Design and Analysis of Cross-Over Trials  book cover
3rd Edition

Design and Analysis of Cross-Over Trials

ISBN 9781439861424
Published October 8, 2014 by Chapman and Hall/CRC
438 Pages 51 B/W Illustrations

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Book Description

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.

Table of Contents

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



View More



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