongs to the location-scale family is a member of the accelerated failure time family.

The more commonly used distributions are the Exponential, Weibull, Lognormal and Log-

logistic.


2.2.1 Inference

We present maximum likelihood estimation for the Weibull case. The Weibull is a very pop-

ular choice of survival distribution and we use this distribution throughout the project for

exemplifying the AFT model. Since the Weibull distribution is a member of the location-

scale family, the model can be written as in (2.7), such that if Ti ∼Weibull, then log(Ti)∼Extreme Value Distribution,

Yi = logTi = zzz′iβββAFT +σεi, (2.11)

where εi follows an extreme value distribution; f (εi)∼ exp(εi− exp(εi)), −∞ < εi < ∞.

The likelihood function is:

LAFT (βββAFT ,σ) = ∏i∈D

1σ

exp[

yi− zzz′iβββAFTσ

− exp(


)]∏i∈C

exp[−exp

(yi− zzz′iβββAFT

σ

)], (2.12)

where the first product is the probability density function of all observed events, D, and the

second product is the survivor function of all censored observations, C.

The log-likelihood function is:

logLAFT (βββAFT ,σ) =−k logσ+ ∑i∈D


−n

∑i=1

exp(


)(2.13)

where k is the number of events or deaths.

Taking the first and second derivatives of the log-likelihood function (2.13) with respect

to the rth element of βββAFT and σ yields,


∂ logLAFT

∂βAFTr

=− 1σ

∑i∈D

zir +1σ

n

∑i=1

zire(yi−zzz′iβAFT

σ) (2.14a)

∂ logLAFT

∂σ=− k

σ− 1

σ∑i∈D

yi− zzz′iβAFT

σ+

1σ

n

∑i=1

yi− zzz′iβAFT

σe(

yi−zzz′iβAFTσ

) (2.14b)

∂2 logLAFT

∂βAFTr∂βAFTs

=− 1σ2

n

∑i=1

zirzise(yi−zzz′iβAFT

σ) (2.14c)

∂2 logLAFT

∂σ2 =k

σ2 +2

σ2 ∑i∈D

yi− zzz′iβAFT

σ−

2σ2

n

∑i=1

yi− zzz′iβAFT

σe(

yi−zzz′iβAFTσ

) 1σ2

n

∑i=1

[yi− z′iβAFT

σ]2e(

yi−zzz′iβAFTσ

) (2.14d)

∂2 logLAFT

∂βAFTr∂σ=

1σ2 ∑

i∈Dzir−

1σ2

n

∑i=1

zire(yi−zzz′iβAFT

σ)− 1

σ2

n

∑i=1

ziryi− zzz′iβAFT

σe(

yi−zzz′iβAFTσ

)

(2.14e)

The maximum likelihood estimates of the parameters βββAFT and σ are obtained by

solving UUU(βββAFT ) = 000, and U(σ) = 0 where UUU(βββAFT ) is the score vector with elements,∂ logLAFT

∂βAFTr, for r = 1, ..., p, and U(σ) = ∂ logLAFT

∂σ. This is easily handled using a Newton-

Raphson algorithm. The asymptotic variance of the estimates of βββ∗AFT =[βββAFT ,σ](p+1)×1 is

the inverse of the information matrix, I(βββ∗AFT ), where I(βββ∗AFT ) = E(− ∂2 logLAFT

∂βββ∗AFT ∂βββ

∗TAFT

), a matrix

with dimensions ((p + 1)× (p + 1)). Similar to the asymptotic properties of the Cox PH

model, it can be shown that:

• The likelihood ratio test statistic, −2log[L(βββ∗AFT )/L(βββ∗AFT )], has a χ2 distribution

with p+1 degrees of freedom.

• The score function, UUU(βββ∗AFT ), has a N(0, I(βββ∗AFT )) distribution.

• The Wald test statistic, (βββ∗AFT −βββ

∗AFT )′I(βββ

∗AFT )(βββ

∗AFT −βββ

∗AFT ), has a χ2 distribution

with p+1 degrees of freedom.


2.3 The Accelerated Hazards Model

The Accelerated Hazards (AH) model, proposed by Chen & Wang (2000), allows for

greater flexibility in the modeling of survivor data. The model for the hazard function

of an individual i with failure time Ti is written as follows:

hAHi(t) = hAH0(tezzz′iβββAH ), (2.15)

where hAH0 is the baseline hazard function. In this model, ezzz′iβββAH characterizes how the

covariates ZZZi alter the time scale of the underlying hazard function. For instance, βββAH > 0

or βββAH < 0 imply acceleration or deceleration of the time scale for the hazard, respectively.

As an example, if there exists one covariate, Zi, that takes a value of 0 for a control group,

and 1 for a treatment group, then eβAH = 12 means that the hazard of the treatment group

progresses in half the time as those in the control group. Similarly, eβAH = 2 means that the

hazard of the treatment group progresses in twice the time as those in the control group;

eβAH = 1 implies no difference between the two groups.

Alternatively, this model can be written in terms of the survival function

SAHi(t) =[

SAH0

(t

ezzz′iβββAH

)]exp(zzz′iβββAH)

, (2.16)

where SAH0 is the survivor function for the baseline group for which all covariates take a

value of 0.

Unlike the PH and the AFT models, the AH model can accommodate crossing of hazard

and survivor curves. Furthermore, the AH model allows the hazard curves of both the

treatment and control groups to start at the same time point. This is particularly useful

in randomized controlled trials where it is more reasonable to assume a comparable risk

or hazard between groups at t = 0. A restriction in the AH model that is not found in

the PH and AFT models is its inability to handle situations where the hazard function is

constant over time (eg. exponential distribution). Therefore, it is imperative to check for

non-constancy of the baseline hazard function before implementing this model.


2.3.1 Inference

When the baseline hazard function, hAH0(t), is fully parameterized, estimation of the pa-

rameters in the AH model can be performed by maximum likelihood. When hAH0(t) is un-

specified, the usual semiparametric estimation procedure for the PH model can be adopted

for estimation of the AH model as will be described in this section. Chen & Wang (2000)

proposed an estimation procedure for the AH model motivated by the fact that the only dif-

ference between the hazard functions, hAHi(t) and hAH0(t) in (2.15) is a time scale change.

Specifically, notice that for a random event time T with a hazard function h(t), its trans-

formation T ∗ = Tezzz′βββaaa , will have a hazard function of the form, h∗(t) = h(te−zzz′βββaaa)e−zzz′βββaaa ,

where βββa is a vector of arbitrary positive real numbers. If the AH model is true for T ,

with the true parameter vector, βββ000, such that h(t) = h0(tezzz′βββ000), the hazard function for the

transformed event time T ∗ then takes the form:

h∗(t) = h(te−zzz′βββaaa)e−zzz′βββaaa

= h0(tezzz′(βββ000−βββaaa))e−zzz′βββaaa. (2.17)

Notice that when βββ000 = βββaaa, the above equation becomes,

h∗(t) = h0(t)e−zzz′βββ000, (2.18)

which implies that the transformed time, T ∗, recovers the proportionality between hazard

functions with a ratio of e−zzz′βββ000 , when the true values of parameter βββ000 are used for trans-

formation. Hence, a partial likelihood method using (2.18) as a working model on the

transformed times may be used for the estimation of the AH model.

The algorithm developed by Chen & Wang (2000) is as follows:

1. Multiply the observed event times xi by a positive number ezzz′iβββaaa , where βββaaa is a p×1

vector of arbitrary real numbers. This transformation allows for a rescaling of the

time axis for individual i, while keeping the individuals in the baseline group on the

original time scale.

2. Consider the working PH model (2.18) on the transformed times. The naive partial


likelihood function takes the form,

LPH(βββaaa,βββ) =n

∏i=1

[e−zzz′iβββ

∑nj=1 I(xie

zzz′jβββaaa > xiezzz′iβββaaa)e−zzz′jβββ

]δi

. (2.19)

The naive partial likelihood estimator of βββ can be defined as values solving the fol-

lowing score equations, by taking the first derivative of the logarithm of (2.19) with

respect to the rth element of βββ,

∂ logLPH(βββaaa,βββ)∂βr

=n

∑i=1

δi

[zir−

∑nj=1 I(x jez jβa > xieziβ)e−z jβz jr

∑nj=1 I(x jez jβa > xieziβ)e−z jβ

]= 0, (2.20)

where βr indicates the parameter associated with the rth covariate, zir, r = 1, ..., p.

3. Update βββaaa in Step 1 to take values that are equivalent to the solution from (2.20) in

Step 2. Repeat Steps 1 and 2 until convergence to obtain the estimate βAFT .

The above algorithm is equivalent to solving a set of equations UUU(βββAH) = 000, where

UUU(βββAH) = (U1(βAH), ...,Up(βAH))′and

Ur(βAH) =n

∑i=1

δi

[zir−

∑nj=1 I(x jez jβAH > xieziβAH )e−z jβAH z jr

∑nj=1 I(x jez jβAH > xieziβAH )e−z jβAH

], r = 1, ..., p. (2.21)

In the two-sample case where there is only one covariate Zi taking a value of 0 or 1 to

indicate two treatment groups, the estimating equation U(βAH) can be written as:

U(βAH) = ∑i1∈D(1)

δi1

[1−

∑nj=1 I(x jez jβAH > xi1eβAH )e−z jβAH z j

∑nj=1 I(x jez jβAH > xi1eβAH )e−z jβAH

]−

∑i0∈D(0)

δi0

[∑

nj=1 I(x jez jβAH > xi0)e

−z jβAH z j

∑nj=1 I(x jez jβAH > xi0)e

−z jβAH

]

= ∑i1∈D(1)

δi1

[∑ j0∈D(0) I(x j0 > xi1eβAH )

∑ j0∈D(0) I(x j0 > xi1eβAH )+∑ j1∈D(1) I(x j1eβAH > xi1eβAH )e−βAH

]−

∑i0∈D(0)

[∑ j1∈D(1) I(x j1eβAH > xi0)e

−βAH

∑ j0∈D(0) I(x j0 > xi0)+∑ j1∈D(1) I(x j1eβAH > xi0)e−βAH

], (2.22)


where D(0) and D(1) indicate the subset of individuals in the control and treatment groups,

respectively.

Define

Y (q)(t) =n

∑i=1

I(Xi ≥ t, Zi = q)

and

N(q)(t) =n

∑i=1

I(Xi ≤ t, ∆i = 1,Zi = q),

where q = 0,1, representing the control and treatment groups respectively. N(q)(t) is the

number of observed deaths or failures up to time t in group q, and Y (q)(t) is the number of

censored observations at time t in group q. The estimating equation (2.22) can be rewritten

in the counting process framework as:

U(βAH) =∫ Y (0)(t)

Y (0)(t)+Y (1)( teβAH

)/eβAHdN(1)

( teβAH

)−

∫ Y (1)( teβAH

)/eβAH

Y (0)(t)+Y (1)( teβAH

)/eβAHdN(0)(t). (2.23)

Notice that in (2.23), the number of individuals at risk in group 1 at time t/eβAH ,

Y (1)( teβAH

), is weighted by a factor of 1/eβAH to put the treatment group at a compara-

ble hazard as the control group at t = 0.

Asymptotically, it can be shown that solving the estimating equation U(βAH) presented

in (2.23) yields estimates for βAH that are normally distributed. A caution in solving the

proposed estimating equation is its tendency to yield multiple solutions. This issue arises

due to the discontinuity of the function. One way of resolving this issue is to take the zero-

crossing of U(βAH) as the estimate for βAH . A zero-crossing is defined as the βAH that

satisfies U(βAH+)U(βAH−)≤ 0 (Tsiatis 1990).

If a solution to (2.23), βAH , is found, the cumulative baseline hazard function, HAH0(t),


in the two-sample case can be estimated by using the Breslow estimator,

HAH0(·; βAH) =∫ ·

0

dN(0)(t)+dN(1)(

teβAH

)Y (0)(t)+Y (1)

(t

eβAH

)/eβAH

(2.24)

where HAH0(·) =∫ ·

0 hAH0(t)dt.

Throughout this project, estimation for βAH is done by solving the estimating equation

shown in (2.23). The corresponding variance estimation procedure for the algorithm above

was also derived by Chen & Wang (2000). Chen & Jewell (2001) presented an alterna-

tive method to estimate the variance without estimating the baseline hazard function using

asymptotic linear theory. The method used in this project to estimate the variance for βAH

from the AH model is based on the development by Chen & Jewell (2001). The theory un-

derpinning variance estimation is challenging and is omitted here. However, an algorithm

for computing the variance estimator, reproduced from Chen & Jewell (2001), is provided

in the Appendix.

2.4 Comparison of the functional forms for the PH, AFT,and AH Models

To compare these models, we focus on a two-sample situation as above, where the scalar

covariate Zi indicates the grouping (control or treatment). When the underlying failure

time distribution is Weibull, the PH, AFT and AH models are equivalent, and the treatment

effects in these models have the relationship,

βPH =−kβAFT = (k−1)βAH , (2.25)

where k is the shape parameter of the Weibull distribution (k = 1/σ), and k 6= 1. When the

underlying distribution is not a Weibull distribution, the three models are not equivalent

- each parameter has its own interpretation, and the choice of model may depend on the

question of interest. Table 2.1 outlines the differences in interpretation of the parameters

of the three models.


Mod

elE

ffec

tIn

terp

reta

tion

PHβ

PH

>0

Trea

tmen

tpro

port

iona

llyin

crea

ses

risk

/haz

ard

bya

fact

orof

eβP

H.

βP

H<

0Tr

eatm

entp

ropo

rtio

nally

decr

ease

sri

sk/h

azar

dby

afa

ctor

ofeβ

PH

.

AFT

βA

FT

>0

Trea

tmen

tdec

eler

ates

failu

retim

eof

the

surv

ivor

func

tion

bya

fact

orof

eβA

FT.

βA

FT

<0

Trea

tmen

tacc

eler

ates

failu

retim

eof

the

surv

ivor

func

tion

bya

fact

orof

eβA

FT.

AH

βA

H>

0Tr

eatm

enta

ccel

erat

esth

eri

sk/h

azar

dby

afa

ctor

ofeβ

AH

.

βA

H<

0Tr

eatm

entd

ecel

erat

esth

eri

sk/h

azar

dby

afa

ctor

ofeβ

AH

.

All

βP

H=

βA

FT

=β

AH

=0

Trea

tmen

tdoe

sno

thav

ean

effe

ct.

Tabl

e2.

1:Pa

ram

eter

inte

rpre

tatio

nfo

rPH

,AFT

and

AH

mod

els


In the proportional hazards model, the relative risk ratio, eβPH , quantifies the magnitude

of the risk ratio between the treatment and control groups. A positive value for βPH (or

eβPH > 1) suggests that the treatment group has a greater risk of failure than the baseline. A

negative value for βPH (or eβPH < 1) suggests a proportionately smaller risk in the treatment

group compared to the baseline. In the accelerated failure time model, a positive value for

βAFT (or eβAFT > 1) can be interpreted as a deceleration of failure time (or a lengthening

of survival time), while a negative value for βAFT (or eβAFT < 1) implies an acceleration of

failure time (or a shortening of the survival time) in the treatment group compared to the

baseline, with respect to the survivor curve. In the accelerated hazards model, interpreta-

tion of βAH depends on the shape of the baseline hazard function. If the hazard function is

increasing over time, a positive value for βAH (or eβAH > 1) implies that the treatment group

has a greater hazard than the control. On the other hand, if the hazard function is decreas-

ing over time, a positive value for βAH (or eβAH > 1) implies that the treatment group has

a reduced hazard compared to the control. The effect of βAH is multiplicative on the time

scale operating on the hazard function.

Figure 2.1 shows the differences in hazard and survivor curves for the three models. The

proportional hazard model is restricted to scenarios with non-crossing hazard and survivor

curves. The accelerated failure time model can handle both crossing and non-crossing of

the hazard curves, but not of the survivor curves. The accelerated hazards model is the only

model that handles crossings of either or both of the hazard and survivor curves.

2.5 Small sample investigation of the performance of theAH estimator

Simulation studies using different baseline hazard functions were conducted to investi-

gate the small sample properties of the accelerated hazards model estimator. Failure times

were generated from a Weibull distribution with shape parameters, k=0.5, 1.5, and 3, and

scale parameters, λ =0.25, 1, and 1.3. The censoring time was simulated from a Uni-

form (0,τ) distribution, where the value of τ was chosen to produce 27% and 53% cen-

sored observations; we also considered the case of no censoring. The covariate (scalar)


0 1 2 3 4 5

0.0

0.2

0.4

0.6

0.8

1.0

Time in Days

Ris

k

Control

PH Treatment

Proportional Hazard

0 1 2 3 4 5

0.0

0.2

0.4

0.6

0.8

1.0

Time in Days

Ris

k

Control

AFT Treatment

Accelerated Failure Time

0 1 2 3 4 5

0.0

0.2

0.4

0.6

0.8

1.0

Time in Days

Ris

k

Control AH Treatment

Accelerated Hazards

0 1 2 3 4 5

0.0

0.2

0.4

0.6

0.8

1.0

Time in Days

P(s

urv

iva

l)

Control

PH Treatment

0 1 2 3 4 5

0.0

0.2

0.4

0.6

0.8

1.0

Time in Days

P(s

urv

iva

l)

Control

AFT Treatment

0 1 2 3 4 5

0.0

0.2

0.4

0.6

0.8

1.0

Time in Days

P(s

urv

iva

l)

Control

AH Treatment

Figure 2.1: Hazard (top row) and survivor (bottom row) functions of PH, AFT, and AH

models. Non-crossing hazard functions for the AFT and AH models are not shown.


was an indicator for a control or a treatment group. Both control and treatment groups

had equal sample sizes, with the total sample size, n, taking values of 100, 500, 1000,

5000, and 10,000. Three magnitudes of treatment effects in the accelerated hazards model

scale, βAH =0, 0.5, and 2, were investigated. A full factorial design was implemented with

the factors k (3 levels), λ (3 levels), censoring percent (3 levels), n (5 levels), and βAH

(3 levels), for 1000 runs of each factor combination. Table 2.2, 2.3, and 2.4 provide the

mean bias of the 1000 estimates of the treatment effect from analyses using PH and AH

models, the empirical variance of the estimators, as well as the ratio of the model-based

variance to the empirical variance for n=100, 500, and 1000. Note that the true values of

the treatment effect are provided in terms of the effect in the AH model. Bias is defined

as the difference between the mean estimate and the true value, βAH . The bias for the PH

model is calculated similarly with the true value, βPH = (k− 1)βAH . The empirical vari-

ance is the standard deviation of the 1000 estimates. The Variance Ratio compares the

model-based variance with the empirical variance. Let s∗ denote the sign of the difference

between the model-based variance and the empirical variance estimate, s∗ = sign(model-

based variance - empirical variance estimate). Then the quantity Variance Ratio is defined

as s∗(model-based variance/empirical variance)s∗ . With this formulation, the range of pos-

sible values of the Variance Ratio is the union of the disjoint set >1 and <-1. In this

formulation, when the ratio of the model-based to empirical variance is less than 1, the

Variance Ratio is the negative of its inverse. A negative ratio implies that the model-based

variance underestimates the empirical variance, while a positive ratio implies an overesti-

mation of the model-based variance. A ratio of 1 implies that both variance estimates are

equivalent.

The results show that the PH and AH estimators are generally unbiased for all scenarios

considered in the study. While the variance ratio is close to 1 for the PH estimator, the AH

variance estimator behaves poorly when n=100, with the model-based variance severely

underestimating the empirical variance, denoted by the negative variance ratios in Tables

2.2, 2.3 and 2.4.

Figure 2.2 shows a level plot of the variance ratios in the accelerated hazards model.

Warm colors (red, darker shades of orange), signifying underestimation of the variance,


are mostly identified in the n=100 case. Figure 2.3 displays the corresponding variance

ratios for the proportional hazards model, and unlike the ratios based from the AH model,

the PH ratios fall close to -1 and 1 (having light green and yellow colors). As the sample

size grows to 500, and 1000, the asymptotic variance estimator for the AH model improves

greatly, showing comparable magnitude to that of the empirical variance. It seems that

model-based variance estimates are only reliable when sample sizes are quite large.

An alternative strategy in estimating the variance for small sample sizes is to use a

bootstrap approach. Here, we propose a non-parametric bootstrap variance estimation pro-

cedure and compare it with the asymptotic variance estimator of Chen & Jewell (2001).

The following outlines the steps in performing a non-parametric bootstrap procedure, with

a bootstrap resampling size of 1000, to obtain an estimate of the variance:

1. Sample with replacement from the dataset to obtain a new (resampled) dataset of the

same size.

2. Fit the AH model as outlined in Section 2.3 to obtain an estimate of the treatment

effect, βAHboot based on these data.

3. Repeat steps 1-2 1000 times to obtain 1000 estimates of βAHboot , and hence an em-

pirical distribution of the estimator. Estimate the variance of βAH by calculating the

variance of the 1000 βAHboot ’s. The 95% confidence interval for ˆVar(βAH) is the 2.5%

and 97.5% quantile of βββAHboot.

Figures 2.4, 2.5, and 2.6, compare the performance of the bootstrap and model-based

variance estimates based on 1000 runs from the same Weibull distribution as the previous

simulation study but focusing on small sample sizes of 100, 300 and 500. (Note again, that

for each run of this study, 1000 bootstrap resamples are generated. As well, note that in a

few of the resampled scenarios the estimating procedure did not converge and these sce-

narios were omitted.) The bootstrapped variance estimates (shown in green circles) when

n=100 are close to the empirical variance estimates (blue circles) in general, except a slight

deviation when the shape parameter, k, is 3. As expected, as the percentage of censoring


gets bigger, the variance increases. As the sample size increases from 100 to 300 and 500,

the model-based variance estimates approach the empirical and bootstrapped estimates. At

n=500, the only scenario where the model-based variance still performs poorly is when the

censoring percentage is high (at 53%), the treatment effect is large (te=2), and the shape

parameter, k, is 1.5.

Figures 2.7, 2.8 and 2.9, display the 95% coverage probabilities of the two variance es-

timators (model-based and bootstrap) based from the AH model, for sample sizes, n=100,

300, and 500, respectively. The bootstrapped estimate of the variance attained coverage

probabilities that are close to 95% in all cases. The AH model-based variance, on the other

hand, had low coverage probabilities in the small sample size case (n=100) but performed

better when the sample size grew to 300 and 500.


Acc

eler

ated

Haz

ards

Mod

elPr

opor

tiona

lHaz

ards

Mod

el(β

PH

=β

AH(k−

1))

Bia

s(E

mpi

rica

lVar

ianc

e)Va

rian

ceR

atio

Bia

s(E

mpi

rica

lVar

ianc

e)Va

rian

ceR

atio

Cen

.%Si

zeSc

ale

(λ)

Shap

e(k

)β

AH

=0β

AH

=0.5

βA

H=2

βA

H=0

βA

H=0

.5β

AH

=2β

AH

=0β

AH

=0.5

βA

H=2

βA

H=0

βA

H=0

.5β

AH

=2

0%10

00.

250.

50.

011

(0.1

83)

0.02

1(0

.187

)0.

105

(0.3

60)

-1.8

9-1

.90

-3.0

0-0

.006

(0.0

41)

-0.0

02(0

.041

)-0

.024

(0.0

53)

1.01

1.04

-1.0

4

1.5

-0.0

05(0

.192

)-0

.002

(0.1

94)

0.05

4(0

.335

)-3

.51

-3.5

7-7

.29

-0.0

02(0

.045

)-0

.001

(0.0

44)

0.02

9(0

.049

)-1

.09

-1.0

51.

04

30.

009

(0.0

10)

0.00

1(0

.011

)0.

000

(0.0

21)

1.39

1.38

1.15

0.01

7(0

.044

)0.

015

(0.0

55)

0.01

0(0

.395

)-1

.05

-1.0

91.

13

10.

50.

011

(0.1

97)

0.03

8(0

.203

)0.

048

(0.3

41)

-2.0

2-2

.14

-3.0

0-0

.003

(0.0

45)

-0.0

11(0

.045

)-0

.007

(0.0

54)

-1.0

9-1

.06

-1.0

6

1.5

0.00

8(0

.180

)0.

019

(0.2

11)

0.02

6(0

.332

)-3

.43

-4.1

6-7

.09

0.00

3(0

.043

)0.

003

(0.0

43)

0.01

8(0

.052

)-1

.04

-1.0

3-1

.03

30.

001

(0.0

11)

-0.0

04(0

.011

)-0

.007

(0.0

19)

1.23

1.29

1.25

0.00

4(0

.046

)0.

006

(0.0

53)

0.09

1(0

.318

)-1

.10

-1.0

51.

23

1.3

0.5

-0.0

16(0

.191

)0.

024

(0.1

87)

0.09

1(0

.361

)-1

.99

-1.9

1-3

.07

0.00

9(0

.043

)-0

.002

(0.0

42)

-0.0

16(0

.053

)-1

.04

1.01

-1.0

4

1.5

-0.0

03(0

.177

)-0

.032

(0.1

77)

0.01

5(0

.333

)-3

.49

-3.5

7-7

.21

-0.0

02(0

.043

)-0

.017

(0.0

42)

0.01

7(0

.050

)-1

.03

-1.0

01.

01

3-0

.003

(0.0

10)

-0.0

03(0

.010

)0.

008

(0.0

20)

1.49

1.33

1.26

-0.0

09(0

.043

)0.

016

(0.0

50)

0.12

5(0

.353

)-1

.04

1.01

1.16

500

0.25

0.5

0.00

3(0

.035

)0.

005

(0.0

36)

0.03

5(0

.065

)1.

061.

061.

10-0

.002

(0.0

09)

-0.0

01(0

.008

)-0

.007

(0.0

10)

-1.0

6-1

.03

1.03

1.5

0.00

2(0

.034

)-0

.006

(0.0

35)

0.00

6(0

.061

)1.

271.

241.

310.

000

(0.0

08)

-0.0

01(0

.008

)0.

006

(0.0

09)

-1.0

5-1

.02

1.06

30.

001

(0.0

02)

-0.0

01(0

.002

)-0

.001

(0.0

04)

1.05

1.10

1.04

0.00

3(0

.008

)-0

.002

(0.0

10)

0.02

1(0

.061

)-1

.00

1.00

-1.0

2

10.

50.

000

(0.0

33)

0.00

4(0

.036

)0.

012

(0.0

65)

1.15

1.11

1.11

0.00

0(0

.008

)0.

000

(0.0

08)

0.00

1(0

.010

)1.

01-1

.02

-1.0

3

1.5

0.00

0(0

.033

)-0

.007

(0.0

37)

0.00

4(0

.067

)1.

331.

201.

15-0

.001

(0.0

08)

-0.0

05(0

.008

)0.

003

(0.0

10)

1.00

-1.0

3-1

.04

3-0

.002

(0.0

02)

0.00

2(0

.002

)-0

.001

(0.0

04)

1.07

1.07

1.07

-0.0

03(0

.008

)0.

006

(0.0

09)

0.01

6(0

.058

)-1

.04

1.07

1.02

1.3

0.5

0.00

0(0

.032

)0.

012

(0.0

33)

0.04

2(0

.074

)1.

231.

17-1

.02

0.00

0(0

.008

)-0

.004

(0.0

08)

-0.0

09(0

.010

)1.

021.

06-1

.06

1.5

-0.0

02(0

.033

)-0

.008

(0.0

35)

0.02

1(0

.065

)1.

271.

221.

25-0

.001

(0.0

08)

-0.0

02(0

.009

)0.

005

(0.0

09)

-1.0

3-1

.05

1.07

30.

003

(0.0

02)

-0.0

02(0

.002

)-0

.001

(0.0

04)

1.10

1.10

1.10

0.00

5(0

.009

)-0

.003

(0.0

09)

0.02

5(0

.056

)-1

.06

1.05

1.07

1000

0.25

0.5

-0.0

05(0

.015

)0.

000

(0.0

17)

0.01

2(0

.029

)1.

171.

051.

150.

003

(0.0

04)

0.00

1(0

.004

)-0

.001

(0.0

05)

1.09

1.01

1.04

1.5

-0.0

01(0

.017

)0.

004

(0.0

17)

0.00

6(0

.033

)1.

081.

121.

12-0

.001

(0.0

04)

0.00

1(0

.004

)0.

000

(0.0

05)

-1.0

61.

01-1

.06

30.

001

(0.0

01)

-0.0

03(0

.001

)0.

001

(0.0

02)

1.09

1.07

-1.0

40.

002

(0.0

04)

-0.0

03(0

.005

)0.

010

(0.0

29)

1.02

1.03

-1.0

1

10.

50.

000

(0.0

15)

0.00

2(0

.018

)0.

012

(0.0

33)

1.14

1.02

-1.0

00.

000

(0.0

04)

0.00

0(0

.004

)-0

.002

(0.0

05)

1.05

-1.0

3-1

.09

1.5

0.00

2(0

.016

)0.

001

(0.0

16)

0.00

9(0

.034

)1.

141.

131.

150.

001

(0.0

04)

0.00

1(0

.004

)0.

002

(0.0

05)

-1.0

11.

051.

06

3-0

.002

(0.0

01)

0.00

0(0

.001

)-0

.002

(0.0

02)

1.07

1.09

-1.0

5-0

.003

(0.0

04)

0.00

2(0

.005

)0.

012

(0.0

30)

1.01

-1.0

1-1

.05

1.3

0.5

0.00

1(0

.017

)0.

006

(0.0

17)

0.01

0(0

.032

)1.

051.

061.

030.

000

(0.0

04)

-0.0

02(0

.004

)0.

000

(0.0

05)

-1.0

21.

04-1

.05

1.5

0.00

2(0

.016

)0.

006

(0.0

18)

0.00

0(0

.032

)1.

171.

061.

150.

001

(0.0

04)

0.00

2(0

.004

)-0

.002

(0.0

05)

1.02

-1.0

21.

00

30.

000

(0.0

01)

0.00

0(0

.001

)0.

001

(0.0

02)

-1.0

51.

031.

100.

000

(0.0

04)

0.00

2(0

.005

)0.

015

(0.0

30)

-1.1

1-1

.01

-1.0

3

Tabl

e2.

2:Su

mm

ary

ofbi

as,v

aria

nce,

and

vari

ance

ratio

sfo

rth

ePH

and

AH

mod

els

fitte

dto

aW

eibu

lldi

stri

butio

nfo

r

the

0%ce

nsor

ing

case

.Not

eth

atβ

PH

=β

AH

(k-1

).


Acc

eler

ated

Haz

ards

Mod

elPr

opor

tiona

lHaz

ards

Mod

el(β

PH

=β

AH(k−

1))

Bia

s(E

mpi

rica

lVar

ianc

e)Va

rian

ceR

atio

Bia

s(E

mpi

rica

lVar

ianc

e)Va

rian

ceR

atio

Cen

.%Si

zeSc

ale

(λ)

Shap

e(k

)β

AH

=0β

AH

=0.5

βA

H=2

βA

H=0

βA

H=0

.5β

AH

=2β

AH

=0β

AH

=0.5

βA

H=2

βA

H=0

βA

H=0

.5β

AH

=2

27%

100

0.25

0.5

0.00

7(0

.260

)0.

028

(0.2

69)

0.05

6(0

.388

)-2

.13

-2.2

3-2

.91

-0.0

01(0

.056

)-0

.002

(0.0

57)

-0.0

08(0

.058

)-1

.01

-1.0

01.

05

1.5

-0.0

09(0

.255

)-0

.017

(0.2

70)

-0.0

06(0

.426

)-4

.65

-4.9

0-8

.88

-0.0

03(0

.058

)-0

.010

(0.0

61)

0.01

1(0

.071

)-1

.01

-1.0

5-1

.08

30.

001

(0.0

15)

0.00

5(0

.016

)0.

001

(0.0

32)

1.30

1.27

-1.0

60.

001

(0.0

64)

0.02

8(0

.071

)0.

092

(0.3

66)

-1.0

9-1

.01

1.29

10.

50.

006

(0.2

58)

0.04

1(0

.276

)0.

049

(0.3

98)

-2.1

5-2

.27

-2.9

2-0

.004

(0.0

58)

-0.0

11(0

.058

)-0

.004

(0.0

60)

-1.0

2-1

.02

1.04

1.5

-0.0

12(0

.266

)0.

001

(0.2

62)

0.06

8(0

.462

)-4

.66

-4.7

7-9

.19

-0.0

03(0

.059

)-0

.003

(0.0

55)

0.02

6(0

.066

)-1

.01

1.04

1.01

30.

000

(0.0

15)

0.00

4(0

.016

)0.

014

(0.0

32)

1.29

1.30

-1.0

30.

001

(0.0

64)

0.02

0(0

.068

)0.

085

(0.3

54)

-1.1

01.

021.

34

1.3

0.5

0.01

8(0

.242

)0.

039

(0.2

72)

0.03

3(0

.392

)-1

.98

-2.2

2-2

.96

-0.0

11(0

.055

)-0

.006

(0.0

57)

-0.0

07(0

.065

)1.

03-1

.02

-1.0

5

1.5

-0.0

20(0

.284

)0.

006

(0.2

71)

0.05

0(0

.494

)-5

.41

-5.2

1-1

0.17

-0.0

08(0

.064

)0.

000

(0.0

56)

0.00

7(0

.067

)-1

.09

1.04

1.01

3-0

.008

(0.0

16)

0.00

1(0

.016

)0.

001

(0.0

32)

1.25

1.33

-1.1

1-0

.015

(0.0

64)

0.02

3(0

.070

)0.

081

(0.3

71)

-1.0

9-1

.00

1.28

500

0.25

0.5

0.00

4(0

.044

)0.

017

(0.0

48)

0.03

8(0

.083

)1.

091.

051.

07-0

.002

(0.0

11)

-0.0

06(0

.011

)-0

.010

(0.0

12)

1.04

-1.0

2-1

.02

1.5

0.00

4(0

.048

)0.

013

(0.0

53)

0.03

7(0

.116

)1.

231.

14-1

.07

0.00

2(0

.012

)0.

005

(0.0

12)

0.00

5(0

.012

)-1

.05

-1.1

01.

02

3-0

.002

(0.0

03)

-0.0

03(0

.003

)0.

004

(0.0

05)

1.20

1.03

1.11

-0.0

04(0

.011

)0.

001

(0.0

14)

0.04

2(0

.075

)1.

04-1

.08

-1.0

5

10.

50.

000

(0.0

46)

0.01

4(0

.050

)0.

036

(0.0

85)

1.07

1.00

1.02

0.00

0(0

.011

)-0

.004

(0.0

12)

-0.0

08(0

.012

)1.

01-1

.04

-1.0

2

1.5

0.01

0(0

.046

)0.

016

(0.0

49)

0.01

9(0

.107

)1.

281.

241.

110.

004

(0.0

11)

0.00

6(0

.011

)0.

001

(0.0

13)

1.00

1.05

-1.0

0

30.

001

(0.0

03)

0.00

2(0

.003

)0.

004

(0.0

06)

1.06

1.08

1.12

0.00

2(0

.011

)0.

005

(0.0

13)

0.04

1(0

.079

)-1

.01

1.03

-1.0

8

1.3

0.5

0.00

5(0

.042

)0.

003

(0.0

47)

0.03

0(0

.075

)1.

131.

071.

20-0

.003

(0.0

10)

0.00

0(0

.011

)-0

.004

(0.0

11)

1.10

1.01

1.05

1.5

0.00

6(0

.048

)0.

009

(0.0

49)

0.02

4(0

.112

)1.

231.

271.

060.

002

(0.0

12)

0.00

2(0

.011

)-0

.004

(0.0

13)

-1.0

21.

02-1

.03

30.

002

(0.0

03)

0.00

0(0

.003

)-0

.003

(0.0

06)

1.07

1.14

1.03

0.00

4(0

.012

)0.

004

(0.0

13)

0.02

4(0

.074

)-1

.07

1.03

-1.0

4

1000

0.25

0.5

-0.0

02(0

.021

)0.

000

(0.0

23)

0.00

7(0

.041

)1.

051.

051.

000.

001

(0.0

05)

0.00

1(0

.005

)0.

001

(0.0

06)

1.03

1.01

-1.0

3

1.5

0.00

2(0

.022

)0.

005

(0.0

25)

0.01

8(0

.049

)1.

181.

091.

280.

001

(0.0

05)

0.00

0(0

.006

)0.

004

(0.0

07)

1.01

1.01

-1.0

8

30.

000

(0.0

01)

0.00

2(0

.001

)0.

000

(0.0

03)

1.16

1.04

1.07

0.00

1(0

.005

)0.

004

(0.0

06)

0.01

6(0

.035

)1.

031.

04-1

.01

10.

5-0

.001

(0.0

23)

-0.0

02(0

.024

)0.

006

(0.0

39)

1.06

-1.0

01.

070.

000

(0.0

06)

0.00

3(0

.006

)0.

002

(0.0

06)

1.00

-1.0

3-1

.02

1.5

0.00

0(0

.023

)-0

.001

(0.0

24)

0.02

6(0

.051

)1.

161.

081.

240.

000

(0.0

06)

-0.0

01(0

.006

)0.

004

(0.0

07)

-1.0

5-1

.01

-1.0

4

30.

000

(0.0

01)

-0.0

01(0

.002

)0.

003

(0.0

03)

1.08

-1.0

1-1

.04

-0.0

01(0

.006

)0.

001

(0.0

07)

0.02

7(0

.038

)-1

.03

-1.0

2-1

.08

1.3

0.5

-0.0

05(0

.023

)0.

009

(0.0

23)

0.01

4(0

.041

)-1

.01

1.06

-1.0

00.

003

(0.0

06)

-0.0

03(0

.005

)-0

.001

(0.0

06)

-1.0

21.

02-1

.01

1.5

0.00

2(0

.023

)0.

010

(0.0

24)

0.00

9(0

.054

)1.

131.

131.

230.

001

(0.0

06)

0.00

4(0

.005

)0.

002

(0.0

07)

-1.0

41.

04-1

.01

30.

000

(0.0

01)

-0.0

01(0

.001

)0.

002

(0.0

03)

1.01

1.06

1.01

0.00

1(0

.006

)0.

003

(0.0

07)

0.01

0(0

.038

)-1

.08

-1.0

2-1

.11

Tabl

e2.

3:Su

mm

ary

ofbi

as,v

aria

nce,

and

vari

ance

ratio

sfo

rth

ePH

and

AH

mod

els

fitte

dto

aW

eibu

lldi

stri

butio

nfo

r

the

27%

cens

orin

gca

se.N

ote

that

βP

H=

βA

H(k

-1).


Acc

eler

ated

Haz

ards

Mod

elPr

opor

tiona

lHaz

ards

Mod

el(β

PH

=β

AH(k−

1))

Bia

s(E

mpi

rica

lVar

ianc

e)Va

rian

ceR

atio

Bia

s(E

mpi

rica

lVar

ianc

e)Va

rian

ceR

atio

Cen

.%Si

zeSc

ale

(λ)

Shap

e(k

)β

AH

=0β

AH

=0.5

βA

H=2

βA

H=0

βA

H=0

.5β

AH

=2β

AH

=0β

AH

=0.5

βA

H=2

βA

H=0

βA

H=0

.5β

AH

=2

53%

100

0.25

0.5

0.02

4(0

.398

)0.

054

(0.3

91)

0.04

0(0

.543

)-2

.63

-2.6

6-3

.18

-0.0

17(0

.087

)-0

.012

(0.0

80)

-0.0

23(0

.095

)-1

.00

1.07

1.03

1.5

0.03

6(0

.437

)0.

055

(0.4

33)

0.01

9(0

.662

)-7

.31

-7.9

3-1

2.05

0.01

9(0

.084

)0.

016

(0.0

85)

0.00

3(0

.109

)1.

051.

02-1

.07

30.

001

(0.0

25)

0.01

1(0

.029

)0.

086

(0.1

91)

1.24

1.01

-4.6

50.

002

(0.1

00)

0.03

6(0

.111

)0.

037

(0.3

28)

-1.0

2-1

.01

1.81

10.

5-0

.004

(0.3

93)

0.03

4(0

.421

)0.

018

(0.5

06)

-2.6

5-2

.75

-2.8

70.

002

(0.0

90)

-0.0

10(0

.089

)-0

.030

(0.0

93)

-1.0

3-1

.04

1.09

1.5

-0.0

30(0

.466

)0.

031

(0.4

91)

0.02

3(0

.704

)-7

.57

-8.4

8-1

2.77

-0.0

10(0

.091

)0.

004

(0.0

92)

0.01

3(0

.099

)1.

011.

011.

01

30.

004

(0.0

23)

0.00

6(0

.025

)0.

081

(0.1

76)

1.32

1.19

-4.4

40.

008

(0.0

93)

0.02

7(0

.106

)0.

038

(0.3

08)

-1.0

3-1

.01

1.84

1.3

0.5

-0.0

20(0

.422

)0.

043

(0.3

97)

-0.0

32(0

.564

)-2

.69

-2.5

6-3

.22

0.00

6(0

.092

)-0

.017

(0.0

85)

0.00

3(0

.110

)-1

.05

1.03

-1.0

9

1.5

-0.0

01(0

.468

)-0

.020

(0.4

90)

-0.0

10(0

.594

)-7

.72

-9.0

1-1

1.44

0.01

8(0

.093

)0.

003

(0.0

97)

0.03

7(0

.108

)-1

.02

-1.0

8-1

.08

30.

006

(0.0

21)

0.00

0(0

.027

)0.

060

(0.1

57)

1.34

1.16

-4.0

80.

012

(0.0

85)

0.01

2(0

.106

)0.

041

(0.3

58)

1.04

-1.0

01.

63

500

0.25

0.5

0.02

0(0

.071

)0.

017

(0.0

71)

0.04

3(0

.145

)1.

041.

06-1

.06

-0.0

10(0

.017

)-0

.006

(0.0

16)

-0.0

08(0

.019

)-1

.02

1.03

1.00

1.5

0.01

6(0

.077

)0.

023

(0.0

82)

0.06

6(0

.235

)1.

151.

12-1

.74

0.00

7(0

.018

)0.

005

(0.0

17)

0.00

1(0

.019

)-1

.05

-1.0

11.

02

3-0

.002

(0.0

04)

-0.0

03(0

.005

)0.

020

(0.0

22)

1.26

1.09

1.30

-0.0

04(0

.017

)-0

.004

(0.0

20)

0.05

4(0

.107

)1.

09-1

.01

-1.1

1

10.

5-0

.005

(0.0

76)

0.02

2(0

.077

)0.

057

(0.1

52)

1.03

1.02

-1.0

90.

002

(0.0

18)

-0.0

06(0

.017

)-0

.008

(0.0

21)

-1.0

6-1

.03

-1.0

7

1.5

0.01

1(0

.074

)0.

023

(0.0

85)

0.05

2(0

.235

)1.

211.

24-1

.53

0.00

5(0

.017

)0.

005

(0.0

17)

0.00

0(0

.019

)1.

021.

031.

01

30.

000

(0.0

04)

0.00

3(0

.005

)0.

010

(0.0

17)

1.12

1.02

1.47

0.00

1(0

.017

)0.

005

(0.0

20)

0.04

6(0

.102

)-1

.02

-1.0

0-1

.09

1.3

0.5

0.01

3(0

.068

)0.

003

(0.0

82)

0.05

2(0

.159

)1.

14-1

.01

-1.1

0-0

.006

(0.0

16)

0.00

3(0

.018

)-0

.005

(0.0

19)

1.05

-1.0

61.

04

1.5

0.00

7(0

.080

)0.

007

(0.0

80)

0.02

4(0

.201

)1.

111.

20-1

.49

0.00

2(0

.019

)-0

.004

(0.0

16)

-0.0

01(0

.019

)-1

.06

1.11

1.01

30.

001

(0.0

04)

0.00

1(0

.005

)0.

010

(0.0

19)

1.19

1.25

1.30

0.00

2(0

.017

)0.

007

(0.0

18)

0.03

0(0

.095

)-1

.01

1.09

-1.0

3

1000

0.25

0.5

-0.0

01(0

.034

)0.

002

(0.0

34)

0.01

3(0

.065

)1.

041.

101.

050.

000

(0.0

08)

0.00

1(0

.008

)-0

.002

(0.0

09)

1.01

1.04

-1.0

2

1.5

0.00

2(0

.036

)0.

007

(0.0

40)

0.05

2(0

.117

)1.

151.

021.

090.

000

(0.0

08)

0.00

1(0

.009

)0.

006

(0.0

10)

1.00

-1.0

4-1

.01

30.

000

(0.0

02)

0.00

1(0

.003

)0.

008

(0.0

10)

1.16

1.02

1.15

0.00

0(0

.009

)0.

002

(0.0

10)

0.02

6(0

.050

)1.

05-1

.00

-1.1

1

10.

50.

000

(0.0

34)

0.00

5(0

.037

)0.

015

(0.0

69)

1.07

1.00

1.08

0.00

0(0

.008

)-0

.001

(0.0

09)

0.00

1(0

.010

)1.

04-1

.05

-1.0

1

1.5

0.00

5(0

.037

)0.

002

(0.0

39)

0.05

8(0

.120

)1.

141.

181.

000.

002

(0.0

09)

-0.0

03(0

.009

)0.

006

(0.0

09)

1.01

1.02

1.01

30.

000

(0.0

02)

0.00

0(0

.003

)0.

009

(0.0

09)

1.06

1.02

1.09

0.00

0(0

.009

)0.

003

(0.0

10)

0.03

1(0

.048

)-1

.04

-1.0

6-1

.10

1.3

0.5

-0.0

07(0

.036

)0.

012

(0.0

34)

0.01

5(0

.070

)1.

031.

091.

020.

003

(0.0

09)

-0.0

04(0

.008

)-0

.001

(0.0

10)

-1.0

41.

05-1

.02

1.5

0.01

0(0

.036

)0.

015

(0.0

40)

0.04

6(0

.123

)1.

161.

14-1

.11

0.00

5(0

.009

)0.

003

(0.0

08)

0.00

4(0

.009

)1.

011.

061.

03

30.

001

(0.0

02)

0.00

1(0

.002

)0.

004

(0.0

09)

1.02

1.08

1.13

0.00

3(0

.009

)0.

006

(0.0

10)

0.01

5(0

.046

)-1

.05

1.04

-1.0

5

Tabl

e2.

4:Su

mm

ary

ofbi

as,v

aria

nce,

and

vari

ance

ratio

fort

hePH

and

AH

mod

els

fitte

dto

aW

eibu

lldi

stri

butio

nfo

rthe

53%

cens

orin

gca

se.N

ote

that

βP

H=

βA

H(k

-1).


No

Cen

sorin

g

λλ

k

0.5

1.53

0.25

11.

3

100

te=

0

0.25

11.

3

500

te=

0

0.25

11.

3

1000

te=

0

0.25

11.

3

5000

te=

0

0.25

11.

3

1000

0te

= 0

0.5

1.53

100

te=

0.5

500

te=

0.5

1000

te=

0.5

5000

te=

0.5

1000

0te

= 0

.5

0.5

1.53

100

te=

250

0te

= 2

1000

te=

250

00te

= 2

1000

0te

= 2

27%

Cen

sorin

g

λλ

k

0.5

1.53

0.25

11.

3

100

te=

0

0.25

11.

3

500

te=

0

0.25

11.

3

1000

te=

0

0.25

11.

3

5000

te=

0

0.25

11.

3

1000

0te

= 0

0.5

1.53

100

te=

0.5

500

te=

0.5

1000

te=

0.5

5000

te=

0.5

1000

0te

= 0

.5

0.5

1.53

100

te=

250

0te

= 2

1000

te=

250

00te

= 2

1000

0te

= 2

Und

eres

t−

5−

4−

3−

2E

ven

(−1)

Eve

n(1)

2

53%

Cen

sorin

g

λλ

k

0.5

1.53

0.25

11.

3

100

te=

0

0.25

11.

3

500

te=

0

0.25

11.

3

1000

te=

0

0.25

11.

3

5000

te=

0

0.25

11.

3

1000

0te

= 0

0.5

1.53

100

te=

0.5

500

te=

0.5

1000

te=

0.5

5000

te=

0.5

1000

0te

= 0

.5

0.5

1.53

100

te=

250

0te

= 2

1000

te=

250

00te

= 2

1000

0te

= 2

Figu

re2.

2:L

evel

plot

ofva

rian

cera

tios

ofth

eem

piri

cala

ndm

odel

-bas

edac

cele

rate

dha

zard

sm

odel

vari

ance

estim

ates

,

fort

reat

men

teff

ects

(te)

0,0.

5,an

d2;

shap

epa

ram

eter

,k=0

.5,1

.5an

d3;

scal

epa

ram

eter

,λ=0

.25,

1,an

d1.

3;sa

mpl

esi

zes,

n=10

0,50

0,10

00,5

000,

and

10,0

00.

Ane

gativ

era

tioim

plie

san

unde

rest

imat

ion

ofth

em

odel

-bas

edva

rian

cede

note

d

bya

war

mor

ange

colo

r,w

hile

apo

sitiv

era

tioim

plie

san

over

estim

atio

nof

the

mod

el-b

ased

vari

ance

deno

ted

bya

dark

gree

nco

lor.


No

Cen

sorin

g

λλ

k

0.5

1.53

0.25

11.

3

100

te=

0

0.25

11.

3

500

te=

0

0.25

11.

3

1000

te=

0

0.25

11.

3

5000

te=

0

0.25

11.

3

1000

0te

= 0

0.5

1.53

100

te=

0.5

500

te=

0.5

1000

te=

0.5

5000

te=

0.5

1000

0te

= 0

.5

0.5

1.53

100

te=

250

0te

= 2

1000

te=

250

00te

= 2

1000

0te

= 2

27%

Cen

sorin

g

λλ

k

0.5

1.53

0.25

11.

3

100

te=

0

0.25

11.

3

500

te=

0

0.25

11.

3

1000

te=

0

0.25

11.

3

5000

te=

0

0.25

11.

3

1000

0te

= 0

0.5

1.53

100

te=

0.5

500

te=

0.5

1000

te=

0.5

5000

te=

0.5

1000

0te

= 0

.5

0.5

1.53

100

te=

250

0te

= 2

1000

te=

250

00te

= 2

1000

0te

= 2

Und

eres

t(−

2)E

ven(

−1)

Eve

n(1)

Ove

rest

(2)

53%

Cen

sorin

g

λλ

k

0.5

1.53

0.25

11.

3

100

te=

0

0.25

11.

3

500

te=

0

0.25

11.

3

1000

te=

0

0.25

11.

3

5000

te=

0

0.25

11.

3

1000

0te

= 0

0.5

1.53

100

te=

0.5

500

te=

0.5

1000

te=

0.5

5000

te=

0.5

1000

0te

= 0

.5

0.5

1.53

100

te=

250

0te

= 2

1000

te=

250

00te

= 2

1000

0te

= 2

Figu

re2.

3:L

evel

plot

ofva

rian

cera

tios

ofem

piri

cala

ndm

odel

-bas

edpr

opor

tiona

lhaz

ard

mod

elva

rian

cees

timat

es,f

or

trea

tmen

teff

ects

(te)

0,0.

5,an

d2;

shap

epa

ram

eter

,k=0

.5,1

.5an

d3;

scal

epa

ram

eter

,λ=0

.25,

1,an

d1.

3;sa

mpl

esi

zes,

n=10

0,50

0,10

00,5

000,

and

10,0

00.A

nega

tive

ratio

impl

ies

anun

dere

stim

atio

nof

the

mod

el-b

ased

vari

ance

deno

ted

by

aye

llow

colo

r,w

hile

apo

sitiv

era

tioim

plie

san

over

estim

atio

nof

the

mod

el-b

ased

vari

ance

deno

ted

bygr

een.


No

Cen

sorin

g

λλ

Variance Estimates

0.1

0.2

0.3

0.4

0.5

0.6

0.25

11.

3

●●

●

●●

●

●●

●

k =

0.5

te =

0

0.25

11.

3

●●

●

●●

●

●●

●

k =

1.5

te =

0

0.25

11.

3●

●●

●●

●●

●●

k =

3te

= 0

0.1

0.2

0.3

0.4

0.5

0.6

●●

●

●●

●

●●

●

k =

0.5

te =

0.5

●●

●

●●

●

●●

●

k =

1.5

te =

0.5

●●

●●

●●

●●

●

k =

3te

= 0

.5

0.1

0.2

0.3

0.4

0.5

0.6

●●

●

●●

●

●●

●

k =

0.5

te =

2

●●

●

●●

●

●●

●

k =

1.5

te =

2

●●

●●

●●

●●

●

k =

3te

= 2

27%

Cen

sorin

g

λλ

0.1

0.2

0.3

0.4

0.5

0.6

0.25

11.

3

●●

●

●●

●

●●

●

k =

0.5

te =

0

0.25

11.

3

●●

●

●●

●

●●

●

k =

1.5

te =

0

0.25

11.

3●

●●

●●

●●

●●

k =

3te

= 0

0.1

0.2

0.3

0.4

0.5

0.6

●●

●

●●

●

●●

●

k =

0.5

te =

0.5

●●

●

●●

●

●●

●

k =

1.5

te =

0.5

●●

●●

●●

●●

●

k =

3te

= 0

.5

0.1

0.2

0.3

0.4

0.5

0.6

●

●

●

●●

●

●●

●

k =

0.5

te =

2

●●

●

●●

●

●●

●

k =

1.5

te =

2

●●

●●

●●

●●

●

k =

3te

= 2

53%

Cen

sorin

g

λλ

0.1

0.2

0.3

0.4

0.5

0.6

0.25

11.

3

●

●●

●●

●

●●

●

k =

0.5

te =

0

0.25

11.

3

●

●●

●●

●

●●

●

k =

1.5

te =

0

0.25

11.

3●

●●

●●

●●

●●

k =

3te

= 0

0.1

0.2

0.3

0.4

0.5

0.6

●●

●

●●

●

●●

●

k =

0.5

te =

0.5

●●

●

●●

●

●●

●

k =

1.5

te =

0.5

●●

●●

●●

●●

●

k =

3te

= 0

.5

0.1

0.2

0.3

0.4

0.5

0.6

●

●●

●●

●

●●

●

k =

0.5

te =

2

●●

●

●●

●

●●

●

k =

1.5

te =

2

●●

●

●●

●

●●

●

k =

3te

= 2

Em

piric

alM

odel

−B

ased

Boo

tstr

ap

● ● ●

Figu

re2.

4:D

otpl

otof

thre

eva

rian

cees

timat

es-

empi

rica

lva

rian

ce(b

lue)

,m

odel

-bas

edva

rian

ce(p

ink)

,an

dno

n-

para

met

ric

boot

stra

pped

vari

ance

(gre

en)

for

data

take

nfr

oma

Wei

bull

dist

ribu

tion

with

n=10

0,sh

ape

(k)=

0.5,

1.5,

and

3,sc

ale

(λ)=

0.25

,1,a

nd1.

3,an

dtr

eatm

ente

ffec

t,β

AH

deno

ted

(te)

=0,

0.5,

and

2.


No

Cen

sorin

g

λλ

Variance Estimates

0.1

0.2

0.3

0.25

11.

3

●●

●●

●●

●●

●

k =

0.5

te =

0

0.25

11.

3

●●

●●

●●

●●

●

k =

1.5

te =

0

0.25

11.

3●

●●

●●

●●

●●

k =

3te

= 0

0.1

0.2

0.3

●●

●●

●●

●●

●

k =

0.5

te =

0.5

●●

●●

●●

●●

●

k =

1.5

te =

0.5

●●

●●

●●

●●

●

k =

3te

= 0

.5

0.1

0.2

0.3

●●

●●

●●

●●

●

k =

0.5

te =

2

●●

●●

●●

●●

●

k =

1.5

te =

2

●●

●●

●●

●●

●

k =

3te

= 2

27%

Cen

sorin

g

λλ

0.1

0.2

0.3

0.25

11.

3

●●

●●

●●

●●

●

k =

0.5

te =

0

0.25

11.

3

●●

●●

●●

●●

●

k =

1.5

te =

0

0.25

11.

3●

●●

●●

●●

●●

k =

3te

= 0

0.1

0.2

0.3

●●

●●

●●

●●

●

k =

0.5

te =

0.5

●●

●●

●●

●●

●

k =

1.5

te =

0.5

●●

●●

●●

●●

●

k =

3te

= 0

.5

0.1

0.2

0.3

●●

●●

●●

●●

●

k =

0.5

te =

2

●●

●

●●

●

●●

●

k =

1.5

te =

2

●●

●●

●●

●●

●

k =

3te

= 2

53%

Cen

sorin

g

λλ

0.1

0.2

0.3

0.25

11.

3

●●

●●

●●

●●

●

k =

0.5

te =

0

0.25

11.

3

●●

●●

●●

●●

●

k =

1.5

te =

0

0.25

11.

3●

●●

●●

●●

●●

k =

3te

= 0

0.1

0.2

0.3

●●

●●

●●

●●

●

k =

0.5

te =

0.5

●●

●●

●●

●●

●

k =

1.5

te =

0.5

●●

●●

●●

●●

●

k =

3te

= 0

.5

0.1

0.2

0.3

●

●●

●●

●

●●

●

k =

0.5

te =

2

●●

●

●●

●

●●

●

k =

1.5

te =

2

●●

●●

●●

●●

●

k =

3te

= 2

Em

piric

alM

odel

−B

ased

Boo

tstr

ap

● ● ●

Figu

re2.

5:D

otpl

otof

thre

eva

rian

cees

timat

es-

empi

rica

lva

rian

ce(b

lue)

,m

odel

-bas

edva

rian

ce(p

ink)

,an

dno

n-

para

met

ric

boot

stra

pped

vari

ance

(gre

en)

for

data

take

nfr

oma

Wei

bull

dist

ribu

tion

with

n=30

0,sh

ape

(k)=

0.5,

1.5,

and

3,sc

ale

(λ)=

0.25

,1,a

nd1.

3,an

dtr

eatm

ente

ffec

t,β

AH

deno

ted

(te)

=0,

0.5,

and

2.


No

Cen

sorin

g

λλ

Variance Estimates

0.1

0.2

0.3

0.25

11.

3

●●

●●

●●

●●

●

k =

0.5

te =

0

0.25

11.

3

●●

●●

●●

●●

●

k =

1.5

te =

0

0.25

11.

3●

●●

●●

●●

●●

k =

3te

= 0

0.1

0.2

0.3

●●

●●

●●

●●

●

k =

0.5

te =

0.5

●●

●●

●●

●●

●

k =

1.5

te =

0.5

●●

●●

●●

●●

●

k =

3te

= 0

.5

0.1

0.2

0.3

●●

●●

●●

●●

●

k =

0.5

te =

2

●●

●●

●●

●●

●

k =

1.5

te =

2

●●

●●

●●

●●

●

k =

3te

= 2

27%

Cen

sorin

g

λλ

0.1

0.2

0.3

0.25

11.

3

●●

●●

●●

●●

●

k =

0.5

te =

0

0.25

11.

3

●●

●●

●●

●●

●

k =

1.5

te =

0

0.25

11.

3●

●●

●●

●●

●●

k =

3te

= 0

0.1

0.2

0.3

●●

●●

●●

●●

●

k =

0.5

te =

0.5

●●

●●

●●

●●

●

k =

1.5

te =

0.5

●●

●●

●●

●●

●

k =

3te

= 0

.5

0.1

0.2

0.3

●●

●●

●●

●●

●

k =

0.5

te =

2

●●

●●

●●

●●

●

k =

1.5

te =

2

●●

●●

●●

●●

●

k =

3te

= 2

53%

Cen

sorin

g

λλ

0.1

0.2

0.3

0.25

11.

3

●●

●●

●●

●●

●

k =

0.5

te =

0

0.25

11.

3

●●

●●

●●

●●

●

k =

1.5

te =

0

0.25

11.

3●

●●

●●

●●

●●

k =

3te

= 0

0.1

0.2

0.3

●●

●●

●●

●●

●

k =

0.5

te =

0.5

●●

●●

●●

●●

●

k =

1.5

te =

0.5

●●

●●

●●

●●

●

k =

3te

= 0

.5

0.1

0.2

0.3

●●

●●

●●

●●

●

k =

0.5

te =

2

●●

●

●●

●

●●

●

k =

1.5

te =

2

●●

●●

●●

●●

●

k =

3te

= 2

Em

piric

alM

odel

−B

ased

Boo

tstr

ap

● ● ●

Figu

re2.

6:D

otpl

otof

thre

eva

rian

cees

timat

es-

empi

rica

lva

rian

ce(b

lue)

,m

odel

-bas

edva

rian

ce(p

ink)

,an

dno

n-

para

met

ric

boot

stra

pped

vari

ance

(gre

en)

for

data

take

nfr

oma

Wei

bull

dist

ribu

tion

with

n=50

0,sh

ape

(k)=

0.5,

1.5,

and

3,sc

ale

(λ)=

0.25

,1,a

nd1.

3,an

dtr

eatm

ente

ffec

t,β

AH

deno

ted

(te)

=0,

0.5,

and

2.


No

Cen

sorin

g

λλ

95% Coverage Probability

0.4

0.5

0.6

0.7

0.8

0.9

0.25

11.

3

●●

●

●●

●

k =

0.5

te =

0

0.25

11.

3

●●

●

●●

●

k =

1.5

te =

0

0.25

11.

3

●●

●

●●

●

k =

3te

= 0

0.4

0.5

0.6

0.7

0.8

0.9

●●

●

●●

●

k =

0.5

te =

0.5

●●

●

●●

●

k =

1.5

te =

0.5

●●

●

●●

●

k =

3te

= 0

.5

0.4

0.5

0.6

0.7

0.8

0.9

●●

●

●●

●

k =

0.5

te =

2

●●

●

●●

●

k =

1.5

te =

2

●●

●

●●

●

k =

3te

= 2

27%

Cen

sorin

g

λλ

0.4

0.5

0.6

0.7

0.8

0.9

0.25

11.

3

●●

●

●●

●

k =

0.5

te =

0

0.25

11.

3

●●

●

●●

●

k =

1.5

te =

0

0.25

11.

3

●●

●

●●

●

k =

3te

= 0

0.4

0.5

0.6

0.7

0.8

0.9

●●

●

●●

●

k =

0.5

te =

0.5

●●

●

●●

●

k =

1.5

te =

0.5

●●

●

●●

●

k =

3te

= 0

.5

0.4

0.5

0.6

0.7

0.8

0.9

●●

●

●●

●

k =

0.5

te =

2

●●

●

●●

●

k =

1.5

te =

2

●●

●

●●

●

k =

3te

= 2

53%

Cen

sorin

g

λλ

0.4

0.5

0.6

0.7

0.8

0.9

0.25

11.

3

●●

●

●●

●

k =

0.5

te =

0

0.25

11.

3

●●

●

●●

●

k =

1.5

te =

0

0.25

11.

3

●●

●

●●

●

k =

3te

= 0

0.4

0.5

0.6

0.7

0.8

0.9

●●

●

●●

●

k =

0.5

te =

0.5

●●

●

●●

●

k =

1.5

te =

0.5

●●

●

●●

●

k =

3te

= 0

.5

0.4

0.5

0.6

0.7

0.8

0.9

●

●●

●●

●

k =

0.5

te =

2

●●

●

●●

●

k =

1.5

te =

2

●●

●

●●

●

k =

3te

= 2

Mod

el−

Bas

edB

oots

trap

● ●

Figu

re2.

7:D

otpl

otof

two

cove

rage

prob

abili

ties

usin

gth

em

odel

-bas

edva

rian

ce(b

lue)

,and

non-

para

met

ric

boot

stra

pped

vari

ance

(pin

k)fo

rda

tata

ken

from

aW

eibu

lldi

stri

butio

nw

ithn=

100,

shap

e(k

)=0.

5,1.

5,an

d3,

scal

e(λ

)=0.

25,1

,and

1.3,

and

trea

tmen

teff

ect,

βA

Hde

note

d(t

e)=

0,0.

5,an

d2.


No

Cen

sorin

g

λλ


0.4

0.5

0.6

0.7

0.8

0.9

0.25

11.

3

●●

●●

●●

k =

0.5

te =

0

0.25

11.

3

●●

●

●●

●

k =

1.5

te =

0

0.25

11.

3

●●

●

●●

●

k =

3te

= 0

0.4

0.5

0.6

0.7

0.8

0.9

●●

●●

●●

k =

0.5

te =

0.5

●●

●

●●

●

k =

1.5

te =

0.5

●●

●

●●

●

k =

3te

= 0

.5

0.4

0.5

0.6

0.7

0.8

0.9

●●

●●

●●

k =

0.5

te =

2

●●

●

●●

●

k =

1.5

te =

2

●●

●

●●

●

k =

3te

= 2

27%

Cen

sorin

g

λλ

0.4

0.5

0.6

0.7

0.8

0.9

0.25

11.

3

●●

●●

●●

k =

0.5

te =

0

0.25

11.

3

●●

●

●●

●

k =

1.5

te =

0

0.25

11.

3

●●

●

●●

●

k =

3te

= 0

0.4

0.5

0.6

0.7

0.8

0.9

●●

●●

●●

k =

0.5

te =

0.5

●●

●

●●

●

k =

1.5

te =

0.5

●●

●

●●

●

k =

3te

= 0

.5

0.4

0.5

0.6

0.7

0.8

0.9

●●

●●

●●

k =

0.5

te =

2

●●

●

●●

●

k =

1.5

te =

2

●●

●

●●

●

k =

3te

= 2

53%

Cen

sorin

g

λλ

0.4

0.5

0.6

0.7

0.8

0.9

0.25

11.

3

●●

●●

●●

k =

0.5

te =

0

0.25

11.

3

●●

●

●●

●

k =

1.5

te =

0

0.25

11.

3

●●

●

●●

●

k =

3te

= 0

0.4

0.5

0.6

0.7

0.8

0.9

●●

●●

●●

k =

0.5

te =

0.5

●●

●

●●

●

k =

1.5

te =

0.5

●●

●

●●

●

k =

3te

= 0

.5

0.4

0.5

0.6

0.7

0.8

0.9

●●

●

●●

●

k =

0.5

te =

2

●●

●

●●

●

k =

1.5

te =

2

●●

●

●●

●

k =

3te

= 2

Mod

el−

Bas

edB

oots

trap

● ●

Figu

re2.

8:D

otpl

otof

two

cove

rage

prob

abili

ties

usin

gth

em

odel

-bas

edva

rian

ce(b

lue)

,and

non-

para

met

ric

boot

stra

pped

vari

ance

(pin

k)fo

rda

tata

ken

from

aW

eibu

lldi

stri

butio

nw

ithn=

300,

shap

e(k

)=0.

5,1.

5,an

d3,

scal

e(λ

)=0.

25,1

,and

1.3,

and

trea

tmen

teff

ect,

βA

Hde

note

d(t

e)=

0,0.

5,an

d2.


No

Cen

sorin

g

λλ


0.4

0.5

0.6

0.7

0.8

0.9

0.25

11.

3

●●

●●

●●

k =

0.5

te =

0

0.25

11.

3

●●

●

●●

●

k =

1.5

te =

0

0.25

11.

3

●●

●●

●●

k =

3te

= 0

0.4

0.5

0.6

0.7

0.8

0.9

●●

●●

●●

k =

0.5

te =

0.5

●●

●●

●●

k =

1.5

te =

0.5

●●

●

●●

●

k =

3te

= 0

.5

0.4

0.5

0.6

0.7

0.8

0.9

●●

●●

●●

k =

0.5

te =

2

●●

●

●●

●

k =

1.5

te =

2

●●

●●

●●

k =

3te

= 2

27%

Cen

sorin

g

λλ

0.4

0.5

0.6

0.7

0.8

0.9

0.25

11.

3

●●

●●

●●

k =

0.5

te =

0

0.25

11.

3

●●

●

●●

●

k =

1.5

te =

0

0.25

11.

3

●●

●

●●

●

k =

3te

= 0

0.4

0.5

0.6

0.7

0.8

0.9

●●

●●

●●

k =

0.5

te =

0.5

●●

●

●●

●

k =

1.5

te =

0.5

●●

●

●●

●

k =

3te

= 0

.5

0.4

0.5

0.6

0.7

0.8

0.9

●●

●●

●●

k =

0.5

te =

2

●●

●

●●

●

k =

1.5

te =

2

●●

●

●●

●

k =

3te

= 2

53%

Cen

sorin

g

λλ

0.4

0.5

0.6

0.7

0.8

0.9

0.25

11.

3

●●

●●

●●

k =

0.5

te =

0

0.25

11.

3

●●

●●

●●

k =

1.5

te =

0

0.25

11.

3

●●

●

●●

●

k =

3te

= 0

0.4

0.5

0.6

0.7

0.8

0.9

●●

●●

●●

k =

0.5

te =

0.5

●●

●●

●●

k =

1.5

te =

0.5

●●

●

●●

●

k =

3te

= 0

.5

0.4

0.5

0.6

0.7

0.8

0.9

●●

●

●●

●

k =

0.5

te =

2

●●

●

●●

●

k =

1.5

te =

2

●●

●

●●

●

k =

3te

= 2

Mod

el−

Bas

edB

oots

trap

● ●

Figu

re2.

9:D

otpl

otof

two

cove

rage

prob

abili

ties

usin

gth

em

odel

-bas

edva

rian

ce(b

lue)

,and

non-

para

met

ric

boot

stra

pped

vari

ance

(pin

k)fo

rda

tata

ken

from

aW

eibu

lldi

stri

butio

nw

ithn=

500,

shap

e(k

)=0.

5,1.

5,an

d3,

scal

e(λ

)=0.

25,1

,and

1.3,

and

trea

tmen

teff

ect,

βA

Hde

note

d(t

e)=

0,0.

5,an

d2.

Chapter 3

Data Analysis

We illustrate the use of the three models presented in Chapter 2 on five data sets - a ran-

domized trial for the treatment of breast cancer, an observational study on coronary artery

bypass graft (CABG) surgery in British Columbia taken from the Cardiac Services BC Reg-

istry, a veteran administration lung cancer study, a study on catheter placement for kidney

dialysis patients, and a study of the administration of carmustine (BCNU) drug for patients

with malignant brain tumours. For simplicity, in all studies, we focus on comparisons of the

treatment and control groups to use these examples for contrasting the PH and AH models.

3.1 Breast Cancer Clinical Trial

A prospective randomized trial for post-menopausal women diagnosed with node-positive

stage I or II breast cancer in British Columbia from 1979-1986 was undertaken to study

the effects of combining radiotherapy with chemotherapy in the treatment of breast cancer

(Ragaz, et al. 1997). All subjects were referred to the British Columbia Cancer Agency

for group randomization and treatment planning. There were 318 subjects in total, 154 of

whom were assigned to the control group (chemotherapy only), while the rest (164) were

assigned to the proposed treatment group (chemotherapy and radiotherapy). Subjects were

followed for 20 years. Failure in this study is defined as either the recurrence of cancer

or death from cancer or other causes; 36% from the control group were right censored,

while about 52% of individuals in the chemotherapy and radiotherapy group were cen-

sored. Overall, there were 177 subjects who failed at the end of the study.

33

CHAPTER 3. DATA ANALYSIS 34

0.0

0.2

0.4

0.6

0.8

1.0

Kaplan−Meier Plot

Time in Years

P(S

urvi

val)

0 2 4 6 8 10 12 14 16 18

Chemotherapy only

Chemotherapy + Radiotherapy

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Cumulative Hazard Plot

Time in Years

Haz

ard

0 2 4 6 8 10 12 14 16 18

Chemotherapy only


Figure 3.1: Left: Kaplan-Meier survivor curves for the control group (black) and treatment

group (red). Right: Cumulative hazard curves for the control group and treatment group.

The Kaplan-Meier survivor curves, displayed in Figure 3.1, suggest that a proportional

hazards model may fit the data well. We fit the three models discussed in Chapter 2 to com-

pare and contrast model fit, parameter estimates and their interpretation. Table 3.1 lists the

estimated covariate effects and their corresponding 95% confidence intervals for the pro-

posed treatment effect (chemotherapy and radiotherapy), using PH, AFT, and AH models.

The PH, Weibull AFT, and the bootstrapped AH models reported p-values < 0.05 in a test

of no difference between treatment and control groups. The bootstrap variance estimates in

the AH model are denoted with an asterisk (*). Note that when the model-based variance

was used in conducting the Wald test for the significance of the treatment effect in the AH

model, the resulting p-value was marginally significant with a much larger value of 0.08.

Using a non-parametric bootstrap approach for variance estimation in the AH model led to

a smaller standard error with a Wald statistic that is significant at α = 5%, and a CI for the

estimate with lower bound that is well above 1. Furthermore, all models agree in the di-

rection of the effect - with all models identifying a favorable outcome to the chemotherapy


plus radiotherapy treatment over the standard chemotherapy only treatment.

In the proportional hazards model, the proposed treatment has a hazard rate that is

proportionately 35% less than the stand-alone chemotherapy treatment. In the accelerated

failure time Weibull model, individuals in the control group age (along the survivor curve)

approximately twice as fast as those in the treatment group. The estimated Extreme Value

scale parameter is 1.38 (SE=0.09), which suggests that the hazard function decreases with

time. In the accelerated hazards model framework, the time scale for risk or hazard pro-

gression for the chemotherapy plus radiotherapy group is about 2.5 times faster than that

for the chemotherapy only group. This acceleration is seen as beneficial because of a gen-

erally decreasing hazard function. A plot of the smoothed hazard function using a kernel

approximation in Figure 3.2 shows such trajectory.

Model β SE(β) p-value eβ 95% CI for eβ

PH -0.437 0.152 0.004 0.646 (0.48, 0.87)

AFT 0.641 0.211 0.002 1.898 (1.25, 2.87)

AH 0.928 0.531 (0.332*) 0.081 (0.005*) 2.529 (0.89, 7.16) (1.32, 4.85)*

Table 3.1: Estimated treatment effects in the analysis of the breast cancer data using PH,

AFT, and AH models. Values with * in the AH model represent bootstrapped estimates.

The p-values correspond to Wald tests of a hypothesis of no treatment effect.

Figure 3.3 shows the non-parametric and fitted cumulative hazard (top panels) and sur-

vivor curves (bottom panels) for the three models overlaid on empirical estimates of these

quantities. The estimated Cox proportional hazard survivor curves seem to fit the data quite

well at all time points for both control and treatment groups. In the accelerated failure time

model, the predicted cumulative hazard curves for the both groups are underestimated up

to about year 10, and overestimated in later years. (The converse can be seen in the es-

timated survivor curve.) The accelerated hazards model sets the risk for both groups to

be equivalent at time 0. However, this feature yielded an undesired overestimation of the

treatment group’s hazard at the beginning of the trial, causing the survivor curve of the

treatment group to be below that of the control group at 2 years. This is not an accurate

description of the data as the treatment group had consistently higher survival rates than


the control group throughout the entire duration of the trial. Overall, the PH model fit the

breast cancer data the best. A plot of the residuals in Figure 3.4 is shown as a visual check

to assess if there is any deviation from proportional hazards. In fact, a linear test that the

slope is 0 gives a p-value of 0.72, providing support for non-violation of the proportional

hazards assumption.

0e+

001e

−04

2e−

043e

−04

Follow−up Time in Years

Haz

ard

Rat

e

0 2 4 6 8 10 12 14 16 18

Smoothed Hazard Curves

Chemotherapy only


Figure 3.2: Smoothed hazard curves for the breast cancer data.


0.00.20.40.60.81.01.2

Tim

e in

Yea

rs

Cumulative Hazard

PH

Mod

el

02

46

810

1214

1618

Kap

lan−

Mei

erF

itted

PH

Che

mot

hera

py

Che

mot

hera

py +

Rad

iatio

n

0.00.20.40.60.81.01.2

Tim

e in

Yea

rs

Cumulative Hazard

AF

T W

eibu

ll M

odel

02

46

810

1214

1618

Kap

lan−

Mei

erF

itted

AF

T W

eibu

ll

Che

mot

hera

py

Che

mot

hera

py +

Rad

iatio

n

0.00.20.40.60.81.01.2

Tim

e in

Yea

rs

Cumulative Hazard

02

46

810

1214

1618

AH

Mod

el

Kap

lan−

Mei

erA

H

Che

mot

hera

py

Che

mot

hera

py +

Rad

iatio

n

0.00.20.40.60.81.0

Tim

e in

Yea

rs

P(Survival)

02

46

810

1214

1618

Kap

lan−

Mei

erF

itted

PH

Che

mot

hera

py

Che

mot

hera

py +

Rad

iatio

n

0.00.20.40.60.81.0

Tim

e in

Yea

rs

P(Survival)

02

46

810

1214

1618

Kap

lan−

Mei

erF

itted

AF

T W

eibu

ll

Che

mot

hera

py

Che

mot

hera

py +

Rad

iatio

n

0.00.20.40.60.81.0

Tim

e in

Yea

rs

P(Survival)

02

46

810

1214

1618

Kap

lan−

Mei

erF

itted

AH

Che

mot

hera

py

Che

mot

hera

py +

Rad

iatio

n

Figu

re3.

3:To

p:C

umul

ativ

eha

zard

curv

esfo

rthe

PH,W

eibu

llA

FT,a

ndA

Hm

odel

s.T

hefit

ted

cum

ulat

ive

haza

rdcu

rves

are

show

nin

solid

lines

vsth

eno

n-pa

ram

etri

cha

zard

curv

esin

dash

edlin

es.

The

base

line

(con

trol

)ha

zard

curv

esar

e

show

nin

blac

k.B

otto

m:

Kap

lan-

Mei

ersu

rviv

orcu

rves

for

the

cont

rol

grou

p(b

lack

dash

ed)

and

trea

tmen

tgr

oup

(red

dash

ed),

with

the

fitte

dsu

rviv

orcu

rves

fort

reat

men

tgro

upsh

own

inre

dso

lidlin

es.


Cox Proportional Hazards Residual Plot

Time

Bet

a(t)

for

trt

230 450 660 860 1200 1700 2900 4300

−2

−1

01

2 ●

●

●

●

●●●

●●

●

●

●

●●

●●

●●

●

●

●

●●

●

●●●●

●

●●●●●

●●

●●

●

●

●

●

●●●

●

●●

●●

●●

●●●

●●

●●

●

●●

●

●

●

●●●

●

●

●●●●●

●

●

●●

●

●●●●●●●

●●

●●

●●

●

●

●●●●●

●●

●●●●●●

●●

●●

●●

●

●

●●●

●

●

●●●

●●

●●●

●●●●●

●

●

●

●●

●●

●

●●

●

●●●

●

●

●●

●●●

●

●

●●●●●●

●●●●●●●

●●●●●

●

●

Figure 3.4: Residual plot from the fit of the proportional hazards model to the breast cancer

data.


3.2 Coronary Artery Bypass Graft Surgery

This study considers the two-year post-surgery survival outcomes of a random sample of

individuals who underwent coronary artery bypass graft (CABG) surgeries from January

1991 to February 2006 in British Columbia. The Cardiac Services BC Registy maintains

a database on all heart-related surgeries performed in BC. Death or mortality data were

taken from the British Columbia Vital Statistics Agency. Our sample consists of 2,644 in-

dividuals, with men comprising about 80% of the sample. Among the individuals sampled,

171 (6.5%) experienced death within 2 years post-surgery. Furthermore, among those who

died, 88 (58 males; 30 females) failed during the first month after having CABG surgery.

This short-term mortality is well-known with CABG surgery, and is defined as the 30-day

operative mortality period. Shown in Figure 3.5 are the 2-year survivor and cumulative

hazard curves for both male and female patients who underwent a CABG surgery. There

is a steep decline in the survival rate, and a corresponding sharp rise in the cumulative

hazard during the first month post-surgery. However, survival rates after the 30-day period

are generally good and optimistic. It should be noted that the Kaplan-Meier curves seem

to suggest that females have a higher mortality rate than males. Indeed, Ghahramani, et

al. (2001) has identified significant gender differences in the modeling of CABG survival

time data. Previous studies (Humphries et al., 2007) found that short-term gender mortality

differences may be attributed to intrinsic factors such as body surface area (BSA) (which

serves as a proxy for size of coronary vessels), with women having lower BSA, and thus

smaller vessels. (Body surface area in m2 units is defined as√

height(cm)weight(kg)3600 ). In this

study, we use gender as the binary covariate in our two-sample comparison.

We performed analyses of gender effects on the CABG data using the proportional haz-

ards, Weibull accelerated failure time, and accelerated hazards models. Table 3.2 shows

the estimates for the female gender effect, β, for the three models considered. Although the

estimates of this effect from all three models have different interpretation, all models show

evidence of a significant difference between males and females, with all p-values < 0.01

under the Wald-type test of no effect. Furthermore, all models find that females perform

worse than males.


0.90

0.92

0.94

0.96

0.98

1.00

Kaplan−Meier Plot

Time in months

P(S

urvi

val)

0 2 4 6 8 10 12 14 16 18 20 22 24

Males

Females

0.00

0.02

0.04

0.06

0.08

0.10


Time in months

Haz

ard

0 2 4 6 8 10 12 14 16 18 20 22 24

Males

Females

Figure 3.5: Left: Kaplan-Meier coronary artery bypass graft surgery 2-year survivor curves

for males (black) and females (red). Right: Two-year cumulative hazard curve for coronary

artery bypass graft surgeries.

In the proportional hazards model, women have a 1.6 times higher risk of mortality

than men during the first 2 years after the surgery. In the accelerated failure time model,

the expected failure rate for females age faster (along the survival scale) by 83% (1-0.169)

compared to males undergoing the same type of surgery. It is estimated that 5% of females

who undergo a CABG surgery will die in about 5 months (162 days), whereas it would

take about 2.6 years (960 days) for males to have an equivalent mortality rate. We note

that the standard error for this estimate is relatively high, resulting in a wide confidence

interval. In the accelerated hazards model, the risk or hazard progression for women un-

dergoing CABG surgery is about 40% slower than that of men undergoing the same type of

surgery during the first 2 years. This deceleration is interpreted as a harmful effect since the

hazard function is decreasing over time, as shown in Figure 3.6. A non-parametric boot-

strap approach was performed to obtain a reliable estimate of the variance of the gender


effect. Bootstrapped variance estimates are denoted with an asterisk (*). The bootstrapped

standard error for βAH is fairly close to the model-based standard error; so too are the cor-

responding 95% confidence interval estimates for eβAH . This close agreement between the

model-based variance and the bootstrapped variance is a reflection of the large sample size

used in this analysis.

Time Period Model β SE(β) P-value eβ 95% CI for eβ

2 years

PH 0.459 0.1672 0.006 1.583 (1.14, 2.20)

AFT -1.776 0.6881 0.0098 0.169 (0.04, 0.65)

AH -0.490 0.18 (0.184*) 0.007 (0.007*) 0.613 (0.43, 0.87) (0.43, 0.88)*

Table 3.2: Estimated gender effects on the coronary artery bypass graft data using PH,

AFT and AH models. Values with * in the AH model represent bootstrapped estimates.

The p-values compared to Wald tests of a hypothesis of no treatment effect.

Plots of the cumulative hazard and survivor curves during the 2-year period are shown

in Figure 3.7. Although the PH model estimates the baseline hazard and survivor curves

quite well, it fails to capture the wide gap in hazards between males and females before

month 8 (shown in Figure 3.6), which results in an overestimation of the survivor rate

for females up to month 8. However, in the succeeding months, the PH model seems to

fit the data well. The Weibull accelerated failure time model does not describe the data

well. Although the estimated gender effect, βAFT , looks reasonable, the estimated baseline

survivor function does not follow the same trajectory as the Kaplan-Meier curves, and is

undershooting the baseline cumulative hazard curve consistently. The estimated scale in the

location-scale framework for the Weibull model is 4.03 (SE=0.30), which implies that the

hazard function decreases with time. In the AH model, the fitted curves overestimate the

survivor rates for females. Visually, the PH model provides the best fit to the data, despite

the slight deviation from a proportional hazards assumption as shown in the residual plot,

Figure 3.8. The deviation from the proportional hazards assumption is detected by the slight

downward slope from time 0 to 240 days, and a positive slope onwards suggesting that the

sharp decline in survival rate early on violates the proportional assumption. However, a test

to see if the slope of the trend in this plot is 0 is non-significant with a p-value of 0.26.


0.00

000

0.00

005

0.00

010

0.00

015

0.00

020

0.00

025

0.00

030

Follow−up Time in months

Haz

ard

Rat

e

0 2 4 6 8 10 12 14 16 18 20 22 24

Smoothed Hazard Curve

Males

Females

Figure 3.6: Smoothed hazard curves for males (black solid) and females (red dashed) who

underwent a coronary artery bypass graft surgery.


0.000.020.040.060.080.10

Tim

e in

mon

ths

Cumulative Hazard

02

46

810

1214

1618

2022

24

Kap

lan−

Mei

erF

itted

PH

PH

Mod

el

Mal

es

Fem

ales

0.000.020.040.060.080.10

Tim

e in

mon

ths

Cumulative Hazard

02

46

810

1214

1618

2022

24

Kap

lan−

Mei

erF

itted

AF

T

AF

T M

odel

Mal

es

Fem

ales

0.000.020.040.060.080.10

Tim

e in

mon

ths

Cumulative Hazard

02

46

810

1214

1618

2022

24

Kap

lan−

Mei

erF

itted

AH

AH

Mod

el Mal

es

Fem

ales

0.900.920.940.960.981.00

Tim

e in

mon

ths

P(Survival)

02

46

810

1214

1618

2022

24

Kap

lan−

Mei

erF

itted

PH

Mal

es

Fem

ales

0.900.920.940.960.981.00

Tim

e in

mon

ths

P(Survival)

02

46

810

1214

1618

2022

24

Kap

lan−

Mei

erF

itted

AF

T

Mal

es

Fem

ales

0.900.920.940.960.981.00

Tim

e in

mon

ths

P(Survival)

02

46

810

1214

1618

2022

24

Kap

lan−

Mei

erF

itted

AH

Mal

es

Fem

ales

Figu

re3.

7:To

p:C

umul

ativ

eha

zard

curv

esfo

rthe

2-ye

arco

rona

ryar

tery

bypa

ssda

taus

ing

the

PH,A

FT,a

ndA

Hm

odel

s.

The

non-

para

met

ric

estim

ates

are

show

nin

dash

edlin

es,w

hile

the

estim

ated

cum

ulat

ive

haza

rdcu

rves

are

show

nin

solid

lines

for

mal

es(b

lack

)an

dfe

mal

es(r

ed).

Bot

tom

:K

apla

n-M

eier

2-ye

arsu

rviv

orcu

rves

for

mal

es(b

lack

dash

ed)

and

fem

ales

(red

dash

ed),

with

the

fitte

dsu

rviv

orcu

rves

show

nin

solid

lines

.



Time

Bet

a(t)

for

sex

0.92 4.6 9.4 24 67 240 490 680

−1

01

23

4

●●●●●

●●●

●

●

●●●

●●

●●●● ●

●

●

●●

●

●● ●●

●

●

●●●

●●

●

●

●● ●●

●

●●●●● ●●●

●●

●●●●● ●●

●

●

●

●●●

●●

●

●

●

●

●●●●●●

●●

●●

●

●

●

●●●●●●●●●●●●●

●●

●●

●

●●

●

●

●●

●●●●●●

●

●●●●●●

●

●●●●●●

●

●●●

●●●

●●●●●

●

●●●●●●

●●●

●●●●●●●●●

●

●●

●●●●

●●●●

Figure 3.8: Residual plot from the fit of the proportional hazards model to the coronary

artery bypass graft surgery data during the first two years post-surgery.


3.3 Veteran Lung Cancer

Kalbfleisch & Prentice (1980, Appendix I, p.223) present data taken from a randomized

clinical trial on the treatment of inoperable advanced lung cancer. A total of 137 male pa-

tients were randomized to either a standard chemotherapy or a new test chemotherapy for

the treatment of the disease. Several covariates were included in the randomization process

at the start of the study including, histology of the cancer cells, Karnofsky score, number

of months from diagnosis, age, and prior therapy. The trial had a high mortality rate with

about 95% of all patients dying within a period of a year and a half. Figure 3.9 shows

the survivor and cumulative hazard curves for the standard and test groups. The estimated

survivor curves for both groups cross at around month 6, with individuals on the standard

treatment performing better earlier in the study, but poorer after 6 months when compared

to the group receiving the test treatment, though these may not be significant differences.

0.0

0.2

0.4

0.6

0.8

1.0

Kaplan−Meier Plot

Follow−up Time in Months

P(S

urvi

val)

0 2 4 6 8 10 12 14 16 18 20

Standard

Test

01

23

4


Follow−up Time in Months

Haz

ard

0 2 4 6 8 10 12 14 16 18 20

Standard

Test

Figure 3.9: Left: Kaplan-Meier survivor curves for the standard (black) and test (red)

chemotherapy groups. Right: Cumulative hazard curves for the two groups.

Table 3.3 lists the parameter estimates of the treatment effect for the three models con-

sidered. All models report a relatively small treatment effect. Furthermore, none of the

models show a significant treatment effect, and there is no agreement in the direction of


this non-significant treatment effect. The estimated scale parameter from the Weibull AFT

model was 1.17 (SE=0.08), signifying that a simpler exponential model may fit the data.

However, a test if the scale parameter can be set to 1 produced a p-value of 0.02. The accel-

erated hazards model estimates the treatment effect as having about a 7% (non-significant)

slower risk progression than the standard therapy. Figure 3.10 shows the smoothed hazard

curves for the standard and test treatment groups. The hazard curve for the standard group

looks relatively flat, while the hazard curve for the test group decreases with time. Recall

that one of the assumptions of the accelerated hazards model is the non-constancy of the

hazard function. Even if the treatment were significant, it would be difficult for the AH

model to capture differences between the two groups.

Model β SE(β) P-value eβ 95% CI for eβ

PH 0.018 0.181 0.920 1.018 (0.71, 1.45)

AFT 0.048 0.208 0.082 1.049 (0.70, 1.58)

AH -0.069 0.105 (1.33*) 0.514 (0.959*) 0.934 (0.76, 1.15) (0.07, 12.75)*

Table 3.3: Estimated treatment effects on the veteran lung cancer data using PH, AFT, and

AH models. Values with * in the AH model represent bootstrapped estimates. The p-values

correspond to Wald tests of a hypothesis of no treatment effect.

Figure 3.11 displays the fitted cumulative hazard and survivor curves for the three mod-

els. The PH and AFT models are restricted to fitting survivor curves that do not cross.

Therefore, both models fail to capture the (perhaps non-significant) crossing at about 6

months displayed in the Kaplan-Meier estimates. Figure 3.12 shows the residual plot from

fitting a proportional hazards model to the data. The curvature in the residual plot suggests

a violation of the proportional hazards assumption. However, a test to see if the slope of

the trend in this plot is 0 is not rejected, with a p-value of 0.74; this may be due to the

non-significant difference between the two groups (and/or lack of sensitivity of the test to

capture quadratic effects).


0.00

00.

005

0.01

00.

015

0.02

0


Follow−up Time in Days

Haz

ard

Rat

e

0 20 40 60 80 100 120 140 160 180 200

Standard

Test

Figure 3.10: Smoothed hazard curves for the standard and test chemotherapy for the treat-

ment of lung cancer.


01234

Fol

low

−up

Tim

e in

Mon

ths

Cumulative Hazard

02

46

810

1214

1618

20

PH

Mod

el

Kap

lan−

Mei

erF

itted

PH

Sta

ndar

d

Tes

t

01234

Fol

low

−up

Tim

e in

Mon

ths

Cumulative Hazard

02

46

810

1214

1618

20

AF

T W

eibu

ll M

odel

Kap

lan−

Mei

erF

itted

AF

T

Sta

ndar

d

Tes

t

01234

Fol

low

−up

Tim

e in

Mon

ths

Cumulative Hazard

02

46

810

1214

1618

20

AH

Mod

el

Kap

lan−

Mei

erF

itted

AH

Sta

ndar

d

Tes

t

0.00.20.40.60.81.0

Fol

low

−up

Tim

e in

Mon

ths

P(Survival)

02

46

810

1214

1618

20

Kap

lan−

Mei

erF

itted

PH

Sta

ndar

d

Tes

t

0.00.20.40.60.81.0

Fol

low

−up

Tim

e in

Mon

ths

P(Survival)

02

46

810

1214

1618

20

Kap

lan−

Mei

erF

itted

AF

T

Sta

ndar

d

Tes

t

0.00.20.40.60.81.0

Fol

low

−up

Tim

e in

Mon

ths

P(Survival)

02

46

810

1214

1618

20

Kap

lan−

Mei

erF

itted

AH

Sta

ndar

d

Tes

t

Figu

re3.

11:

Top:

Cum

ulat

ive

haza

rdcu

rves

for

the

vete

ran

lung

canc

erda

taus

ing

the

PH,A

FT,a

ndA

Hm

odel

s.T

he

non-

para

met

ric

estim

ates

are

show

nin

dash

edlin

es,

whi

leth

ees

timat

edcu

mul

ativ

eha

zard

curv

esar

esh

own

inso

lid

lines

fors

tand

ard

trea

tmen

t(bl

ack)

and

test

trea

tmen

t(re

d).B

otto

m:K

apla

n-M

eier

surv

ivor

curv

esfo

rsta

ndar

dtr

eatm

ent

(bla

ckda

shed

)and

test

trea

tmen

t(re

dda

shed

),w

ithth

efit

ted

surv

ivor

curv

essh

own

inso

lidlin

es.



Time

Bet

a(t)

for

trt

8.2 19 32 54 99 130 220 390

−3

−2

−1

01

2

●●●

●●●

●●

●●

●●

●●●●●●

●●●

●●●

● ●●

●

●

●

●

●

●●●●●

●

●

●

●

●

●●

●

●

●

●●●●●

●

●●

●

●● ●

●●●●

●

●●

●●●

●

●●●

●●

●●●

●●●●

●●●

●●●●●●

●

●

●

●●●●●●●

●

●

●

●

●

●●

●

●

●●●

●

●

●

●●

●

●

●

●

●

●

●

●●●

Figure 3.12: Residual plot from the fit of the proportional hazards model to the veteran

lung cancer data.


3.4 Kidney Catheter Data

McGilchrist & Aisbett (1991) discuss a clinical trial to assess the time to infection, at the

catheter insertion site, of patients receiving kidney dialysis treatment. A total of 119 pa-

tients were grouped according to either a surgical or percutaneous insertion of the catheter.

Patients were followed until an infection was observed; individuals who needed removal of

the catheter for reasons other than an infection were treated as censored events. About 22%

(26) developed an infection over the 30-month course of the study. Figure 3.13 displays

the survivor and cumulative hazard curves for the two groups we consider here, namely

those undergoing surgical or percutaneous catheter placement. Notice that the estimated

survivor curves for the two groups cross early on, with the percutaneous group having a

higher infection rate (low survival) than the surgical group before 5 months but remaining

relatively flat after.

0 5 10 15 20 25

0.0

0.2

0.4

0.6

0.8

1.0

Survivor Curves

Time in months

P(in

fect

ion)

Surgically

Percutaneously

0 5 10 15 20 25

0.0

0.5

1.0

1.5

Cumulative Hazard Curves

Time in months

Haz

ard

Surgically

Percutaneously

Figure 3.13: Left: Kaplan-Meier survivor curves for surgical (black) and percutaneous

placement (red) of the catheter for patients undergoing kidney dialysis. Right: Cumulative

hazard curves for the two groups.


Table 3.4 shows estimates of the effects of catheters placed percutaneously over surgi-

cal placement for the three models. All models agree in the direction of the effect, with

the percutaneous catheter placement preferred over the surgical placement of catheters, al-

though none of the p-values suggest a significant difference. The PH model predicts the

risk for patients who had a percutaneous-placed catheter to be half that of developing an

infection against catheters placed surgically. Similarly, the AFT model estimates that the

percutaneously-placed catheter group have a 1.8 times decelerated (failure time) pace than

the surgically-placed catheter group. It is estimated that 50% of individuals in the surgical

group will develop an infection in 24 months (2 years), while it would take 45 months in the

percutaneous group for half of the subjects to get an infection. The AH model estimates a

deceleration in the time of risk progression in the percutaneous group by 80%. This is seen

as a favorable procedure because the estimated underlying baseline (surgically-treated) haz-

ard curve is shown to be increasing (see Figure 3.14). Using the model-based variance to

calculate the Wald test statistic for the AH model resulted in a significant p-value (0.0014);

however results using the bootstrapped estimate resulted in a non-significant effect. Sub-

stantial differences between model-based and resampling variance estimates are observed

in this study with a small sample size of 119.


PH -0.618 0.398 0.121 0.539 (0.25, 1.18)

AFT 0.623 0.468 0.184 1.865 (0.75, 4.67)

AH -1.564 0.4904 (1.1143*) 0.0014 (0.1605*) 0.209 (0.08, 0.55) (0.02, 1.86)*

Table 3.4: Estimated treatment effects on the percutaneous catheter placement using PH,

AFT, and AH models. Values with * in the AH model represent bootstrapped estimates.


Figure 3.15 shows the non-parametric and fitted survivor and cumulative hazard curves

based on the fitted PH, AFT Weibull, and AH models. Neither the PH, nor any parametric

AFT model can handle a cross in the survivor curves. It is therefore not surprising to see

that both PH and AFT models failed to capture this feature in the data, while the AH model

was able to offer a reasonably good fit to the data. The proportional hazards residual plot

in Figure 3.16 verifies the violation of the proportionality assumption in the early months


(p-value=0.003).

0 5 10 15

0.00

0.02

0.04

0.06

0.08

Follow−up Time in months

Haz

ard

Rat

eSmoothed Hazard Curves

Surgically

Percutaneously

Figure 3.14: Smoothed hazard curves for surgically and percutaneously-placed catheters

for patients undergoing kidney dialysis.


05

1015

2025

0.00.51.01.5

PH

Mod

el

Tim

e in

mon

ths

Cumulative Hazard

Kap

lan

Mei

erF

itted

PH

Sur

gica

lly

Per

cuta

neou

sly

05

1015

2025

0.00.51.01.5

AF

T W

eibu

ll M

odel

Tim

e in

mon

ths

Cumulative Hazard

Kap

lan

Mei

erF

itted

AF

T

Sur

gica

lly

Per

cuta

neou

sly

05

1015

2025

0.00.51.01.5

AH

Mod

el

Tim

e in

mon

ths

Cumulative Hazard)

Kap

lan

Mei

erF

itted

AH

Sur

gica

lly

Per

cuta

neou

sly

05

1015

2025

0.00.20.40.60.81.0

Tim

e in

mon

ths

P(Infection)

Kap

lan

Mei

erF

itted

PH

Sur

gica

lly

Per

cuta

neou

sly

05

1015

2025

0.00.20.40.60.81.0

Tim

e in

mon

ths

P(Infection)

Kap

lan

Mei

erF

itted

AF

T

Sur

gica

lly

Per

cuta

neou

sly

05

1015

2025

0.00.20.40.60.81.0

Tim

e in

mon

ths

P(Infection)

Kap

lan

Mei

erF

itted

AH

Sur

gica

lly

Per

cuta

neou

sly

Figu

re3.

15:

Top:

Cum

ulat

ive

haza

rdcu

rves

for

the

kidn

eyca

thet

erpl

acem

entd

ata

usin

gth

ePH

,AFT

,and

AH

mod

els.

The

non-

para

met

ric

estim

ates

are

show

nin

dash

edlin

es,w

hile

the

estim

ated

cum

ulat

ive

haza

rdcu

rves

are

show

nin

solid

lines

for

the

surg

ical

lypl

acem

entg

roup

(bla

ck)

and

the

perc

utan

eous

lypl

acem

entg

roup

(red

).B

otto

m:

Kap

lan-

Mei

er

surv

ivor

curv

esfo

rth

esu

rgic

ally

plac

emen

tgro

up(b

lack

dash

ed)

and

the

perc

utan

eous

lypl

acem

entg

roup

(red

dash

ed),

with

the

fitte

dsu

rviv

orcu

rves

show

nin

solid

lines

.



Time

Bet

a(t)

for

V3

1 3.6 6.5 10 16 17 22 25

−6

−4

−2

02

●●●●●●

●

●●

●

●

●● ●

●

●● ● ● ● ●

●

● ● ●●

Figure 3.16: Residual plot from the fit of the proportional hazards model to the kidney

catheter placement data.


3.5 Brain Tumor Data

Brem et al (1995) conducted a randomized, prospective clinical trial on the implantation of

carmustine (also known as bis-chloronitrosourea or BCNU) polymer discs in patients with

recurrent malignant brain gliomas. The carmustine in the polymer disc is a chemother-

apeutic drug for the treatment of brain tumours. The drug in the disc is slowly released

into the brain over a 2 to 3 week period from the day of surgery or implantation. The trial

consisted of 222 patients who were randomly assigned with equal probability to either a

placebo group (112), or the BCNU group (110), the group receiving the carmustine poly-

mer discs, with the placebo group receiving empty polymer implants. The mortality rate

for this trial was high, with about 93% of the patients dying within the 4-year time period.

However, in this study, we will only consider the first 52 weeks of the trial, as this is where

the marked differences between the placebo and BCNU groups occur. At week 52, about

80% (176 out of 222) of the patients had died, corresponding to a censoring percentage

of 20%. Displayed in 3.17 are the Kaplan-Meier survivor curves and the corresponding

cumulative hazard curves for the two groups. The placebo group seems to perform worse

than the BCNU group after about week 12.

0.0

0.2

0.4

0.6

0.8

1.0

Follow−up Time in Weeks

P(S

urvi

val)

0 4 8 12 16 20 24 28 32 36 40 44 48 52

Kaplan−Meier Survival Curves

Placebo

BCNU

0.0

0.5

1.0

1.5


Haz

ard

0 4 8 12 16 20 24 28 32 36 40 44 48 52

Cumulative Hazard Curves

Placebo

BCNU

Figure 3.17: Left: Kaplan-Meier survivor curves for the placebo (black) and BCNU poly-

mer (red) groups. Right: Cumulative hazard curves for the two groups.


Table 3.5 provides the estimated treatment effect using the three models under study.

Although all p-values under a Wald test report a non-significant treatment effect, all models

agree on the direction of the treatment estimate, with both PH and AH models having β < 0

and the AFT model having a β > 0, implying a better prognosis on patients who received

the BCNU polymer disc. The hazard risk ratio of 0.83 from the PH model means that the

BCNU group has a 13% less risk than the placebo group. The AFT estimate of 1.25 implies

that the BCNU group age slower (along the survival scale) by 25% when compared to the

individuals in the placebo group. It estimates that 50% of individuals in the placebo group

will die in about 32 weeks, while it would take about 40 weeks in individuals receiving

BCNU to have the same mortality rate. Similarly, the AH estimate of 0.54 suggests that the

BCNU group decelerates the hazard progression of the placebo group by 46%. This is seen

as a beneficial effect since the hazard function increases after the initial decline in weeks 1

to 4.


PH -0.191 0.151 0.207 0.827 (0.62, 1.11)

AFT 0.223 0.164 0.170 1.249 (0.91, 1.72)

AH -0.609 0.1004 (1.0985*) 0.0545 (0.5791*) 0.544 (0.45, 0.66) (0.06, 4.68)*

Table 3.5: Estimated treatment effects on the carmustine (BCNU) polymer disc data using

PH, AFT, and AH models. Values with * in the AH model represent bootstrapped estimates.


Figure 3.19 displays the non-parametric cumulative hazard (top) and the survivor curves

(bottom), overlaid on the fitted curves from the three models. The PH model does not cap-

ture the gap between the two curves from weeks 20 to 32 very well. In fact, the residual plot

on Figure 3.20 shows a deviation from the proportionality assumption. The AFT Weibull

model fails to capture the shape of the survivor curves, with the estimated Weibull shape

parameter being 0.92 (SE=0.05). Interestingly, the AH model gives a fairly good fit to the

data except from weeks 36 onwards, where the model displays a crossing of the survivor

curves when it should not.


0.00

0.01

0.02

0.03

0.04

0.05

0.06


Haz

ard

Rat

e

0 4 8 12 16 20 24 28 32 36 40 44 48 52


Placebo

BCNU

Figure 3.18: Smoothed hazard curves for the placebo (black solid) and BCNU polymer

(red dashed) for the treatment of brain tumor.


0.00.51.01.5

Fol

low

−up

Tim

e

Hazard

04

812

1620

2428

3236

4044

4852

PH

Mod

el

Kap

lan

Mei

erF

itted

PH

Pla

cebo

BC

NU

0.00.51.01.5

Fol

low

−up

Tim

e

Hazard

04

812

1620

2428

3236

4044

4852

Kap

lan

Mei

erF

itted

AF

T

AF

T M

odel P

lace

bo

BC

NU

0.00.51.01.5

Fol

low

−up

Tim

e

Hazard

04

812

1620

2428

3236

4044

4852

Kap

lan

Mei

erF

itted

AH

AH

Mod

el

Pla

cebo

BC

NU

0.00.20.40.60.81.0

Fol

low

−up

Tim

e

P(Survival)

04

812

1620

2428

3236

4044

4852

PH

Mod

el

Kap

lan

Mei

erF

itted

PH

Pla

cebo

BC

NU

0.00.20.40.60.81.0

Fol

low

−up

Tim

e

P(Survival)

04

812

1620

2428

3236

4044

4852

Kap

lan

Mei

erF

itted

AF

T

Pla

cebo

BC

NU

0.00.20.40.60.81.0

Fol

low

−up

Tim

e

P(Survival)

04

812

1620

2428

3236

4044

4852

Kap

lan

Mei

erF

itted

AH

Pla

cebo

BC

NU

Figu

re3.

19:

Top:

Cum

ulat

ive

haza

rdcu

rves

for

the

BC

NU

poly

mer

disc

data

usin

gth

ePH

,AFT

,and

AH

mod

els.

The

non-

para

met

ric

estim

ates

are

show

nin

dash

edlin

es,w

hile

the

estim

ated

cum

ulat

ive

haza

rdcu

rves

are

show

nin

solid

lines

fort

hepl

aceb

ogr

oup

(bla

ck)a

ndth

eB

CN

Upo

lym

ergr

oup

(red

).B

otto

m:K

apla

n-M

eier

surv

ivor

curv

esfo

rthe

plac

ebo

grou

p(b

lack

dash

ed)a

ndth

eB

CN

Upo

lym

ergr

oup

(red

dash

ed),

with

the

fitte

dsu

rviv

orcu

rves

show

nin

solid

lines

.



Time

Bet

a(t)

for

trea

t

6.4 12 15 20 25 30 37 44

−2

−1

01

2

●●

●

●

●

●

●●●

●

●

●●

●

●

●●

●●

●

●●

●

●●●

●●

●●

●

●

●●

●●

●

●● ●●●

●

●

●

●●●● ●

●●●

●●

●●

●●

● ●●●

●●

●

●●●

●●

●●

●●●

●●●

●

●●

●

●●●●●

●●

●

●

●●

●

●●

●

●

●●●

●●

●●

● ●●●

●●

●●●

●

●

●

●●

●●●●●

●●

●

●

●

●●

●● ●●●●

●

●

●●

●●●●

●

●

●

●

●

●●●

●●

●●●

●

●

●

●

●

●● ●●

●

●

●●

●

●

●●

Figure 3.20: Residual plot from the fit of the proportional hazards model to the brain tumor

data.


3.6 Summary AH Parameter Interpretation

Parameter interpretation for the AH model is more complicated than the PH or AFT mod-

els because it relies on a careful assessment of the underlying hazard curve. In practice,

the hazard function can be estimated by the use of kernel smoothing methods, which can

give slightly different shapes depending on the bandwidth chosen. The sign of the estimate

cannot in itself determine a favourable or harmful treatment outcome. Instead, it needs to

be interpreted with the understanding of the shape of the underlying baseline hazard func-

tion. In cases where the baseline hazard function is non-monotone, interpretation may be

difficult.

Table 3.6 shows a summary of features of the five datasets. All of the datasets presented

had crossing hazards, except for the CABG data, which make the AH model a plausible

choice in data fitting. In general, when the proportionality assumption is not grossly vio-

lated, the proportional hazards model gives reasonable estimates, and is the preferred model

choice. The AH model is superior only in the case when there is crossing in both hazard and

survivor functions, and when the hazard function is not flat (eg. kidney catheter data). In

this particular case, the AH model was able to capture the crossing of the survivor curves,

which neither the PH nor the AFT model can accommodate. In the brain tumor data, both

the placebo and BCNU groups have similar hazards at time 0. The AH model gave a good

fit to the BCNU group but the fit to the placebo group had problems past week 28.

Dataset Hazard same at t=0? Hazards Cross? Survivor Curves Cross? Best Model

Breast Cancer No Yes (tail) No PH

CABG No No No PH

Veteran Lung Cancer No Yes Yes None

Kidney Catheter No Yes Yes AH

Brain Tumor Yes Yes No AH (treatment)

Table 3.6: Summary of features of the five datasets considered in the chapter, and the best

model fitted in each dataset. The column headers reflect features seen from the estimated

Kaplan-Meier curves, without regard for significance of these effects.

Chapter 4

Exploring the fit of the PH and AHsurvivor curves when the model hascrossing hazards

We investigate the performance of the proportional hazards and the accelerated hazards

models for a two-sample scenario when the true underlying hazard and survivor curves

cross. Our interest is mainly to see how well the accelerated hazards model can fit data

with crossing curves, but which are not of the AH family. Although we know that the pro-

portional hazards model cannot handle data with crossing hazards and/or survivor curves,

we use it here as a reference since the PH model is widely used in practice, at times even

when violations in the proportionality assumption is observed. This limited investigation

will then also provide some insight on the behaviour of PH regression when the PH model

is false.

We induce a non-AH model with crossing survivor curves through the use of a log-

logistic distribution with density function, f (t) = as (

ts)

a−1/(1 + ( ts)

a)2, hazard function,

h(t) = as (

ts)

a−1/(1+( ts)

a), and corresponding survivor function, S(t) = 1/(1+( ts)

a). Here,

a is the shape parameter, while s is the scale parameter of the distribution. In a two-sample

(treatment/control) situation, survivor curves for the two groups cross if the treatment ef-

fect (denoted as β) is incorporated in the shape parameter of the loglogistic distribution.

Therefore, the shape and scale parameters for the control group are a and s, while the

61

CHAPTER 4. EXPLORING THE FIT OF THE PH AND AH SURVIVOR CURVES 62

corresponding parameters for the treatment group are aeβ and s, respectively. The hazard

function of the loglogistic distribution is monotonically decreasing when the shape param-

eter is ≤ 1, and increases to a peak then decreases when the shape parameter is > 1 (see

Figure 4.1). The scale parameter, s, modulates the narrowing or widening of the hazard

curve on the time scale.

0 2 4 6 8 10

0.2

0.4

0.6

0.8

1.0

Time

Haz

ard

Loglogistic Hazard Curves

a=1

a=0.5

a=1.5

Figure 4.1: Hazard curves for the loglogistic distribution with shape parameters, a=0.5, 1,

and 1.5 and scale parameter, s=1.

We conducted a simulation study with a sample size of 100, with half of the observa-

tions in each of the control and treatment groups, using two different scenarios - a case


where both control and treatment groups do not have equal hazards at time 0, and a case

where they do. The fits of the models are compared by computing the mean square error

of the mean estimated survivor curves from 1000 runs. In both cases, to make a fair com-

parison of the treatment curves, the fitted curve for the PH model is truncated to match that

of the AH model. Recall that the AH model scales the time, and therefore, the timeline

for the predicted treatment survivor curve is a scaled function of that of the control group’s

survivor curve.

4.1 Case I

We explore the performance of the AH model when the control and treatment groups do

not have equivalent risk at the start of the study. A loglogistic distribution having a shape

parameter of 1.5 and a scale parameter of 4 was chosen for the control group and a shape

of 1.5eβ for the treatment group, with β = −1. The hazard and survivor curves for both

groups are displayed in Figure 4.2, with the control group denoted as the baseline. We note

that a crossing occurs at around t=4, which is about the median (50th) percentile for both

groups. This implies that half of the subjects in each group have failed by time=4. This

particular choice of parameters also induces a sharper decline in the survival curve for the

treatment group early in the study, but the rate of failure tapers off as time progresses.

Figure 4.3 shows the predicted baseline and treatment survivor curves of the two mod-

els, with the mean fitted curves from the PH model displayed in the panels on the left, and

the mean fitted curves from the AH model displayed in the panels on the right, for three

effect sizes, β = −0.5,−1,−1.5. Both models failed to capture the crossing of the sur-

vivor curves even when the effect size is large at -1.5. Alternatively, Figure 4.4 displays a

comparison of the mean fitted PH and AH baseline and treatment curves against the true

loglogistic survivor curves. When the effect size is small (-0.5), both the PH and AH mod-

els have comparable mean squared errors (MSE). Mean squared error is defined here as

the average of the sum of the squared difference between the true survivor curve and the

estimated mean survivor curve at some grid of values (ie. ∑ni=1(true−estimate)2

m−1 , where m is the

number of grid points). For a moderate effect size (-1), both models give a similar fit, by


0 5 10 15 20

0.00

0.05

0.10

0.15

0.20

0.25

0.30

Hazard Curves

Time

Ris

k

Baseline

Treatment

0 5 10 15 20

0.0

0.2

0.4

0.6

0.8

1.0

Survivor curves

TimeP

(Sur

viva

l)

Baseline

Treatment

Figure 4.2: Hazard (left) and survivor (right) curves for a loglogistic distribution with

shape=1.5 and scale=4 for a treatment effect size of -1.

underestimating the survival rate in the earlier times, and overestimating the survival rates

at later times. This pattern is consistent when the effect size is larger, at -1.5. The MSEs of

the mean fitted curves for the treatment group based on the AH model are smaller than the

corresponding values from the fit of the PH model.

The boxplots in Figure 4.5 show the distribution of the MSEs of the estimates of the

survivor function for the control and treatment groups, from each of the 1000 runs based

on both AH and PH models. The distribution of these MSEs based on the AH model has

a consistently heavier tail than that based on the PH model in all scenarios but the median

MSE values based on the AH model are generally similar or smaller, compared to those

based on the PH model for the estimated survivor function of the treatment group, and

larger for the estimated survivor function of the control group.


0 10 20 30 40

0.0

0.2

0.4

0.6

0.8

1.0

Time

P(S

urvi

val)

BaselineTreatmentFitted PH BaselineFitted PH Treatment

ββ == −− 0.5

PH

0 10 20 30 40

0.0

0.2

0.4

0.6

0.8

1.0

Time

P(S

urvi

val)

BaselineTreatmentFitted AH BaselineFitted AH Treatment

ββ == −− 0.5

AH

0 10 20 30 40

0.0

0.2

0.4

0.6

0.8

1.0

Time

P(S

urvi

val)


ββ == −− 1

0 10 20 30 40

0.0

0.2

0.4

0.6

0.8

1.0

Time

P(S

urvi

val)


ββ == −− 1

0 10 20 30 40

0.0

0.2

0.4

0.6

0.8

1.0

Time

P(S

urvi

val)


ββ == −− 1.5

0 10 20 30 40

0.0

0.2

0.4

0.6

0.8

1.0

Time

P(S

urvi

val)


ββ == −− 1.5

Figure 4.3: Comparison of the PH and AH fits for effect sizes of -0.5 (top), -1 (middle),

and -1.5 (bottom) when the hazards do not start at the same point at t=0. The dashed curves

display true survivor functions corresponding to the baseline and treatment groups.


0 10 20 30 40

0.0

0.2

0.4

0.6

0.8

1.0

Time

P(S

urvi

val)

BaselineTreatmentFitted PH (MSE=8.0 E−4)Fitted AH (MSE=5.8 E−4)

ββ == −− 1 2

0 10 20 30 40

0.0

0.2

0.4

0.6

0.8

1.0

Time

P(S

urvi

val)


ββ == −− 1 2

0 10 20 30 40

0.0

0.2

0.4

0.6

0.8

1.0

Time

P(S

urvi

val)


ββ == −− 1

0 10 20 30 40

0.0

0.2

0.4

0.6

0.8

1.0

Time

P(S

urvi

val)


ββ == −− 1

0 10 20 30 40

0.0

0.2

0.4

0.6

0.8

1.0

Time

P(S

urvi

val)


ββ == −− 1.5

0 10 20 30 40

0.0

0.2

0.4

0.6

0.8

1.0

Time

P(S

urvi

val)


ββ == −− 1.5

Figure 4.4: Comparison of the fitted PH and AH curves with the true survivor curves

(shown with dashed lines) for three effect sizes (-0.5, -1, -1.5), with the fitted baseline

curves on the left panels, and the fitted treatment curves on the right panels.


●●● ●● ● ● ●●● ● ●●●● ●●●● ● ●● ●● ●● ●● ● ●●● ●● ● ● ● ●● ●●●● ● ● ● ● ● ● ●●●● ●● ● ●

●●●● ● ●●●● ●●● ● ●●● ●● ●●● ● ● ●●●● ● ●● ●● ●●● ●● ●● ● ● ●●●● ● ●●● ●● ●● ●● ●

0.0000.0050.0100.0150.0200.0250.030

ββ==

−−1

2

MSE

AH

PH

●● ● ●● ●●● ● ●●● ●● ●●● ●● ● ● ● ●● ●● ●

●●● ●● ● ●●● ● ●● ● ●● ● ●● ●●●● ● ●● ● ●● ●●● ● ●●●

0.0000.0050.0100.0150.0200.0250.030

ββ==

−−1

MSE

AH

PH

Bas

elin

e cu

rve

MS

E

● ● ● ●● ● ●● ●●● ● ●● ● ●● ●● ●● ●● ●●

●●●● ● ●● ●● ● ●●●●● ●●● ● ●● ● ● ●● ●●● ●● ●● ●

0.0000.0050.0100.0150.0200.0250.030

ββ==

−−1.

5

MSE

AH

PH

●●●● ●● ● ● ● ●● ● ●●● ●● ●● ● ●● ● ●●● ● ●● ●●● ●●● ● ● ●● ●● ●●● ●● ● ●● ●●● ●● ● ●● ● ●● ● ●● ●●●● ● ●

● ●●● ● ●●●● ●● ●● ●● ● ● ●●● ●● ● ● ●● ●● ●●●● ● ●● ●● ●● ●● ● ●●● ●● ● ● ●● ● ●● ●● ●●● ● ●● ●● ●● ●● ●

0.000.010.020.030.040.050.06

ββ==

−−1

2

MSE

AH

PH

●● ●●● ●●● ● ●● ●● ●●●● ●● ●● ● ●●● ● ●● ● ●● ●● ●●● ●● ●● ●● ●●● ●●● ● ●● ● ●● ●●● ● ●● ●● ●●● ●● ● ● ●●●● ●●

●● ●●● ●● ● ● ●● ●●● ●●● ●● ●●●● ●●●● ● ● ●● ● ● ●●●● ●●● ● ●● ● ●● ● ●●● ●●● ● ●● ● ●● ●● ● ●● ● ●●●● ●●

0.000.010.020.030.040.050.06

ββ==

−−1

MSE

AH

PH

Tre

atm

ent c

urve

MS

E

● ● ●● ●● ●● ●● ● ●● ●● ●●●●● ●●● ●● ●● ●● ● ●● ● ● ●●● ● ●●● ● ● ●● ● ● ●●● ● ●● ●●● ● ● ●● ● ●● ● ●●●●● ●● ●●● ● ● ●● ●●●● ● ●●● ●●● ●●● ● ●●● ●●●●●

● ● ●●● ●● ●●●● ●●●●● ●● ● ●● ●●● ●● ●●●● ●● ●● ●● ●●● ● ●●● ●● ●●●● ●●● ●● ● ● ● ●●● ● ●●● ● ●● ● ●● ● ● ●● ●● ●●

0.000.010.020.030.040.050.06

ββ==

−−1.

5

MSE

AH

PH

Figu

re4.

5:B

oxpl

ots

ofm

ean

squa

red

erro

rsfo

rthe

AH

and

PHm

odel

sfo

rβ=

-0.5

,-1,

-1.5

inC

ase

I,fo

rasa

mpl

esi

ze

of10

0,fo

r100

0si

mul

atio

nru

ns.


4.2 Case II

This section explores the performance of the AH model when the control and treatment

groups have equivalent risk at the beginning of the study. A loglogistic distribution having

a shape parameter of 4 and a scale parameter of 50 has hazard and survivor curves that cross

when the treatment curve is taken from the same distribution but with a shape parameter

of 4eβ, β 6= 0. Figure 4.6 displays the hazard and survivor curves for the loglogistic distri-

bution with β =−1. We restrict our attention to the time interval (0,100] by incorporating

fixed censoring at t=100. This censoring scheme corresponds to about a 20% censoring

percentage. This allows both an incorporation of censoring in this study, as well as a fo-

cus on the interesting feature of this scenario, namely the time of crossing of the survivor

curves.

0 50 100 150 200 250 300

0.00

0.01

0.02

0.03

0.04

0.05

Hazard Curves

Time

Ris

k

Baseline

Treatment

0 50 100 150 200 250 300

0.0

0.2

0.4

0.6

0.8

1.0

Survivor curves

Time

P(S

urvi

val)

Baseline

Treatment

Figure 4.6: Hazard (left) and survivor (right) curves for a loglogistic distribution with shape

parameters, a=4 and scale parameter, s=50 for a treatment effect size of β=-1.

The mean fitted baseline and treatment curves based on both the PH (left panels) and

AH (right panels) models are shown in Figure 4.7. Although the AH model was able to

capture the crossing of the survivor curves, the predicted curves do not provide an adequate

fit to the data. Both models had trouble fitting the earlier part of the survivor curves but


0 20 40 60 80 100

0.0

0.2

0.4

0.6

0.8

1.0

Time

P(S

urvi

val)

PH Model


ββ == −− 0.5

0 20 40 60 80 100

0.0

0.2

0.4

0.6

0.8

1.0

Time

P(S

urvi

val)

AH Model


ββ == −− 0.5

0 20 40 60 80 100

0.0

0.2

0.4

0.6

0.8

1.0

Time

P(S

urvi

val)


ββ == −− 1

0 20 40 60 80 100

0.0

0.2

0.4

0.6

0.8

1.0

Time

P(S

urvi

val)


ββ == −− 1

0 20 40 60 80 100

0.0

0.2

0.4

0.6

0.8

1.0

Time

P(S

urvi

val)

PH Model


ββ == −− 1.5

0 20 40 60 80 100

0.0

0.2

0.4

0.6

0.8

1.0

Time

P(S

urvi

val)

AH Model


ββ == −− 1.5

Figure 4.7: Comparison of the PH and AH fits for effect sizes of -0.5 (top), -1 (middle), and

-1.5 (bottom) when the hazards start at the same point at t=0. The dashed curves display

true survivor functions corresponding to the baseline and treatment groups.


gave a reasonable fit in the latter part of the curve. Alternatatively, Figure 4.8 gives a com-

parison of the mean fitted PH and AH baseline and treatment survivor curves against the

true loglogistic survivor curves. It is interesting to note that the AH model fits the baseline

curve reasonably well, having lower MSEs than the PH model. However, the opposite is

observed, with the PH model having lower MSEs than the AH model, in estimating the

treatment survivor curve.

The boxplots on Figure 4.9 show the distribution of the MSEs of the predicted base-

line and treatment survivor curves for the 1000 simulation runs based on both AH and PH

models. In all scenarios, the distribution of MSEs for the fits based on the AH model gives

a consistently heavier tail than that from the PH model. The median MSE values from the

fitted baseline curves based from the AH model are generally similar or smaller compared

to those from the fitted PH model but are bigger when looking at the estimated survivor

function of the treatment group. This pattern is opposite to what was observed in Case I.


0 20 40 60 80 100

0.0

0.2

0.4

0.6

0.8

1.0

Time

P(S

urvi

val)

ββ == −− 1 2BaselineTreatmentFitted PH (MSE=1.3 E−3)Fitted AH (MSE=1.4 E−3)

0 20 40 60 80 100

0.0

0.2

0.4

0.6

0.8

1.0

Time

P(S

urvi

val)

ββ == −− 1 2BaselineTreatmentFitted PH (MSE=1.8 E−3)Fitted AH (MSE=3.9 E−3)

0 20 40 60 80 100

0.0

0.2

0.4

0.6

0.8

1.0

Time

P(S

urvi

val)

ββ == −− 1


0 20 40 60 80 100

0.0

0.2

0.4

0.6

0.8

1.0

Time

P(S

urvi

val)

ββ == −− 1


0 20 40 60 80 100

0.0

0.2

0.4

0.6

0.8

1.0

Time

P(S

urvi

val)

ββ == −− 1.5


0 20 40 60 80 100

0.0

0.2

0.4

0.6

0.8

1.0

Time

P(S

urvi

val) ββ == −− 1.5


Figure 4.8: Comparison of the fitted PH and AH curves with the true survivor curves

(shown with dashed lines) for three effect sizes (-0.5, -1, -1.5), with the fitted baseline

curves on the left panels, and the fitted treatment curves on the right panels. Fixed censoring

was done at t=100.


●● ● ●● ●●● ● ● ●●●● ●●● ●● ●●●● ● ●● ●● ●● ●● ●●●● ●● ●● ●●● ● ● ●●● ●●●● ● ●● ●● ● ●●● ●●● ●●●● ● ● ●● ● ●● ●● ●●

0.000.010.020.030.04

ββ==

−−1

2

MSE

AH

PH

●● ● ●●●●● ●● ● ●● ●●

● ● ●● ●● ●● ●● ●● ●● ● ●● ●● ●● ●●●● ●● ●●

0.000.010.020.030.04

ββ==

−−1

MSE

AH

PH

Bas

elin

e cu

rve

MS

E

●●● ●● ●● ● ● ●● ● ●● ●● ●● ● ● ●●● ●●● ● ●●● ●● ● ●●● ●●● ●●

● ● ●● ● ●● ● ●

0.000.010.020.030.04

ββ==

−−1.

5

MSE

AH

PH

● ● ●● ●● ●● ●● ●●● ●●●● ● ●● ● ● ●● ● ● ●●● ● ● ●● ●● ●●● ●● ●● ●●●● ● ●●●● ●● ●●●● ●

●●● ●●● ● ●● ● ●● ● ● ●●●● ● ●● ● ●●● ●● ●● ● ●● ●●● ● ●●● ● ● ●● ●● ● ● ●● ●● ● ● ●● ● ●● ●●● ●●● ●● ●● ●

0.000.010.020.030.04

ββ==

−−1

2

MSE

AH

PH

●●●●●● ● ● ●● ●●● ● ●●

●●● ●● ● ● ●● ●● ●●● ●●● ●● ●●● ● ●● ●● ●●● ●●● ●●● ●● ● ●● ●● ●● ●● ● ●● ●● ●

0.000.010.020.030.04

ββ==

−−1

MSE

AH

PH

Tre

atm

ent c

urve

MS

E●● ● ●● ●●● ●● ● ●● ● ●● ●

● ●●● ● ●● ● ●● ●● ● ●● ●●●● ● ● ●●● ● ● ●●● ●●● ● ●

0.000.010.020.030.04

ββ==

−−1.

5

MSE

AH

PH

Figu

re4.

9:B

oxpl

ots

ofm

ean

squa

red

erro

rsfo

rthe

AH

and

PHm

odel

sfo

rβ=

-0.5

,-1,

-1.5

inC

ase

II,f

ora

sam

ple

size

of10

0,fo

r100

0si

mul

atio

nru

ns.


4.3 Conclusion

Both cases investigated in this chapter allowed for a crossing between the baseline and

treatment survivor curves. In Case I, we showed a scenario where the differences between

the survivor rates for the baseline and treatment groups before the crossing are relatively

small, and may not be clinically significant. In this case, both the PH and AH models

captured the main feature of the data - the wide gap between the baseline and treatment

survivor curves after the cross-over. In Case II, the gaps between the baseline and treat-

ment survivor functions before and after the cross-over are both substantially large. The PH

model essentially fits the data by smoothing out the before and after effect, and may lead to

no evidence of a difference between groups. The AH model acknowledges the cross-over

effect but does not estimate it well.

In our exploration, we found that the AH model had difficulties fitting crossing survivor

curves which are not from the AH family. Even when the true survivor curves cross, the

fitted AH model may not. The PH model tends to form a middle ground between the

baseline and treatment groups, under(over)-estimating the higher (lower) survivor curve

before and after the cross-over. We comment that this suggests an important need for

development and use of powerful tests of PH forms before adopting a PH model.

Chapter 5

Discussion

This project investigated performance of the Accelerated Hazards model relative to the

more commonly-used Proportional Hazards and Accelerated Failure Time models. Unlike

the PH Model, the Accelerated Hazards model allows for cross-overs of the survivor func-

tions of the cohorts under study. Since the AH model assumes that hazards for different

cohorts are time-scaled versions of the same function, it is useful to first empirically exam-

ine the shape of the hazard curves in a k-sample problem before utilization of this model

to assess this assumption. The simulation studies performed in Chapter 2 proposed the use

of the non-parametric bootstrap as an alternative method for estimating the variance for the

AH model in small sample sizes, as the performance of the Wald statistic was quite poor.

Analyses on five different datasets were performed to explore how the fit of the AH

model compared to the PH and AFT models. In the case when the shapes of the hazard

curves show the same pattern but have different time scales, as seen in the brain tumor

example, and when the hazards of the groups at time 0 are similar, this model may provide

a useful alternative to the other, more frequently used, models. However, when the shapes

of the estimated hazard functions differ substantially such that one group’s curve cannot

be empirically described as a scaled version of the other (eg. lung cancer example), and

when the estimated hazards at time 0 vary greatly between groups, as seen in the breast

cancer study, then difficulties may arise when attempting to fit the AH model. Note also

that interpretation of parameters of the AH model may be difficult or not particularly in-

formative when the hazard function is non-monotone, especially when either of the hazard

74

CHAPTER 5. DISCUSSION 75

or survivor curves cross. For example, in the case where the hazard curve decreases at the

beginning of the study and then increases later, the treatment group may be preferable (hav-

ing lower risk) over the control group in the beginning but has higher risk in the latter part

of the study. In such cases, the researcher’s interest may lie on estimating the time when

the hazard curves reach their peaks or troughs rather than quantifying how much faster the

hazard of one group is accelerated over the other.

In Chapter 4, simulation studies were conducted to quantify the goodness of fit of the

AH model both when the hazards of the control and treatment groups did not have similar

values at time 0 (Case I), and when they did (Case II). In Case I, the shapes of the underly-

ing hazard curves were very different from each other. Although both PH and AH models

failed to capture the crossing in the beginning, the AH model had smaller MSE than the PH

model. In Case II, the shapes of the true hazard functions were similar but one group had

a considerably higher peak than the other, making it difficult for the AH model to adapt,

although the crossing of the survivor curves was captured slightly when the treatment effect

was large.

Chen & Jewell (2001) proposed a general class of model that captures the proportional

hazards, accelerated failure time and accelerated hazards models. For the two-sample prob-

lem, for example, the model includes two parameters - one parameter (β1) quantifies the

time scale change of the hazard function for the treatment group, relative to the control,

and the other parameter (β2) measures the proportionality of the hazard curves of the two

groups. The model is written as h1(t) = h0(teβ1)eβ2 , where h1 is the hazard function for

the treatment group and h0 is the hazard function for the control group. Though fitting this

model is not straightforward, it would be interesting to explore if a scenario as described

in Case II of Chapter 4 can be estimated well by the above-mentioned more general model.

Another idea might be to permit individual-specific flexibility in the AH model through

frailty terms which operate on the time scale for the hazard function, in a similar manner

as the covariate effects. As well, though semiparametric methods are quite popular, splines

are becoming increasingly used for modeling baseline intensity functions; in the AH model,

using a simple cubic spline, say, with one or two linear knots might be useful.

CHAPTER 5. DISCUSSION 76

As an attempt to describe the goodness of fit of the PH and AH models, we examined the

mean squared error of the mean estimate of the survivor curves. Formal goodness of fit tests

were not explored in this project but literature by Chen (2001) has explored tests for model

adequacy such as the Gill-Schumacher and Kolmogorov-Smirnov test. We suggest that the

topic of goodness of fit for these models, particularly with regard to simple diagnostics

and plots for assessing departures from model assumptions, deserves greater study. The

incorporation of weight functions in the estimating equation presented in Chapter 2 for

the AH model might also be given further consideration. We conclude by noting that the

PH model, however, tends to fit data reasonably well provided that the proportionality

assumption is not grossly violated.

Appendix A

Appendix

Chen & Wang (2000) showed that the solution to (2.23), βAH , yields asymptotically normal

estimates with variance in the form of,

Σ =β2

AH∫ T0

0 g(t,βAH)h0(t)dt

{∫ T0

0 g(t,βAH)h′0(t)tdt}2,

g(t,βAH) = [1−π(t,βAH)]π(t,βAH){ρS∗0(t)+(1−ρ)S∗1(t/βAH)/βAH},

π(t,βAH) =(1−ρ)S∗1(t/βAH)/βAH

ρS∗0(t)+(1−ρ)S∗1(t/βAH)/βAH,

ρ = limn→∞

n0

n0 +n1> 0.

The estimation procedure for Σ as outlined in the Appendix of Chen & Wang (2000)

requires an estimate of the first and second derivatives of the unknown baseline hazard

function, hAH0(t). In practice, this may be difficult to implement, although several strategies

have been put forward by Tsiatis (1990) and Lin et al (1998). In a succeeding paper by Chen

& Jewell (2001), the authors suggested a variance estimation method that relies on large

sample size approximation and does not need an estimate of the baseline hazard function.

The method was adapted from a technical report by Huang and was subsequently published

(Huang, 2002). The method can be performed as follows:

77

APPENDIX A. APPENDIX 78

1. Find a solution for βAH by solving U(βAH−) U(βAH+) ≤ 0 (zero crossings of

U(βAH), where U(βAH) is defined as in (2.23).

2. Decompose the estimate of the variance Σ into σσ′, where

Σ = n−1n

∑i=1

∫ t

0(zi− z(t))2dNi(te−ziβAH ), (A.1)

and

z(t) = ∑ni=1Yi(te−ziβAH zi)

∑ni=1Yi(te−ziβAH )

= ∑ni=1 I(Xi ≥ te−ziβAH )zi

∑ni=1 I(Xi ≥ te−ziβAH )

3. Solve for b such that U(b) = σ.

4. A variance estimate of√

n(βAH −βAH) is (b− βAH)2, where βAH is the true β, and

βAH is an estimate of βAH from Step 1.

Bibliography

[1] H. Brem, S. Piantadosi, P. Burger, M. Walker, R. Selker, N. Vick, K. Black, M. Sisti,

S. Brem, G. Mohr, P. Muller, and R. Morawetz. Placebo-controlled trial of safety

and efficacy of intraoperative controlled delivery by biodegradable polymers of

chemotherapy for recurrent gliomas. The Lancet, 345:1008–1012, 1995.

[2] N. Breslow. Covariance analysis of censored survival data. Biometrics, 30:89–99,

1974.

[3] Y.Q. Chen and N.P. Jewell. On a general class of semiparametric hazards regression

models. Biometrika, 88:687–702, Sept 2001.

[4] Y.Q. Chen and M.C. Wang. Analysis of accelerated hazards model. Journal of the

American Statistical Association, 95(450):608–618, June 2000.

[5] Y.Q. Chen and M.C. Wang. Estimating a treatment effect with the accelerated hazards

model. Controlled Clinical Trials, 21:369–380, 2000.

[6] D.R. Cox. Regression models and life-tables. Journal of the Royal Statistical Society

Series B, 33:187–220, 1972.

[7] M. Ghahramani, C. Dean, and J. Spinelli. Simultaneous modelling of operative

mortality and long-term survival after coronary artery bypass surgery. Statistics in

Medicine, 20:1931–1945, 2001.

[8] Y.J. Huang. Calibration regression of censored lifetime medical cost. Journal of the

American Statistical Association, 97:318–327, 2002.

79

BIBLIOGRAPHY 80

[9] K. Humphries, M. Gao, A. Pu, S. Lichtenstein, and C. Thompson. Significant

improvement in short-term mortality in women undergoing coronary artery bypass

surgery (1991 to 2004). Journal of the American College of Cardiology, 49:1552–

1558, 2007.

[10] J.D. Kalbfleisch and R.L. Prentice. The Statistical Analysis of Failure Time Data.

Wiley: New York, 1980.

[11] J. Lawless. Statistical Models and Methods for Lifetime Data. John Wiley and Sons:

Hoboken, 2003.

[12] D. Lin, L. Wei, and Z Ying. Accelerated failure time models for counting processes.

Biometrika, 85:605–618, 1998.

[13] C. McGlichrist and C. Aisbett. Regression with frailty in survival analysis. Biomet-

rics, 47:461–466, 1991.

[14] S. Piantadosi. Clinical Trials: A Methodological Perspective. Wiley: New York,

1997.

[15] J. Ragaz, S. Jackson, N. Le, I. Plenderleith, and J. Spinelli. Adjuvant radiotherapy

and chemotherapy in node-positive premenopausal women with breast cancer. The

New England Journal of Medicine, 337:956–962, 1997.

[16] T. Therneau and P. Grambsch. Modeling survival data: extending the Cox model.

Springer: New York, 2000.

[17] A. A. Tsiatis. Estimating regression parameters using linear rank tests for censored

data. The Annals of Statistics, 18:354–372, 1990.

[18] L.J. Wei, Z.. Ying, and D.Y. Lin. Linear regression analysis of censored survival data

based on rank tests. Biometrika, 77:845–851, 1990.

APPROVAL - Summitsummit.sfu.ca/system/files/iritems1/13564/etd6333_CCo.pdf · ing me direction in...

Documents

Transcript of APPROVAL - Summitsummit.sfu.ca/system/files/iritems1/13564/etd6333_CCo.pdf · ing me direction in...