STATISTICS APPLIED TO CLINICAL TRIALS FOURTH EDITION Statistics Applied to Clinical Trials Fourth edition by TON J CLEOPHAS, MD, PhD, Professor Statistical Consultant, Circulation, Boston, USA, Co-Chair Module Statistics Applied to Clinical Trials, European Interuniversity College of Pharmaceutical Medicine, Lyon, France, Internist-clinical pharmacologist, Department Medicine, Albert Schweitzer Hospital, Dordrecht, The Netherlands AEILKO H ZWINDERMAN, MathD, PhD, Professor Co-Chair Module Statistics Applied to Clinical Trials, European Interuniversity College of Pharmaceutical Medicine, Lyon, France, Professor of Statistics, Department Biostatistics and Epidemiology, Academic Medical Center, Amsterdam, The Netherlands TOINE F CLEOPHAS, BSc Department of Research, Damen Shipyards, Gorinchem, The Netherlands and EUGENE P CLEOPHAS, BSc Technical University, Delft, The Netherlands Prof Ton J Cleophas Albert Schweitzer Hospital Dordrecht The Netherlands Prof Aeilko H Zwinderman Academic Medical Center Amsterdam The Netherlands Toine F Cleophas Damen Shipyards Gorinchem The Netherlands Eugene P Cleophas Technical University Delft The Netherlands ISBN 978-1-4020-9522-1 e-ISBN 978-1-4020-9523-8 Library of Congress Control Number: 2008939866 © Springer Science + Business Media B.V 2009 No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work Printed on acid-free paper springer.com TABLE OF CONTENTS PREFACES xvii FOREWORD xxi CHAPTER / HYPOTHESES, DATA, STRATIFICATION General considerations Two main hypotheses in drug trials: efficacy and safety Different types of data: continuous data Different types of data: proportions, percentages and contingency tables Different types of data: correlation coefficient Stratification issues Randomized versus historical controls Factorial designs Conclusions 10 References 11 13 14 15 15 16 CHAPTER / THE ANALYSIS OF EFFICACY DATA Overview The principle of testing statistical significance The t-value = standardized mean result of study Unpaired t-test Null-hypothesis testing of or more unpaired samples Three methods to test statistically a paired sample Null-hypothesis testing of or more paired samples Null-hypothesis testing with complex data Paired data with a negative correlation 10 Rank testing 11 Rank testing for or more samples 12 Conclusions 13 References 17 18 21 22 24 25 29 30 31 37 40 42 42 CHAPTER / THE ANALYSIS OF SAFETY DATA Introduction, summary display Four methods to analyze two unpaired proportions Chi-square to analyze more than two unpaired proportions McNemar’s test for paired proportions Survival analysis Odds ratio method for analyzing two unpaired proportions Odds ratios for group, two treatments Conclusions 45 46 52 55 56 58 61 61 v vi TABLE OF CONTENTS CHAPTER / LOG LIKELIHOOD RATIO TESTS FOR SAFETY DATA ANALYSIS Introduction Numerical problems with calculating exact likelihoods The normal approximation and the analysis of clinical events Log likelihood ratio tests and the quadratic approximation More examples Discussion Conclusions References 63 63 64 66 68 69 70 70 CHAPTER / EQUIVALENCE TESTING Introduction Overview of possibilities with equivalence testing Calculations Equivalence testing, a new gold standard? Validity of equivalence trials Special point: level of correlation in paired equivalence studies Conclusions 73 75 76 77 77 78 79 CHAPTER / STATISTICAL POWER AND SAMPLE SIZE What is statistical power Emphasis on statistical power rather than null-hypothesis testing Power computations Examples of power computation using the t-table Calculation of required sample size, rationale Calculations of required sample size, methods Testing inferiority of a new treatment (the type III error) Conclusions References 81 82 84 85 91 91 95 97 97 CHAPTER / INTERIM ANALYSES Introduction Monitoring Interim analysis Group-sequential design of interim analysis Continuous sequential statistical techniques Conclusions References 99 99 100 103 103 105 105 CHAPTER / CONTROLLING THE RISK OF FALSE POSITIVE CLINICAL TRIALS Introduction Bonferroni test Least significant difference test (LSD) test Other tests for adjusting the p-values 107 108 109 109 TABLE OF CONTENTS vii Composite endpoint procedures No adjustments at all, and pragmatic solutions Conclusions References 110 110 111 111 CHAPTER / MULTIPLE STATISTICAL INFERENCES Introduction Multiple comparisons Multiple variables Conclusions References 113 113 118 121 121 CHAPTER 10 / THE INTERPRETATION OF THE P-VALUES Introduction Renewed attention to the interpretation of the probability levels, otherwise called the p-values Standard interpretation of p-values Common misunderstandings of the p-values Renewed interpretations of p-values, little difference between p = 0.06 and p = 0.04 The real meaning of very large p-values like p > 0.95 P-values larger than 0.95, examples (Table 2) The real meaning of very small p-values like p < 0.0001 P-values smaller than 0.0001, examples (Table 3) 10 Discussion 11 Recommendations 12 Conclusions 13 References 127 128 129 130 131 131 133 133 CHAPTER 11 / RESEARCH DATA CLOSER TO EXPECTATION THAN COMPATIBLE WITH RANDOM SAMPLING Introduction Methods and results Discussion Conclusions References 137 138 139 142 142 CHAPTER 12 / STATISTICAL TABLES FOR TESTING DATA CLOSER TO EXPECTATION THAN COMPATIBLE WITH RANDOM SAMPLING Introduction Statistical tables of unusually high p-values How to calculate the p-values yourself Additional examples simulating real practice, multiple comparisons Discussion Conclusions References 145 147 147 150 152 153 153 123 123 124 126 126 viii TABLE OF CONTENTS CHAPTER 13 / PRINCIPLES OF LINEAR REGRESSION Introduction More on paired observations Using statistical software for simple linear regression Multiple linear regression Multiple linear regression, example Purposes of linear regression analysis Another real data example of multiple linear regression (exploratory purpose) It may be hard to define what is determined by what, multiple and multivariate regression Limitations of linear regression 10 Conclusions 155 156 159 162 164 168 169 171 172 173 CHAPTER 14 / SUBGROUP ANALYSIS USING MULTIPLE LINEAR REGRESSION: CONFOUNDING, INTERACTION, SYNERGISM Introduction Example Model (figure 1) (I.) Increased precision of efficacy (figure 2) (II.) Confounding (III.) Interaction and synergism Estimation, and hypothesis testing Goodness-of-fit Selection procedures 10 Main conclusion 11 References 175 175 176 178 179 180 181 182 183 183 184 CHAPTER 15 / CURVILINEAR REGRESSION Introduction Methods, statistical model Results Discussion Conclusions References 185 186 188 194 196 196 CHAPTER 16 / LOGISTIC AND COX REGRESSION, MARKOW MODELS, LAPLACE TRANSFORMATIONS Introduction 199 Linear regression 199 Logistic regression 203 Cox regression 209 Markow models 212 Regression-analysis with Laplace transformations 213 TABLE OF CONTENTS Discussion Conclusions References ix 217 218 219 CHAPTER 17 / REGRESSION MODELING FOR IMPROVED PRECISION Introduction Regression modeling for improved precision of clinical trials, the underlying mechanism Regression model for parallel-group trials with continuous efficacy data Regression model for parallel-group trials with proportions or odds as efficacy data Discussion Conclusions References 225 227 227 CHAPTER 18 / POST-HOC ANALYSES IN CLINICAL TRIALS, A CASE FOR LOGISTIC REGRESSION ANALYSIS Multiple variables methods Examples Logistic regression equation Conclusions References 229 229 232 233 234 CHAPTER 19 / CONFOUNDING Introduction First method for adjustment of confounders: subclassification on one confounder Second method for adjustment of confounders: regression modeling Third method for adjustment of confounders: propensity scores Discussion Conclusions References CHAPTER 20 / INTERACTION Introduction What exactly is interaction, a hypothesized example How to test interaction statistically, a real data example with a concomitant medication as interacting factor, incorrect method Three analysis methods Using a regression model for testing interaction, another real data example Analysis of variance for testing interaction, other real data examples Discussion Conclusions References 221 221 223 224 235 236 237 238 241 242 243 245 245 248 248 252 254 259 260 261 x TABLE OF CONTENTS CHAPTER 21 / META-ANALYSIS, BASIC APPROACH Introduction Examples Clearly defined hypotheses Thorough search of trials Strict inclusion criteria Uniform data analysis Discussion, where are we now? Conclusions References 263 264 266 266 266 267 275 276 276 CHAPTER 22 / META-ANALYSIS, REVIEW AND UPDATE OF METHODOLOGIES Introduction Four scientific rules General framework of meta-analysis Pitfalls of meta-analysis New developments Conclusions References 277 277 278 281 284 285 285 CHAPTER 23 / CROSSOVER STUDIES WITH CONTINUOUS VARIABLES Introduction Mathematical model Hypothesis testing Statistical power of testing Discussion Conclusion References 289 290 291 293 296 297 298 CHAPTER 24 / CROSSOVER STUDIES WITH BINARY RESPONSES Introduction Assessment of carryover and treatment effect Statistical model for testing treatment and carryover effects Results Examples Discussion Conclusions References 299 300 301 302 304 305 306 306 CHAPTER 25 / CROSS-OVER TRIALS SHOULD NOT BE USED TO TEST TREATMENTS WITH DIFFERENT CHEMICAL CLASS Introduction 309 Examples from the literature in which cross-over trials are correctly used 311 TABLE OF CONTENTS Examples from the literature in which cross-over trials should not have been used Estimate of the size of the problem by review of hypertension trials published Discussion Conclusions References CHAPTER 26 / QUALITY-OF-LIFE ASSESSMENTS IN CLINICAL TRIALS Introduction Some terminology Defining QOL in a subjective or objective way? The patients’ opinion is an important independent-contributor to QOL Lack of sensitivity of QOL-assessments Odds ratio analysis of effects of patient characteristics on QOL data provides increased precision Discussion Conclusions References CHAPTER 27 / STATISTICAL ANALYSIS OF GENETIC DATA Introduction Some terminology Genetics, genomics, proteonomics, data mining Genomics Conclusions References xi 313 315 316 317 318 319 319 321 322 323 324 327 328 328 331 332 334 335 339 339 CHAPTER 28 / RELATIONSHIP AMONG STATISTICAL DISTRIBUTIONS Introduction Variances The normal distribution Null-hypothesis testing with the normal or t-distribution Relationship between the normal-distribution and chi-square-distribution, null-hypothesis testing with chi-square distribution Examples of data where variance is more important than mean Chi-square can be used for multiple samples of data Discussion Conclusions 10 References 348 349 352 353 354 CHAPTER 29 / TESTING CLINICAL TRIALS FOR RANDOMNESS Introduction Individual data available 355 355 341 341 342 344 346 BIAS DUE TO CONFLICTS OF INTERESTS 543 Sleight P, Yusuf S, Pogue J, Tsuyuki R, Diaz R, Probsfield J Blood pressure reduction and cardiovascular risk in HOPE Study Lancet 2001; 358: 2130-1 7.PROGRESS Collaborative Group Randomised trial of a perindopril-based blood-pressure lowering regimen among 6105 individuals with previous stroke or transient ischaemic attack Lancet 2001; 358: 1033-41 Meyer FP, Cleophas TJ Meta-analysis of beta-blockers in heart failure Int J Clin Pharmacol Ther 2001; 39: 561-563 and 39: 383-8 Cleophas TJ, Zwinderman AH Efficacy of HMG-CoA reductase inhibitors dependent on baseline cholesterol levels, secondary analysis of the Regression Growth Evaluation Statin Study (REGRESS) Br J Clin Pharmacol 2003; 56: 465-6 10 Relman AJ, Cleophas TJ, Cleophas GI The pharmaceutical industry and continuing medical education JAMA 2001; 286: 302-4 Appendix 545 546 APPENDIX APPENDIX Chi-square distribution 547 548 F-distribution APPENDIX APPENDIX Paired non-parametric test: Wilcoxon signed rank test, the table uses smaller of the two ranknumbers N pairs P