Table Of ContentSpringer Series in Pharmaceutical Statistics
Meinhard Kieser
Methods and
Applications
of Sample Size
Calculation and
Recalculation in
Clinical Trials
Springer Series in Pharmaceutical Statistics
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Meinhard Kieser
Methods and Applications
of Sample Size Calculation
and Recalculation in Clinical
Trials
123
Meinhard Kieser
Institute of Medical Biometry
andInformatics
University of Heidelberg
Heidelberg, Baden-Württemberg, Germany
ISSN 2366-8695 ISSN 2366-8709 (electronic)
SpringerSeries inPharmaceutical Statistics
ISBN978-3-030-49527-5 ISBN978-3-030-49528-2 (eBook)
https://doi.org/10.1007/978-3-030-49528-2
MathematicsSubjectClassification: 62P10
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To Yvonne
Preface
Correctcalculationoftherequiredsamplesizeisessentialformedicalresearchpro-
jects.Inclinicaltrials,againinknowledgecanonlybeachievedifthesamplesize
ofthestudyissufficientlyhigh.Atthesametime,thesamplesizeshouldnotbelarger
thannecessaryduetoethicalandeconomicreasons.Therefore,samplesizecalcula-
tionisoneofthemostfrequentandimportanttopicsofbiostatisticalconsultingaswell
asamajorissuewhenassessingclinicaltrialprotocolsinethicscommittees.
This book provides methods for sample size calculation and recalculation for a
wide range of scenarios encountered when planning clinical trials. To facilitate a
deeper understanding of the procedures, the derivations of the methods are pre-
sented within a common general framework. Furthermore, application of the
methods isillustrated by real clinical trial examples and source code written in the
software R implementing the presented approaches is provided. In addition,
numeroushintsaregiventopublishedcontributionsonthetopic.Bythis,thebook
aimsatgivinganup-to-dateoverviewonsample sizecalculationandrecalculation
addressing both application-oriented practitioners and biostatisticians who are also
interested in the theoretical background of the methods they apply.
Numerouscolleagueshavecontributedtothesuccessofthisbook.Firstofall,I
would like to thank Andrea Wendel for her extremely patient and careful writing
of the manuscript. I gratefully acknowledge the very valuable contributions made
by Dr. Marietta Kirchner and Dr. Katrin Jensen that considerably improved the
presentationandcontent.Dr.LauraBennerandDr.SvenjaSchülerkindlyprepared
the R code implementing the presented sample size methods, performed compu-
tations,andreviewedvariouschapters.Largepartsofthebookarebasedoncourses
I held within the Master Program “Medical Biometry/Biostatistics” conducted by
the Institute of Medical Biometry and Informatics (IMBI) at the University of
Heidelberg and within summer schools on the topic. The book profited from dis-
cussions with the students as well as from the thorough review of draft versions
of the book by my colleagues at the IMBI Lukas Baumann, Rouven Behnisch,
Samuel Kilian, Johannes Krisam, Jan Meis, Maximilian Pilz, and Stella Erdmann,
who also contributed R code, performed computations, and prepared figures. In
addition,IamgratefultoSpringerPublishersandtheEditorsoftheSpringerSeries
vii
viii Preface
in Pharmaceutical Statistics for giving me the opportunity to publish this book.
Aspecialthank goesto Dr. Eva Hiripi from Springerfor herimportant advice and
support. Definitely, I would not have been able to prepare the work without the
continuing encouragement by my wife Yvonne. Her unique ability to inspire and
motivate and to ensure at the same time welcome distractions created perfect
conditions for writing this book.
Heidelberg, Germany Meinhard Kieser
April 2020
Contents
Part I Basics
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1 Background and Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.1 The ChroPac Trial . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.2 The Parkinson Trial . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 General Considerations When Calculating Sample Sizes . . . . . . 7
2 Statistical Test and Sample Size Calculation. . . . . . . . . . . . . . . . . . 11
2.1 The Main Principle of Statistical Testing . . . . . . . . . . . . . . . . . 11
2.2 The Main Principle of Sample Size Calculation . . . . . . . . . . . . 14
Part II Sample Size Calculation
3 Comparison of Two Groups for Normally Distributed Outcomes
and Test for Difference or Superiority . . . . . . . . . . . . . . . . . . . . . . 19
3.1 Background and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 z-Test. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.3 t-Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.4 Analysis of Covariance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.5 Bayesian Approach. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.5.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.5.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4 Comparison of Two Groups for Continuous and Ordered
Categorical Outcomes and Test for Difference or Superiority. . . . . 37
4.1 Background and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.2 Continuous Outcomes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
ix
x Contents
4.3 Ordered Categorical Outcomes . . . . . . . . . . . . . . . . . . . . . . . . 40
4.3.1 Assumption-Free Approach. . . . . . . . . . . . . . . . . . . . . 40
4.3.2 Assuming Proportional Odds. . . . . . . . . . . . . . . . . . . . 43
5 Comparison of Two Groups for Binary Outcomes
and Test for Difference or Superiority . . . . . . . . . . . . . . . . . . . . . . 47
5.1 Background and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.2 Asymptotic Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.2.1 Difference of Rates as Effect Measure . . . . . . . . . . . . . 48
5.2.2 Risk Ratio as Effect Measure . . . . . . . . . . . . . . . . . . . 53
5.2.3 Odds Ratio as Effect Measure. . . . . . . . . . . . . . . . . . . 55
5.2.4 Logistic Regression . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.3 Exact Unconditional Tests. . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5.3.1 Basic Concepts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5.3.2 Fisher-Boschloo Test . . . . . . . . . . . . . . . . . . . . . . . . . 61
6 Comparison of Two Groups for Time-to-Event Outcomes
and Test for Difference or Superiority . . . . . . . . . . . . . . . . . . . . . . 65
6.1 Background and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
6.1.1 Time-to-Event Data . . . . . . . . . . . . . . . . . . . . . . . . . . 65
6.1.2 Sample Size Calculation for Time-to-Event Data . . . . . 67
6.2 Exponentially Distributed Time-to-Event Data . . . . . . . . . . . . . 68
6.3 Time-to-Event Data with Proportional Hazards. . . . . . . . . . . . . 70
6.3.1 Approach of Schoenfeld . . . . . . . . . . . . . . . . . . . . . . . 71
6.3.2 Approach of Freedman. . . . . . . . . . . . . . . . . . . . . . . . 72
7 Comparison of More Than Two Groups and Test
for Difference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
7.1 Background and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
7.2 Normally Distributed Outcomes. . . . . . . . . . . . . . . . . . . . . . . . 79
7.3 Continuous Outcomes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
7.4 Binary Outcomes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
7.4.1 Chi-Square Test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
7.4.2 Cochran-Armitage Test. . . . . . . . . . . . . . . . . . . . . . . . 84
7.5 Time-to-Event Outcomes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
8 Comparison of Two Groups and Test for Non-Inferiority . . . . . . . 91
8.1 Background and Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
8.2 Normally Distributed Outcomes. . . . . . . . . . . . . . . . . . . . . . . . 92
8.2.1 Difference of Means as Effect Measure . . . . . . . . . . . . 92
8.2.2 Ratio of Means as Effect Measure. . . . . . . . . . . . . . . . 95
8.3 Continuous and Ordered Categorical Outcomes . . . . . . . . . . . . 97
