Sample Size Tables for Clinical Studies, Third Edition

263 Pages·2008·3.505 MB·English

by  David Machin, Michael J. Campbell, Say Beng Tan, Sze Huey Tan(auth.)

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Sample Size Tables
for Clinical Studies
Sample Size Tables for Clinical Studie s , 3rd edition. D. M., M. J. Campbell, S. B. Tan, and S. H. Tan
© 2009 D. Machin, M.J. Campbell, S.B. Tan, S.H. Tan. ISBN: 978-1-4051-4650-0
9781405146500_1_pre.qxd 9/10/08 13:32 Page ii
To
Oliver, Joshua and Sophie
David, John and Joseph
Lisa and Sophie
Kim San, Geok Yan and Janet
Companion CD-ROM
A companion CD-ROM is included with this book.
The CD contains software that will enable you to implement the sample size calculation
methods discussed in the book.
The CD content is referenced throughout the text where you see this symbol SS
S
See Chapter 17 for full details of the software on the CD and how to install and use it.
Technical requirements:
Operating System: Windows XP, 2000;
CPU: 500 MHz;
RAM: 128 MB;
Disc drive: 16 ×CD drive;
Hard drive: 10 MB of free space
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Sample Size Tables
for Clinical Studies
David Machin
Children’s Cancer and Leukaemia Group
University of Leicester, UK;
Division of Clinical Trials and Epidemiological Sciences
National Cancer Centre, Singapore;
Medical Statistics Unit
School of Health and Related Sciences
University of Sheffield, UK
Michael J. Campbell
Medical Statistics Unit
School of Health and Related Research
University of Sheffield, UK
Say Beng Tan
Singapore Clinical Research Institute, Singapore
Duke–NUS Graduate Medical School, Singapore
Sze Huey Tan
Division of Clinical Trials and Epidemiological Sciences
National Cancer Centre, Singapore
THIRD EDITION
A John Wiley & Sons, Ltd., Publication
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This edition first published 2009, © 2009 by D. Machin, M.J. Campbell, S.B. Tan, S.H. Tan
Previous editions: 1987, 1997
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Library of Congress Cataloging-in-Publication Data
Sample size tables for clinical studies / David Machin...[et al.].a3rd ed.
p. ; cm.
Includes bibliographical references and index.
ISBN 978-1-4051-4650-0
1. Clinical trialsaStatistical methodsaTables. I. Machin, David, 1939–
[DNLM: 1. Research DesignaTables. 2. Clinical Trials as TopicaTables. 3. Sample SizeaTables.
4. Statistics as TopicaTables. W 16 S192 2008]
R853. C55S36 2008
615.5072′4adc22
2008030332
ISBN: 978-1-4051-4650-0
A catalogue record for this book is available from the British Library.
Set in 9.5/13pt Minion by Graphicraft Limited, Hong Kong
Printed and bound in Singapore by COS Printers Pte Ltd
1 2009
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Contents
Preface, viii
1 Basic design considerations, 1
2 Distributions and confidence intervals, 14
Table 2.1 The Normal distribution functionaprobability that a Normally
distributed variable is less than z, 27
Table 2.2 Percentage points of the Normal distribution for αand 1−β, 28
Table 2.3 Values of θ(α, β)=(z +z )2, 28
1−α/2 1−β
Table 2.4 The t-distribution, 29
3 Comparing two independent groups for binary data, 30
Table 3.1 Sample size for the comparison of two proportions, 38
Table 3.2 Sample size for the comparison of two proportions using the odds
ratio (OR), 40
4 Comparing two independent groups for ordered
categorical data, 42
5 Comparing two independent groups for continuous data, 47
Table 5.1 Sample sizes for the two sample t-test with two-sided α=0.05, 54
Table 5.2 Sample sizes for the two sample t-test with unequal variances, 55
Table 5.3 Sample sizes for the one sample t-test with two-sided α=0.05, 57
6 Cluster designs, repeated measures data and more than
two groups, 58
Table 6.1 Multiplying factor for repeated measures designs, 66
7 Comparing paired groups for binary, ordered categorical and
continuous outcomes, 67
Table 7.1 Sample sizes for paired binary data, 82
Table 7.2 Sample sizes for paired continuous data with two-sided α=0.05, 83
8 Comparing survival curves, 84
Table 8.1 Number of critical events for comparison of survival rates (Logrank test), 97
Table 8.2 Number of subjects for comparison of survival rates (Logrank test), 99
Table 8.3 Number of critical events for comparison of two exponential survival
distributions with two-sided α=0.05, 101
v
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vi Contents
9 Equivalence, 102
Table 9.1 Sample sizes for bioequivalence studiesadifference between two means or
ratio of two means, 117
Table 9.2 Sample sizes for testing the equivalence of two means, 118
Table 9.3 Sample sizes for testing the equivalence of two proportions, 120
10 Confidence intervals, 122
Table 10.1 Sample sizes required to observe a given confidence interval width for a
given proportion in a sample from a large population, 136
Table 10.2 Sample sizes required to observe a given confidence interval width for the
difference between two proportionsaindependent groups, 137
Table 10.3 Sample sizes required to observe a proportionate confidence interval width
for the difference between two groups expressed via the odds ratio (OR), 138
Table 10.4 Sample sizes required to observe a given confidence interval width for the
difference between two proportions from paired or matched groups, 139
Table 10.5 Sample sizes required to observe a given confidence interval width to
estimate a single mean or the difference between two means for independent or
matched groups, 141
11 Post-marketing surveillance, 142
Table 11.1 Sample sizes required to observe a total of aadverse reactions with a given
probability 1−βand anticipated incidence λ, 149
Table 11.2 Sample sizes required for detection of a specific adverse reaction with
background incidence, λ, known, 150
0
Table 11.3 Sample sizes required for detection of a specific adverse reaction with
background incidence unknown, 151
Table 11.4 Number of cases to be observed in a case-control study, 152
12 The correlation coefficient, 153
Table 12.1 Sample sizes for detecting a statistically significant correlation coefficient,
157
13 Reference intervals and receiver operating curves, 158
Table 13.1 Sample sizes in order to obtain a required reference intervalaNormal
distribution, 169
Table 13.2 Sample sizes in order to obtain a required reference intervalanon-Normal
distribution, 170
Table 13.3 Sample sizes required to observe a given sensitivity or specificity in
diagnostic accuracy studiesasingle sample, 171
Table 13.4 Sample sizes required to observe a given sensitivity or specificity in
diagnostic accuracy studiesatwo sample unpaired design, 173
Table 13.5 Sample sizes required to observe a given sensitivity or specificity in
diagnostic accuracy studiesatwo sample matched paired design, 175
Table 13.6 Sample sizes required to observe a given confidence interval width for
receiver operating curves (ROC), 177
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Contents vii
14 Observer agreement studies, 179
Table 14.1 Sample sizes required to observe a given confidence interval to estimate
the proportion of disagreements between two observers, 189
Table 14.2 Sample sizes required to observe a given confidence interval to estimate
the within observer variation, 190
Table 14.3 Sample sizes required to observe a given confidence interval to minimise
the number of subjects required to achieve the desired precision in the probability of
their disagreement, Θ , 191
Dis
Table 14.4 Sample sizes required to observe a given confidence interval width for
inter-observer agreement using Cohen’s Kappa, κ, 192
Table 14.5 Sample sizes required to observe a given intra-class correlation, ρ, using
the confidence interval approach, 193
Table 14.6 Sample sizes required to observe a given intra-class correlation using the
hypothesis testing approach with two-sided α=0.05, 194
15 Dose finding studies, 195
16 Phase II trials, 207
Table 16.1 Fleming–A’Hern single-stage Phase II design, 225
Table 16.2 Gehan two-stage Phase II designaStage 1, 226
Table 16.3 Gehan two-stage Phase II designaStage 2, 227
Table 16.4 Simon Optimal and Minimax designs, 228
Table 16.5 Bayesian single threshold design (STD), 229
Table 16.6 Bayesian dual threshold design (DTD), 230
Table 16.7 Case and Morgan EDA design with α=0.05, 231
Table 16.8 Case and Morgan ETSL design with α=0.05, 232
Table 16.9 Simon, Wittes and Ellenberg design, 233
Table 16.10 Bryant and Day design, 235
17 Sample size software SS , 237
S
Cumulative references, 239
Author index, 249
Subject index, 253
A companion CD-ROM of the sample size software SS is included in the inside back cover
S
of this book
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Preface
It is now more than 20years since the first edition of this book and 10years from the second.
The need for evidence-based estimates of the required size of a study is now universally
recognized. Since the second edition the methodology for sample-size calculation has been
widely extended, which is the main reason for a third edition. A second reason is the vastly
improved computing power available. For the first edition, the tabulations were extensive
to obviate separate calculations. A computer program to extend the range of the tables was
available for the second edition.
This edition comes with sample size software SS , which we hope will give the user even
S
greater flexibility and easy access to a wide range of designs, and allow design parameters to be
tailored more readily to specific problems. Further, as some early phase designs are adaptive
in nature and require knowledge of earlier patients’ response to determine the relevant options
for the next patient, a (secure) database is provided for these.
Designing modern clinical research studies requires the involvement of multidisciplinary
teams, with the process of sample size determination not being something that can be done
by the statistician alone. So while software is available that can compute sample sizes (even
from the internet), we feel that it is necessary that such software be complemented with
a book that clearly explains and illustrates the methodology, along with tables. Feedback
from users of earlier editions suggests that this can facilitate planning discussions within the
research team.
Thus a major consideration has been to present the details, which are often complex,
as clearly as possible and to illustrate these with appropriate examples. One objective of this
approach is to encourage the wider use of sample size issues at the design stage in areas such
as laboratory studies, which have been relatively neglected compared to epidemiological
studies and clinical trials.
David Machin
Michael J. Campbell
Say Beng Tan
Sze Huey Tan
Singapore; Leicester and Sheffield, UK
viii
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1
Basic design considerations
SUMMARY
This chapter reviews the reasons why sample-size considerations are important when planning a clinical
study of any type. The basic elements underlying this process including the null and alternative study
hypotheses, effect size, statistical significance level and power are described. We introduce the notation
to distinguish the population parameters we are trying to estimate from the study, from their anticipated
value at the design stage, and finally their estimated value once the study has been completed. In the
context of clinical trials, we emphasise the need for randomised allocation of subjects to treatment.
1.1 Why sample size calculations?
To motivate the statistical issues relevant to sample-size calculations, we will assume that we
are planning a two-group clinical trial in which subjects are allocated at random to one of two
alternative treatments for a particular medical condition and that a single binary endpoint
(success or failure) has been specified in advance. However, it should be emphasised that the
basic principles described, the formulae, sample-size tables and associated software included
in this book are equally relevant to a wide range of design types covering all areas of medical
research: ranging from the epidemiological, to clinical and laboratory-based studies.
Whatever the field of enquiry a well-designed study will have considered the questions posed
carefully and, what is the particular focus for us, formally estimated the required sample size
and will have recorded the supporting justification for the choice. Awareness of the import-
ance of these has led to the major medical and related journals demanding that a detailed
justification of the study size be included in any submitted article as it is a key component for
peer reviewers to consider when assessing the scientific credibility of the work undertaken.
For example, the General Statistical Checklist of the British Medical Journal, asks: ‘Was a
pre-study calculation of study size reported?’
In any event, at a more mundane level, investigators, grant-awarding bodies and medical
product development companies will all wish to know how much a study is likely to ‘cost’ both
in terms of time and resource consumed as well as monetary terms. The projected study size
will be a key component in this ‘cost’. They would also like to be reassured that the allocated
resource will be well spent by assessing the likelihood that the study will give unequivocal
results. In addition, the regulatory authorities, including the Food and Drug Administration
(FDA 1988) in the USA and the Committee for Proprietary Medicinal Products (CPMP 1995)
in the European Union, require information on planned study size. These are encapsulated in
Sample Size Tables for Clinical Studie s , 3rd edition. D. M., M. J. Campbell, S. B. Tan, and S. H. Tan
© 2009 D. Machin, M.J. Campbell, S.B. Tan, S.H. Tan. ISBN: 978-1-4051-4650-0
1
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2 Chapter 1
the guidelines of the International Conference on Harmonisation of Technical Requirements
for Registration of Pharmaceuticals for Human Use (1998) ICH Topic E9.
If too few subjects are involved, the study is potentially a misuse of time because realistic
medical differences are unlikely to be distinguished from chance variation. Too large a study
can be a waste of important resources. Further, it may be argued that ethical considerations
also enter into sample size calculations. Thus a small clinical trial with no chance of detect-
ing a clinically useful difference between treatments is unfair to all the patients put to the
(possible) risk and discomfort of the trial processes. A trial that is too large may be unfair if
one treatment could have been ‘proven’ to be more effective with fewer patients as, a larger
than necessary number of them has received the (now known) inferior treatment.
Providing a sample size for a study is not simply a matter of giving a single number from a
set of tables. It is, and should be, a several-stage process. At the preliminary stages, what is required
are ‘ball-park’ figures that enable the investigators to judge whether or not to start the detailed
planning of the study. If a decision is made to proceed, then the later stages are to refine the
supporting evidence for the early calculations until they make a persuasive case for the final
patient numbers chosen which is then included (and justified) in the final study protocol.
Once the final sample size is determined, the protocol prepared and approved by the
relevant bodies, it is incumbent on the research team to expedite the recruitment processes
as much as possible, ensure the study is conducted to the highest of standards possible and
eventually reported comprehensively.
Cautionary note
This book contains formulae for sample-size determination for many different situations. If
these formulae are evaluated with the necessary input values provided they will give sample sizes
to a mathematical accuracy of a single subject. However, the user should be aware that when
planning a study of whatever type, one is planning in the presence of considerable uncertainty
with respect to the eventual outcome. This suggests that, in the majority of applications, the
number obtained should be rounded upwards to the nearest five, 10 or even more to establish
the required sample size. We round upwards as that would give rise to narrower confidence
intervals, and hence more ‘convincing’ evidence.
In some cases statistical research may improve the numerical accuracy of the formulae which
depend on approximations (particularly in situations with small sample sizes resulting), but
these improvements are likely to have less effect on the subsequent subject numbers obtained
than changes in the planning values substituted into the formulae. As a consequence, we have
specifically avoided using these refinements if they are computationally intensive. In contrast,
and as appropriate, we do provide alternative methods which can easily be evaluated to give
the design team a quick check on the accuracy of their computations and some reassurance on
the output from SS and the tables we provide.
S
1.2 Design and analysis
Notation
In very brief terms the (statistical) objective of any study is to estimate from a sample the
value of a population parameter. For example, if we were interested in the mean birth