New Developments in Biostatistics and Bioinformatics (Frontiers of Statistics)

FRONTIERS OF STATISTICS FRONTIERS OF STATISTICS The book series provides an overview on the new developments in the f...

Author: Jianqing Fan

122 downloads 1154 Views 15MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form

DOWNLOAD PDF

FRONTIERS OF STATISTICS

FRONTIERS OF STATISTICS The book series provides an overview on the new developments in the frontiers of statistics. It aims at promoting statistical research that has high societal impacts and offers a concise overview on the recent developments on a topic in the frontiers of statistics. The books in the series intend to give graduate students and new researchers an idea where the frontiers of statistics are, to learn common techniques in use so that they can advance the fields via developing new techniques and new results. It is also intended to provide extensive references so that researchers can follow the threads to learn more comprehensively what the literature is, to conduct their own research, and to cruise and lead the tidal waves on the frontiers of statistics.

SERIES EDITORS Jianqing Fan

ZhimingMa

Frederick L. Moore' 18 Professor of Finance.

Academy of Math and Systems Science,

Director of Committee of Statistical Studies,

Institute of Applied Mathematics,

Department of Operation Research and

Chinese Academy of Science,

Financial Engineering,

No.55, Zhong-guan-cun East Road,

Princeton University, NJ 08544, USA.

Beijing 100190, China.

EDITORIAL BOARD Tony Cai, University of Pennsylvania Min Chen, Chinese Academy of Science Zhi Geng, Peking University Xuming He, University of Illinois at Urbana-Champaign Xihong Lin, Harvard University Jun Liu, Harvard University Xiao-Ji Meng, Harvard University Jeff Wu, Georgia Institute of Technology Heping Zhang, Yale University

evv Developments in Biostatistics and Bioinformatics Editors

Jianqing Fan Princeton University, USA

Xihong Lin Harvard University, USA

Jun S. Liu Harvard University, USA

Volume 1

Higher Education Press

World Scientific NEW JERSEY. LONDON· SINGAPORE· BEIJING· SHANGHAI· HONG KONG· TAIPEI· CHENNAI

Jianqing Fan

XihongLin

Department of Operation Reasearch and

Department of Biostatistics of the School of

Financial Engineering

Public Health

Princeton University

Harvard University

Jun Liu Department of Statistics Harvard University

Copyright @ 2009 by

Higher Education Press 4 Dewai Dajie, 1000 II, Beijing, P.R. China and

World Scientific Publishing Co Pte Ltd 5 Toh Tuch Link, Singapore 596224

All rights reserved. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without permission in writing from the Publisher.

ISBN 13: 978-981-283-743-1 ISBN 10: 981-283-743-4 ISSN 1793-8155

Printed in P. R. China

Preface

The first eight years of the twenty-first century has witted the explosion of data collection, with relatively low costs. Data with curves, images and movies are frequently collected in molecular biology, health science, engineering, geology, climatology, economics, finance, and humanities. For example, in biomedical research, MRI, fMRI, microarray, and proteomics data are frequently collected for each subject, involving hundreds of subjects; in molecular biology, massive sequencing data are becoming rapidly available; in natural resource discovery and agriculture, thousands of high-resolution images are collected; in business and finance, millions of transactions are recorded every day. Frontiers of science, engineering, and humanities differ in the problems of their concerns, but nevertheless share a common theme: massive or complex data have been collected and new knowledge needs to be discovered. Massive data collection and new scientific research have strong impact on statistical thinking, methodological development, and theoretical studies. They have also challenged traditional statistical theory, methods, and computation. Many new insights and phenomena need to be discovered and new statistical tools need to be developed. With this background, the Center for Statistical Research at the Chinese Academy of Science initiated the conference series "International Conference on the Frontiers of Statistics" in 2005. The aim is to provide a focal venue for researchers to gather, interact, and present their new research findings, to discuss and outline emerging problems in their fields, to lay the groundwork for future collaborations, and to engage more statistical scientists in China to conduct research in the frontiers of statistics. After the general conference in 2005, the 2006 International Conference on the Frontiers of Statistics, held in Changchun, focused on the topic "Biostatistics and Bioinformatics". The conference attracted many top researchers in the area and was a great success. However, there are still a lot of Chinese scholars, particularly young researchers and graduate students, who were not able to attend the conference. This hampers one of the purposes of the conference series. However, an alternative idea was born: inviting active researchers to provide a bird-eye view on the new developments in the frontiers of statistics, on the theme topics of the conference series. This will broaden significantly the benefits of statistical research, both in China and worldwide. The edited books in this series aim at promoting statistical research that has high societal impacts and provide not only a concise overview on the recent developments in the frontiers of statistics, but also useful references to the literature at large, leading readers truly to the frontiers of statistics. This book gives an overview on recent development on biostatistics and bioinformatics. It is written by active researchers in these emerging areas. It is intended v

VI

Preface

to give graduate students and new researchers an idea where the frontiers of biostatistics and bioinformatics are, to learn common techniques in use so that they can advance the fields via developing new techniques and new results. It is also intended to provide extensive references so that researchers can follow the threads to learn more comprehensively what the literature is and to conduct their own research. It covers three important topics in biostatistics: Analysis of Survival and Longitudinal Data, Statistical Methods for Epidemiology, and Bioinformatics, where statistics is still advancing rapidly today. Ever since the invention of nonparametric and semiparametric techniques in statistics, they have been widely applied to the analysis of survival data and longitudinal data. In Chapter 1, Jianqing Fan and Jiancheng Jiang give a concise overview on this subject under the framework of the proportional hazards model. Nonparametric and semiparametric modeling and inference are stressed. Dongling Zeng and Jianwen Cai introduce an additive-accelerated rate regression model for analyzing recurrent event in Chapter 2. This is a flexible class of models that includes both additive rate model and accelerated rate models, and allows simple statistical inference. Longitudinal data arise frequently from biomedical studies and quadratic inference function provides important approaches to the analysis of longitudinal data. An overview is given in Chapter 3 on this topic by John Dziak, Runze Li, and Annie Qiu. In Chapter 4, Yi Li gives an overview on modeling and analysis of spatially correlated data with emphasis on mixed models. The next two chapters are on statistical methods for epidemiology. Amy Laird and Xiao-Hua Zhou address the issues on study designs for biomarker-based treatment selection in Chapter 5. Several trial designs are introduced and evaluated. In Chapter 6, Jinbo Chen reviews recent statistical models for analyzing twophase epidemiology studies, with emphasis on the approaches based on estimatingequation, pseudo-likelihood, and maximum likelihood. The last four chapters are devoted to the analysis of genomic data. Chapter 7 features protein interaction predictions using diverse data sources, contributed by Yin Liu, Inyoung Kim, and Hongyu Zhao. The diverse data sources information for protein-protein interactions is elucidated and computational methods are introduced for aggregating these data sources to better predict protein interactions. Regulatory motif discovery is handled by Qing Zhou and Mayetri Gupta using Bayesian approaches in Chapter 8. The chapter begins with a basic statistical framework for motif finding, extends it to the identification of cis-regulatory modules, and then introduces methods that combine motif finding with phylogenetic footprint, gene expression or ChIP-chip data, and nucleosome positioning information. Cheng Li and Samir Amin use single nucleotide polymorphism (SNP) microarrays to analyze cancer genome alterations in Chapter 9. Various methods are introduced, including paired and non-paired loss of heterozygosity analysis, copy number analysis, finding significant altered regions across multiple samples, and hierarchical clustering methods. In Chapter 10, Evan Johnson, Jun Liu and Shirley Liu give a comprehensive overview on the design and analysis of ChIPchip data on genome tiling microarrays. It spans from biological background and ChIP-chip experiments to statistical methods and computing. The frontiers of statistics are always dynamic and vibrant. Young researchers

Preface

vii

are encouraged to jump into the research wagons and cruise with tidal waves of the frontiers. It is never too late to get into the frontiers of scientific research. As long as your mind is evolving with the frontiers, you always have a chance to catch and to lead next tidal waves. We hope this volume helps you getting into the frontiers of statistical endeavors and cruise on them thorough your career. Jianqing Fan, Princeton Xihong Lin, Cambridge Jun Liu, Cambridge August 8, 2008

This page intentionally left blank

Contents

Preface

Part I

Analysis of Survival and Longitudinal Data

Chapter 1 Non- and Semi- Parametric Modeling in Survival Analysis Jianqing Fan, Jiancheng Jiang. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3 1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2 Cox's type of models............................................... 3 Multivariate Cox's type of models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4 Model selection on Cox's models.................................. 5 Validating Cox's type of models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 6 Transformation models. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 7 Concluding remarks............................................... References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..

3 4 14 24 27 28 30 30

Chapter 2 Additive-Accelerated Rate Model for Recurrent Event Donglin Zeng, Jianwen Cai . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 35 1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2 Inference procedure and asymptotic properties. . . . . . . . . . . . . . . . . . .. 3 Assessing additive and accelerated covariates .. . . . . . . . . . . . . . . . . . . .. 4 Simulation studies. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 5 Application....................................................... 6 Remarks.......................................................... Acknowledgements ................................................. ;. Appendix............................................................ References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..

35 37 40 41 42 43 44 44 48

Chapter 3

An Overview on Quadratic Inference Function Approaches for Longitudinal Data John J. Dziak, Runze Li, Annie Qu.................................. 49

1 Introduction...................................................... 2 The quadratic inference function approach. . . . . . . . . . . . . . . . . . . . . . .. 3 Penalized quadratic inference function. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4 Some applications of QIF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 5 Further research and concluding remarks. . . . . . . . . . . . . . . . . . . . . . . . .. Acknowledgements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..

49 51 56 60 65 68

Contents

x

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 68 Chapter 4

Modeling and Analysis of Spatially Correlated Data Yi Li................................................................ 73

1 Introduction .................................................... " 2 Basic concepts of spatial process. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3 Spatial models for non-normal/discrete data ....................... 4 Spatial models for censored outcome data ....................... " 5 Concluding remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..

Part II

73 76 82 88 96 96

Statistical Methods for Epidemiology

Chapter 5

Study Designs for Biomarker-Based Treatment Selection Amy Laird, Xiao-Hua Zhou. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 103

1 Introduction..................................................... 2 Definition of study designs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3 Test of hypotheses and sample size calculation. . . . . . . . . . . . . . . . . .. 4 Sample size calculation ......................................... " 5 Numerical comparisons of efficiency. . . . . . . . . .. . . . . . . . . . . . . . . . . . .. 6 Conclusions...................................................... Acknowledgements.................................................. Appendix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. References ......................................................... "

103 104 108 111 116 118 121 122 126

Chapter 6

Statistical Methods for Analyzing Two-Phase Studies Jinbo Chen......................................................... 127

1 Introduction... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2 Two-phase case-control or cross-sectional studies. . . . . . . . . . . . . . . .. 3 Two-phase designs in cohort studies ............................ " 4 Conclusions .................................................... " References...........................................................

Part III

127 130 136 149 151

Bioinformatics

Chapter 7

Protein Interaction Predictions from Diverse Sources Yin Liu, Inyoung Kim, Hongyu Zhao............................... 159

1 Introduction..................................................... 159 2 Data sources useful for protein interaction predictions .......... " 161 3 Domain-based methods.......................................... 163 4 Classification methods ......................................... " 169

Contents 5 Complex detection methods ..................................... , 6 Conclusions...... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. Acknowledgements ................................................. , References .......................................................... ,

xi 172 175 175 175

Chapter 8

Regulatory Motif Discovery: From Decoding to Meta-Analysis Qing Zhou, Mayetri Gupta ...... ................................... , 179

1 Introduction..................................................... 2 A Bayesian approach to motif discovery. . . . . . . . . . . . . . . . . . . . . . . . .. 3 Discovery of regulatory modules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4 Motif discovery in multiple species. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Motif learning on ChIP-chip data ............................... , 6 Using nucleosome positioning information in motif discovery.. . .. 7 Conclusion......... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..

179 181 184 189 195 201 204 205

Chapter 9

Analysis of Cancer Genome Alterations Using Single Nucleotide Polymorphism (SNP) Microarrays Cheng Li, Samir Amin .............................................. 209

1 Background. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2 Loss of heterozygosity analysis using SNP arrays. . . . . . . . . . . . . . . .. 3 Copy number analysis using SNP arrays ........................ , 4 High-level analysis using LOH and copy number data............ 5 Software for cancer alteration analysis using SNP arrays. . . . . . . . .. 6 Prospects............................. . . . . . . . . . . . . . . . . . . . . . . . . . .. Acknowledgements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..

209 212 216 224 229 231 231 231

Chapter 10

Analysis of ChIP-chip Data on Genome Tiling Microarrays W. Evan Johnson, Jun S. Liu, X. Shirley Liu....................... 239

1 Background molecular biology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 2 A ChIP-chip experiment......................................... 3 Data description and analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4 Follow-up analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 5 Conclusion................................... . . . . . . . . . . . . . . . . . . .. References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..

239 241 244 249 254 254

Subject Index. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 259 Author Index . ......................................................... , 261


Part I

Analysis of Survival and Longitudinal Data


Chapter 1 N on- and Semi- Parametric Modeling in Survival Analysis * Jianqing Fan t

Jiancheng Jiang +

Abstract In this chapter, we give a selective review of the nonparametric modeling methods using Cox's type of models in survival analysis. We first introduce Cox's model (Cox 1972) and then study its variants in the direction of smoothing. The model fitting, variable selection, and hypothesis testing problems are addressed. A number of topics worthy of further study are given throughout this chapter. Keywords: Censoring; Cox's model; failure time; likelihood; modeling; non parametric smoothing.

1

Introduction

Survival analysis is concerned with studying the time between entry to a study and a subsequent event and becomes one of the most important fields in statistics. The techniques developed in survival analysis are now applied in many fields, such as biology (survival time), engineering (failure time), medicine (treatment effects or the efficacy of drugs), quality control (lifetime of component), credit risk modeling in finance (default time of a firm). An important problem in survival analysis is how to model well the conditional hazard rate of failure times given certain covariates, because it involves frequently asked questions about whether or not certain independent variables are correlated with the survival or failure times. These problems have presented a significant challenge to statisticians in the last 5 decades, and their importance has motivated many statisticians to work in this area. Among them is one of the most important contributions, the proportional hazards model or Cox's model and its associated partial likelihood estimation method (Cox, 1972), which stimulated -The authors are partly supported by NSF grants DMS-0532370, DMS-0704337 and NIH ROl-GM072611. tDepartment of ORFE, Princeton University, Princeton, NJ 08544, USA, E-mail: jqfan@ princeton.edu tDepartment of Mathematics and Statistics, University of North Carolina, Charlotte, NC 28223, USA, E-mail: [email protected]

3

Jianqing Fan, Jiancheng Jiang

4

a lot of works in this field. In this chapter we will review related work along this direction using the Cox's type of models and open an academic research avenue for interested readers. Various estimation methods are considered, a variable selection approach is studied, and a useful inference method, the generalized likelihood ratio (GLR) test, is employed to address hypothesis testing problems for the models. Several topics worthy of further study are laid down in the discussion section. The remainder of this chapter is organized as follows. We consider univariate Cox's type of models in Section 2 and study multivariate Cox's type of models using the marginal modeling strategy in Section 3. Section 4 focuses on model selection rules, Section 5 is devoted to validating Cox's type of models, and Section 6 discusses transformation models (extensions to Cox's models). Finally, we conclude this chapter in the discussion section.

2

Cox's type of models

Model Specification. The celebrated Cox model has provided a tremendously successful tool for exploring the association of covariates with failure time and survival distributions and for studying the effect of a primary covariate while adjusting for other variables. This model assumes that, given a q-dimensional vector of covariates Z, the underlying conditional hazard rate (rather than expected survival time T),

A(tlz)=

. hm

D.t--+o+

1 ut

AP{t:::;T'0 (u) du is the cumulative baseline hazard function. Equation (2.3) allows one to estimate the function III using regression techniques if >'o(t) is known. The likelihood function can also be derived. When 8i = 0, all we know is that the survival time Ti ;? Ci and the probability for getting this is

whereas when 8i = 1, the likelihood of getting Ti is !(TiIZi ) = !(XiIZi). Therefore the conditional (given covariates) likelihood for getting the data is

and using (2.2), we have

L = I : 10g(>'(XiIZi )) - I:A(XiIZ i ) 8;=1

i

(2.5) For proportional hazards model (2.1), we have specifically (2.6) Therefore, when both 'ljJ(-) and >'00 are parameterized, the parameters can be estimated by maximizing the likelihood (2.6).

Estimation. The likelihood inference can be made about the parameters in model (2.1) if the baseline >'00 and the risk function 'ljJ(.) are known up to a vector of unknown parameters {3 (Aitkin and Clayton, 1980), i.e. >'00

= >'o{-; (3)

and

'ljJ(-)

= 'ljJ(.; (3).

When the baseline is completely unknown and the form of the function 'ljJ(.) is given, inference can be based on the partial likelihood (Cox, 1975). Since the full likelihood involves both (3 and >'o(t), Cox decomposed the full likelihood into a product of the term corresponding to identities of successive failures and the term corresponding to the gap times between any two successive failures. The first term inherits the usual large-sample properties of the full likelihood and is called the partial likelihood.


6

The partial likelihood can also be derived from counting process theory (see for example Andersen, Borgan, Gill, and Keiding 1993) or from a profile likelihood in Johansen (1983). In the following we introduce the latter. Example 1 [The partial likelihood as profile likelihood; Fan, Gijbel, and King (1997)] Consider the case that 'Ij;(z) = 'Ij;(z; (3). Let tl < ... < tN denote the ordered failure times and let (i) denote the label of the item failing at k Denote by Ri the risk set at time ti-, that is Ri = {j : Xj ~ td. Consider the least informative nonparametric modeling for Ao('), that is, Ao(t) puts point mass (}j at time tj in the same way as constructing the empirical distribution: N

Ao(t; (})

= L (}jI(tj

~ t).

(2.7)

j=l Then N

AO(Xi ;{}) = L{}jI(i E Rj ).

(2.8)

j=l Under the proportional hazards model (2.1), using (2.6), the log likelihood is n

logL

L[6i{logAo(Xi;{}) +'Ij;(Zi;(3)} i=l -AO(Xi; (}) exp{ 'Ij;(Zi; (3))].

=

(2.9)

Substituting (2.7) and (2.8) into (2.9), one establishes that n

logL

=

L[1og{}j j=l n

+ 'Ij;(Z(j);(3)]

N

- LL{}jI(i E Rj)exp{'Ij;(Zi;(3)}. i=l j=l

(2.10)

Maximizing log L with respect to {}j leads to the following Breslow estimator of the baseline hazard [Breslow (1972, 1974)]

OJ

=

[L exp{'lj;(Zi; (3)} iERj

rl.

(2.11)

Substituting (2.11) into (2.10), we obtain

~~logL =

n

L('Ij;(Zei);(3) -log[L exp {'Ij;(Zj;(3)}]) - N. t=l JERi

This leads to the log partial likelihood function (Cox 1975) n

£((3)

=L t=l

('Ij;(Zei); (3) -

log

[L exp{'lj;(Zj; (3)}]). JERi

(2.12)

Chapter 1 Non- and Semi- Parametric Modeling in Survival Analysis

7

An alternative expression is n

n

R({3) = I:('Iji(ZCi);,8) -log[I:}j(Xi )exP {'Iji(Zj;,8)}]), i=l

j=l

where }jet) = I(Xj ~ t) is the survival indicator on whether the j-th subject survives at the time t. The above partial likelihood function is a profile likelihood and is derived from the full likelihood using the least informative nonparametric modeling for Ao(·), that is, Ao(t) has a jump (h at k

/3

Let be the partial likelihood estimator of,8 maximizing (2.12) with respect to,8. By standard likelihood theory, it can be shown that (see for example Tsiatis 1981) the asymptotic distribution y'n(/3 - (3) is multivariate normal with mean zero and a covariance matrix which may be estimated consistently by (n- 11(/3))-1, where

1(,8) =

r J

T

o

and for k

[S2(,8,t) _ (S1(,8,t))®2] dN(t) So (,8, t) So (,8, t)

= 0, 1, 2, n

Sk(,8,t) = I:Yi(t)'ljiI(Zi;{3)®kexp{'Iji(Zi;,8)}, i=1

where N(t) = I(X ~ t,o = 1), and x®k = 1,x,xxT , respectively for k = 0,1 and 2. Since the baseline hazard Ao does not appear in the partial likelihood, it is not estimable from the likelihood. There are several methods for estimating parameters related to Ao. One appealing estimate among them is the Breslow estimator (Breslow 1972, 1974) (2.13)

Hypothesis testing. After fitting the Cox model, one might be interested in checking if covariates really contribute to the risk function, for example, checking if the coefficient vector ,8 is zero. More generally, one considers the hypothesis testing problem Ho: ,8 = ,80· From the asymptotic normality of the estimator null distribution of the Wald test statistic

/3, it follows that the asymptotic

8


is the chi-squared distribution with q degrees of freedom. Standard likelihood theory also suggests that the partial likelihood ratio test statistic (2.14) and the score test statistic Tn

=

U(!3of rl(!3o)U(!3 o)

have the same asymptotic null distribution as the Wald statistic, where U(!3o)

=

C'(!3o) is the score function (see for example, Andersen et al., 1993). Cox's models with time-varying covariates. The Cox model (2.1) assumes that the hazard function for a subject depends on the values of the covariates and the baseline. Since the covariates are independent of time, the ratio of the hazard rate functions oftwo subjects is constant over time. Is this assumption reasonable? Consider, for example, the case with age included in the study. Suppose we study survival time after heart transplantation. Then it is possible that age is a more critical factor of risk right after transplantation than a later time. Another example is given in Lawless (1982, page 393) with the amount of voltage as covariate which slowly increases over time until the electrical insulation fails. In this case, the impact of the covariate clearly depends on time. Therefore, the above assumption does not hold, and we have to analyze survival data with time-varying covariates. Although the partial likelihood in (2.12) was derived for the setting of the Cox model with non-time-varying covariates, it can also be derived for the Cox model with time-varying covariates if one uses the counting process notation. For details, see marginal modeling of multivariate data using the Cox's type of models in Section 3.1. More about Cox's models. For the computational simplicity of the partial likelihood estimator, Cox's model has already been a useful case study for formal semiparametric estimation theory (Begun, Hall, Huang, and Wellner 1982; Bickel, Klaassen, Ritov, and Wellner 1993; Oakes 2002). Moreover, due to the derivation of the partial likelihood from profile likelihood (see Example 1), Cox's model has been considered as an approach to statistical science in the sense that "it formulates scientific questions or quantities in terms of parameters 'Y in a model f(y; 'Y) representing the underlying scientific mechanisms (Cox, 1997)j partition the parameters 'Y = ((), TJ) into a subset of interest () and other nuisance parameters TJ necessary to complete the probability distribution (Cox and Hinkley, 1974),develops methods of inference about the scientific quantities that depend as little as possible upon the nuisance parameters (Barndorff-Nielsen and Cox, 1989),- and thinks critically about the appropriate conditional distribution on which to base inferece" (Zeger, Diggle, and Liang 2004). Although Cox's models have driven a lot of statistical innovations in the past four decades, scientific fruit will continue to be born in the future. This motivates us to explore some recent development for Cox's models using the nonparametric idea and hope to open an avenue of academic research for interested readers.


2.1

9

Cox's models with unknown nonlinear risk functions

Misspecification of the risk function 'IjJ may happen in the previous parametric form 'IjJ(.; {3), which could create a large modeling bias. To reduce the modeling bias, one considers nonparametric forms of 'IjJ. Here we introduce such an attempt from Fan, Gijbels, and King (1997). For easy exposition, we consider only the case with q = 1:

A(tlz)

= Aa(t) exp{'IjJ(z)},

(2.15)

where z is one dimensional. Suppose the form of 'IjJ(z) in model (2.15) is not specified and the p-th order derivative of 'IjJ(z) at the point z exists. Then by the Taylor expansion,

'IjJ(Z) ~ 'IjJ(z)

+ 'IjJ'(z)(Z -

z)

'IjJ(p) (z)

+ ... + - - I-(Z p.

z)P,

for Z in a neighborhood of z. Put Z

= {I, Z -

Z,··· ,(Z - Z)P}T and Zi

= {I, Zi -

Z,··· ,(Zi - z)p}T,

where T denotes the transpose of a vector throughout this chapter. Let h be the bandwidth controlling the size of the neighborhood of x and K be a kernel function with compact support [-1,1] for weighting down the contribution of remote data points. Then for IZ - zl ::;:; h, as h - t 0,

'IjJ(Z) ~ ZT a , where a = (aa,a1,··· ,apf = {'IjJ(z),'IjJ'(z),· .. ,'IjJ(p)(z)/p!}T. By using the above approximation and incorporating the localizing weights, the local (log) likelihood is obtained from (2.9) as n

£n({3,8)

=

n- 1

L: [8i {log Aa(Xi ; 8) + zf a} i=l

-Aa(Xi ; 8) exp(Zf a)] Kh(Zi - x),

(2.16)

where Kh(t) = h- 1 K(t/h). Then using the least-informative nonparametric model (2.7) for the baseline hazard and the same argument as for (2.12), we obtain the local log partial likelihood N

L:Kh(Z(i) - z)(Z0)a i=l

-IOg[L: exp{ZG)a}Kh(Zj -

z)]).

(2.17)

JERi

Maximizing the above function with respect to a leads to an estimate ex of a. Note that the function value 'IjJ(z) is not directly estimable; (2.17) does not involve the intercept aa since it cancels out. The first component a1 = (f;'(z) estimates 'IjJ'(z).

10


It is evident from model (2.15) that 'ljJ(z) is only identifiable up to a constant. By

imposing the condition 'ljJ(0)

= 0, the function 'ljJ(z) can be estimated by ,(}(z)

=

l

z

,(}'(t) dt.

According to Fan, Gijbels, and King (1997), under certain conditions, the following asymptotic normality holds for ,(}' (z):

where

and

v~,(z) = a2(z)J-1(z)

JK~(t)2

dt

with K;(t) = tK(t)/ J t 2K(t) dt and a 2(z) = E{8IZ = z} -1. With the estimator of 'ljJO, using the same argument as for (2.13), one can estimate the baseline hazard by (2.18) Inference problems associated with the resulting estimator include constructing confidence intervals and hypothesis tests, which can be solved via standard nonparametric techniques but to our knowledge no rigor mathematical theory exists in the literature. A possible test method can be developed along the line of the generalized likelihood ratio (GLR) tests in Section 5, and theoretical properties of the resulting tests are to be developed. For multiple covariates cases, the above modeling method is applicable without any difficulty if one employs a multivariate kernel as in common nonparametric regression. See Section 2.2 for further details. However, a fully nonparametric specification of 'ljJ(.) with large dimensionality q may cause the "curse of dimensionality" problem. This naturally leads us to consider some dimension reduction techniques.

2.2

Partly linear Cox's models

The partly linear Cox's model is proposed to alleviate the difficulty with a saturated specification of the risk function and takes the form (2.19) where Ao 0 is an unspecified baseline hazard function and


11

where the form of the function '1/JI(zl;{3) is known up to an unknown vector of finite parameters (3, and 'l/J2(-) is an unknown function. This model inherents nice interpretation of the finite parameter {3 in model (2.1) while modeling possible nonlinear effects of the d x 1 vector of covariates Z2. In particular, when there is no parametric component, the model reduces to the aforementioned full nonparametric model in Section 2.1. Hence, in practice, the number of components in Z2 is small. The parameters (3 and function 'l/J2(Z2) can be estimated using the profile partial likelihood method. Specifically, as argued in the previous section, the function 'l/J2 admits the linear approximation

'l/J2(Z2) ~ 'l/J2(Z2)

+ 'I/J~(z2f(Z2 -

Z2)

== a T Z2

when Z2 is close to Z2, where a = {'l/J2(Z2), 'I/J~(Z2)TV and Z2 = {l, (Z2 - Z2)TV. Given (3, we can estimate the function 'l/J2(-) by maximizing the local partial likelihood N

in(a)

=

:L KH(Z2(i) - Z2) ('I/J~ (Zl(i); (3)

+ Zr(i) a

i=l

-log[:L exp{'l/Jl(Zl(j);(3)

+ Zr(j)a}K H(Z2j

- Z2)]) ,

(2.20)

JERi

where K H(Z2) = \HI- 1 K(H- 1 Z2) with K(·) being a d-variate probability density (the kernel) with unique mode 0 and J uK(u)du = 0, H is a nonsingular d x d matrix called the bandwidth matrix (see for example Jiang and Doksum 2003). For expressing the dependence of the resulting solution on (3, we denote it by &(Z2; (3) = {~2 (Z2; (3), ~~ (Z2; (3)). Substituting ~2(-; (3) into the partial likelihood, we obtain the profile partial likelihood of (3 n

i n (!3)

=

:L('l/Jl(Zl(i);f3)

+~2(Z2(i);13)

i=l

-log[:L exp{'I/Jl(Zlj; (3)

+ ~2(Z2j; {3)}]).

(2.21 )

JERi

.a

.a.

Maximizing (2.21) with respect to will lead to an estimate of We denote by 13 the resulting estimate. The estimate of function 'l/J2(-) is simply ~2('; 13). By an argument similar to that in Cai, Fan, Jiang, and Zhou (2007), it can be shown that the profile partial likelihood estimation provides a root-n consistent estimator of (see also Section 3). This allows us to estimate the nonparametric component 'l/J2 as well as if the parameter (3 were known.

.a

2.3

Partly linear additive Cox's models

The partly linear model (2.19) is useful for modeling failure time data with multiple covariates, but for high-dimensional covariate Z2, it still suffers from the so-called "curse-of-dimensionality" problem in high-dimensional function estimation. One


12

of the methods for attenuating this difficulty is to use the additive structure for the function 'ljJ2(·) as in Huang (1999), which leads to the partly linear additive Cox model. It specifies the conditional hazard of the failure time T given the covariate value (z, w) as

A{tlz, w} = AO(t) exp{ 'ljJ(z; 13) + ¢(w)},

(2.22)

where ¢(w) = ¢l(wd + ... + ¢J(wJ). The parameters of interest are the finite parameter vector 13 and the unknown functions ¢j's. The former measures the effect of the treatment variable vector z, and the latter may be used to suggest a parametric structure of the risk. This model allows one to explore nonlinearity of certain covariates, avoids the "curse-of-dimensionality" problem inherent in the saturated multivariate semiparametric hazard regression model (2.19), and retains the nice interpretability of the traditional linear structure in Cox's model (Cox 1972). See the discussions in Hastie and Tibshirani (1990). Suppose that observed data for the i-th subject is {Xi, lSi, Wi, Zi}, where Xi is the observed event time for the i-th subject, which is the minimum of the potential failure time Ti and the censoring time Gi , lSi is the indicator of failure, and {Zi' Wi} is the vector of covariates. Then the log partial likelihood function for model (2.22) is n

C(j3, ¢)

=

L lSi { 'ljJ(Zi; 13) + ¢(Wi ) -log L

rj(j3, ¢)},

(2.23)

JERi

i=l

where

rj(j3,¢) = exp{'ljJ(Zj;j3)

+ ¢(Wj)}.

Since the partial likelihood has no finite maximum over all parameters (13, ¢), it is impossible to use the maximium partial likelihood estimation for (13, ¢) without any restrictions on the function ¢. Now let us introduce the polynomial-spline based estimation method in Huang (1999). Assume that W takes values in W = [0, Let

IF.

< 6 < ... < ~K < ~K+l = I} be a partition of [0, IJ into K subintervals IKi = [~i' ~i+l) i = 0, ... ,K - 1, and IKK = [~K' ~K+1J, where K == Kn = O(nV) with 0 < v < 0.5 being a positive {= {O =

~o

integer such that

h ==

max

l~k~K+1

I~k

-

~k-ll =

O(n-V).

Let S( C, ~) be the space of polynomial splines of degree C ~ 1 consisting of functions s(·) satisfying: (i) the restriction of s(·) to IKi is a polynomial of order C- 1 for 1 ~ i ~ K; (ii) for C ~ 2, sis C- 2 times continuously differentiable on [O,IJ. According to Schumaker (1981, page 124), there exists a local basis B i (·), 1 ~ i ~ qn for S(C, {) with qn = Kn + C, such that for any ¢nj (.) E S(C, {), qn

¢nj(Wj) =

L bjiBi(Wj), i=l

1 ~ j ~ J.


13

Put

B(w) = (Bl(w),· .. ,Bqn(w))T, B(w) = (BT(Wr), ... ,BT(wJ)f,

b j = (bjl , ... ,bjqn)T, b = (bi, ... ,b'})T.

bf

Then cPnj(Wj) = B(wj) and cPn(w) == 2:,1=1 cPnj(Wj) = bTB(w). Under regular smoothness assumptions, cPj's can be well approximated by functions in S(C, ~). Therefore, by (2.23), we have the logarithm of an approximated partial likelihood n

C({J, b)

L b"i{ ¢(Zi;}3) + cPn(Wd -log L exp[¢(Zj;}3) + cPn(Wj )]},

=

.=1

(2.24)

JERi

where J

L cPnj(Wji )

cPn(Wi) =

j=l

with Wji being the j-th component of Wi, for i = 1,··· ,n. Let (/3, b) maximize the above partial likelihood (2.24). Then an estimator of cP(-) at point w is simply ,

the cP(w)

J'

= 2:,j=l cPj(Wj)

.'

wIth cPj(Wj)

=

,T

b j B(wj).

As shown in Huang (1999), when ¢(z; f3) = zT f3, the estimator vn-consistency. That is, under certain conditions,

vn(/3 -

f3) = n- l / 2 I- l (f3)

/3 achieves

n

L l~(Xi' b"i, Zi, Wi) + Op(l) i=l

where I(f3)

= E[l~(X,

b., Z, W)]02 is the information bound and

l~(X, 8, Z, W) = IT (Z -

a*(t) - h*(W)) dM(t)

is the efficient score for estimation of f3 in model (2.22), where h*(w) = hiCwr) + ... + h j (w J) and (a * , hi, . .. ,h j) is the unique L2 functions that minimize

where

M(t)

=

M{X

~ t} - I

t

I{X

~ u} exp[Z'f3 + cP(W)] dAo(u)

is the usual counting process martingale. Since the estimator, achieves the semiparametric information lower bound and is asymptotically linear, it is asymptotically efficient among all the regular estimators (see Bickel, Klaassen, Ritov, and Wellner 1993). However, the information lower bound cannot be consistently estimated, which makes inference for f3 difficult in practice. Further, the asymptotic distribution of the resulting estimator

/3,


14

¢ is hard to derive.

This makes it difficult to test if ¢ admits a certain parametric

form. The resulting estimates are easy to implement. Computationally, the maximization problem in (2.24) can be solved via the existing Cox regression program, for example coxph and bs in Splus software [for details, see Huang (1999)]. However, the number of parameters is large and numerical stability in implementation arises in computing the partial likelihood function. An alternative approach is to use the profile partial likelihood method as in Cai et al. (2007) (see also Section 3.2). The latter solves many much smaller local maximum likelihood estimation problems. With the estimators of (3 and ¢('), one can estimate the cumulative baseline hazard function Ao(t) = J~ Ao(u)du by a Breslow's type of estimators:

1(8 t

Ao(t)

=

n

Yi (u)eX P{¢(Zi;,B)

+ ¢(Wi

where Yi(u) = I(Xi ~ u) is the at-risk indicator and Ni(u) is the associated counting process.

3

n

nr 8 dNi(u), 1

= I(Xi
'j (l(3j I), j=1

(4.3)

where P>'j (l(3j I) is a given nonnegative function called a penalty function with Aj as a regularisation or tuning parameter. The tuning parameters can be chosen subjectively by data analysts or objectively by data themselves. In general, large values of Aj result in simpler models with fewer selected variables. When Ki = 1, J = 1, dn = d, and Aj = A, it reduces to the penalized partial likelihood in Fan and Li (2002). Many classical variable selection criteria are special cases of (4.3). An example is the Lo penalty (or entropy penalty)

p>.(IBI) = 0.5A 2 I(IBI =I- 0). In this case, the penalty term in (4.3) is merely 0.5nA 2 k, with k being the number of variables that are selected. Given k, the best fit to (4.3) is the subset of k variables having the largest likelihood R({3) among all subsets of k variables. In other words, the method corresponds to the best subset selection. The number of variables depends on the choice of A. The AIC (Akaike, 1973), BIC (Schwarz, 1978), qr.criterion (Shibata, 1984), and RIC (Foster & George, 1994) correspond to A = (2/n)1/2, {10g(n)/n}1/2, [log{log(n)}] 1/2, and {10g(dn )/n}1/2, respectively. Since the entropy penalty function is discontinuous, one requires to search over all possible subsets to maximise (4.3). Hence it is very expensive computationally. Furthermore, as analysed by Breiman (1996), best-subset variable selection suffers from several drawbacks, including its lack of stability. There are several choices for continuous penalty functions. The L1 penalty, defined by p>.(IBI) = AIBI, results in the LASSO variable selector (Tibshirani, 1996). The smoothly clipped absolute deviation (SCAD) penalty, defined by

p~(B)

= Al(IOI ::;; A)

+ (aA -

0)+ l(O > A),

a-I

(4.4)


26

for some a > 2 and A > 0, with PA(O) = O. Fan and Li (2001) recommended a = 3.7 based on a risk optimization consideration. This penalty improves the entropy penalty function by saving computational cost and resulting in a continuous solution to avoid unnecessary modelling variation. Furthermore, it improves the L1 penalty by avoiding excessive estimation bias. The penalised pseudo-partial likelihood estimator, denoted by maximises (4.3). For certain penalty functions, such as the L1 penalty and the SCAD penalty, maximising L(f3) will result in some vanishing estimates of coefficients and make their associated variables be deleted. Hence, by maximising L(f3), one selects a model and estimates its parameters simultaneously. Denote by f30 the true value of f3 with the nonzero and zero components f310 and f320. To emphasize the dependence of Aj on the sample size n, Aj is written as Ajn. Let Sn be the dimension of f31O,

/3,

As shown in Cai, Fan, Li, and Zhou (2005), under certain conditions, if 0, bn ----f 0 and d~jn ----f 0, as n ----f 00, then with probability tending to one, there exists a local maximizer /3 of L(f3), such that an

----f

Furthermore, if Ajn ----f 0, JnjdnAjn ----f 00, and an = O(n- 1 / 2 ), then with probability tending to 1, the above consistent local maximizer /3 = (/3f, /3'f)T must be such that (i) /32 = 0 and (ii) for any nonzero constant Sn x 1 vector Cn with c;cn

r::: Tr-1/2(A 11 11

ynC n

+ ~ ){ f31 A

- f310

+

(

All

= 1,

+ ~ )-1 b}

D N(O, 1),

-----7

where All and r ll consist of the first Sn columns and rows of A(f3lO, 0) and r(f3lO, 0), respectively (see the aforementioned paper for details of notation here). The above result demonstrates that the resulting estimators have the oracle property. For example, with the SCAD penalty, we have an = 0, b = 0 and ~ = 0 for sufficiently large n. Hence, by the above result,

VnC;:r~11/2 All (/31

-

f31O) ~ N(O, 1).

The estimator /31 shares the same sampling property as the oracle estimator. Furthermore, /32 = 0 is the same as the oracle estimator that knows in advance that f32 = O. In other words, the resulting estimator can correctly identify the true model, as if it were known in advance. Further study in this area includes extending the above model selection method to other Cox's type of models, such as the partly linear models in Sections 2.3, 3.3 and 3.4.


5

27

Validating Cox's type of models

Even though different Cox's type of models are useful for exploring the complicate association of covariates with failure rates, there is a risk that misspecification of a working Cox model can create large modeling bias and lead to wrong conclusions and erroneous forecasting. It is important to check whether certain Cox's models fit well a given data set. In parametric hypothesis testing, the most frequently used method is the likelihood ratio inference. It compares the likelihoods under the null and alternative models. See for example the likelihood ratio statistic in (2.14). The likelihood ratio tests are widely used in the theory and practice of statistics. An important fundamental property of the likelihood ratio tests is that their asymptotic null distributions are independent of nuisance parameters in the null model. It is natural to extend the likelihood ratio tests to see if some nonparametric components in Cox's type of models are of certain parametric forms. This allows us to validate some nested Cox's models. In nonparametric regression, a number of authors constructed the generalized likelihood ratio (GLR) tests to test if certain parametric/nonparametric null models hold and showed that the resulting tests share a common phenomenon, the Wilks phenomenon called in Fan, Zhang, and Zhang (2001). For details, see the reviewing paper of Fan and Jiang (2007). In the following, we introduce an idea of the GLR tests for Cox's type of models. Consider, for example, the partly linear additive Cox model in (2.22):

.A{ tlz, w}

=

.Ao(t) exp{ zT,B + (h (WI) + ... + t),

we obtain

E[dN(te- XT (3) _ Y(te- XT (3)dte- XT {3 ZT ,IX, Z, C 1\ T > te- XT {3J

Donglin Zeng, Jianwen Cai

38

Therefore, for fixed (3 and " we can estimate /-l(t) by rescaling N(t) and CAT with a factor exTf3 ; thus, an estimator for /-l(t) is the Breslow-type estimator given by A

/-l(t;{3,,) =

lt 2:;=1 o

x x {dNj (se- XJ f3) - Yj(se- Jf3)dse- Jf3 ZJ,} Y( -XT(3) . uj=1 j se J

"n

To estimate (3 and" we consider the following estimating equations

After substituting il(t;{3,,) into the left-hand side, we have

_ 2:;=1 Yj(te- XJf3)(dNj (te- XJf3) -

ZJ,e- XJ f3 dt )} = O.

(2.1)

2:;=1 Yj(te- XJf3 )

The re-transformation of the numerator on the left-hand side yields

Equivalently,

n

2:jYi(t)

i=1

Z

{ (x.)-

"n )} uJ=1 y. (tex,[ f3-xJ (3) (Zj X· J

XTf3-XTf3 ' 2 : j =1 Yj(te' J) n

J

(dNi(t)-zT,dt) =0.

(2.2) We wish to solve equation (2.2), or equivalently equation (2.1). However, the estimating function in (2.2) is not continuous in {3. Therefore, we suggest minimizing the norm of the left-hand side of (2.2) to obtain the estimators. Particularly, we implement the Neider-Meader simplex method to find the minimum in our numerical studies. At the end of this section, we will show y'n(S - (30,.:y - ,0) converges in distribution to a multivariate normal distribution with mean zeros and covariance in a sandwiched form A -1 B (A -1 f, where

Chapter 2

A

~ V."

Additive-Accelerated Rate Model for Recurrent Event

J+,(t){(~)

En

Y.(teX'{f3-xJf3) ( Zj )

- J;n~ y.(tex,{f3-xJf3~j J

and B

=

39

1

} (dNi(t) - zT ,dt)

J

II

f3=f30 ,'Y=,o

E [SiST] with Si equal to

x (dNi (te- X'{f30) - zT ,e-X'{f3odt) -

J

Yo (te •

_XT (Zi) E [Y(te,f30 ) Xi

+

XTf30

) {dN(te-

XTf30

) - ZT ,oeXTf3o E [Y(te)]

J

x XTf3O }i(te- '{f3O)E [Y(te)

XTf30

dt}]

(i)]

E [Y(te- XTf30 ) {dN(te- XTf30 ) - ZT ,oe- XTf30 dt}] E [Y(te- XTf30 )] 2

x

(2.3) .

Therefore, to estimate the asymptotic covariance, we can consistently estimate Si by Hi , simply replacing (30, ,0 and f..Lo by their corresponding estimators and replacing expectation with empirical mean in the expression of Si. To estimate A, we let () denote ((3,,) and define

Un ((})

= n- 1

tJ

_ E7=1

X [}i(te- '{f3)

(i:) {

dNi(te- X'{f3) - ZT,dte- x '{f3

[}j(te-XJf3)(dNj(te-XJf3) - Z!,e- XJ f3 dt)]

E7=1

}l.

[}j(te- XJ f3)]

We also let A be the numerical derivative of A with appropriate choice of perturbation size hn, i.e., the k-th column of A is given by (Un(B + hnek) - Un(B))/h n where ek is the k-th canonical vector. In Theorem 2 given later, we will show A is a consistent estimator for A. Thus, the asymptotic covariance is estimated by

40


We start to state the asymptotic properties for (/3/'1). We assume the following conditions. (AI) Assume X and Z are bounded and with positive probability, [1, X, Z) are linearly independent. (A2) Matrix A is assumed to be non-singular. (A3) P(C > TIX, Z) > 0 and C given X, Z has bounded density in [0, T). (A4) !-lo(t) is strictly increasing and has bounded second derivative in [0, T]. Condition (AI) is necessary since otherwise, f3 and "( assumed in the model may not be identifiable. Both conditions (A2) and (A3) are standard. Under these conditions, the following theorems hold. These conditions are similar to conditions 1~4 in Ying (1993). Theorem 1. Under conditions (Al)~(A4), there exists (/3, i) locally minimizing the norm of the estimating function in (2) such that

Theorem 2. Under conditions (Al)~(A4) and assuming h n ---t 0 and fohn E (co, Cl) for two positive constants Co and Cl, A-I B(A -1) T is a consistent estimator for A- 1B(A-1 )T.

Thus, Theorem 1 gives the asymptotic normality of the proposed estimators and Theorem 2 indicates that the proposed variance estimator is valid. Using the estimators for f3 and ,,(, we then estimate the unknown function p,O with fl(t; /3, i), i.e.,

The following theorem gives the asymptotic properties for fl(t). Theorem 3. Let Dx be the support of X. Then under conditions (Al)~(A4), xT for any to < TSUPXED x e- {3o, fo(fl(t) - P,o(t)) converges weakly to a Gaussian process in loo [0, to].

The upper bound for to in Theorem 3 is determined by the constraint that the study duration time is T. The proofs of all the theorems utilize the empirical process theory and are given in the appendix.

3

Assessing additive and accelerated covariates

One important assumption in the proposed model is that covariates X have accelerated effect while covariates Z have additive effect. However, it is unknown in practice which covariates should be included as part of X or Z. In this section, we propose some ways for assessing these covariates. From the proposed model, if X and Z are determined correctly, we then expect that the mean of Y(t)(N(t)* - p,(te XT {3) - tZT "() is zero. As the result,

Chapter 2


41

correct choices of X and Z should minimize the asymptotic limit of n-

1

i

T

o

n

2

L"Yi(t) {Nt(t) - f-t(te XTf3 ) - tz[ 'Y} dt. i=l

The following result also shows that if we use wrong choice of X or Z, this limit cannot be minimized.

Proposition 1. Suppose Xc and Zc are correct covariates with non-zero accelerated effect f3c and non-zero additive effect 'Yc respectively. Further assume (C1) the domain of all the covariates is offull rank; (C2) f-tc(t) is continuous and satisfies f-tc(O) = 0 and f-t~(0) > 0; (C3) Xc contains more than 2 covariates; or, Xc contains one covariate taking at least 3 different values; or, Xc is a single binary variable but f-tc is not a linear function. Then if Xw and Zw are the wrong choices of the accelerated effect and the additive effect respectively, i.e., Xw #- Xc or Zw #- Zc, then for any non-zero effect f3w and 'Yw and function f-tw,

1T E [Y(t) {N*(t) - f-tw(te X;f3w) - tZ;;''Yw } 2] dt > 1T E[Y(tHN*(t)-f-tc(te X;f3C)-tZ';'Ycf]dt. The proof of Proposition is given in the appendix. Condition (C1) says that the covariates can vary independently; Condition (C2) is trivial; Condition (C3) says that the only case we exclude here is the f-t(t) = At and Xc is binary. The excluded case is unwanted since f-t(te Xcf3c ) = A{{ef3c - l)Xc + l}t resulting that Xc can also be treated as additive covariate. From Proposition 1, it is concluded that whenever we misspecify the accelerated covariates or additive covariates which has non-zero effect, we cannot minimize the limit of the square residuals for all observed t. Hence, if we find X and Z minimizing the criterion, it implies that the differences between X and Xc and Z and Zc are only those covariates without effect.

4

Simulation studies

We conduct two simulation studies to examine the performance of the proposed estimators with moderate sample sizes. In the first simulation study, we generate recurrent event using the following intensity model

E[dN(t)IN(t-), X, z, ~J = ~(df-t(tef3x)

+ 'YZdt),

where Z is a Bernoulli random variable and X = Z + N(O, 1), f-t(t) = O.Se t , and ~ is a gamma-frailty with mean 1. It is clear this model implies the marginal model as given in Section 2. Furthermore, we generate censoring time from a uniform distribution in [0,2J so that the average number of events per subject is about


42

1.5. In the second simulation study, we consider the same setting except that /-L(t) = 0.8t2 and the censoring distribution is uniform in [0,3]. To estimate (3 and " we minimize the Euclidean norm of the left-hand side in equation (2). Our starting values are chosen to be close to the true values in order to avoid local minima. To estimate the asymptotic covariance of the estimators, we use the numerical derivative method as suggested in Theorem 2. Particularly, we choose h n to be n- 1 / 2 , 3n- 1 / 2 and 5n- 1 / 2 but find the estimates robust to these choices. Therefore, our table only reports the ones associated with h n = 3n- 1 / 2 . Our numerical experience shows that when sample size is as small as 200, the minimization may give some extreme value for (3 in a fraction of 4% in the simulations; the fraction of bad cases decreases to about 1% with n = 600. Hence, our summary reports the summary statistics after excluding outliers which are 1.5 inter-quartile range above the third quartile or below the first quartile. Table 1 gives the summary of the simulation studies from n = 200, 400, and 600 based on 1000 repetitions. Particularly, "Est" is the median of the estimates; "SEE" is the estimate for the standard deviation from all the estimates; "ESE" is the mean of the estimated standard errors; "CP" is the coverage probability of the 95% confidence intervals. From the table, we conclude that the proposed estimators perform reasonably well with the sample size considered in terms of small bias and accurate inference.

Table 1: Summary of Simulation Studies /-L(t) = 0.8e t

n 200

parameter

f3 'Y

400

(3

600

(3

'Y 'Y

true -0.5 1 -0.5 1 -0.5 1

Est -0.501 0.991 -0.496 0.979 -0.490 1.000

SEE

ESE

CP

0.366 0.414 0.246 0.296 0.202 0.240

0.381 0.437 0.252 0.311 0.207 0.255

0.94 0.96 0.93 0.96 0.94 0.96

/-L(t) = 0.8t2

n 200

parameter (3

400

(3

600

(3

'Y 'Y 'Y

5

true -0.5 1 -0.5 1 -0.5 1

Est -0.478 0.970 -0.498 0.992 -0.493 0.998

SEE

ESE

CP

0.331 0.307 0.240 0.227 0.192 0.185

0.344 0.328 0.237 0.235 0.195 0.195

0.94 0.95 0.92 0.95 0.93 0.96

Application

We apply the proposed method to analyze two real data sets. The first data arise from a chronic granulomatous disease (CGD) study which was previously analyzed by Therneau and Hamilton (1997). The study contained a total 128 patients with 65 patients receiving interferon-gamma and 63 receiving placebo. The number of recurrent infections was 20 in the treatment arm and was 55 in the placebo arm.

Chapter 2


43

Table 2: Results from Analyzing Vitamin A Trial Data Covariate vitamin A vs placebo age in years

Estimate -0.0006 -0.0040

Std Error 0.0411 0.0008

p-value 0.98 < 0.001

We also included the age of patients as one covariate. The proposed additiveaccelerated rate model was used to analyze this data. To assess whether the treatment variable and the age variable have either additive or accelerated effect, we utilized the criterion in Section 3 and considered the four different models: (a) both treatment and age were additive effects; (b) both treatment and age were accelerated effects; (c) treatment was additive and age was accelerated; (d) treatment was accelerated and age was additive. The result shows that the model with minimal error is model (b), i.e., the accelerated rate model. Such a model was already fitted in Ghosh (2004) where it showed that the treatment had a significant benefit in decreasing the number of infection occurrences. A second example is from the Vitamin A Community Trial conducted in Brazil (Barreto et aI, 1994). The study was a randomized community trial to examine the effect of the supplementation of vitamin A on diarrhea morbidity and diarrhea severity in children living in areas where its intake was inadequate. Some previous information showed that the vitamin A supplement could reduce the child mortality by 23% to 34% in populations where vitamin deficiency was endemic. However, the effect on the morbidity was little known before the study. The study consisted of young children who were assigned to receive either vitamin A or placebo every 4 months for 1 year in a small city in the Northeast of Brazil. Since it was indicated before that treatment effect might be variable over time, for illustration, we restrict our analysis to the first quarter of the follow-up among boys, which consists of 486 boys with an average number of 3 events. Two covariates of interest are the treatment indicator (vitamin A vs placebo) and the age. As before, we use the proposed method to select among the aforementioned models (a)-(d). The final result shows that model (c) yields the smallest prediction error. That is, our finding shows that the treatment effect is accelerative while the age effect is additive. Particularly, Table 2 gives the results from model (c). It shows that the treatment effect is not significant; however, younger boys tended to experience significantly more diarrhea episodes than older boys. Figure 2 gives the estimated function for J.L(t).

6

Remarks

We have proposed a flexible additive-accelerated model for modelling recurrent event. The proposed model includes the accelerated rate model and additive rate model as special cases. The proposed method performs well in simulation studies. We also discuss the method for assessing whether the covariates are additive covariates or accelerated covariates. When the model is particularly used for prediction for future observations, overfitting can be a problem. Instead of using the criterion function in assessing

Donglin Zeng, lianwen Cai

44

o

20

40

80

60

100

120

Days

Figure 2: Estimated function for J-l(t) in the Vitamin A trial

additive covariates or accelerated covariates, one may consider a generalized crossvalidation residual square error to assess the model fit. Theoretical justification of the latter requires more work. Our model can be generalized to incorporate time-dependent covariates by assuming

E[N(t)\X, Zl = {L(te X (t)Tf3)

+ Z(tf 'Y-

The same inference procedure can be used to estimate covariate effects. However, the nice result on assessing additive covariates or accelerated covariates as Proposition 1 may not exist due to the complex relationship among time-dependent covariates. Finally, our model can also be generalized to incorporate time-varying coefficient.

Acknowledgements We thank Professor Mauricio Barreto at the Federal University of Bahia and the Vitamin A Community Trial for providing the data. This work was partially supported by the National Institutes of Health grant ROI-HL57444.

Appendix Proof of Theorems 1-3. We prove Theorem 1 and Theorem 2. For convenience, we let 0 denote ({3, ,) and use the definition of Un(O). For 0 in any compact set, it is easy to see the class of functions {Y(te-

XT

(3)}, {ZT ,e-xT f3}, {N(te- XT (3) }

Chapter 2


45

are P-Donsker so P-Glivenko-Cantelli. Therefore, in the neighborhood of 80 , Un (8) uniformly converges to

U(8) == E [I Yi(te- x [i3)

(i:) {dN i(te- X[i3) - Z'[-r dte - x [i3

_ E [Yj (te- XJi3) (dNj (te-xJ.B) - ZJ"{e-XJ.Bdt)] }] E [Yj(te-XJ.B)]

.

Furthermore, we note that uniformly in 8 in a compact set,

v'n(Un(8) - U(8))

~ Go

[f Y(te-XT~)

(;.) {dN(e-XT't) -

zr7 e- XT 'dt

P n [Y(te-XTi3)(dN(e-XTi3t) - ZT"{e-XT.Bdt )] }] P n [Y(te-XT.B)]

(i) J]

-G n

[IYCte-XT.B)CdN(e-XT.Bt) _ ZT"{e-XT.Bdt) P [Y(te-XT.B) P n [YCtcXT.B)]

+G n [I Y(te-

XTi3

)p [Y Cte-

XTi3

)

(i)]

x P [Y(te-XTi3)CdN(e-XTi3t) - ZT"{e- XTi3 dt)] P n [Y(te-XT.B)] P [Y(te- XTi3 )]

1 ,

where P nand P refer to the empirical measure and the expectation respectively, and G n = y'n(P n - P). Hence, from the Donsker theorem and the definition of 8 i , we obtain n

sup Iv'nUn(8) - v'nU(8) \lJ-Oo\:;;;M/v'n

n- 1 / 2

L 8 l = op(l). i

i=1

On the other hand, conditions (A2)-(A4) implies that U(8) is continuously differentiable around 80 and U(8) = U(80 ) + A(8 - 80 ) + 0(18 - 80 1). Additionally, it is straightforward to verify U(80) = O. As the result, it holds n

sup Iv'nUn (8) - v'nA(8 - 80 ) \O-Oo\:;;;M/v'n

-

1 2

n- /

L 8 l = op(l). i

i=1

We obtain a similar result to Theorem 1 in Ying (1993). Following the same proof as given in Corollary 1 in Ying (1993), we prove Theorem 1.


46

To prove Theorem 2, from the previous argument, we obtain uniformly in a

y'n- neighbor hood of 00 , n

:L S

..;nun(e) = ..;nA(O -eo) +n- 1/ 2

i

+op(1).

i=l

Thus, for any canonical vector e and h n

-+

0,

As the result,

Un(O + hne) - Un(O) _ A (_1_) h - e + op y'nh . n n Since y'nh n is bounded away from zero,

We have shown that the estimator based on the numerical derivatives, i.e., A, converges in probability to A. The consistency of Si to Si is obvious. Therefore,

A-1B(A-l)T

A-1B(A-1)T.

-+

To prove Theorem 3, from fl(t)'s expression, we can obtain

Y(se-xTi3)E[dN(se-xTi3) - Y(se-xTi3)dse-xTi3ZTil o E[Y(se- XT ,8)J2 E[dN(se- xTi3 ) - Y(se-xTi3)dse-xTi3 ZTil +..;n io E[Y(se-XTt3)] - J.L(t) + op(l).

-G n

[

i

t

A

1

r

Clearly, N(te- XT ,8), Y(te- XT ,8), ZT'Y all belong to some Donsker class. Additionally, when t ~ to, E[Y(te- XTt3 )] > O. Hence, after the Taylor expansion of the third term and applying the Donsker theorem, we obtain A

..;n(J.L(t)

_

r{dN(se-XT,80)-Y(se-XT,80)dSCXT,8oZT'Yo} n J.L(t)) - G io E[Y(se-XT,8o)] _

Chapter 2 Additive-Accelerated Rate Model for Recurrent Event

47

Therefore, using the asymptotic expansions for (J and l' as given in proving Theorem 1, we can show vfn(P,(t) - Ito(t)) converges weakly to a Gaussian process in

lOO[O, to]. Proof of Proposition 1. We prove by contradiction. Assume

faT E

[Y(t) {N*(t) - Itw(te x J;f3w) _ tZ'{;/yw}

~ faT E

[Y(t) {N*(t) -

lte(teX~f3c) -

tZ'[ "Ie}

2] dt 2] dt.

Notice E [Y(t) {N*(t) - gt(X, Z)}2] is minimized for gt(X, Z) = E[N*(t)IY(t), X,

Z] = lte(teX~ f3c) + tZ'[ "Ie and such gt(X, Z) is unique almost surely. Therefore, for any t E [0, rJ, we conclude Itw(te XJ;f3 w) + tZJ;"Iw = lte(teX~f3c) with probability one. We differentiate (A.l) at t =

1t~(O)exJ;f3w

+ tZ'[ "Ie

(A.l)

°and obtain

+ ZJ;"Iw = 1t~(O)eX~f3c + Z'[ "Ie.

(A.2)

We show Xc must be included in Xw. Otherwise, there exists some covariate Xl in Xc but in Zw. We will show this is impossible by considering the following two cases. Case I. Suppose Xl has at least three different values and f3el is the coefficient of Xl. Fix the values of all the other covariates. Then equation (A.2) gives

aef3cIXl - bXl

=

d

for some constants a > 0, b, and d. However, since f3el i- 0, the function on the left-hand side of the above equation is strictly convex in Xl so the equation has at most two solutions. Since Xl has at least three different values, we obtain the contradiction. Case II. Suppose Xl has only two different values. Without loss of generality, we assume Xl = or 1. Then we obtain

°

for some function g. Here, Xc,-l means covariates in Xc except Xl and the same is defined for f3e,-l. Consequently, X'[_lf3e,-1 =constant, which implies that Xe,-l has to be empty. That is, Xc only contains a single binary variable. Thus, equation (A.l) gives If ef3c < 1, we replace t by ef3c k t and sum over k = 0,1,2, .... This gives Ite(t) = Act for some constant Ae. If ef3c > 1, we replace t by e-f3c k t and sum over k = 1,2, ... and obtain the same conclusion. Since we assumed that J-le(t) is not a

48


linear function of t when Xc is a binary covariate, P,c(t) = Act introduces the contradiction. As the result, Xc must be included in Xw. Thus, it is easy to see p,:"(0) =1= o. We repeat the same arguments but switch Xw and Xc. We conclude Xw is also included in Xc. Therefore, Xw = Xc. Clearly, f3w = f3c and p,:"(0) = p,~(0) by the fact that Zw and Zc are different from Xw and Xc and Condition (A.1). This further gives z'E,w = z'[ Ic. Since the covariates are linearly independent and IW =1= 0 and IC =1= 0, we obtain Zw = ZC. This contradicts with the assumption Xw =1= Xc or Zw =1= ZC·

References [1] Andersen, P. K. and Gill, R. D. (1982). Cox's regression model for counting processes: a large sample study. Annals of Statistics, 10, 1100-1120. [2] Cox, D. R. and Oakes, D. (1984). Analysis of Survival Data. Chapman and Hall: London. [3] Ghosh, D. (2004). Accelerated rates rgression models for recurrent failure time data. Lifetime Data Analysis, 10, 247-26l. [4] Kalbfleisch, J. D. and Prentice, R. L. (1980). The Statistical Analysis of Failure Time Data. Wiley: New York. [5] Lawless, J. F. and Nadeau, C. (1995). Some simple and robust methods for the analysis of recurrent events. Technometrics, 37, 158-168. [6] Pepe, M. S. and Cai, J. (1993). Some graphical displays and marginal regression analysis for recurrent failure times and time dependent covariates. Journal of the American Statistical Association, 88, 811-820. [7] Schaubel, D. E., Zeng, D., and Cai, J. (2006). A Semiparametric Additive Rates Model for Recurrent Event Data. Lifetime Data Analysis, 12,386-406. [8] Therneau, T. M. and Hamilton, S. A. (1997). RhDNase as an example of recurrent event analysis. Statistics in Medicine, 16, 2029-2047. [9] Ying, Z. (1993). A large sample study of rank estimation for censored regression data. Annals of Statistics, 21, 76-99.

Chapter 3 An Overview on Quadratic Inference Function Approaches for Longitudinal Data John J. Dziak * Runze Li t

Annie Qu

:j:

Abstract Correlated data, mainly including longitudinal data, panel data, functional data and repeated measured data, are common in the fields of biomedical research, environmental studies, econometrics and the social sciences. Various statistical procedures have been proposed for analysis of correlated data in the literature. This chapter intends to provide a an overview of quadratic inference function method, which proposed by Qu, Lindsay and Li (2000). We introduce the motivation of both generalized estimating equations method and the quadratic inference method. We further review the quadratic inference method for time-varying coefficient models with longitudinal data and variable selection via penalized quadratic inference function method. We further outline some applications of the quadratic inference function method on missing data and robust modeling.

Keywords: Mixed linear models; hierarchical linear models; hierarchical generalized linear models; generalized estimating equations; quadratic inference function; dispersion parameter; GEE estimator; QIF estimator; modified Cholesky decomposition; time-varying coefficient model; penalized QIF; smoothly clipped absolute; LASSO.

1

Introduction

Correlated data occurs almost everywhere, and is especially common in the fields of biomedical research, environmental studies, econometrics and the social sciences (Diggle et al. 2002, Davis 2002, Hedeker & Gibbons 2006). For example, to achieve sufficient statistical power using a limited number of experiment units in clinical trials, the outcome measurements are often repeatedly obtained from the *The Methodology Center, The Pennsylvania State University, 204 E. Calder Way, Suite 400 State College, PA 16801, USA. E-mail: [email protected] tDepartment of Statistics and The Methodology Center, The Pennsylvania State University, University Park, PA 16802-2111, USA. E-mail: [email protected] tDepartment of Statistics, Oregon State University, Corvallis, OR 97331-4606, USA. E-mail: [email protected]

49

50

John J. Dziak, Runze Li, Annie Qu

same subject at different time points; in education, students' achievements are more likely to be similar if they are from the same class, and the class, school or community are treated as natural clusters; in spatial environmental studies, researchers often have no control over spatially correlated field samples such as streams or species abundance. There are two major existing approaches to modeling and analyzing correlated data. One is the subject-specific approach including mixed linear models (Laird & Ware, 1982), hierarchical linear models (Bryk & Raudenbush, 1992), and hierarchical generalized linear models (Lee & NeIder, 1996). The other is a population average approach including generalized estimating equations (Liang & Zeger, 1986). The former approach emphasizes modeling heterogeneity among clusters which induces a correlation structure among observations. The latter models the correlation structure directly. These two approaches yield regression estimators with different interpretations, and different values in practice (Zeger, Liang & Albert, 1988; Neuhaus, Kalbfleisch & Hauck, 1991; Lee & NeIder 2004). The major drawback to subject-specific approaches is that random effects are assumed to follow an explicit distribution, and typically a normal random effects distribution is assumed. Lee & NeIder (1996) allow a broader class of parametric models for random effects. However, in general there is not enough information for goodness-of-fit tests for random effects distributions. Neuhaus, Hauck & Kalbfleish (1992) show that when the distributions of the random effects are misspecified, the estimator of fixed effects could be inconsistent for logistic models. On the other hand, the generalized estimating equations approach has an advantage in that it requires only the first two moments of the data, and a misspecified working correlation does not affect the root n consistency of the regression parameter estimation; though the misspecification of working correlation does affect the efficiency of the regression parameter estimation (see Liang & Zeger 1986, Fitzmaurice 1995, Qu, Lindsay & Li, 2000). In general, the generalized estimating equation approach lacks a probabilistic interpretation since the estimating functions are not uniquely determined. Therefore, it is not obvious how to do model checking or goodness-of-fit test based on likelihood functions such as the likelihood ratio test. For these reasons, Heagerty & Zeger (2000) developed marginalized multilevel models, a likelihood based method which is less sensitive to misspecified random effects distributions. However, this might be computational complex and difficult since the marginal approach requires integrations, and, if a high dimension of random effects is involved, the estimates may not be analytically tractable. In order to overcome the limitations of the above approaches, Qu, Lindsay & Li (2000) proposed the quadratic inference function (QIF) method to analyze longitudinal data in a semiparametric framework defined by a set of mean zero estimating functions. This approach has the advantages of the estimating function approach, as it does not require the specification of the likelihood function and does not involve intractable computations. It also overcomes the limitation of estimating function approach such as lacking of inference functions. The difference in QIF between two nested models is analogous to the difference in minus twice the log likelihood, so it may provide a semiparametric analog to the likelihood ratio

Chapter 3

Quadratic Inference Function Approaches

51

test. Specifically, it provides an asymptotic chi-squared distribution for goodnessof-fit tests and hypothesis testing for nested regression parameters. Since the QIF method was proposed, this method has been further developed to cope with various difficulties in the analysis of longitudinal data. Its applications have been published in the various statistical journals. This chapter aims to provide a partial review on this topic. This chapter is organized as follows. Section 2 gives background and motivation of the quadratic inference function approach, and presents some theoretic properties of the quadratic inference function estimator. In Section 3, we introduce penalized quadratic inference functions for time-varying coefficient models and variable selection for longitudinal data. Section 4 presents some main ideas about how to apply the quadratic inference function approach for testing whether missing data is ignorable for an estimating equation approach CQu & Song, 2002). Section 4 also demonstrates that quadratic inference function estimation is potentially more robust to outliers than the ordinary generalized estimating equations estimation (Qu & Song, 2004). Some research topics that need to further study are presented in Section 5.

2

The quadratic inference function approach

Suppose that we collect a covariate vector Xij and a response Yij for individual = 1,···,J and i = 1,··· ,no Denote Yi = CYil,··· ,YiJ)T and Xi = (XiI, ... , XiJ ) T. Start with a simple continuous response case. The following linear regression model is useful to explore the relationship between the covariates and the continuous response.

i at time tj, j

(2.1) where ei is assumed to be an independent and identically random error with mean o. It is well known that when the within subject random errors are correlated, the ordinary least squares estimator for j3 is not efficient. To improve efficiency of the ordinary least squares estimator, consider the weighted least squares estimator (WLSE)

i=1

i=1

If the covariance matrix of ei is of the form (7"2~ with a known ~, but unknown (7"2, then WLSE with Wi = ~-1 is the best linear unbiased estimator (BLUE). In practice, the covariance matrix of ei is typically unknown, and therefore the WLSE requires us to specify a covariance structure. In practice, the true covariance structure is generally complicated and unknown, except in very simple situations. How to construct a good estimator for f3 when the covariance structure is misspecified? This poses a challenge in the analysis of longitudinal data.

52

2.1


Generalized estimating equations

The outcome variable could be discrete in many longitudinal studies. Naturally, we consider a generalized linear model for a discrete response. For iid data, the generalized linear model assumes that given the covariates, the conditional distribution of the response variable belongs to the exponential family (McMullagh & NeIder, 1989). In some situations, specification of a full likelihood function might be difficulty. It might be more desirable to assume the first two moments instead. The quasi-likelihood approach can be used to develop inference procedures for the generalized linear model (Wedderburn, 1974). Unfortunately, it is still a challenge for correlated discrete responses, such as binary or count response. As done for weighted least squares approach, the generalized estimating equation (GEE) approach (Liang & Zeger, 1986) assumes only the mean structure and variance structure along with the working correlation for the discrete longitudinal/repeated measurement data. In what follows, we briefly introduce the GEE approach. As in the generalized linear models, it is assumed that

E(YijIXij)

g-l(x'f;{3),

=

Var(Yijlxij)

=

1>V(P,ij),

where P,ij = E(Yijlxij), g(.) is a known link function, 1> is called a dispersion parameter, and V (.) is called a variance function. Let us present a few examples. Example 1. For continuous response variable, the normal error linear regression model assumes that the random error in (2.1) follows a normal distribution. In other words, given Xij, the conditional distribution of Yij is N(xI;i3,u 2 ). Then g(p,) = p" the identity link, and 1> = u 2 and V(p,) = 1. Example 2. For binary outcome variable, the logistic regression model assumes that given Xij, Yij follows a Bernoulli distribution with success probability

p(xd = J

Then g(p)

= p/(l - p),

exp(xfi3) J. 1 + exp(xI;{3)

the logit link, 1>

=

1 and V(p)

= p(l - p).

Example 3. For count outcome variable, the log-linear Poisson regression model assumes that given Xij, Yij follows a Poisson distribution with mean

>'(Xij) = exp(x'f;{3). Then g(>.)

= log(>.),

the log link, 1> = 1 and V(>.)

= >..

For these examples, it is easy to specify the first two moments for Yij marginal ly. Thus, we may further construct quasi-likelihood when the conditional distribution of Yij is not available. It is quite difficult to specify the joint distribution of binary or count responses Yij, j = 1, ... ,J here.

Chapter 3 Let Pi

=


(Pil,··· ,PiJ

f,

53

be a J x d matrix with the j-th row being

Di

8g- l (xT;(3)/8(3, Ai be a J x J diagonal matrix with the j-th diagonal element

V(pij). For a given working correlation matrix R i , the GEE estimator is the solution of the following estimation equation: (2.2) i==l

It can be verified that for model (2.1), the WLSE with weight matrix Wi = A;/2RiA;/2 is the solution of (2.2). As demonstrated in Liang & Zeger (1986), the choice of the working correlation matrix Ri does not affect the consistency of the GEE estimator, but could affect its efficiency; and if Ri is correctly specified, then the resulting estimation is the most efficient.

2.2

Quadratic inference functions

The quadratic inference function (QIF) approach (Qu, Lindsay & Li, 2000) shares the same asymptotic efficiency of the estimator as that of GEE when the correlation matrix equals the true one, and is the most efficient in asymptotic sense for a given class of correlation matrices. Thus, the QIF estimator is at least as asymptotic efficient as the corresponding GEE for a given working correlation. Before we describe the ideas of QIF, we first briefly discuss the common choices of the working correlation matrix Ri in the GEE method. There are two commonly used working correlation matrices: equicorrelated matrix and AR working correlation structures. The equicorrelated (also known as exchangeably correlated or compound symmetric) matrix is defined as

1 p ... P] pl··· p

R=

. . .. ..

.. [ ..

'.

p p ... 1

In the implementation of the GEE method, the inverse of R is used here. For equicorrelated working correlation structure, we have

where 1 is the J x 1 vector with all elements equaling 1, and al = 1/(1 - p) and a2 = - p/ { (1 - p) (1 - P + J p)}. This indicates that the inverse of equicorrelated matrix can be represented as a linear combination of nonnegative definite matrices. For the AR correlation structure, assume that Zt, t = 1,··· ,J is an AR sequence with order q. In other words, we can represent Zj as min{t-l,q} Zt

=

L

j==l

CPjZt-j

+ et·

(2.3)

54


where et's are independent white noise with mean zero and variance (J2. Denote z = (Zl,'" ,zJ)Y, e = (el,'" ,eJ)Y and L to be a lower triangular matrix having ones on its diagonal and (i, i - j)-element -CPj, for i = 1"" ,J and j = 1,'" ,min{i - 1, q}. Then (2.3) can be rewritten as Lz = e. Thus, Lcov(z)L T = cov(e) = (J2 I. This indeed is the modified Cholesky decomposition of cov(z) (see (2.7) below). Therefore, cov-l(z)

= (J-2L T L

Denote U j is a J x J matrix with (i, i - j)-element being 1 and all other elements being O. Note that UfUk = 0 for j -1= k. Then

j

j

j

j

Thus, the inverse of the covariance matrix of an AR sequence can be also represented as a linear combination of symmetric matrices. Based on these observations, the QIF approach assumes that the inverse of a within-subject working correlation matrix R can be expressed as a linear combination 2::~=1 bkMk, where the bk's are unknown constants, and bkMk'S are known, symmetric matrices. Reexpressing (2.2), the GEE estimate is then the value of {3 which sets the following quasi-score to zero: M s = n -1 DiAi-1/2 ( blb l 1

+ ... + br M r ) A-i l / 2(yi

Notice that s is a linear combination of the "extended score" where mi

=

[Dr

A~1/2hMl~~1/2(Yi -

-

J-Li ) .

mn =

(2.4)

~ 2::~=1 bimi,

J-Li)] . (2.5)

Dr A~1/2brMrA~1/2(Yi - J-Li) Since mn contains more estimating equations than the dimension of {3, it could be impossible to set all equations to be zero. Hansen (1982) proposed the generalized method of moment (GMM) which attempts to combine these estimating equations optimally. The GMM could be traced back from the minimum X2 method introduced by Neyman (1949) which is further developed by Ferguson (1958,1996). The GMM estimator 7J is obtained by minimizing m~C-l mn, where C is a weight matrix, instead of solving s = O. Hansen (1982) has shown that the best weight matrix C is the covariance matrix of mn . In practice the covariance matrix of mn is often unknown. Qu, Lindsay & Li (2000) suggested taking the weight matrix C to be its sample counterpart, i.e. n- 2 2::~1 MiMr, and defined the quadratic inference function (2.6)

Chapter 3 and the QIF estimator


fj is

55

defined to be

13 =

argmin(3Q(f3).

The idea of QIF approach is to use data-driven weights, which gives less weight to the estimating equations with larger variances, rather than setting the weights via ad hoc estimators of the parameters of the working correlation structure as in (2.4). There is no explicit form for fj. A numerical algorithm, such as NewtonRaphson algorithm or Fisher scoring algorithm, should be used to minimization Q(f3). See Qu, Lindsay & Li (2000) for details. One of the main advantage of the QIF approach is that if the working correlation is correctly specified, the QIF estimator has an asymptotic variance as low as the GEE. If the working structure is incorrect, the QIF estimator is still optimal among the same linear class of estimating equations, while the GEE estimator with the same working correlation is not. See Qu, Lindsay & Li (2000) for some numerical comparisons. The asymptotic property of the QIF estimator fj has been studied under the framework of the GMM. It has been shown that no matter whether C is consistent for C or not, fj is root n consistent and asymptotic normal, provided that C is positive definite. As shown in Qu & Lindsay (2003), if the true score function is included in M, then the QrF estimator in the context of a parametric model, is asymptotically equivalent to the MLE and thus shares its first-order asymptotic optimality. Unlike GEE, the QIF estimation minimizes a clearly defined objective function. The quadratic form Qn itself has useful asymptotic properties, and is directly related to the classic quadratic-form test statistics (see Hansen 1982, Greene 2000, Lindsay & Qu 2003), as well as somewhat analogous to the quadratic GEE test statistics of Rotnitzsky & Jewell (1990) and Boos (1992). When p < q, Qn can be used as an asymptotic X2 goodness-of-fit statistic or a test statistic for hypotheses about the parameters 13k, it behaves much like a minus twice of log-likelihood. ~

L

• Q(f3o) - Q(f3) ---t X~ under the null hypothesis Ho : 13 = 130· • More generally, if 13 = ['I,b, ~lT where 'I,b is an r-dimensional parameter of interest and ~ is a (p - r)-dimensional nuisance parameter, then the profile test statistic Qn('l,bo,[O) - Qn(;j),[) is asymptotically X; for testing Ho : 'I,b = 'l,bo. This could be used for testing the significance of a block of predictors. Thus, the QIF plays a similar role to the log-likelihood function of the parametric models. For the two primary class of working correlation matrices, the equicorrelated matrices and AR-1 working correlation structure, their inverse can be expressed as a linear combination of several basis matrices. One can construct various other working correlation structure through the linear combination 2::~=1 bkMk. For example, we can combine equi-correlation and AR-1 correlation structures together by pooling their bases together. One may find such linear combination of given correlation structure through the Cholesky decomposition. It is known that a positive definite matrix I; has a modified Cholesky decomposition: I;-l

= LTDL,

(2.7)

56


where L is a lower triangular matrix having ones on its diagonal and typical element -¢ij in the (i, j) position for 1 :::; j < i :::; m, and D is a diagonal matrix with positive elements. As demonstrated for the AR working correlation structure, the modified Cholesky decomposition may be useful to find such a linear combination. In the recent literature, various estimation procedures have been suggested for covariance matrices using the Cholesky decomposition. Partial references on this topic are Barnard et al. (2000), Cai & Dunson (2006), Dai & Guo (2004), Daniels & Pourahmadi (2002), Houseman, et al. (2004), Huang, et al (2006), Li & Ryan (2002), Pan & Mackenzie (2003), Pourahmadi (1999, 2000), Roverato (2000), Smith & Kohn (2002), Wang & Carey (2004), Wu & Pourahmadi (2003) and Ye & Pan (2006). We limit ourselves in this section with the setting in which the observation times tj'S are the same for all subjects. In practice, this assumption may not always be valid. For subject-specific observation times tij'S, we may bin the observation times first, and then apply the QIF approach to improve efficiency based on the binned observation times. From our experience, this technique works well for functional longitudinal data as discussed in the next section.

3

Penalized quadratic inference function

We now introduce the penalized QIF method to deal with high-dimensionality of parameter space. We first show how to apply the QIF for time-varying coefficient models.

3.1

Time-varying coefficient models

Varying coefficient models have become popular since the work by Hastie & Tibshirani (1993). For functional longitudinal data, suppose that there are n subjects, and for the i-th subject, data {Yi(t), XiI (t), ... ,Xid(t)} were collected at times t = tij, j = 1, ... ,ni. To explore possible time-dependent effects, it is natural to consider d

Yi(t)

=

!3o(t)

+L

Xik (t)!3k (t)

+ Ei(t).

(3.1)

k=I

This is called a time-varying coefficient model. In these models, the effects of predictors at any fixed time are treated as linear, but the coefficients themselves are smooth functions of time. These models enable researchers to investigate possible time-varying effects of risk factors or covariates, and have been popular in the literature of longitudinal data analysis. (See, for example, Hoover, et al., (1998), Wu, Chiang & Hoover (1998), Fan & Zhang (2000), Martinussen & Scheike (2001), Chiang, Rice & Wu (2001), Huang, Wu & Zhou (2002,2004) and references therein.) For discrete response, a natural extension of model (3.1) is d

E{Yi(t)lxi(t)} = g-I{!30(t)

+L k=l

Xik(t)!3k(t)}.

(3.2)

Chapter 3


57

With slight abuse of terminology, we shall still refer this model to as a timevarying coefficient model. Researchers have studied how to incorporate the correlation structure to improve efficiency of estimator of the functional coefficients. For a nonparametric regression model, which can be viewed as a special case of model (3.2), Lin & Carroll (2000) demonstrated that direct extension of GEE from parametric models to nonparametric models fails to properly incorporate the correlation structure. Under the setting of Lin & Carroll (2000), Wang (2003) proposed marginal kernel GEE method to utilize the correlation structure to improve efficiency. Qu & Li (2006) proposed an estimation procedure using penalized QIF with L2 penalty. The resulting penalized QIF estimator may be applied to improve efficiency of kernel GEE type estimators (Lin & Carroll, 2000; Wang, 2003) when the correlation structure is misspecified. The penalized QIF method originates from the penalized splines (Ruppert, 2002). The main idea of the penalized QIF is to approximate each (3k(t) with a truncated spline basis with a number of knots, thus creating an approximating parametric model. Let /'O,Z, l = 1"" ,K be chosen knots. For instance, if we parametrize (3k with a power spline of degree q = 3 with knot /'O,z's we have (3k(t)

= 'YkO + 'Yk1t + 'Yk2 t2 + 'Yk3 t3 +

2: 'Yk,l+3(t -

/'O,z)t·

z

Thus, we can reexpress the problem parametrically in terms of the new spline regression parameters 'Y. To avoid large model approximation error (higher bias), it might be necessary to take K large. This could lead to an overfitting model higher variance). To achieve a bias-variance tradeoff by reducing overfitting, Qu & Li (2006) proposed a penalized quadratic inference function with the L2 penalty: K

QC'Y) + n)..

L L 'Y~,l+3' k

(3.3)

Z=l

where).. is a tuning parameter. In Qu & Li (2006), the tuning parameter).. is chosen by minimizing a modified GCV statistic: GCV

=

(1 _ n-1dfQ)2

(3.4)

with dfQ = tr( (Q+n).. Nf1Q), where N is a diagonal matrix such that Dii = 1 for knot terms and 0 otherwise. The GCV statistic is motivated by replacing RSS in the classic GCV statistic (Craven & Wahba, 1979) with Q. For model (3.2), it is of interest to test whether some coefficients are timevarying or time-invariant. Furthermore, it is also of interest to delete unnecessary knots in the penalized splines because it is always desirable to have parsimonious models. These issues can be formulated as statistical hypothesis tests of whether the corresponding 'Y's are equal to zero or not. Under the QIF approach this can be done by comparing Q for the constrained and unconstrained models. See Qu & Li (2006) for a detailed implementation of penalized spline QIF's.

58

3.2


Variable selection for longitudinal data

Many variables are often measured in longitudinal studies. The number of potential predictor variables may be large, especially when nonlinear terms and interactions terms between covariates are introduced to reduce possible modeling biases. To enhance predictability and model parsimony, we have to select a subset of important variables in the final analysis. Thus, variable selection is an important research topic in the longitudinal data analysis. Dziak & Li (2007) gives an overview on this topic. Traditional variable selection criteria, such as AIC and BIC, for linear regression models and generalized linear models sometimes are used to select significant variables in the analysis of longitudinal data (see Zucchini, 2000, Burnham & Anderson 2004, Kuha 2004, Gurka 2006 for general comparisons of these criteria). These criteria are not immediately relevant to marginal modeling because the likelihood function is not fully specified. In the setting of GEE, some workers have recently begun proposing analogues of AIC (Pan, 2001, Cantoni et al., 2005) and BIC (Jiang & Liu, 2004), and research here is ongoing. Fu (2003) proposed penalized GEE with bridge penalty (Frank & Friedman, 1993) for longitudinal data, Dziak (2006) carefully studied the sampling properties of the penalized GEE with a class of general penalties, including the SCAD penalty (Fan & Li, 2001). Comparisons between penalized GEE and the proposals of Pan (2001) and Cantoni, et al (2005) are given in Dziak & Li (2007). This section is focus on summarizing the recent development of variable selection for longitudinal data by penalized QIF. A natural extension of the AIC and BIC is to replace the corresponding negative twice log-likelihood function by the QIF. For a given subset M of {1, ... ,d}, denote Me to be its complement, and i3M to be the minimizer of Qf((3), the QIF for the full model, viewed as a function of (3M by constraining (3Mc = O. Define a penalized QIF (3.5) where #(M) is the cardinality of M, Qf(i3M) is the Qf((3) evaluated at (3M = i3 and (3Mc = O. The AIC and BIC correspond to A = 2 and log(n), respectively. Wang & Qu (2007) showed that the penalized QIF with BIC penalty enjoys the well known model selection consistency property. That is, suppose that the true model exists, and it is among a fixed set of given candidate models, then with probability approaching one, the QIF with BIC penalty selects the true model as the sample size goes to infinity. The best subset variable selection with the traditional variable selection criteria becomes computationally infeasible for high dimensional data. Thus, instead of the best subset selection, stepwise subset selection procedures are implemented for high dimensional data. Stepwise regression ignores the stochastic errors inherited in the course of selections, and therefore the sampling property of the resulting estimates is difficult to understand. Furthermore, the subset selection approaches lack of stability in the sense that a small changes on data may lead to a very different selected model (Breiman, 1996). To handle issues related with high dimensionality, variable selection procedures have been developed to select significant variables and estimate their coefficients simultaneously (Frank & Friedman,

Chapter 3


59

1993, Breiman, 1995, Tibshirani, 1996, Fan & Li, 2001 and Zou, 2006). These procedures have been extended for longitudinal data via penalized QrF in Dziak (2006). Define penalized QrF d

QJ(3)

+n LPAj(l,6jl),

(3.6)

j=l

where PAj (-) is a penalty function with a regularization parameter Aj. Note that different coefficients are allowed to have different penalties and regularization parameters. For example, we do not penalize coefficients 1'0, ... ,1'3 in (3.3); and in the context of variable selection, data analysts may not want to penalize the coefficients of certain variables because in their professional experience they believe that those variables are especially interesting or important and should be kept in the model. Minimizing the penalized QrF (3.6) yields an estimator of,6. With proper choice of penalty function, the resulting estimate will contain some exact zeros. This achieves the purpose of variable selection. Both (3.3) and (3.5) can be re-expressed in the form of (3.6) by taking the penalty function to be the L2 penalty, namely, PAj (l,6j I) = Aj l,6j 12 and the Lo penalty, namely, PAj (I,6j I) = AjI(I,6j I =I- 0), respectively. Frank & Friedman (2003) suggested using Lq penalty, PAj (l,6j I) = Aj l,6j Iq (0 < q < 2). The L1 penalty was used in Tibshirani (1996), and corresponds to the LASSO for the linear regression models. Fan & Li (2001) provides deep insights into how to select the penalty function and advocated using a nonconvex penalty, such as the smoothly clipped absolute deviation (SCAD) penalty, defined by

AI,6I, 2

(.II) _

PA (I f-/

-

(a -1)A2-(Ii3I_aA)2

{

2(a-1) (a+1)A2 2

'

,

if 0 :::; 1,61 < A; if A :::; 1,61 < aA; if 1,61 ? aA.

Fan & Li (2001) suggested fixing a = 3.7 from a Bayesian argument. Zou (2006) proposed the adaptive LASSO using weighted L1 penalty, PAj (l,6j I) = AjWj l,6j I with adaptive weights Wj. The L 1 , weighted L 1 , L2 and SCAD penalty functions are depicted in Figure 1. The SCAD estimator is similar to the LASSO estimator since it gives a sparse and continuous solution, but the SCAD estimator has lower bias than LASSO. The adaptive LASSO uses the adaptive weights to reduce possible bias of LASSO. Dziak (2006) showed that, under certain regularity conditions, nonconvex penalized QIF can provide a parsimonious fit and enjoys something analogous to the asymptotic "oracle property" described by Fan & Li (2001) in the least-squares context. Comparisons of penalized QIF estimator with various alternatives can be found in Dziak (2006).


60

Penalty Functions 3.5 ~--------.------,------,------., , - ,L 1 - - Weighted L 1 3 III'~

2.5 ...

C

2

]

p...1.5

"

. •~

','., '. , , , ... ... ,

, ~: ,

"

~

'", "

~ ~

......

~

,,

",

';""

' OL-----________ .-,.,~~

~~~~

-5

o

...

,

.'

~

"

0.5

, ~ ,"

... ,

,

,

... ... ,

______________

~

5

b

Figure 1: Penalty Functions. The values for>. are 0.5,0.5,0.125 and 1, respectively. The adaptive weight for the weighted L1 penalty is 3

4

Some applications of QIF

It is common for longitudinal data to contain missing data and outlying observations. The QIF method has been proposed for testing whether missing data are ignorable or not in the estimating equation setting (Qu & Song, 2002). Qu & Song (2004) also demonstrated that the QIF estimator is more robust than the ordinary GEE estimator in the presence of outliers. In this section, we outline the main ideas of Qu & Song (2002, 2004).

4.1

Missing data

Qu & Song (2002) define whether missing data is ignorable in the context of the estimating equation setting based on whether the estimating equations satisfy the mean-zero assumption. This definition differs somewhat from Rubin's (1976) definition which is based on the likelihood function. Qu & Song's (2002) approach shares the same basis as Chen & Little (1999) on decomposing data based on missing-data patterns. However, it avoids exhaustive parameter estimation for each missing pattern as in Chen & Little (1999). The key idea of Qu & Song's approach is that if different sets of estimating equations created by data sets with different missing patterns are compatible, then the missing mechanism is ignorable. This is equivalent to testing whether different sets of estimating equations satisfy the zero-mean assumption under common parameters, i.e., whether E(s) = O. This can be carried out by applying an over-identifying

Chapter 3


61

restriction test, which follows a chi-squared distribution asymptotically. For example, suppose each subject has three visits or measurements, then there are four possible patterns of missingness. The first visit is mandatory; then subjects might show up for all appointments, miss the second, miss the third, or miss the second and third. We construct the QIF as

(4.1)

where 8j and 6 j are the estimating functions for the j-th missing pattern group and its empirical variance respectively. If 81, ... ,84 share the same mean structure, then the test statistic based on the QIF above will be relatively small compared to the cut-off chi-squared value under the null. Otherwise the QIF will be relatively larger. Clearly, if 81, ... ,84 do not hold mean-zero conditions under common parameters, the missingness might not be ignorable, since estimating functions formulated by different missing patterns do not lead to similar estimators. It might be recommended to use working independence here, or combine several similar missing patterns together, to keep the dimension to a reasonable size if there are too many different missing patterns. This approach is fairly simple to apply compared to that of Chen & Little (1999), since there is no need to estimate different sets of parameters for different missing patterns. Another advantage of this approach can be seen in the example of Rotnitzky & Wypij (1994), where the dimension of parameters for different missing patterns are different. The dichotomous response variables record asthma status for children at ages 9 and 13. The marginal probability is modeled as a logistic regression (Rotnitzky & Wypij, 1994) with gender and age as covariates: logit{Pr(Yit = I)} = /30

+ /31I(male) + /32I(age =

13),

where Yit = 1 if the i-th child had asthma at time t = 1,2 and I(E) is the indicator function for event E. About 20% of the children had asthma status missing at age 13, although every child had his or her asthma status recorded at age 9. Note that there are three parameters /30, /31 and /32 in the model when subjects have no missing data, but only two identifiable parameters, /30 and /31, for the incomplete case. Since the dimension of parameters is different for different missing patterns, Chen & Little's (1999) approach requires a maximum identifiable parameter transformation in order to perform the Wald test. However, the transformation might not be unique. Qu & Song (2002) do not require such a transformation, but show that the QIF goodness-of-fit test and the Wald test are asymptotically equivalent.

4.2

Outliers and contamination

The QIF estimator is more robust against outlying observations than the ordinary GEE estimator. Both GEE and QIF asymptotically solve equations which lead to

62


an M-estimator. A robust estimator has a bounded influence function (Hampel et al., 1986). The influence function of the GEE is not bounded, while the influence function of the QIF is bounded (Qu & Song, 2004). This could explain why the QIF is more robust than the GEE for contaminated data. Hampel (1974) defines the influence function of an estimator as IF(z, Pj3)

=

inf

~((1- c)Pj3 + c~j3) - ~(Pj3)

E:~O

C

(4.2)

where Pj3 is the probability measure of the true model and ~z is the probability measure with mass 1 at contaminated data point z. If (4.2) is not bounded as a function of z, then the asymptotic bias in ~ introduced by contaminated point z could be unbounded, that is, one could move ~ to infinite value by allowing the contaminated point z to be infinite. Hampel et al. (1986) show that for an M-estimator, solving the estimating equation L~=l Si(Zi, (3), the influence function is proportional to Si(Z, (3). Thus the influence function is bounded if and only if the contribution of an individual observation to the score function is bounded. If the influence function is not bounded, then the asymptotic "breakdown point" is zero and the corresponding estimator could be severely biased even with a singlegross outlier. Consider a simple case of GEE with a linear model using working independence. An individual observation's contribution to the score function is Xit (Yit x~(3), which diverges if Yit is an outlier relative to the linear model. Qu & Song (2004) showed that the QIF does not have this kind of problem. In fact, the QIF has a "redescending" property whereby the contribution of a single anomalous observation to the score function goes to zero as that outlying observation goes to infinity. This is because the weighting matrix C is an empirical variance estimator of the extended score, and the inverse of C plays a major role in the estimation as it assigns smaller weights for dimensions with larger variance. Thus, the QIF automatically downweights grossly unusual observations. This result, however, would not hold for the working-independence structure since in that case the QIF is equivalent to the GEE estimator. Qu & Song (2004) show in their simulation how sufficiently large changes in a few observations can cause drastic effects on the GEE estimator but have only minor effects on the QIF estimator. The ordinary GEE can be made robust by downweighting unusual clusters and/or unusual observations (Preisser & Qaqish, 1996, 1999; He et al. 2002; Cantoni 2004; Cantoni et al. 2005). However, it could be difficult to identify outliers, since if some data do not fit the model well, it is not necessary that they are outliers. In addition, the choice of weighting scheme might not be obvious, and therefore difficult to determine.

4.3

A real data example

In this section we demonstrate how to use the QIF approach in real data analysis. We consider the CD4 data set, described in Kaslow et al. (1987) and a frequentlyused data set in the literature of varying-coefficient modeling (Wu, Chiang & Hoover, 1998; Fan & Zhang 2000; Huang, Wu & Zhou 2002, 2004; Qu & Li

Chapter 3


63

2006). The response of interest is CD4 cell level, a measure of immune system strength, for a sample of HIV-positive men. Covariates include time in years from start of study (TIME), age at baseline (AGE, in years), smoking status at baseline (SMOKE; binary-coded with 1 indicating yes), and CD4 status at baseline (PRE). Measurements were made up to about twice a year for six years, but with some data missing due to skipped appointments or mortality. There were 284 participants, each measured at from 1 to 14 occasions over up to 6 years (the median number of observations was 6, over a median time of about 3.4 years). Some previous work has modeled this dataset with four time-varying linear coefficients:

y(t)

=

(3o(t)

+ (3s(t)SMOKE + (3A(t)AGE + (3p(t)PRE + c:(t).

(4.3)

We chose to extend this model to check for possible second-order terms:

y(t)

=

(3o(t) + (3s(t)SMOKE + (3A(t)AGE + (3p(t)PRE +(3SA(t)SMOKE' AGE + (3sp(t)SMOKE' PRE +(3AP(t)AGE' PRE + (3AA(t)AGE 2 + (3pp(t)PRE2

(4.4)

+ c:(t),

and to center the time variable at 3 to reduce correlation among the resulting terms. In fitting the model, AGE and PRE were also centered at their means to reduce collinearity. We initially modeled each (3 coefficient using a quadratic spline with two evenly spaced knots:

We did not apply a penalty. Combining (4.4) with (4.5) we effectively have a linear model with 45 parameters. Fitting this model under, say, AR-1 working QIF structure, would be very challenging because the nuisance C matrix would be of dimension 90 x 90 and would have to be empirically estimated and inverted, resulting in high sampling instability and perhaps numerical instability. Therefore, we instead start with a working independence structure. The estimated coefficients are depicted in Figure 2. As one might expect, the wobbly curves suggest that this large model overfits, so we will proceed to delete some of the 45 ,),'s to make the fit simpler. If the last four ,),'s for a particular (3 are not significantly different (as a block) from zero we will consider the (3 to be essentially constant in time (a time-invariant effect). If the last three are not significantly different from zero, we will consider the coefficient to be at most a linear function in time (this is equivalent to simply having a linear interaction term between the predictor and time). Otherwise, we will retain the spline form. To test whether a set of ')"s may be zero, we fit the model with and without the added constraint that they be zero, and compare the difference in Q between the models to its null-hypothesis distribution, a chi-squared with degrees of freedom equal to the number of parameters constrained to zero. Because we are using working-independence QIF (for which the number of equations exactly equals the number of coefficients), the full-model QIF will be zero, so the constrained-model QIF serves as the test statistic. It was concluded after some exploratory testing


64

~A

~S

~o

I' /

Vl

J \ ,. - -

0

co.

M

co.

0

\

/

, Vl-

o

2 4 Time

o

6

co.

/,

_

-

"\

\ I

I

I

2

~p ~_~ -

,

\

0 N

4

6

024

6

Time

Time

~SA

~SP

:3-~1 -

I

,"'-"

/

\

/

/

co. 0,

where 8 < O. Rejection of the null hypothesis signifies non-inferiority of the mean of W as compared to the mean of X. In the test for superiority, we want to know if the mean of W is greater than the mean of X by a significant amount, so we can use these same hypotheses with 0 > O. In this case, rejection of the null hypothesis indicates superiority of the mean of W as compared to the mean of X. If we know the variances (J"f and (J"~, then we reject Ho at significance level 0: if

x-w-o

Amy Laird, Xiao-Hua Zhou

110 If E

= Ea > 8, then the power of the above test for rejecting a false

null hypothesis

is given by

We set the above expression in parentheses equal to Zl-{3 and solve for n2 in terms of nl. Therefore the sample sizes necessary in each part of the trial to reject a false null hypothesis are given asymptotically by and

3.3

Test for equivalence

Suppose we want to know whether the means of X and W differ in either direction by some significant amount 8. We might have this question if we were interested in proving that there is no significant difference in the outcome variable between the two arms of the trial. We use the test for equivalence and consider the hypotheses versus

Ha:

lEI < o.

Note that rejection of the null hypothesis indicates equivalence of the means of X and W. If we know the variances O"f and O"~, then we reject Ho at significance level a if or If lEI = lEal < 0, then the power ofthe above test for rejecting a false null hypothesis is given by

So to find the sample size needed to achieve power 1- f3 at solve

lEI = lEal < 8 we must

and hence the sample size necessary in each part of the trial to reject a false null hypothesis are given asymptotically by and

Chapter 5 Study Designs for Biomarker-Based Treatment Selection

111

By comparison with the formula for n2 in the test for equality, we see that the roles of a and (3 have been reversed. Note that in the usual case, in which 8 > 0 and 1 - (3 > a, this hypothesis test requires a greater sample size than do the other three.

4

Sample size calculation

We use the test for equality to compute the number of patients necessary to include in each arm of the trial to reject a false null hypothesis of zero difference between the means of response in the two arms of the trial, with type I error a and type II error 1 - (3. Here we derive this total number for each of the five designs under consideration, similar to [5], except equal variances in each marker-treatment subgroup and 1:1 randomization ratios are not assumed. Similarly to the notation in the last section, let X be the continuous-valued response of a patient in the trial. Let Z be the treatment assignment (A or B) of a patient. Let D be the true (unmeasurable) binary-valued marker status of a patient, and let R be the result of the marker assay. We denote by X ZR the response of a subject with marker assay R who is receiving treatment Z. Let /LZR and a~R denote the mean and variance of response, respectively, among patients with marker assay R and receiving treatment Z. For ease of computation we let p denote a combination of Z- and R-values, which we use for indexing. Finally we let 'Y denote the proportion of people in the general population who are truly marker-negative, that is, 'Y = P(D = 0). To relate Rand D, we let Asens = peR = liD = 1) denote the sensitivity of the assay in diagnosing marker-positive patients and Aspec = peR = OlD = 0) denote the specificity of the assay in diagnosing marker-negative patients. On the other hand, we let w+ = P(D = 11R = 1) denote the positive predictive value of the assay, and B_ = P(D = OIR = 0) denote the negative predictive value. By Bayes' rule,

w+=P(D=lIR=l) P(D = 1,R= 1) peR = 1) peR = liD = l)P(D = 1) - peR = liD = l)P(D = 1) + peR = liD = O)P(D = 0) Asens (1 - 'Y) - Asens(l - 'Y) + (1 - Aspech and similarly we have

B_ = P(D = OIR = 0)

=

Aspec'Y (1 - Asens)(l - 'Y) + Aspec'Y


112

General formula for variance To calculate the variance of patient response in one arm of a trial, we use the general formula

where Varp(E(Xlp)) = Ep(E(Xlp) - E(X))2 = ~all

kP(p = k)(E(Xlp) - E(X)?

and Ep(Var(Xlp))

4.1

=

~all kP(p

= k)Var(Xlp = k).

Traditional design

Note that since the traditional design does not involve a marker assay, the sample size calculation is independent of the assay properties. In the test for equality we test the null hypothesis of equality of means of patient response, Ho : /-LA = /-LB. The expected response in arm 1 of the trial, in which all patients receive treatment A is given by /-LA

= E(XIZ = A) = /-LAO"! + /-LAl(l- "!)

/-LB

= E(XIZ = B) = /-LBO"! + /-LBl(l- ,,!).

and similarly

We calculate the components of the variance of response using the general formula:

We obtain

and

See the Appendix for details of the calculations throughout this section. Therefore the number of patients needed in each arm of the trial to have a test of equality with type I error Q and type II error 1 - f3 are given by and so the total number of patients necessary is n

= nl + n2 = ('" + 1)n2.

Chapter 5

4.2

Study Designs for Biomarker-Based Treatment Selection

113

Marker by treatment interaction design

Let 1/1 denote the difference in means of response between treatments B and A within assay-positive patients, and I/o denote the difference in means of response between the treatments within assay-negative patients. As the primary question that this design addresses is whether there is a marker-treatment interaction, the null hypothesis under consideration is Ho : 1/1 = I/o. Since the quality of the assay determines in which group a given patient will be, the imperfection of the assay will have an impact on the sample size calculation in this design. The expected difference in response among assay-positive and assay-negative patients are given respectively by

1/1

= E(XIZ = B, R = 1) - E(XIZ = A, R = 1) = (/.LBI - /.LAdw+

+ (/.LBO -

/.LAo)(I - w+)

and

Hence we have

We calculate the variance of response among assay-positive patients as such:

Tf = Var(XIZ = B,R = 1) + Var(XIZ = A, R = 1), by independence, and using the general formula

Var(XIZ

= B,R = 1) = VarD(E(XIZ = B,R = I,D)) +ED(Var(XIZ = B,R = I,D)).

Hence the variances of patient response in each marker-treatment group are given by

+ (/.LBo - 1/81)2(1 - w+) + O"~lW+ + O"~o(I - w+) + (/.LBO - 1/81)20_ + O"~l (1 - 0_) + 0"~00I/Al?W+ + (/.LAO - I/Al)2(1 - w+) + O"~lW+ + O"~o(I - w+) I/Ad(I- 0-) + (/.LAO - I/Ado- + O"~l(I- 0_) + 0"~00-.

T11

=

(/.LBI - 1/81)2W+

T10

=

(/.L81 - I/Bd 2 (1 - 0_)

Til = (/.LAI Tio = (/.LAI -

Therefore we have 2 2 2 Tl = TB1 + TAl = [(/.L81 - 1/81)2

+ (/.LAl - I/Ad 2]w+ + [(/.LBO +[0"~1 + 0"~11w+ + [O"~o + O"~ol(I - w+),

1/81)2

+ (/.LAO

- I/Ar)2](1 - W+)

and in the same way, 2

TO

2 2 = TBO + TAO = [(/.LBI - 1/81? + (/.LAl

+[O"~l

+ O"~ll (1 -

0_)

- I/Adl(I - 0_)

+ [(/.LBO -

+ [O"~o + O"~olO-.

1/81)2

+ (/.LAO -

I/Al)2]e_


114

In a large trial, if each patient screened for the trial is randomized, we may expect that the proportion of assay-positive patients in the trial reflects the proportion of assay-positive people we would see in the general population. Alternatively, an investigator may wish to have balanced marker status groups, in which case the prevalence of the marker in the trial is 0.50. In the second case the number of patients needed to be randomized in the trial is, as in [5],

so that the total number needed is n

4.3

= n1 + n2 = 2n2.

Marker-based strategy design

Let lim denote the mean of response in the marker-based arm (M = 1), and lin denote the mean ofresponse in the non-marker-based arm (M = 2). In the test for equality we test the null hypothesis Ho : lim = lin. Note that the imperfection of the assay will affect the assignment of patients to treatments in the marker-based arm, and hence the sample size. The mean of response in the marker-based arm is given by

lim

= E(XIM = 1) = [/tBlW+ + /tBo(l - W+)][Asens(1- "Y) + (1 - Aspech] +[/tA1(1- 0_)

+ /tA oO_][l- Asens(1- "Y) -

(1 - Aspech].

We observe that lin is ItA from the traditional design:

We calculate the variance of response in the marker-based arm using the general formula:

7~

= Var(XIM = 1) =

+ ER[Var(XIM =

VarR[E(XIM = 1, R)]

1, R)].

Then using properties of the study design, we have

VarR[E(XIM

= 1, R)]

= [/tBlW+ + /tBo(l - w+)]2[Asens(1- "Y + (1 - Aspech] +[/tAl(l - 0_)

+ /tA oO_]2[(1- Asens)(l - "Y) + Aspec"Y]

and

ER[Var(XIM =

=

[(/tBl - lIBl)2W+

1, R)]

+ (/tBO -

lIBl)2(1 - w+)

+ (1 - Aspech] +[(/tBl - lIB1)2(1 - 0_) + (/tBO *[(1 - Asens)(l - "Y) + Aspec"Y].

+ 0'11W+ + 0'10(1- W+)]

*f.~sens(1- "Y)

- lIBl)20_

+ 0'11 (1 -

0_)

+ 0'10 0-1

Chapter 5

Study Designs for Biomarker-Based Treatment Selection

115

Now, we observe that the variance of response in the non-marker-based arm is T1 from the traditional design: T;

= T1 = ,(1-,)(/LAO - /LAd 2 + ,O"~o + (1-')0"~1'

Hence, the number of patients needed in each of the trial is and so the total number of patients needed is n

4.4

= nl

+ n2 = ('x + l)n2'

Modified marker-based strategy design

We again let Vm denote the mean of response in the marker-based arm (M = 1) and we now let Vnr denote the mean of response in the non-marker-based arm (M = 2). In the test for equality we test the null hypothesis Ho : Vm = Vnr . Note that the imperfection of the assay will affect the assignment of patients to treatments in each of the arms, and hence the total sample size. Note that Vm is the same as in the marker-based strategy design. We calculate the mean Vnr of response in the non-marker-based arm to be Vnr

= E(XIM = 2) 1

= '" + 1 [(1 -,)(/LBI

+ "'/LAd + ,(/LBO + "'/LAO)],

and we calculate the variance of response in this arm using the general formula: T;r

= Var(XIM = 2) = Var[E(XIM = 2, Z)] + E[Var(XIM = 2, Z)],

which gives T;r

=

_1_[[/LBl(1-,) ",+1

+ /LBo,J 2 + "'[/LAl(1 -I) + /LAo,f + T~ + M1J·

Hence, the number of patients needed in each arm of the trial is and

n2

=

(Zl-/2

so the total number of patients needed is n

4.5

2

2 + Zl-,B) 2 (T + Tnr) 7

(Vm

= nl

- Vnr )2

'

+ n2 = (,x + l)n2'

Targeted design

The means and variances of response in the targeted design are exactly VBl, VAl, T~l' and T1l from the marker by treatment interaction design:

+ (/LBO - /LAo)(1 - W+) /LAd(1 - (L) + (/LBO - /LAO)eVBl)2w+ + (/LBO - vBl)2(1 - W+) + O"~lW+ + O"~o(1- W+) VAd 2w+ + (/LAO - vAd 2(1- W+) + O"~lW+ + O"~o(1 - w+).

VBl = (/LBI - /LAl)W+ VAl = (/LBI T~l = (/LBI T1l = (/LAI -


116

To reject the null hypothesis Ho : VEl = of patients needed in each arm is

V Al

using the test for equality, the number 2

and

n2

=

(ZI-a/2

+ Zl-,B?(~ + Tin) (VEl - VAr)2

and the total number necessary in the trial is n = nI + n2 = (r;, + 1)n2. Our sample size computations pertain only to the test of equality of means, and we would have attained different formulae had we used a different type of hypothesis test.

5

Numerical comparisons of efficiency

We calculated the efficiency of each "alternative" design relative to the traditional design with regard to the number of patients needed to be randomized for each design to have a test of equality with type I error rate DO = 0.05 and type II error rate 1-,8 = 0.80. We evaluated this quantity, the ratio of the number of patients in the traditional design to the number in the alternative design, as a function of the true prevalence (1- ')') of the marker among patients in the population of interest (on the x-axis). In this calculation we considered various values of the sensitivity and specificity of the assay, and the size of the treatment effect for marker-negative patients relative to marker-positive patients. Specifically, we evaluated this quantity for each combination of sensitivity and specificity equal to 0.6, 0.8, and 1.0, and for the cases in which there is no treatment effect in marker-negative patients, and that in which the treatment effect for marker-negative patients is half that of the marker-positive patients. As in [3] and [5], there is no loss of generality in choosing specific values for the means of response. We present results with the following values: Scenario 1:

/-LBI =

2

1, /-LBO

0, /-LAI

=

2

2

= 2

aBI = aBO = aAI = aAO

Scenario 2:

/-LBI =

1, /-LBO

=

0.5, /-LAI

2 2 2 2 a BI = aBO = a Al = a AD

0, /-LAO 1 = ; =

=

0, /-LAO

0, =

0,

1

= .

We assumed here the variance of patient response was constant across all marker and treatment subgroups. Results shown in Figures 6-9 are alterations of those in

[5].

5.1

Marker by treatment interaction design

When there is no treatment effect among marker-negative patients, relative efficiency depends heavily on marker prevalence: for low prevalence, the interaction design is more efficient unless the assay is very poor, while for high prevalence, the traditional design is more efficient. When the treatment effect among markernegative patients is half that of marker-positive patients, the interaction design

Chapter 5 Study Designs for Biomarker-Based Treatment Selection

117

requires a very large number of patients, and the traditional design is much more efficient. Recall that in this calculation we have assumed balanced marker subgroups. Results are very similar if the proportion of marker-positive patients included in the trial reflects the proportion of marker-positive people we would find in the general population, as seen in [5].

5.2

Marker-based strategy design

When there is no treatment effect among marker-negative patients, we see that the traditional design is at least as efficient as the marker-based strategy design, and that the efficiency has very little dependence on the marker prevalence. When the assay is perfectly sensitive, the two designs require the same number of patients, regardless of the specificity. When the assay has imperfect sensitivity however, the traditional design requires fewer patients. On the other hand, when the treatment effect among marker-negative patients is half that of marker-positive patients, the traditional design requires fewer patients regardless of the properties of the assay, and the efficiency depends heavily on the marker prevalence. These results are not surprising since the treatment effect is diluted in the marker-based strategy design.

5.3

Modified marker-based strategy design

The modified marker-based strategy design is much less efficient than the traditional design in each of the situations in the simulation. When there is no treatment effect among marker-negative patients, marker prevalence has almost no bearing on the relative efficiency, while prevalence and efficiency have a more complex relationship in the case where the treatment effect among marker-negative patients is half that of marker-positive patients. As in the marker-based strategy design, the treatment effect is diluted in the modified marker-based strategy design relative to the traditional design.

5.4

Targeted design

The targeted design requires fewer patients to be randomized than the traditional design for every combination of prevalence, sensitivity, and specificity in each of the two scenarios. This result is what we might expect since the targeted design includes only those patients for whom we expect to see a large treatment effect. When there is no treatment effect in marker-negative patients, the relative efficiency gain for the targeted design is especially pronounced, particularly when the sensitivity and specificity of the assay are close to one. The efficiency gain for the targeted design is also greater when the true marker prevalence is low; when the prevalence is 100%, the two designs are identical, and very little efficiency is gained from the targeted design for a marker with a high prevalence in the population. When the treatment effect among marker-negative patients is half that of marker-positive patients, these effects are subdued due to the decreased


118

ability of the marker to divide patients into groups of sharply-differing treatment effect; the marker has smaller predictive value. Not surprisingly, there is very little efficiency gain for the targeted design when the assay is poor.

5 4 3

2

5

0

+

4

'0

+0

3

+0

2

'+0

·+°0

';., and T that vary in dimensionality when a putative motif in V is included or excluded from the CRM. We therefore integrate out the continuous parameters analytically and condition on variables a and T when updating the CRM composition. Let n(u) = (8, p, T, >.) denote the set of all parameters in the model, for a fixed u. Then, the marginalized conditional posterior probability for a module configuration u is:

where only 8 and T are dependent on u; and a and T are the sets of locations and types, respectively, of all putative motif sites (for all the D motifs in V). Thus, only when the indicator Ui for the weight matrix 8 i is 1, do its site locations and types contribute to the computation of (3.2). When we modify the current u by

Chapter 8

Regulatory Motif Discovery: Prom Decoding to Meta-Analysis

187

excluding a motif type, its site locations and corresponding motif type indicators are removed from the computation of (3.2). For EMC, we need to prescribe a set of temperatures, t1 > t2 > ... > tM = 1, one for each member in the population. Then, we define ¢i(Ui) ex: exp[logP(ui I a,T,S)/td, and ¢(U) ex: rr~l¢i(Ui). The "population" U = (U1,··· ,UM) is then updated iteratively using two types of moves: mutation and crossover. In the mutation operation, a unit Uk is randomly selected from the current population and mutated to a new vector Vk by changing the values of some of its bits chosen at random. The new member Vk is accepted to replace Uk with probability min(l, r m ), where rm = ¢k(Vk)/¢k(Uk). In the crossover step, two individuals, Uj and Uk, are chosen at random from the population. A crossover point x is chosen randomly over the positions 1 to D, and two new units Vj and Vk are formed by switching between the two individuals the segments on the right side of the crossover point. The two "children" are accepted into the population to replace their parents Uj and Uk with probability . (1 ,rc) , were h CMV;),7r,~), and let Z = (ZI,'" ,ZN) and L = (L 1 ,'" ,LN) be latent variables denoting the state identity and state lengths. Li is a non-zero number denoting the state length if it is a point where a run of states ends, i.e. if Zj=k, (i-l+1):::;;j:::;;i; Zi+l,Zi-!#·k; l:::;;k:::;;K otherwise.

Chapter 8

Regulatory Motif Discovery: From Decoding to Meta-Analysis

203

The observed data likelihood then may be written as:

L(O; Y) = L LP(YIZ, L, O)P(LIZ, O)P(ZIO). Z L

(6.1)

The likelihood computation (6.1) is analytically intractable, involving a sum over all possible partitions of the sequence Y with different state conformations, and different state lengths (under the state restrictions). However, one can formulate a data augmentation algorithm which utilizes a recursive technique to efficiently sample from the posterior distributions of interest, as shown below. The key is to update the states and state length durations in an recursive manner, after calculating the required probability expressions through a forward summation step. Let an indicator variable It take the value 1 if a segment boundary is present at position t of the sequence, meaning that a state run ends at t (It = 1 {=> L t -=I- 0). In the following, the notation Y[l:tj is used to denote the vector {Yl, Y2, ... ,yd· Define the partial likelihood of the first t probes, with the state Zt = k ending at t after a state run length of L t = l, by the "forward" probability:

at(k, l) = P(Zt = k, L t = l,It = 1, Y[l:tj)' Also, let the state probability marginalized over all state lengths be given by f3t(k) = L,;~1'kat(k,l). Let d(l) = min{D 1 ,'" ,DK} and d(K) = max{D 1 , " ' , D K }. Then, assuming that the length spent in a state and the transition to that state are independent, i.e. P(l, kll', k') = P(L t = llZt = k)Tk1k = Pk(l)Tk1k, it can be shown that

at(k,l)

= P(Y[t-I+1:tjIZt = k)Pk(l) LTk1kf3t-l(k'),

(6.2)

k'#k for 2 ~ t ~ N; 1 ~ k ~ K; l E {d(1),d(l)+l, .. · ,min[d(K),t]}. The boundary conditions are: at(k,l) = 0 for t < l < d(l), and al(k,l) = 7l'kP(Y[1:ljIZI = k)Pk(l) for d(l) ~ l ~ d(K), k = 1,'" ,K. Pk(') denotes the k-th truncated negative binomial distribution. The states and state duration lengths (Zt, Ld (1 ~ t ~ N) can now be updated, for current values of the parameters 0 = (IL, T, cp, 7r, ~), using a backward sampling-based imputation step:

1. Set i = N. Update ZNly,O using P(ZN = kly,O) = 2. Next, update LNIZN = k, y, 0 using

P(LN = llZN = k, y, 0) =

~~~~k)'

P(LN = l, ZN = kly, 0) _ aN(k, l) P(ZN = kly, 0) - f3N(k) .

3. Next, set i=i - LN, and let LS(i) = LN. Let D(2) be the second smallest value in the set {Dl' ... ,DK}' While i > D(2), repeat the following steps: • Draw Zily, 0, ZHLS(i) , LHLS(i) using

where k E {I"" ,K} \ ZHLS(i)'

Qing Zhou, Mayetri Gupta

204

• Draw LilZi , y, 0 using P(L i = llZi, y, 0) • Set LS(i - L i ) = L i , i = i - L i .

6.3

= Q~:f~}

.

Application to a yeast data set

The HGHMM algorithm was applied to the normalized data from the longest contiguous mapped region, corresponding to about 61 Kbp (chromosomal coordinates 12921 to 73970), of yeast chromosome III [47]. The length ranges for the three states were: (1) linker: Dl = {I, 2, 3, ... }, (2) delocalized nucleosome: D2 = {9,'" ,30}, and (3) well-positioned nucleosome: D3 = {6, 7, 8}. It is of interest to examine whether nucleosome-free state predictions correlate with the location of TFBSs. Harbison et al. (2004) used genomewide location analysis (ChIP-chip) to determine occupancy of DNA-binding transcription regulators under a variety of conditions. The ChIP-chip data give locations of binding sites to only a lKb resolution, making further analysis necessary to determine the location of binding sites at a single nucleotide level. For the HGHMM algorithm, the probabilities of state membership for each probe were estimated from the posterior frequencies of visiting each state in M iterations (excluding burn-in). Each region was assigned to the occupancy state k, for which the estimated posterior state probability P(Zi = klY) = E~l J(Z}j) = k)/M was maximum. For all probes, this probability ranged from 0.5 to 0.9. Two motif discovery methods SDDA [16] and BioProspector [30] were used to analyze the sequences for motif lengths of 8 to 10 and a maximum of 20 motifs per set. Motif searches were run separately on linker (L), nucleosomal (N) and delocalized nucleosomal (D) regions predicted by the HGHMM procedure. The highest specificity (proportion of regions containing motif sites corresponding to high binding propensities in the Harbison et al. (2004) data) was for the linker regions predicted by HGHMM: 61% by SDDA and 40% by BP (Table 4). Sensitivity is defined as the proportion of highly TF-bound regions found when regions were classified according to specific state predictions. The highest overall specificity and sensitivity was observed for the linker regions predicted with HGHMM, indicating nucleosome positioning information may aid significantly in motif discovery when other information is not known. Table 4: Specificity (Spec) and Sensitivity (Sens) of motif predictions compared to data from Harbison et al SDDA Linker Deloc Nucl Nucleosomal

7

Spec 0.61 0.19 0.16

BP Sens 0.7 0.8 0.5

Spec 0.40 0.15 0.09

Sens 0.87 0.63 0.43

Conclusion

In this article we have tried to present an overview of statistical methods related to the computational discovery of transcription factor binding sites in genomic DNA

Chapter 8


205

sequences, ranging from the initial simple probabilistic models to more recently developed tools that attempt to use auxiliary information from experiments, evolutionary conservation, and chromatin structure for more accurate motif prediction. The field of motif discovery is a very active and rapidly expanding area, and our aim was to provide the reader a snapshot of some of the major challenges and possibilities that exist in the field, rather than give an exhaustive listing of work that has been published (which would in any case be almost an impossible task in the available space). With the advent of new genomic technologies and rapid increases in the volume, diversity, and resolution of available data, it seems that in spite of the considerable challenges that lie ahead, there is strong promise that many exciting discoveries in this field will continue to be made in the near future.

References [1) Auger, 1. E. and Lawrence, C. E. (1989). Algorithms for the optimal identification of segment neighborhoods. Bull. Math. BioI., 51(1):39-54. [2) Bailey, T. and Elkan, C. (1994). Fitting a mixture model by expectation maximization to discover motifs in biopolymers. Proceedings of the Second International Conference on Intelligent Systems for Molecular Biology, 28-36. [3) Baldi, P., Chauvin, Y., Hunkapiller, T., and McClure, M.A. (1994). Hidden Markov models of biological primary sequence information. Proc. Natl. Acad. Sci. USA, 91, 1059-1063. [4) Barash, Y., Elidan, G., Friedman, N., and Kaplan, T. (2003). Modeling dependencies in protein-DNA binding sites. In RECOMB proceedings, 28-37. [5J Boyer, L.A., Lee, T.L, Cole, M.F., Johnstone, S.E., Levine, S.S., Zucker, J.P., et al. (2005). Core transcriptional regulatory circuitry in human embryonic stem cells. Cell, 122, 947-956. [6) Bussemaker, H. J., Li, H., and Siggia, E. D. (2000). Building a dictionary for genomes: Identification of presumptive regulatory sites by statistical analysis. Proc. Natl. Acad. Sci. USA, 97(18):10096-10100. [7} Bussemaker, H.J., Li, H., and Siggia, E.D. (2001). Regulatory element detection using correlation with expression, Nat. Genet., 27, 167-17l. [8} Chipman, H.A., George, E.1., and McCulloch, R.E. (2006). BART: Bayesian additive regression trees, Technical Report, Univ. of Chicago. [9J Conlon, E.M., Liu, X.S., Lieb, J.D., and Liu, J.S. (2003). Integrating regulatory motif discovery and genome-wide expression analysis, Proc. Natl. Acad. Sci. USA, 100, 3339-3344. [lOJ Das, D., Banerjee, N., and Zhang, M.Q. (2004). Interacting models of cooperative gene regulation, Proc. Natl. Acad. Sci. USA, 101, 16234-16239. [11J Dempster, A. P., Laird, N. M., and Rubin, D. B. (1977). Maximum likelihood from incomplete data via the EM algorithm. J. Roy. Stat. Soc. B,39(1):1-38. [12] Durbin, R., Eddy, S., Krogh, A., and Mitchison, G. (1998). Biological sequence analysis. Cambridge University Press. [13J Felsenstein, J. (1981). Evolutionary trees from DNA sequences: A maximum likelihood approach. J. Mol. Evol., 17, 368-376.

206


[14] Friedman, J.H. (1991). Multivariate adaptive regression splines, Ann. Statist., 19, 1-67. [15] Green, P. J. (1995). Reversible jump MCMC and Bayesian model determination. Biometrika, 82,711-732. [16] Gupta, M. and Liu, J. S. (2003). Discovery of conserved sequence patterns using a stochastic dictionary model. J. Am. Stat. Assoc., 98(461):55-66. [17] Gupta, M. and Liu, J. S. (2005). De-novo cis-regulatory module elicitation for eukaryotic genomes. Pmc. Nat. Acad. Sci. USA, 102(20):7079-7084. [18] Gupta, M. (2007). Generalized hierarchical Markov models for discovery of length-constrained sequence features from genome tiling arrays. Biometrics, in press. [19] Jensen, S.T., Shen, L., and Liu, J.S. (2006). Combining phylogenetic motif discovery and motif clustering to predict co-regulated genes. Bioinjormatics, 21, 3832-3839. [20] Keles, S. et al., van der Laan, M., and Eisen, M.B. (2002). Identification of regulatory elements using a feature selection method, Bioinjormatics, 18, 1167-1175. [21] Kellis, M., Patterson, N., Endrizzi, M., Birren, B., and Lander, E. (2003). Sequencing and comparison of yeast species to identify genes and regulatory elements. Nature, 423,241-254. [22] Krogh, A., Brown, M., Mian, L.S., Sjoander, K., and Haussler, D. (1994). Hidden Markov models in computational biology: Applications to protein modeling. J. Mol. Biol., 235, 1501-153l. [23] Lawrence, C.E., Altschul, S.F., Boguski, M.S., Liu, J.S., Neuwald, A.F., and Wooton, J.C. (1993). Detecting subtle sequence signals: a Gibbs sampling strategy for multiple alignment. Science, 262, 208-214. [24] Lawrence, C. E. and Reilly, A. A. (1990). An expectation-maximization (EM) algorithm for the identification and characterization of common sites in biopolymer sequences. Proteins, 7,41-5l. [25] Li, X., and Wong, W.H. (2005). Sampling motifs on phylogenetic trees. Pmc. Natl. Acad. Sci. USA, 102,9481-9486. [26] Liang, F. and Wong, W. H. (2000). Evolutionary Monte Carlo: applications to cp model sampling and change point problem. Statistica Sinica, 10,317-342. [27] Liu, J.S. and Lawrence, C. E. (1999). Bayesian inference on biopolymer models, Bioinjormatics, 15, 38-52. [28] Liu, J. S., Neuwald, A. F., and Lawrence, C. E. (1995). Bayesian models for multiple local sequence alignment and Gibbs sampling strategies. J. Am. Stat. Assoc., 90,1156-1170. [29] Liu, J. S., Wong, W. H., and Kong, A. (1994). Covariance structure of the Gibbs sampler with applications to the comparisons of estimators and augmentation schemes. Biometrika, 81,27-40. [30] Liu, X., Brutlag, D. L., and Liu, J. S. (2001). Bioprospector: discovering conserved DNA motifs in upstream regulatory regions of co-expressed genes. Pacific Symposium on Biocomputing, 127-138. [31] Liu, Y., Liu, X.S., Wei, L., Altman, R.B., and Batzoglou, S. (2004). Eukaryotic regulatory element conservation analysis and identification using com-

Chapter 8


207

parative genomics. Genome Res. 14, 451-458. [32] Luger, K. (2006). Dynamic nucleosomes. Chromosome Res, 14,5-16. [33] Neuwald, A. F., Liu, J. S., and Lawrence, C. E. (1995). Gibbs Motif Sampling: detection of bacterial outer membrane protein repeats. Protein Science, 4,1618-1632. [34] Moses, A.M., Chiang, D.Y., and Eisen, M.B. (2004). Phylogenetic motif detection by expectation-maximization on evolutionary mixtures. Pac. Smp. Biocomput., 9, 324-335. [35] Rabiner, L. R (1989). A tutorial on hidden markov models and selected applications in speech recognition. Proceedings of the IEEE, 77, 257-286. [36] Sabatti, C. and Lange, K. (2002). Genomewide motif identification using a dictionary model. IEEE Proceedings, 90,1803-1810. [37] Sandelin, A., Alkema, W., Engstrom, P., Wasserman, W.W., and Lenhard, B. (2004). JASPAR: an open-access database for eukaryotic transcription factor binding profiles, Nucleic Acids Res., 32, D91-D94. [38] Schneider, T.D. and Stephens, RM. (1990). Sequence logos: a new way to display consensus sequences. Nucleic Acids Res., 18,6097-6100. [39] Siddharthan, R, Siggia, E.D., and van Nimwegen, E. (2005). PhyloGibbs: A Gibbs sampling motif finder that incorporates phylogeny. PLoS Comput. Biol., 1, e67. [40] Sinha, S., Blanchette, M., and Tompa, M. (2004). PhyME: A probabilistic algorithm for finding motifs in sets of orthologous sequences. BMC Bioinformatics, 5, 170. [41] Sinha, S. and Tompa, M. (2002). Discovery of novel transcription factor binding sites by statistical overrepresentation. Nucleic Acids Research, 30,55495560. [42] Stormo, G. D. and Hartzell, G. W. (1989). Identifying protein-binding sites from unaligned DNA fragments. Proc. Natl. Acad. Sci. USA, 86,1183-1187. [43] Tanner, M. and Wong, W. H. (1987). The calculation of posterior distributions by data augmentation. J. Am. Stat. Assoc., 82,528-550. [44] Thompson, W., Palumbo, M. J., Wasserman, W. W., Liu, J. S., and Lawrence, C. E. (2004). Decoding human regulatory circuits. Genome Research, 10,1967-1974. [45] Wang, T. and Stormo, G.D. (2003). Combining phylogenetic data with coregulated genes to identify regulatory motifs. Bioinformatics, 19, 2369-2380. [46] Wingender, E., Chen, X., Hehl, R, Karas, H., Liebich, 1., Matys, V., Meinhardt, T., Pruss, M., Reuter, I., and Schacherer, F. (2000). TRANSFAC: An integrated system for gene expression regulation. Nucleic Acids Res., 28, 316-319. [47] Yuan, G.-C., Liu, y'-J., Dion, M. F., Slack, M. D., Wu, L. F., Altschuler, S. J. and Rando, O. J. (2005). Genome-scale identification of nucleosome positions in S. cerevisiae. Science, 309, 626-630. [48] Zhao, X., Huang, H., and Speed, T. P. (2004). Finding short DNA motifs using permuted markov models. In RECOMB proceedings, 68-75. [49] Zhou, Q. and Liu, J. S. (2004). Modeling within-motif dependence for transcription factor binding site predictions. Bioinformatics, 20(6):909-916.

208


[50] Zhou, Q. and Liu, J.S. (2008). Extracting sequence features to predict proteinDNA interactions: a comparative study. Nucleic Acids Research, in press. [51] Zhou, Q. and Wong, W.H. (2004). CisModule: De novo discovery of cisregulatory modules by hierarchical mixture modeling. Proc. Natl. Acad. Sci. USA, 101, 12114-12119. [52] Zhou, Q. and Wong, W.H. (2007). Coupling hidden Markov models for the discovery of cis-regulatory modules in multiple species. Ann. Appl. Statist. to appear.

Chapter 9 Analysis of Cancer Genome Alterations U sing Single Nucleotide Polymorphism (SNP) Microarrays Cheng Li *

Samir Amin t

Abstract Loss of heterozygosity (LOR) and copy number changes of chromosomal regions bearing tumor suppressor genes or oncogenes are key events in the evolution of cancer cells. Identifying such regions in each sample accurately and summarizing across multiple samples can suggest the locations of cancerrelated genes for further confirmatory experiments. Oligonucleotide SNP microarrays now have up to one million SNP markers that can provide genotypes and copy number signals simultaneously. In this chapter we introduce SNP-array based genome alteration analysis methods of cancer samples, including paired and non-paired LOR analysis, copy number analysis, finding significant altered regions across multiple samples, and hierarchical clustering methods. We also provide references and summaries to additional analysis methods and software packages. Many visualization and analysis functions introduced in this chapter are implemented in the dChip software (www.dchip.org), which is freely available to the research community.

1 1.1

Background Cancer genomic alterations

A normal human cell has 23 pairs of chromosomes. For the autosomal chromosomes (1 to 22), there are two copies of homologous chromosomes inherited respectively from the father and mother of an individual. Therefore, the copy number of all the autosomal chromosomes is two in a normal cell. However, in a cancer cell the copy number can be smaller or larger than two at some chromosomal regions due to chromosomal deletions, amplifications and rearrangement. These alterations start *Department of Biostatistics and Computational Biology, Harvard School of Public Health, Dana-Farber Cancer Institute, 44 Binney St. Boston, MA, 02115, USA, Email: eli @ hsph. harvard. edu tDepartment of Medical Oncology, Dana-Farber Cancer Institute, 44 Binney St. Boston, MA, 02115, USA.

209

210

Cheng Li, Samir Amin

to happen randomly in a single cell but are subsequently selected and inherited in a clone of cells if they confer growth advantage to cells. Such chromosomal alterations and growth selections have been associated with the conversion of ancestral normal cells into malignant cancer cells. The most common alterations are loss-of-heterozygosity (LOH), point mutations, chromosomal amplifications, deletions, and translocations. LOH (the loss of one parental allele at a chromosomal segment) and homozygous deletion (the deletion of both parental alleles) can disable tumor suppressor genes (TSG) [1]. In contrast, chromosomal amplifications may increase the dosage of oncogenes that promote cell proliferation and inhibit apoptosis. The detection of these alterations may help identify TSGs and oncogenes and consequently provide clues about cancer initiation or growth [2, 3]. Analyzing genomic alteration data across multiple tumor samples may distinguish chromosomal regions harboring cancer genes from regions with random alterations due to the instability of cancer genome, leading to identification of novel cancer genes [4, 5]. Defining cancer subtypes based on genomic alterations may also provide insights into the new classification and treatment of cancer [6, 7].

1.2

Identifying cancer genomic alterations using oligonucleotide SNP microarrays

Several experimental techniques are currently used to identify genomic alterations at various resolutions and aspects. The cytogenetic methods range from fluorescence in situ hybridization (FISH) to spectral karyotyping (SKY) [8] and provide global view of chromosomal organizations at the single cell level. At individual loci, low-throughput methods measure allelic polymorphisms or DNA copy numbers using locus-specific primers, while high-throughput methods, such as digital karyotyping [9] and comparative genomic hybridization (CGH) using chromosomes or micro arrays [10-12], measure DNA copy number changes at many loci simultaneously. Single nucleotide polymorphisms (SNP) are single base pair variations that occur in the genome of a species. They are the most common genetic variations in the human genome and occur on average once in several hundred basepairs. The NCB! dbSNP database (http://www.ncbi.nlm.nih.gov/projects/SNP /) stores about 10 million human SNPs identified by comparing the DNA of different individuals. High-density oligonucleotide SNP micro arrays have also been developed by Affymetrix for high-throughput genotyping of human SNPs. The SNP marker density has increased rapidly during the recent years, from Mapping 10K array (10,000 markers) and Mapping lOOK array set (116,204 markers spaced at 23.6 Kb) to the latest SNP 6.0 array with near one million SNPs [13-16]. The experimental assays require only 250 ng starting DNA sample and one primer to amplify the portions of the genome containing the SNPs interrogated by the array. This complexity-reduced sample is then fragmented, labeled and hybridized to an oligonucleotide SNP array containing probes complementary to the sequences surrounding the SNP loci. The Affymetrix analysis software is commonly used to analyze the scanned array image and compute genotype calls of all the SNPs in a

Chapter 9

Analysis of Cancer Genome Alterations US'ing Single N~Lcleotide ... 211

C

CCTCGGACTAATGGCCATT Probe Sequence

CGGAGCCTGACTACCGGTAA

:::::: gl::I:::1111I11111:11111111'111!1111l:::I::1:':11lllll111:111111111111'11111111\: (A)

(B)

1: (A) The designing scheme of a probe set on the HuSNP array, which contains 1400 SNPs. Five quartets (columns) of oligonucleotide probes interrogate the genotype of a SNP. In the central quartet, the perfect match (PM) probes are complementary to the reference DNA sequences surrounding the SNP with two alleles (denoted as allele A and B). The mismatch (MM) probes have substituted central base pair compared to the corresponding PM probes and they control for cross-hybridization signals. Shifting the central quartet by -1, 1 and 4 basepairs forms additional four quartets. The newer generations of SNP arrays have selected quartets from both forward and reverse strands Figure 6). (B) A genotyping algorithm makes genotype calls for this SNP in three different based on hybridization patterns. For a diploid genome, three genotypes are possible in different individuals: AA, AB and BB

sample with high accuracy (Figure 1) [17,79]. Several groups have pioneered the use of SNP arrays for LOR analysis of cancer [6, 13, 18]. These studies compared the SNP genotype calls or probe signals of a cancer sample to that of a paired normal sample from the same patient (Figure 2), and identified chromosomal regions with shared LOH across multiple samples of a cancer type. They provided a proof of principle that SNP arrays can identify genomic alterations at comparable or superior resolution to microsatelite markers. Additional studies have applied SNP arrays to identify LOH in variouscancer tissue types [7, 19]. In addition, The SNP arrays have been utilized for copy number analysis of cancer samples [20, 21]. probe level signals on the SNP arrays identified genomic amplifications and homozygous deletions, which were confirmed by Q-PCR and arrays CGH on the same samples. These analysis methods have been implemented in several software packages to be used by the research community (see section 5). In this chapter, we will review the analysis methods and software for SNP array applications in cancer genomic studies. We will focus on the methods and software developed by our group since they are representative, but we will also discuss related methods developed by others and future directions.

Cheng Li, 8amir Amin

212

Tumor

Normal

A

A

A B

LOR (subject #58)

B

A

Retention

B

(subject #57)

Non-informative B

(subject #55)

2: Comparing the genotypes of a SNP in paired normal and tumor samples gives LOH calls. Only SNP markers that are heterozygous in normal samples are informative to provide LOH or retention calls

2 2.1

Loss of heterozygosity analysis using SNP arrays LOH analysis of paired normal and tumor samples

For each SNP, there are two alleles existing in the population and are usually represented by A and B (corresponding to two of the four different nucleotides: A/T/G/C). Each person has two copies of every chromosome, one inherited from each parent. Therefore, the genotype of an SNP in one sample can be homozygous (AA or BB) if the SNP alleles in the two chromosomes are the same, or heterozygous (AB) if they are different. Loss of heterozygosity (LOH) refers to the loss of contribution of one parent in selected chromosome regions, due to hemizygous deletion or mitotic gene conversion [23]. LOH events frequently occur when somatic normal cells undergo transformations to become cancerous cells. By comparing the normal and tumor cells from the same patient, the SNPs can be classified as LOH or retention of heterozygosity (or simply "retention") (Figure 2). Afterwards, the LOH data can be visualized along a chromosome (Figure 3), and non-informative calls may be inferred to have LOR or retention calls through the "Nearest neighbor" or "The same boundary" method [22].

2.2

Tumor-only LOH inference

Often paired normal samples for primary tumors or cell lines are not available, and for hematologic malignancies, blood-derived normal DNA is difficult to obtain. As higher density SNP arrays become available, it is possible to use only the tumor

Chapter 9

Analysis of Cancer Genome Alterations Using Single Nu.cleotide ... 213

Figure 3: Observed LOH calls by comparing normal (N) and tumor (T) genotypes at the same SNP. The sample pairs are displayed on the columns (column names only show tumor samples) and the SNPs are ordered on the rows by their p-arm to q-arm positions in one chromosome. Yellow: retention (AB in both N and T), Blue: LOH (AB in N, AA or BB in T), Red: Conflict (AAjBB in N, BBjAA or AB in T), Gray: non-informative (AAjBB in both Nand T), White: no call (no call in N or T)

sample to identify LOH regions. For example, when we see a long stretch of homozygous calls in a chromosome region of a tumor sample, it is highly likely that LOH has happened in this region. If we assume the average heterozygosity is 0.3 and the SNPs are independent the probability of observing N consecutive homozygous SNP calls when no LOH happens is (1 -- O.3)N, which can be very small when N is large, favoring the hypothesis that the region has undergone LOR. Previously, the homozygosity mapping of deletion (HOMOD) method identified regions of hemizygous deletion in unmatched tumor cell lines by defining a region of LOR as more than five consecutive homozygous microsatellite markers [24J. However, applying this approach to SNP array data is not trivial since SNP markers are less polymorphic, the inter-marker distances are variable, and the haplotype structure may lead to interdependence in the genotype calls of neighboring SNPs. In addition, as with other methods there is a small rate of genotyping error. These multiple sources of information are not comprehensively accounted for in a


214

simple probability or counting method. We utilized a Hidden Markov Model (HMM; [25]) to formalize this inference and use these multiple sources of information [26]. HMMs have been used for modeling biological data in diverse applications such as protein and DNA sequence analysis [27-29], linkage studies [30, 31], and array CGH analysis [32]. SNP genotype or signal data along a chromosome are chain-like, have SNP-specific genotype or signal distributions, and have variable inter-marker distances. For tumor-only LOH analysis, we conceptualize the observed SNP calls (Homozygous call A or B, Heterozygous call AB) as being generated by the underlying unobserved LOH states (either Loss or Retention) of the SNP markers (Figure 4). Under the LOH Retention state, we observe homozygous or heterozygous SNP calls with probabilities determined by the allele frequency of an SNP, Under the LOH Loss state, we will almost surely observe homozygous SNP call. However, we need to account for possible SNP genotyping errors at< 0.01 rate [14] and also for wrong mapping of a few markers. These distributions give the "Emission probabilities" of the HMM. haplotype-related SNP-SNP dependence

observed SNPcall

~0 -.@-

~~~b~b7lity-1

/1

undenlying LOH state

RET

I

-.0-·-.0 -.0- -.0- ~

1--8-1

I RET

1--1 t

I RET

I

I

I--~I LOSS 1--

transition probability

Figure 4: Graphical depiction of the elements comprising the Hidden Markov Model (HMM) for LOH inference. The underlying unobserved LOR states (Loss or Retention (RET)) of the SNP markers generate the observed SNP calls via emission probabilities. The solid arrows indicate the transition probabilities between the LOR states. The broken arrows indicate the dependencies between the consecutive SNP genotypes within a block of linkage disequilibrium, which are handled in an expanded RMM model (not shown)

In order to determine the HMM initial probabilities, we next estimated the proportion of the genome that is retained by assuming that no heterozygous markers should be observed in a region of Loss except in the case of genotyping error. To this end, the proportion of the retention genome in a tumor sample was estimated by dividing the proportion of heterozygous markers in the tumor sample by the average rate of heterozygosity of the SNPs in the population (0.35 for the 10K arrays and 0.27 for the lOOK array [14, 15]. This retention marker proportion was used as the HMM initial probability, specifying the probability of observing the Retention state at the beginning of the p-arm, and was also used as the background LOH state distribution in a sample. Lastly, we specified the transition probabilities that describe the correlation between the LOH states of adjacent markers. In a particular sample, the larger the

Chapter 9

Analysis of Cancer Genome Alterations Using Single Nucleotide ... 215

distance between the two markers is, the more likely genetic alteration breakpoints will happen within the interval and make the LOH states of the two markers independent. We use the Haldane's map function B = (1 - e- 2d )/2 [31] to convert the the physical distance d (in the unit of 100 Megabases :::::: 1 Morgan) between two SNP markers to the probability (2B) that the LOH state of the second marker will return to the background LOH state distribution in this sample and thus independent from the LOH state of the first marker. Although Haldane's map function is traditionally used in linkage analysis to describe meiotic crossover events, the motivations for applying it in the context of LOH events are the following: (1) LOH can be caused by mitotic recombination events [33] and mitotic recombination events may share similar initiation mechanisms and hot spots with the meiotic crossover events [34]; (2) the empirical transition probabilities estimated from observed LOH calls based on paired normal and tumor samples agree well with Haldane's map function (see Figure 2 in [26]). We used this function to determine the transition probabilities as follows. If one marker is Retention, the only way for the next adjacent marker to be Loss is that the two markers are independent and the second marker has background LOH state distribution. This happens with probability 2() . Po (Loss), where Po (Loss) is the background LOH Loss probability described previously. If the first marker is Loss, the second could be Loss either due to the dependence between the two markers (occurring with the probability 1- 2()), or due to that the second marker has background LOH state distribution (occurring with the probability 2()· Po (Loss)). Therefore, the probabilities of the second marker being Loss given the LOH status of the current marker are: P (LossJLoss) = 2B· Po (Loss) + (1 - 2B), { P (LossJRetention) = 2() . Po (Loss). These equations provided the marker-specific transition probabilities of the HMM, determining how the LOH state of one marker provides information about the LOH state of its adjacent marker. The HMM with these emission, initial and transition probabilities specifies the joint probability of the observed SNP marker calls and the unobserved LOH states in one chromosome of a tumor sample. The Forward-backward algorithm [29] was then applied separately to each chromosome of each sample to obtain the probability of the LOH state being Loss for every SNP marker, and the inferred LOH probabilities can be displayed in dChip and analyzed in downstream analyses. To evaluate the performance of the HMM and tune its parameters, we applied it to the 10K array data of 14 pairs of matched cancer (lung and breast) and normal cell lines [21]. Only data from autosomes were used. The average observed sample LOH proportion of these paired samples is 0.52 ± 0.15 (standard deviation). We assessed the performance of the HMM by comparing the LOR events inferred using only the tumor data to the LOH events observed by comparing tumor data to their normal counterparts. Since the tumor-only HMM provides only the probability for each SNP being Loss or Retention, rather than making a specific, we used the least stringent threshold to make LOH calls, in which a SNP is called Loss if it has a probability of Loss greater than 0.5, and Retention otherwise. Using


216

this threshold, very similar results were achieved when comparing the inferred LOR from unmatched tumor samples to the observed LOR from tumor/normal pairs (Figure 5). Specifically, we found that 17, 105 of 17, 595 markers that were observed as Loss in tumor/normal pairs were also called Loss in unmatched tumors (a sensitivity of 97.2%), and 15, 845 of 16, 069 markers that were observed as Retention in tumor/normal pairs were also called Retention in unmatched tumors (for a specificity of 98.6%). However, comparison of the tumor only inferences (Figure 5B) to the observed LOH calls (Figure 5A) identifies occasional regions that are falsely inferred as LOH in the tumor only analysis (indicated by arrows), We found that these regions are due to haplotype dependence of SNP genotypes, which can be addressed by extending the HMM model to consider the genotype dependence between neighboring SNPs (Figure 4; see [26] for details). Tumor/normal observations Chro 10 I II III IV

Tumor only inferences I II III IV

1".,,' .

"' :.·1•.·• .•·• 1 10) normal reference samples to analyze together with tumor samples, we propose to use a trimming method to obtain reference signal distribution. Specifically, we assume that, in a set of reference normal samples (or a set of tumor samples when normal samples are not available or too few), for any SNP, at most a certain percent of the samples have abnormal copy numbers at the SNP locus (the abnormal percentage P is user defined, such as 10%). Then for each SNP, (P/2)% of samples with extreme signals are trimmed from the high and low end, and the rest samples are used to estimate the mean and standard deviation of the signal distribution of normal copy number 2 at this SNP. This trimming method is designed to accommodate the small amount of CNVs in reference normal samples. To address the issue (2) above and also test the feasibility of the method proposed in (1), we used a public lOOK SNP array dataset of 90 normal individuals of European ethnicity (https: / / www.affymetrix.com/support / technical/sample_data / hapmap_trio_data. affx). We set abnormal percentage as 5% to search for CNVs in these samples, and this method is able to identify 39 CNVs in the 90 normal samples with size ranging from 120Kb to 16Mb (4 to 411 SNPs) (Figure 9). Some of the CNVs are recurrent in related family members. These identified CNVs in normal samples can be excluded when analyzing lOOK SNP data of tumor samples of Caucasian origin, so that the same CNVs found in tumor samples are not considered in downstream analysis. If some studies contain paired normal samples for tumors, one can screen the normal samples for CNVs and the CNVs found in both normal and tumor will be excluded from cancer locus analysis. In addition, public databases such as Database of Genomic Variants (http://projects.tcag.ca/variationj) and Human Structural Variation Database [48] contain the known CNVs from various studies. The information of the genome position, size, and individual ethnicity of the CNVs in these databases can be used to exclude the known CNVs in normal individuals when analyzing the cancer samples of the corresponding ethnic groups.

Figure 9: The CNVs of chromosome 22 identified in 90 normal individuals using the lOOK Xba SNP array. The rows are ordered SNPs and the columns are samples. The white or light red colors correspond to lower than 2 copies (homozygous or hemizygous deletions). The CNVs are between the region 20.7 - 22 Mb (22qll.22 - ql1.23). The CNV in this region have been reported in other studies [42, 48]. Samples F4 and F6 are from the same family and share a CNV region

224


Furthermore, the SNP-array based methods have also been used to find genomic alterations in other diseases involving chromosomal changes, such as Down syndrome (chromosome 21 has 3 copies), autism spectrum disorder [49, 50] and mental retardation [51]. Similar trimming method as above may be used to identify copy number changes in the samples of these diseases. The DECIPHER database (https:/ / decipher.sanger.ac. uk!) can be queried to check if a CNV is related to a disease phenotype.

3.6

Other copy number analysis methods for SNP array data

Another major research topic in copy number analysis using SNP or CGH arrays is how to infer the underlying real copy numbers in cells based on the raw copy number or log ratios observed from arrays. In addition to the HMM method introduced, median smoothing is a simple but intuitive method. One can set an SNP marker window size (e.g 10) so that a SNP's smoothed copy number is the median of the raw copy numbers of the SNPs in the surrounding window. Compared to the HMM-inferred copy number, this method performs faster and gives closer result to the raw copy numbers. It is also more robust to outliers in raw copy numbers, and does not need parameter specifications as in the HMM method. However, median-smoothed copy numbers are not as smooth as HMMinferred copy numbers, and copy changes smaller than half of the window size will be smoothed out. The circular binary segmentation method has been introduced to analyze array CGH data and identify copy number change points in a recursive manner [52]. This algorithm utilizes the permutation of probe orders to test the significance of candidate change points, based on which the mean and variance of chromosomal segments are estimated piecewise. Furthermore, a regression model is used to formalize the detection of DNA copy number alterations as a penalized least squares regression problem [53]. One study compared 11 different algorithms for smoothing array CGH data, including methods using segmentation detection, smoothing, HMM, regression, and wavelets [54]. These authors also provided a website allowing analysis of array CGH data using multiple algorithms (http://compbio.med.harvard.edu/CGHweb!).

4 4.1

High-level analysis using LOH and copy number data Finding significantly altered chromosome regions across multiple samples

After inferring LOH and copy number altered chromosome regions in individual samples, we are interested in defining the regions of alteration shared by multiple tumors more than random chance. Under the clonal selection process that generates the tumor cells, the alterations at cancer gene loci are more probable to

Chapter 9


be selected in the tumor samples than the alterations elsewhere in the genome. Therefore, the non-randomly shared alteration regions are likely to harbor cancercausing genes or are more susceptible to chromosomal damages in cancer samples, while the rest of the genome contains random alterations due to the genetic instability of tumor. There are existing methods to infer cancer-related chromosome regions from LOH data based on microsatellite markers or array CGH data of multiple samples. The instability-selection modeling of allelic loss and copy number alteration data has been used to locate cancer loci and discover combinations of genetic events associated with cancer sample clusters [57,58]. This method has been applied to a meta-analysis of 151 published studies of LOH in breast cancer [59]. A tree-based multivariate approach focuses on a set of loci with marginally most frequent alterations [60]. The pairwise frequencies among these loci are then transformed into distances and analyzed by tree-building algorithms. The "Cluster along chromosomes" (CLAC) algorithm constructs hierarchical clustering trees for a chromosome and selects the interesting clusters by controlling the False Discovery Rate (FDR) [61]. It also provides a consensus summary statistic and its FDR across a set of arrays. We have developed a scoring method to identify significantly shared LOH regions unlikely to be due to random chance [22]. For paired normal and tumor samples, the first step is to define an LOH scoring statistic across samples. For a particular chromosomal region containing one or more SNPs, we defined a score in each individual to quantify the region's likelihood of being real LOH: the proportion of LOH markers among all the informative markers in this region with some penalty given to conflict LOH calls. We used the proportion of LOH markers rather than the actual counts of LOH markers due to different marker densities at different chromosomal regions. The scores of all individuals are then summed up to give a summary LOH score for this chromosomal region. The second step is to use permutation to assess the significance of the score. Under the null hypothesis that there are no real LOH regions for the entire chromosome (all the observed LOH markers come from measurement error), one can generate the null distribution of the LOH statistic by permuting the paired normal and tumor samples and then obtain LOH scoring statistics based on the permuted datasets. If all the observed LOH events are due to call errors and thus are not cancer-related (the null hypothesis), then the paired normal and tumor samples are conceptually indistinguishable, and the observed differences between them represent the background noise from which we would like to distinguish potentially real LOH events. Therefore, we can create the background noise by permuting the "normal" and "tumor" labels on each pair. We then compare the LOH events in the original dataset with the LOH events in a large number of such permuted datasets to assess the statistical significance of the former. The MaxT procedure [62] is used to adjust the p-values for multiple testing. For each permuted dataset, we obtain the maximal score among all the regions in the genome or a chromosome. The adjusted p-value of a specific region is the proportion of the permuted maximal scores that are greater than the observed LOH score. The significantly shared LOH regions in an SNP dataset of 52 pairs of normal and prostate cancer

226


(A)

(13)

Figure 10: (A) In the dChip LOR view, the tumor-only LOR inference calls LOR (blue color) in multiple prostate samples. The curve and the vertical line on the represent LOR scores and a significance threshold. (B) The 3 Mb region of the LOR scores in (A) contains the known TSG retinoblastoma 1. The cytoband and gene information are displayed on the left. Data courtesy of Rameen Beroukhim

samples harbor the known PTEN tumor suppressor gene [22]. We also tested the method on an breast cancer data where LOR events occur much more frequently [7]. The p-value curve for all chromosomes is able to capture the where the LOR events occur frequently across multiple tumors and the LOR T'lAltl',p,'n cluster samples into two meaningful subtypes. The permutation approach above tests the null hypothesis that the observed shared LOR is due to SNP genotyping or mapping errors and that the normal and tumor labeling is non-informative for producing genomic alterations. This is reflected in the permutation scheme used to randomly switch normal and tumor sample within a sample pair to generate permutation datasets under the null hypothesis. A more direct null hypothesis is that there is no region in the tumor genome that is selected to have more shared LOR events than the rest of the genome regions. In addition, when the paired normal sample is not available for LOH or copy number analysis, we can still use a simple scoring method such as the proportion of samples having LOH or copy alterations to quantify the sharing of LOH (Figure lOA), but permuting paired samples is not an option anymore, In such situations, we propose to permute SNP loci in each sample while preserving most dependence structure of neighboring SNPs. Specifically, for each sample with SNPs ordered first by chromosomes and then by positions within chromosome, we randomly partition the whole genome into K 2) blocks, and randomly switch the order of these blocks but preserving the order of SNPs in each block. In this way the SNPs associated with LOH or copy number alterations in a sample are randomly relocated in blocks to new positions in the genome, while only minimally perturbing the dependence of the LOH or copy number data of neighboring markers. The same permutation applies to all samples using a different random partition for each sample. The LOH or copy number alteration score at

Chapter 9


each SNP locus can then be computed for the permuted dataset, and the MaxT method can be to assess the significance of the original scores. A LOH scoring curve can be computed and displayed next to the LOH data along with a significance threshold to help researchers locate and explore significantly shared LOH regions (Figure 10). Another cancer alteration scoring and permutation method GIS TIC [63] had been successfully applied to SNP array data of glioma and lung cancer samples to identify novel cancer-related genes that were subsequently experimentally confirmed [5].

4.2

Hierarchical clustering analysis

Clustering and classification methods have been very popular and widely used in microarray data analysis since the introduction of gene expression microarrays [64, 65]. Increasingly SNP and array CGH data have been analyzed by these methods since they offer a global view of the genome alterations in cancer samples. It is often interesting to look for the subsets of tumor samples harboring similar alteration events across the genome and correlate such tumor sub clusters with clinical outcomes. Clustering chromosome loci may also reveal genes on different chromosomes that undergo genomic alterations simultaneously and therefore possibly belong to the same oncogenic pathways [66,67]. We have implemented a sample clustering algorithm using the LOH data of one or all chromosomes [22] (Figure llA). The sample distances are defined as the proportion of markers that have discordant Loss or Retention status. This method can discover subtypes of breast cancer and lung cancer based on LOH profiles [6, 7]. However, we have noticed that when using the LOH data of all SNPs for clustering, the result tends to be driven by the SNP markers that are in the "Retention" status across all samples. Therefore, a filtering procedure similar to that of gene expression clustering can be applied to only use those SNPs that have significant alteration scores (Section 4.1) These SNPs have excessive alterations and may harbor known or potential cancer genes, so their LOH status may be more informative for sample clustering. Following similar reasoning, we have implemented a function to filter the SNPs that are located within a certain distance to a list of specified genes (such as all known cancer genes [68] or kinase genes [69]) and use their LOH or copy number status for clustering. Similarly, chromosome regions can be clustered based on LOH or copy number data. A chromosome region can be a whole chromosome, cytoband or the transcription and promoter region of a gene. For the 500K SNP array, a single gene may contain multiple SNPs. The inferred LOH or copy number data for SNPs in a region can be averaged to obtain the genomic alteration data of the region, and the data for regions in turn will be used for region filtering to obtain interesting regions to cluster region and samples, similar to the SNP-wise filtering above. The advantage of this region-based clustering is that we will cluster hundreds of regions instead of 500K to one million SNPs, and chromosome regions can be defined by selected or filtered genes or cytobands to interrogate the genome at various resolutions or according to specific set of genes. For LOH data, we can define the distance between two regions as the average absolute value of

228

Cheng Li, Sarnir Arnin

(A)

(B)

Figure 11: Clustering the same set of breast cancer samples using SNP LOH data (A) and expression data (B). A similar sample cluster emerge in both clustering figures (note the left-most sample branches highlighted in blue). The blue/yellow colors indicate LOH/retention events (A), and the red/blue colors indicate high or low expression level (B). The labels below sample name are lymph node status (p: positive, n: np
>,

0

u

7

6 5 4 3 2 I 0 -I 0

-

......---=

~

!!!!

-

%::1[= =

2

4

6

-

copy number by SNP aITay

CAl

(B)

(C)

F igure 8 of Chapter 9: (A) The agreement between the inferred copy number fro m SNP array (X-ax is) and the Q-PCR based copy number (Y-ax is) o n the sa me sa mples. The error bars represent the standard deviation of Q-PCR based copy number of several loci hav ing the sa me infe rred copy nu mber. T he figure is courtesy of X iaojun Zhao. (B) The LOH events on chromosome 8 of six tumor samples are hig hlighted in blue/dark color.

(C) The correspondi ng raw copy numbers of both n0l111al and tumor samples are shown for the same samples. In the sampl e 1648 (the I st co lumn), the who le chromosome LOH is due to hemi zygous deletio n in the p-arm but gene conversio n in the q-alTl1

Figure 9 of Chapter 9: The CNVs of chromosome 22 identifi ed in 90 normal indi viduals using the l OOK Xba SNP array. The rows are ordered SNPs and the co lumn s are samp les. The white or light red co lors cOITespond to lower than 2 cop ies (homozygous or hemi zygo us de letions). The CNVs are between the region 20.7 - 22 Mb (22q 11.22 -- q 11.23). The CNV in thi s region have been reported in other studies [42 , 48]. Samp le F4 and F6 are from the same fa mil y and share a CNV region

!\:.:!~\;;!i~~,,~g}~

)

(/ ~

I< I

New Developments in Biostatistics and Bioinformatics (Frontiers of Statistics)

Statistics In Drug Research: Methodologies and Recent Developments (Biostatistics, 10)

Frontiers in Statistics

Frontiers in Statistics

Frontiers of Bioinformatics: Unsolved Problems And Challenges

New Frontiers in Economics

New Frontiers in Angiogenesis

New Frontiers in Angiogenesis

New frontiers in economics

New Frontiers

The new avenues in bioinformatics

New Frontiers in Cognitive Aging

New Frontiers in Regenerative Medicine

New developments in parsing technology

New Developments in Differential Equations

New Developments in Biomedical Engineering

New Developments in Advanced Welding

New Frontiers In Polymer Synthesis

New Frontiers in Cognitive Aging

New Frontiers in Asymmetric Catalysis

New Frontiers in Asymmetric Catalysis

New Developments in Parsing Technology

New Developments in Catalysis Research

New Frontiers in Environmental Research

New Developments in Differential Equations

New Developments in Liquid Crystals

New Developments in Asthma Research

New Trends and Developments in Automotive Industry

New developments in formal languages and applications

New Developments and Applications in Sensing Technology

New Developments in Biostatistics and Bioinformatics (Frontiers of Statistics)

Statistics In Drug Research: Methodologies and Recent Developments (Biostatistics, 10)

Frontiers in Statistics

Frontiers in Statistics

Frontiers of Bioinformatics: Unsolved Problems And Challenges

New Frontiers in Economics

New Frontiers in Angiogenesis

New Frontiers in Angiogenesis

New frontiers in economics

New Frontiers

The new avenues in bioinformatics

New Frontiers in Cognitive Aging

New Frontiers in Regenerative Medicine

New developments in parsing technology

New Developments in Differential Equations

New Developments in Biomedical Engineering

New Developments in Advanced Welding

New Frontiers In Polymer Synthesis

New Frontiers in Cognitive Aging

New Frontiers in Asymmetric Catalysis

New Frontiers in Asymmetric Catalysis

New Developments in Parsing Technology

New Developments in Catalysis Research

New Frontiers in Environmental Research

New Developments in Differential Equations

New Developments in Liquid Crystals

New Developments in Asthma Research

New Trends and Developments in Automotive Industry

New developments in formal languages and applications

New Developments and Applications in Sensing Technology

Recommend Documents