Statistics in Practice

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Statistics in Practice is an important international series of texts which provide detailed coverage of statistical concepts, methods and worked case studies in specific fields of investigation and study.

With sound motivation and many worked practical examples, the books show in down-to-earth terms how to select and use an appropriate range of statistical techniques in a particular practical field within each title's special topic area.

The books provide statistical support for professionals and research workers across a range of employment fields and research environments. Subject areas covered include medicine and pharmaceutics; industry, finance and commerce; public services; the earth and environmental sciences, and so on.

The books also provide support to students studying statistical courses applied to the above areas. The demand for graduates to be equipped for the work environment has led to such courses becoming increasingly prevalent at universities and colleges. It is our aim to present judiciously chosen and well-written workbooks to meet everyday practical needs. Feedback of views from readers will be most valuable to monitor the success of this aim.

A complete list of titles in this series appears at the end of the volume.

This edition first published 2011
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Library of Congress Cataloging-in-Publication Data

Lui, Kung-Jong, author.
Binary data analysis of randomized clinical trials with noncompliance/Kung-Jong Lui, Department of Mathematics and Statistics, San Diego State University, USA.
p. ; cm.
Includes bibliographical references and index.
ISBN 978-0-470-66095-9 (cloth) – ISBN 978-1-119-99160-1 (epdf) – ISBN 978-1-119-99161-8 (obook) – ISBN 978-1-119-99390-2 (epub) – ISBN 978-1-119-99391-9 (mobi)
1. Clinical trials–Statistical methods. 2. Drugs–Testing–Statistical methods. I. Title.
[DNLM: 1. Randomized Controlled Trials as Topic–methods. 2. Medication Adherence. 3. Statistics as Topic–methods. QV 771]
RM301.27.L85 2011
615.5072$′$4–dc22
2010051072

A catalogue record for this book is available from the British Library.

Print ISBN: 978-0-470-66095-9
ePDF ISBN: 978-1-119-99160-1
oBook ISBN: 978-1-119-99161-8
ePub ISBN: 978-1-119-99390-2
Mobi ISBN: 978-1-119-99391-9

Dedicated to

Professors William G. Cumberland and Abdelmonem A. Afifi at UCLA, as well as Professor Daniel McGee at Florida State University

Preface

In a randomized clinical trial (RCT), it is quite common to encounter patients who do not comply with their assigned treatment due to ethical reasons, patient's decision or the feature of a study design (such as pre-randomized consent designs). Since noncompliance often occurs non-randomly, the commonly-used subgroup analyses, including as-treated (AT) analysis and as-protocol (AP) analysis, are well known to produce a possibly biased inference of treatment efficacy due to the incomparability of the underlying prognostic conditions for patients between two comparison groups. To alleviate this concern, the intent-to-treat (ITT) (or as-randomized (AR)) analysis has been often suggested for a RCT with noncompliance. However, the ITT analysis estimates the programmatic effectiveness rather than the treatment efficacy. Although ITT analysis may provide us with unbiased test for assessing the superiority of an experimental treatment to a standard treatment, the ITT analysis tends to underestimate the relative treatment effect in the presence of noncompliance under certain commonly-assumed conditions. Thus, how to assess the treatment efficacy in a RCT with noncompliance becomes practically useful and important.

The analysis of data for a RCT with noncompliance is generally quite complicated even for the simplest case of a simple noncompliance RCT, in which only patients assigned to the experimental treatment can have access to the experimental treatment. Furthermore, the frequent involvement of sophisticated numerical iterative procedures based on likelihoods to obtain parameter estimates makes this topic even more challenging and difficult for many clinicians and data analysts to appreciate. This book is to focus attention on the level which clinicians with one year of solid training in biostatistics can comprehend, and provides readers with a simple, systematic, and organized approach to study treatment effect for a RCT with noncompliance when the patient response is dichotomous and the noncompliance status is all-or-none in a variety of situations. This book adopts an instructive and easily-understood approach by using contingency tables to explicitly lay down the latent probability structure of observed data so that readers can easily visualize the logics and the ideas behind the development of the proposed test procedures and estimators in a one unified model frame. By contrast, when using the proportion difference (PD) to measure the relative treatment effect, we assume the structural risk additive model based on the model-based approach. While using the proportion ratio (PR) to measure the relative treatment effect, we assume the structural risk multiplicative model. Furthermore, this book presents all test procedures, estimators and sample size calculation procedures in closed forms. Readers may simply use a hand calculator to calculate all the test statistics, interval estimators or sample size calculation formulae without the need of employing any iterative numerical procedures in the situations considered here. For the easy access of the particular topic of reader's interest, this book is written in such a constructive structure that the underlying assumptions, notation, test procedures and formulae in each chapter are self-contained. Readers may directly refer to the particular chapter without the need of reading the details in all the preceding chapters, although I must admit that some assumptions, definitions in notation, and important notes are repeated to avoid confusions in narrative or ambiguities in formulae and findings. Through some real-life examples and computer-simulated data, readers can appreciate the practical usefulness of the test procedures and estimators discussed in this book. The exercises given at the end of each chapter can further help readers better understand the underlying assumptions, the theory and limitations of the proposed test procedures and estimators, as well as other relevant issues and extensions. To facilitate the use of sample size determination presented here, we include in Appendix SAS programs that can be easily modified by readers to accommodate the situations in which they are interested. Despite the book generally adopting the principal stratification approach to account for the effect due to noncompliance, this book also briefly addresses use of the model-based approach (which is related to a quite general class of the structural mean models (SMMs) proposed elsewhere) and notes the relations of parameters and estimators between these two approaches. Because the discussion on the SMM is truly beyond the modest scope of this book, the SMM is not discussed in the book. Readers who are interested in this area may begin with reading a few key references regarding the SMMs cited here.

This book is intended for postgraduates, clinicians, biostatisticians and data analysts. This book can be used as supplemental material for an introductory-level course focusing on clinical statistics or experimental trials in Epidemiology, Psychology and Sociology. This book may also be used as a desk reference for clinicians or biostatisticians when they come across binary data in the presence of noncompliance. To clarify the main issues raised by noncompliance and strengthen the narrative, we explicitly define and discuss some common assumptions and terms encountered in a RCT with noncompliance, as well as include numerical examples to illustrate the bias of most commonly-used subgroup analyses in Chapter 1. Because testing superiority, non-inferiority and equivalence, interval estimation and sample size calculation are all the most fundamental statistical topics for analyzing clinical data, this book concentrates discussions on these when we use the PD, the PR and the odds ratio (OR) to measure treatment efficacy under various frequently-encountered situations. These include parallel groups design (Chapter 2), stratified sampling (Chapter 3), cluster sampling (Chapter 4), parallel sampling with subsequent missing outcomes (Chapter 5) and data in repeated binary measurements (Chapter 6). Clinicians and biostatisticians should find that this book is useful and handy.

I wish to express my indebtedness to my colleagues Drs. Richard Levine, Barbara Bailey and Kristin Duncan at San Diego State University and the five anonymous reviewers who generously provided valuable comments and suggestions on an early draft and outlines of contents of the manuscript. I also wish to thank my wife Jen-Mei, whose continued patience and understanding have endured throughout so many years and made the work much more pleasant than it otherwise would have been. I want to thank my brothers Dan-Yang, Kung-Yi and Kung-Jen for their encouragements in many years. Finally I want to express my deepest appreciation to my parents, Shung-Wu and Li-Ching for their endless love, spiritual support and guidance, which continue to last in my memory.

Kung-Jong Lui
San Diego

About the Author

Kung-Jong Lui is a professor in the Department of Mathematics and Statistics at San Diego State University. He obtained his Ph.D. in biostatistics in 1982, M.S. in biostatistics in 1979, M.A. in Mathematics in 1977, all from UCLA, and B.S. in Mathematics in 1975 at Fu-Jen Catholic University at Taipei, Taiwan. He has had 150 publications in peer-reviewed journals, including Biometrics, Statistics in Medicine, Biometrical Journal, Computational Statistics and Data Analysis, Psychometrika, Journal of Biopharmaceutical Statistics, Drug Information Journal, Contemporary Clinical Trials, Journal of Applied Statistics, Statistical Methodology, Communications in Statistics, Theory and Methods, Science, Nature, Proceedings of National Academy of Sciences, Journal of Official Statistics, IEEE Transactions on Reliability, Environmetrics, Test, American Journal of Epidemiology, American Journal of Public Health, New England Journal of Medicine, Journal of the American Medical Association, etc. He is the author of the book Statistical Estimation of Epidemiological Risk published by Wiley in 2004. He is an Associate Editor for Biometrical Journal. He is a Fellow of the American Statistical Association, a Fellow of the American College of Epidemiology, and a life member of International Chinese Statistical Association.

1

Randomized clinical trials with noncompliance: issues, definitions and problems of commonly used analyses

When comparing an experimental treatment with a standard treatment (or placebo), we often employ a randomized clinical trial (RCT), in which eligible patients (after obtaining their informed consents) are randomized to one of the two treatments under comparison. One of the most fundamental ideas behind use of the RCT is, as shown in Figure 1.1, that all (known or unknown) covariates affecting patients’ responses are expected to balance through randomization. Thus, when there is a difference in the distribution of patient responses between two treatments under perfect compliance, we may attribute this to different treatments they receive between the two randomized groups.

Figure 1.1 Schema for a RCT.

However, noncompliance can often occur in a RCT. When a patient feels that the burden of taking his/her assigned treatment is not worth its perceived benefits, the patient may decide not to comply with his/her assigned treatment (Heitjan, 1999). Noncompliance can also occur as a result of a negative experience of taking a treatment, drug sharing among participated patients, an error in treatment administration by study staff, or even the feature of a pre-randomized study design (Zelen, 1979, 1982, 1986, 1990). Because noncompliance often occurs nonrandomly, simply excluding patients who do not comply with their assigned treatment from data analysis may produce a misleading inference. For convenience in the following discussion, we call the RCT with noncompliance, in which only patients assigned to an experimental treatment group can have access to the experimental treatment, the simple noncompliance RCT.

Example 1.1 Consider the simple noncompliance RCT, in which children who resided in 225 villages randomly selected out of 450 villages were assigned to the intervention group of receiving two large oral doses of vitamin A supplementation, while children who resided in the remaining 225 villages were assigned to the control group of receiving no vitamin A supplementation (Sommer and Zeger, 1991; Sommer, Tarwotjo and Djunaedi et al., 1986). Approximately 20 % of children assigned to the intervention group did not receive vitamin A supplementation, but children assigned to the control group were all assumed to receive no vitamin A supplementation. To investigate whether there was a self-selection bias due to noncompliance, we might compare the mortality rate of children who were assigned to the intervention group but received no vitamin A supplementation with the mortality rate of children who were assigned to the control group. We summarize these data in Table 1.1 (Sommer and Zeger, 1991). For the purpose of illustration, we ignore the intraclass correlation (which will be discussed in Chapter 4) of survival outcomes between children within villages here. When employing the commonly used two independent sample-proportion statistics to test the equality of mortality rates (Fleiss, 1981), we find strong statistical significance (p-value < 0.001) based on the data in Table 1.1. In other words, there is strong evidence that children who declined receiving vitamin A supplementation tended to be in poorer health or at a higher risk of mortality. To help readers easily appreciate the schema of this simple noncompliance RCT, we may use the schema as shown in Figure 1.2.

Table 1.1 The observed cell frequency and the corresponding cell proportion (in parenthesis) in preschool children who were assigned to the intervention group but did not receive the vitamin A supplementation versus those in preschool children who were assigned to the control group and assumed to all receive no vitamin A supplementation.

| Patients assigned to the intervention group but received no vitamin A supplementation | Patients assigned to the control group
Death | 34 (1.4 %) | 74 (0.6 %)
Survival | 2385 (98.6 %) | 11514 (99.4 %)

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