Design and

Analysis of
Cross-Over Trials
Second Edition

Byron Jones
and
Michael G. Kenward

CHAPMAN & HALL/CRC

A CRC Press Company
Boca Raton London New York Washington, D.C.

©2003 CRC Press LLC

disclaimer Page 1 Tuesday, May 29, 2001 9:21 AM

Library of Congress Cataloging-in-Publication Data

Jones, Byron and Kenward, Michael G.

Design and Analysis of Cross-Over Trials: Second Edition / Byron Jones and
Michael G. Kenward

p. cm.
Includes bibliographical references.
ISBN 0-41260-640-2

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 Contents

List of figures

List of tables

Preface to the second edition.

1 Introduction
1.1 What is a cross-over trial?
1.2 With which sort of cross-over trial are we concerned?
1.3 Why do cross-over trials need special consideration?
1.4 A brief history
1.5 Notation, Models and Analysis
1.6 Aims of this book
1.7 Structure of the book

2 The 2 × 2 cross-over trial

2.1 Introduction
2.2 Plotting the data
2.3 The analysis using t-tests
2.4 Sample size calculations
2.5 The analysis of variance
2.6 Aliasing of effects
2.7 Consequences of preliminary testing
2.8 Analyzing the residuals
2.9 A Bayesian analysis of the 2 × 2 trial
2.10 The use of baseline measurements
2.11 The use of covariates
2.12 Nonparametric analysis
2.13 Binary data

3 Higher-order designs for two treatments

3.1 Introduction
3.2 ‘Optimal’ designs
3.3 Balaam’s design for two treatments

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 3.4 The effect of preliminary testing in Balaam’s design
3.5 Three-period designs with two sequences
3.6 Three-period designs with four sequences
3.7 A three-period six-sequence design
3.8 Which three-period design to use?
3.9 Four-period designs with two sequences
3.10 Four-period designs with four sequences
3.11 Four-period designs with six sequences
3.12 Which four-period design to use?
3.13 Which two-treatment design to use?

4 Designing cross-over trials for three or more treat-

ments
4.1 Introduction
4.2 Variance-balanced designs
4.3 Optimality results for cross-over designs
4.4 Which variance balanced design to use?
4.5 Partially balanced designs
4.6 Comparing test treatments to a control
4.7 Factorial treatment combinations
4.8 Extending the simple model for carry-over effects
4.9 Computer search algorithms

5 Analysis of Continuous Data

5.1 Introduction
5.2 The fixed subject effects model
5.3 The random subject effects model
5.4 Analyses for higher-order two treatment designs
5.5 The general linear mixed model
5.6 Analysis of repeated measurements within periods
5.7 Cross-over data as repeated measurements
5.8 Case study: an analysis of a trial with many periods

6 Analysis of Categorical Data

6.1 Introduction
6.2 Binary data: subject effect models
6.3 Binary data: marginal models
6.4 Categorical data
6.5 Further topics

7 Bioequivalence trials
7.1 What is bioequivalence
7.2 Testing for average bioequivalence

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 7.3 Power and sample size for ABE in the 2 × 2 design
7.4 Individual bioequivalence
7.5 Population bioequivalence
7.6 ABE for a replicate design
7.7 Kullback–Leibler divergence for evaluating bioequivalence
7.8 Modelling pharmacokinetic data

A Least squares estimation

B SAS code for assessing ABE, IBE and PBE in replicate

cross-over trials

Bibliography

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 List of figures

2.1 Period 2 vs Period 1 plots.

2.2 Period 2 vs Period 1 plot with centroids.
2.3 Profiles plot for PEFR data.
2.4 Group-by-periods plot for PEFR data.
2.5 Mean differences vs totals
2.6 Histograms
2.7 Quantile-Quantile plots
2.8 Probabilities of Type I and Type II errors
2.9 Mean squared errors of PAR, CROSS and TS
2.10 Coverage probability
2.11 Power curves for PAR,
CROSS and TS (left panel
λd = 0,
middle panel λd = σ/ (n)),right panel λd = 5σ/ (n))
2.12 Contours of power for TS (ρ = 0.6)
2.13 Diagnostic plots (period 1 only)
2.14 Posterior distributions for treatment and carry-over
effects for the PEFR data: P (R|y) - dotted curve,
P (T |R, y) - solid curve, P (T |y) - dashed curve.
2.15 WinBUGS doodle for 2 × 2 cross-over trial
2.16 Scatter plots of log EFT time against IQ for each
group/period category.
2.17 Scatter plots of the subject totals of log EFT time against
IQ for each group.
2.18 Scatter plots of within-subject log EFT differences against
IQ for each group.

3.1 Mean squared errors of PAR, CROSS and TS

3.2 Coverage probability
3.3 Power curves for PAR,
λd = 0,
CROSS and TS (left panel

middle panel λd = 1σ/ (n),right panel λd = 5σ/ (n)))
3.4 Contours of power for TS

5.1 Example 5.3: Subject profiles for each group

5.2 Example 5.3: mean treatment profiles.
5.3 Example 5.5: individual treatment contrast profiles.

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 5.4 Example 5.5: mean treatment contrast profiles.
5.5 Example 5.5: difference in mean treatment contrast
profiles.
5.6 Example 5.6: mean treatment profiles.
5.7 Example 5.6: empirical distribution of the treatment F
statistics.
5.8 Example 5.7: plot of Display means at each Speed
5.9 Example 5.7: fitted period profiles.

7.1 Observed concentration-time profile for a single subject

in a given period
7.2 Subject profile plots for 2 × 2 trial
7.3 Ratios (T/R) for AUC and Cmax for 2 × 2 trial
7.4 Histograms and normal probability plots for 2 × 2 trial
7.5 Concentration-time profile for a single subject in a given
period

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 List of tables

1.1 Treatment sequences used in Example 1.2

2.1 Group 1(AB) Mean morning PEFR (L/min).

2.2 Group 2(BA) Mean morning PEFR (L/min).
2.3 The group-by-period means for the mean PEFR data.
2.4 Group 1(AB) Subject totals and differences.
2.5 Group 2(BA) Subject totals and differences.
2.6 The fixed effects in the full model.
2.7 Total number of subjects required for 2 × 2 cross-over
2.8 Analysis of variance for full model: Sums of squares
2.9 Analysis of variance for full model: Expected mean
squares
2.10 Analysis of variance for the PEFR data
2.11 Power for CROSS, TS and TS corrected
2.12 PEFR data: Studentized residuals for Group 1, Period 1
2.13 PEFR data: Studentized residuals for Group 2, Period 1
2.14 Patel’s (1983) data F EV1 measurements
2.15 Expectations of the responses in each group
2.16 Contrasts for effects of interest
2.17 Skeleton analysis of variance for a 2 × 2 cross-over trial
with an additional categorical covariate
2.18 Log(EFT) and IQ values.
2.19 Log(EFT) and IQ values.
2.20 Log(EFT) and IQ values.
2.21 Data from mouthwash trial.
2.22 Subject totals and differences and their ranks.
2.23 Subject totals and differences and their ranks (first-half
data).
2.24 Group 1(AB) Percentage of nights with no additional
medication.
2.25 Group 2(BA) Percentage of nights with no additional
medication.
2.26 Group 1(AB) Night-time symptom score.
2.27 Group 2(BA) Night-time symptom score.

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 2.28 Centre 1 data from clinical trial for relief of heartburn.
2.29 Centre 2 data from clinical trial for relief of heartburn.
2.30 PROC FREQ results for each centre and combined data
for heartburn trial.
2.31 Duration of exercise in seconds from patients suffering
ischaemic heart disease.
2.32 Columns of combined data for exercise duration trial.
2.33 Values of U-statistics for exercise duration trial.
2.34 Parameter estimates for exercise duration trial (with
carry-over).
2.35 Parameter estimates for exercise duration trial (no
carry-over).
2.36 2 × 2 Binary Cross-over Trial.
2.37 Data from a two-centre 2 × 2 trial on cerebrovascular
deficiency. Outcomes 0 and 1 correspond to abnormal
and normal electrocardiogram readings.
2.38 2 × 2 Binary Cross-over Trial.
2.39 2 × 2 contingency table.
2.40 Mainland-Gart Contingency Table.
2.41 Mainland-Gart Table for Example 2.5.
2.42 Contingency table for Prescott’s test.

3.1 Design 3.1

3.2 Design 3.2
3.3 Design 3.3
3.4 Design 3.4
3.5 Design 3.5
3.6 Expectations of ȳij. for direct-by-carry-over interaction
model
3.7 Design 3.2.1
3.8 Design 3.2.2
3.9 Design 3.2.3
3.10 The expectations of ȳij. for Design 3.2.1
3.11 The variances and covariances for the three-period designs
2
(in multiples of σn )
3.12 Design 3.4.1
3.13 Design 3.4.2
3.14 Design 3.4.3
2
3.15 Variances of the estimators (in multiples of σn )
3.16 Design 3.6.1
3.17 The variances of the estimators (in multiples of σ 2 /N )
3.18 The variances of the estimators (in multiples of σ 2 /N )

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 3.19 The variances of the estimators (in multiples of σ 2 /N )
assuming that a treatment cannot carry over into itself
3.20 Design 4.2.1
3.21 Design 4.2.2
3.22 Design 4.2.3
3.23 Design 4.2.4
3.24 Design 4.2.5
3.25 Design 4.2.6
3.26 Design 4.2.7
3.27 Variances (in multiples of σ 2 /N ) of effects obtained from
Designs 4.2.1 – 4.2.7
3.28 Design 4.4.13
3.29 Variances of the estimators (in multiples of σ 2 /n)
3.30 Variances of the estimators (in multiples of σ 2 /n)
3.31 Design 4.6.136
3.32 Design 4.6.146
3.33 Variances (in multiples of σ 2 /n) of effects obtained from
Designs 4.6.136 and 4.6.146
3.34 Variances (in multiples of σ 2 /n) of effects obtained from
Designs 4.6.136 and 4.6.146
3.35 The variances of the estimators (in multiples of σ 2 /N )
3.36 The variances of the estimators (in multiples of σ 2 /N )

4.1 Latin square design for four treatments (18.18, 12.50)

4.2 Orthogonal Latin square design for three treatments
(80.00, 44.44)
4.3 Orthogonal Latin square design for four treatments
(90.91, 62.50)
4.4 Balanced Latin square design for four treatments
(90.91,62.50)
4.5 Two balanced Latin squares for six treatments (96.55,
77.78)
4.6 Balanced Latin square design for nine treatments (98.59,
86.42)
4.7 Russell nearly balanced Latin square design for five
treatments (83.86 or 78.62, 63.73 or 59.75)
4.8 Russell nearly balanced Latin square design for seven
treatments (92.62, 92.55 or 90.18, 77.50, 77.44 or 75.46)
4.9 Prescott triple Latin square design for five treatments
(94.94, 72.00)
4.10 Prescott triple Latin square design for seven treatments
(97.56, 81.63)

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 4.11 Anderson and Preece locally balanced design for seven
treatments (97.56, 81.63)
4.12 One of Patterson’s incomplete designs for seven treat-
ments (79.84, 57.03)
4.13 Incomplete design for four treatments obtained using a
BIB (71.96, 41.98)
4.14 Balanced extra-period design for three treatments (93.75,
75.00)
4.15 Balanced extra-period design for four treatments (96.00,
80.00)
4.16 Quenouille design for t = 3 (100.00, 80.56)
4.17 Berenblut’s design for t = 3 (100.00, 85.94)
4.18 Anderson training-schedule design for seven treatments
(97.41, 89. 95)
4.19 Nearly strongly balanced design for t = 8 (99.90 or 99.79,
93.26 or 93.16)
4.20 Nearly strongly balanced design for t = 5 (99.87 or 99.75,
88.89 or 88.78)
4.21 Balanced designs for t = 3 and 4
4.22 Balanced designs for t = 5 and 6
4.23 Balanced designs for t = 7, 8 and 9
4.24 Biswas and Raghavarao design for t = 4, p = 3
4.25 PB design constructed from PBIB(2) design S1
4.26 PB cyclic design generated by (0132) and (0314)
4.27 Numbers of treatments, periods and subjects
4.28 Iqbal and Jones cyclic design generated by (06725381)
4.29 Some PB cross-over designs for t ≤ 9 and p ≤ t
4.30 Some PB cross-over designs for t ≤ 9 and p ≤ t, continued
4.31 A control balanced design for t = 3
4.32 Variances of contrasts in Pigeon’s designs
4.33 Generalized cyclic design for the 2 × 2 factorial
4.34 Split-plot design for the 2 × 2 factorial
4.35 Designs found using John/Russell/Whitaker algorithm
4.36 Variance (in multiples of σ 2 /n) of treatment and carry-
over parameters
4.37 Variance (in multiples of σ 2 /n) of treatment and carry-
over parameters
4.38 Variance (in multiples of σ 2 /n) of treatment and carry-
over parameters

5.1 Example 5.1: treatment occurrence by period in the

INNOVO design.

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 5.2 Example 5.1: Post-ductal arterial oxygen tension (kPa)
from the INNOVO trial (Base: baseline measurement,
Resp: post-treatment measurement.)
5.3 Example 5.2: data from the three-treatment two-period
design.
5.4 Example 5.2: recovery of between-subject information
using simple weighted and REML estimates, with carry-
over effects in the model.
5.5 Example 5.2: small sample adjusted standard errors and
degrees of freedom.
5.6 Morning PEFR from the COPD trial, with created
missing values.
5.7 Example 5.3: average scores for amantadine trial.
5.8 Example 5.3: group-by-period means
5.9 Example 5.4: systolic blood pressures from a three-period
design with four groups – sequence groups ABB and
BAA.
5.10 Example 5.4: systolic blood pressures from a three-period
design with four groups – sequence groups ABA and
BAB.
5.11 Example 5.4: period means for each sequence group.
5.12 Example 5.5: blood sugar levels (mg %)
5.13 Example 5.5: calculated treatment contrasts.
5.14 Example 5.6: systolic blood pressures (in mm Hg) from
a three-period cross-over trial; S: subject, P: period, T:
treatment.
5.15 Example 5.6: Wald tests from analyzes and with fixed,
and random, subject effects.
5.16 Example 5.4: subject contrasts for λ | τ and τ – Group
ABB.
5.17 Example 5.4: subject contrasts for λ | τ and τ – Group
BAA.
5.18 Example 5.4: subject contrasts for Group 3 (ABA).
5.19 Example 5.4: subject contrasts for Group 4 (BAB).
5.20 Example 5.6: trial on intermittent claudication, design
and LVET measurements (ms)
5.21 Example 5.6: conventional analysis with fixed subject
effects.
5.22 Example 5.6: variance-correlation matrix (REML esti-
mate)
5.23 Example 5.6: analysis using an unstructured covariance
matrix.

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 5.24 Example 5.6: variance-correlation matrix (OLS based
estimate).
5.25 Example 5.7: Williams Latin Square used in McNulty’s
Experiment.
5.26 Example 5.7: Treatment labels in McNulty’s Experiment.
5.27 Example 5.7: Complete design for Experiment 1.
5.28 Example 5.7: data from McNulty’s Experiment 1 (S:
subject, R:replicate).
5.29 Example 5.7: McNulty’s analysis of variance (data
averaged over replicates).
5.30 Example 5.7: Treatment combination means (each of 16
observations).
5.31 Example 5.7: fitting treatment and carry-over effects
(Type II sums of squares).
5.32 Example 5.7: F tests after dropping carry-over effects.
5.33 Example 5.7: factorial effects for treatments (Type I SS).
5.34 Example 5.7: factorial effects for treatments and carry-
overs (Type I SS).
5.35 Example 5.7: factorial effects for treatments and carry-
over of Display (Type I SS).
5.36 Example 5.7: least squares means, adjusted for carry-overs
effect of Display.
5.37 Example 5.7: random subjects effects model.
5.38 Example 5.7: random subjects and AD(1) covariance
structure.
5.39 Example 5.7: quadratic period profile.
5.40 Example 5.7: cubic spline period profile.

6.1 Example 6.1: Binary data from a four-period cross-over

trial
6.2 Example 2.5, Centre 2: significance probabilities from
asymptotic and randomization versions of the Mainland-
Gart test. (1) and (2) conventional chi-squared tests,
without and with Yates’ correction; (3) and (4) Wald
tests based on tables with one half added to (3) all cells,
and (4) only zero cells.
6.3 Example 6.2: inferences for the B-A treatment difference.
6.4 Example 6.1: results from the subject specific analyzes.
6.5 Example 6.1: raw marginal probabilities.
6.6 Example 2.5, both centres: subject-specific and marginal
analyzes.
6.7 Example 6.1: treatment effects from the subject specific
and marginal analyzes.

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 6.8 Example 6.2: data from the trial on pain relief.
6.9 Example 6.2: results from the random subject effects and
marginal analyzes.
6.10 Example 6.2: data set restructured for fitting a partial
proportional odds model.

7.1 Bioequivalence trial: RT sequence

7.2 Bioequivalence trial: TR sequence
7.3 Replicate design for bioequivalence trial
7.4 Replicate design cross-over study with Test (T) and
Reference (R) formulations
7.5 Replicate design cross-over study with Test (A) and
Reference (B) formulations (cont’d)

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 This book is Dedicated to To Hilary, Charlotte and Alexander
and to Pirkko

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 Preface to the second edition.

This book is concerned with a particular sort 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, which appeared
in 1989, was the first to be wholly devoted to the subject. We remarked
in the preface to the first edition that there existed a “large and growing
literature.”. This growth has continued during the intervening years, and
includes the appearance of three other books devoted to the topic. Nat-
urally the newer developments have not been spread uniformly across
the subject, but have reflected both the areas where the design has re-
mained of continued in widespread use and new areas where it has grown
in importance. Equally naturally, some of the literature also reflects the
particular interests and idiosyncracies of key researchers. This new edi-
tion of the book reflects those areas of development that we regard as
important from a practical perspective, but where material is as ap-
propriate and relevant now as it was before, we have kept the previous
structure more or less intact.
In the first edition we started with a chapter wholly devoted to the
two-period two-treatment design. The aim was that this should be, to
a large extent, self-contained and mostly at a level that was accessible
to the less statistically experienced. It was intended that this chapter
alone would be sufficient for those who had no need to venture beyond
this simple design. We have kept this structure, and aim, for the second
edition but have enlarged the chapter somewhat, both to incorporate
intervening developments, and to increase the degree of self-sufficiency
by including more on nonparametric analyzes and simple analyzes for
binary data.
One major change over the last 13 years has been the development of
very general tools for the analysis both for continuous and discrete de-
pendent data that are available in widely used computer packages. This
has meant that we are now able to bring virtually all the analyzes in the
book into a small number of general frameworks, and in turn, this has
allowed a more coherent development of the methods for analysis. This is
reflected in two new chapters (5 and 6) on the analysis of continuous and

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categorical cross-over data, respectively. These have absorbed the previ-
ous chapters and sections on analysis that were scattered throughout the
first edition of the book. Our approach to the analysis of categorical data
in particular has been much affected by the many developments of the
past 13 years. The wide availability of the necessary software tools has
also meant that we have been able to provide the required statements
to allow the reader to apply the methods described in these packages.
The two-treatment cross-over trials have become increasingly widely
used in recent years in bio-equivalence trials. The special nature of the
analyzes associated with such trials have evolved to the point that we
have included an entirely new chapter to this topic. The methods used
there rest heavily on the framework developed in Chapter 5.
We remain indebted to those who helped with their advice on the
first edition of the book: David Fletcher, Peter Freeman, Andy Grieve,
Matthew Koch, Sue Lewis, John Matthews, and Harji Patel; and to those
who gave us permission to use their unpublished data: Peter Blood, Ed
Bryant, Arend Heyting, John Lewis, Jorgen Seldrup and Robert Wood-
field. We remain grateful to the companies, Ciba-Geigy Pharmaceuticals,
ICI Pharmaceuticals, Duphar B.V, the Ortho Pharmaceuticals Corpora-
tion and the Upjohn company for permission to use their data. For this
second edition we must also thank Diana Elbourne, GlaxoSmithKline
Pharmaceuticals and Pauline McNulty for permission to use data.
We are also grateful to the following for the various sorts of help they
have given us: to Gary Koch and Gail Tudor for reading and comment-
ing on the sections on nonparametric methods, to Gary Koch for many
helpful suggestions and support over the years, to Scott Patterson for
reading and commenting on Chapter 7, for providing the datasets used
in Chapter 7, for providing the SAS code for analyzing bioequivalence
data and for sharing his extensive knowledge and experience of bioequiv-
alence trials, to Alan Owen for writing the basic BUGS code on which
the WinBUGS code in Chapter 2 was based, John Cotton for many
helpful discussions over the years and for introducing us to Pauline Mc-
Nulty’s PhD thesis, to David Spiegelhalter for permission to reproduce
the WinBUGS doodle in Chapter 2, to Maiser Asghar for help in process-
ing the data from the COPD trial used in Chapter 2 and to Nye John
and David Whitaker for allowing us access to their unpublished work
on the construction of optimal cross-over designs and access to their
computer package CrossOver that searches for optimal designs. Finally
Byron Jones is grateful to Frank Rockhold, Darryl Downing and Va-
lerii Fedorov at GlaxoSmithKline Pharmaceuticals for the support given
during the writing of this book.
This book as been typeset using the LaTeX system, and we are grateful
to the staff at Chapman& Hall/CRC for their help with this.

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 Of course, we take full responsibility for any errors or omissions in the
text.

Computer software mentioned in the text

CrossOver Version 1.0: D. Whitaker and J. A. John. A package for

the computer generation of crossover designs. Department of Statis-
tics, University of Waikato, New Zealand.
GenStat - Sixth Edition: Lawes Agricultural Trust. Supplied by VSN
International, Wilkinson House, Jordan Hill Road, Oxford, UK.
LogXact: Cytel Software Corporation, 675 Massachusetts Ave., Cam-
bridge, MA 02139, USA.
MLwiN: Centre for Multilevel Modelling, Institute of Education, 20
Bedford Way, London WC1H 0AL, UK.
SAS: SAS Institute Inc., SAS Campus Drive, Cary, North Carolina
27513, USA.
Splus 6.1 for Windows: Insightful Corporation, 1700 Westlake Av-
enue N, Suite 500, Seattle, Washington 98109, USA.
Stata: Stata Corporation, 702 University Drive East, College Station,
Texas 77840, USA.
StatXact: Cytel Software Corporation, 675 Massachusetts Ave., Cam-
bridge, MA 02139, USA.
WinBUGS: MRC Biostatistics Unit, Institute of Public Health, Uni-
versity Forvie Site, Robinson Way, Cambridge CB2 2SR, UK.

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