3rd Edition

Design and Analysis of Cross-Over Trials

By Byron Jones, Michael G. Kenward Copyright 2015
    438 Pages 51 B/W Illustrations
    by Chapman & Hall

    Design and Analysis of Cross-Over Trials is concerned with a specific kind of comparative trial known as the cross-over trial, in which subjects receive different sequences of treatments. Such trials are widely used in clinical and medical research, and in other diverse areas such as veterinary science, psychology, sports science, and agriculture.

    The first edition of this book was the first to be wholly devoted to the subject. The second edition was revised to mirror growth and development in areas where the design remained in widespread use and new areas where it had grown in importance. This new Third Edition:

    • Contains seven new chapters written in the form of short case studies that address re-estimating sample size when testing for average bioequivalence, fitting a nonlinear dose response function, estimating a dose to take forward from phase two to phase three, establishing proof of concept, and recalculating the sample size using conditional power
    • Employs the R package Crossover, specially created to accompany the book and provide a graphical user interface for locating designs in a large catalog and for searching for new designs
    • Includes updates regarding the use of period baselines and the analysis of data from very small trials
    • Reflects the availability of new procedures in SAS, particularly proc glimmix
    • Presents the SAS procedure proc mcmc as an alternative to WinBUGS for Bayesian analysis

    Complete with real data and downloadable SAS code, Design and Analysis of Cross-Over Trials, Third Edition provides a practical understanding of the latest methods along with the necessary tools for implementation.

    List of Figures

    List of Tables

    Preface to the Third Edition


    What Is a Cross-Over Trial?

    With Which Sort of Cross-Over Trial Are We Concerned?

    Why Do Cross-Over Trials Need Special Consideration?

    A Brief History

    Notation, Models, and Analysis

    Aims of This Book

    Structure of the Book

    The 2×2 Cross-Over Trial


    Plotting the Data

    Analysis Using T-Tests

    Sample Size Calculations

    Analysis of Variance

    Aliasing of Effects

    Consequences of Preliminary Testing

    Analyzing the Residuals

    A Bayesian Analysis of the 2×2 Trial

    Bayes Using Approximations

    Bayes Using Gibbs Sampling

    Use of Baseline Measurements

    Use of Covariates

    Nonparametric Analysis

    Testing λ1 =λ2

    Testing t1 =t2, Given that λ1 =λ2

    Testing π1 =π2, Given that λ1 =λ2

    Obtaining the Exact Version of the Wilcoxon Ranksum Test Using Tables

    Point Estimate and Confidence Interval for Δ =t1 −t2

    A More General Approach to Nonparametric Testing

    Nonparametric Analysis of Ordinal Data

    Analysis of a Multicenter Trial

    Tests Based on Nonparametric Measures of Association

    Binary Data


    McNemar’s Test

    The Mainland–Gart Test

    Fisher’s Exact Version of the Mainland–Gart Test

    Prescott’s Test

    Higher-Order Designs for Two Treatments


    "Optimal" Designs

    Balaam’s Design for Two Treatments

    Effect of Preliminary Testing in Balaam’s Design

    Three-Period Designs with Two Sequences

    Three-Period Designs with Four Sequences

    A Three-Period Six-Sequence Design

    Which Three-Period Design to Use?

    Four-Period Designs with Two Sequences

    Four-Period Designs with Four Sequences

    Four-Period Designs with Six Sequences

    Which Four-Period Design to Use?

    Which Two-Treatment Design to Use?

    Designing Cross-Over Trials


    Variance-Balanced Designs

    Designs with p = t

    Designs with p < t

    Designs with p > t

    Designs with Many Periods

    Optimality Results for Cross-Over Designs

    Which Variance-Balanced Design to Use?

    Partially Balanced Designs

    Comparing Test Treatments to a Control

    Factorial Treatment Combinations

    Extending the Simple Model for Carry-Over Effects

    Computer Search Algorithms

    Analysis of Continuous Data


    Example: INNOVO Trial: Dose–Response Study

    Fixed Subject Effects Model

    Ignoring the Baseline Measurements

    Adjusting for Carry-Over Effects

    Random Subject Effects Model

    Random Subject Effects

    Recovery of Between-Subject Information

    Small Sample Inference with Random Effects

    Missing Values

    Use of Baseline Measurements

    Introduction and Examples

    Notation and Basic Results

    Pre-Randomization Covariates

    Period-Dependent Baseline Covariates

    Baselines as Response Variables

    Incomplete Data

    Analyses for Higher-Order Two-Treatment Designs

    Analysis for Balaam’s Design

    General Linear Mixed Model

    Analysis of Repeated Measurements within Periods

    Example: Insulin Mixtures

    Cross-Over Data as Repeated Measurements

    Allowing More General Covariance Structures

    Robust Analyses for Two-Treatment Designs

    Higher-Order Designs

    Case Study: An Analysis of a Trial with Many Periods

    Example: McNulty’s Experiment

    McNulty’s Analysis

    Fixed Effects Analysis

    Random Subject Effects and Covariance Structure

    Modeling the Period Effects

    Analysis of Discrete Data


    Modeling Dependent Categorical Data

    Types of Model

    Binary Data: Subject Effect Models

    Dealing with the Subject Effects

    Conditional Likelihood

    Binary Data: Marginal Models

    Marginal Model

    Categorical Data

    Example: Trial on Patients with Primary Dysmenorrhea

    Types of Model for Categorical Outcomes

    Subject Effects Models

    Marginal Models

    Further Topics

    Count Data

    Time to Event Data

    Issues Associated with Scale

    Bioequivalence Trials

    What Is Bioequivalence?

    Testing for Average Bioequivalence

    Case Study: Phase I Dose–Response Noninferiority Trial


    Model for Dose Response

    Testing for Noninferiority

    Choosing Doses for the Fifth Period

    Analysis of the Design Post-Interim

    Case Study: Choosing a Dose–Response Model


    Analysis of Variance

    Dose–Response Modeling

    Case Study: Conditional Power


    Variance Spending Approach

    Interim Analysis of Sleep Trial

    Case Study: Proof of Concept Trial with Sample Size Re-Estimation


    Calculating the Sample Size

    Interim Analysis

    Data Analysis

    Case Study: Blinded Sample Size Re-Estimation in a Bioequivalence Study


    Blinded Sample Size Re-Estimation (BSSR)


    Case Study: Unblinded Sample Size Re-Estimation in a Bioequivalence Study That Has a Group Sequential Design


    Sample Size Re-Estimation in a Group Sequential Design

    Modification of Sample Size Re-Estimation in a Group Sequential Design

    Case Study: Various Methods for an Unblinded Sample Size Re-Estimation in a Bioequivalence Study




    Appendix A: Least Squares Estimation

    Case 1

    Case 2

    Case 3




    Byron Jones is a senior biometrical fellow and executive director in the Statistical Methodology Group at Novartis Pharmaceuticals. Previously he was a senior statistical consultant/senior director at Pfizer and a senior director and UK head of the Research Statistics Unit at GlaxoSmithKline. In addition to 14 years of experience in the pharmaceutical industry, he has 25 years of experience in academia, ultimately holding the position of professor of medical statistics at De Montfort University. Currently he is an honorary professor at the London School of Hygiene and Tropical Medicine, visiting professor at University College London and at the University of Leicester, and a visiting professorial fellow at Queen Mary, University of London.

    Michael G. Kenward is GlaxoSmithKline professor of biostatistics at the London School of Hygiene and Tropical Medicine. Previously he held positions at the Universities of Kent and Reading in the UK, and at research institutes in the UK, Iceland, and Finland. He has acted as a pharmaceutical industry consultant in biostatistics for more than 25 years. His research interests include the analysis of longitudinal data and cross-over trials, and modeling in biostatistics, with a particular interest in the problem of missing data. He has co-authored three textbooks and is well known for his 1994 Royal Statistical Society read paper on missing data.

    "Jones and Kenward added several valuable case studies to the third edition of their book. The case studies illustrate elegantly the applications of recent innovations in statistical methodologies to cross-over trials. The new edition is an excellent reference for scientists who want to understand cross-over trials or are interested in learning how statistical advancements in the last decade could be used to expand the versatility of cross-over trials."
    Christy Chuang-Stein, Ph.D., Vice President, Head of Statistical Research and Consulting Center, Pfizer Inc.

    "As in the previous two editions, this edition offers a comprehensive coverage on the design and analysis of cross-over trials. With several major noteworthy updates, it will assist statisticians to conveniently tackle practical issues that arise in a cross-over trial… The most substantial update is the addition of seven new chapters (Chapters 8–14) in the form of short case studies. These real-world examples cover a wide range of issues and solutions above and beyond what is commonly encountered in a cross-over trial and significantly broaden the book…the third edition of Design and Analysis of Cross-Over Trials remains an outstanding reference for statisticians who work on cross-over trials, whether occasionally or frequently."
    —Haiying Chen, Wake Forest School of Medicine, in Journal of the American Statistical Association, Volume 111, 2016

    "Jones and Kenward present students, academics, and researchers with the third edition of their text, dedicated to an understanding of a comparative trait known as the cross-over trial, through which patients involved in a study received different sequences of treatments. New for the third edition, the text includes seven new chapters devoted to case studies, coverage of the R package Crossover, updates related to the use of period baselines and the analysis of very small trials, and a variety of other features."
    Ringgold, Inc. Book News, February 2015

    Praise for the Second Edition:
    "In the second edition, updated from the original published in 1989, the authors have added discussions of new, more comprehensive (downloadable) datasets and some additional topics. ... Substantially updated with more than 130 new references, the book has been thoroughly modernized to reflect new developments in this area. Among the new material added to the book is a chapter on bioequivalence and a discussion of new methods for longitudinal and categorical data. This book continues to be a recommended choice as a valuable reference for clinical statisticians and those who study medical trials where treatments through cross-over design are a feasible approach. For those who already own the first edition, updating to the second will help keep you current on recent developments in this area."
    Journal of the American Statistics Association