Power Analysis of Trials with Multilevel Data © 2016 Taylor & Francis Group, LLC CHAPMAN & HALL/CRC Interdisciplinar y Statistics Series Series editors: N. Keiding, B.J.T. Morgan, C.K. Wikle, P. van der Heijden Published titles AGE-PERIOD-COHORT ANALYSIS: NEW MODELS, METHODS, AND EMPIRICAL APPLICATIONS Y. Yang and K. C. Land ANALYSIS OF CAPTURE-RECAPTURE DATA R. S. McCrea and B. J. T. Morgan AN INVARIANT APPROACH TO STATISTICAL ANALYSIS OF SHAPES S. Lele and J. Richtsmeier ASTROSTATISTICS G. Babu and E. Feigelson BAYESIAN ANALYSIS FOR POPULATION ECOLOGY R. King, B. J. T. Morgan, O. Gimenez, and S. P. Brooks BAYESIAN DISEASE MAPPING: HIERARCHICAL MODELING IN SPATIAL EPIDEMIOLOGY, SECOND EDITION A. B. Lawson BIOEQUIVALENCE AND STATISTICS IN CLINICAL PHARMACOLOGY S. Patterson and B. Jones CLINICAL TRIALS IN ONCOLOGY, THIRD EDITION S. Green, J. Benedetti, A. Smith, and J. Crowley CLUSTER RANDOMISED TRIALS R.J. Hayes and L.H. Moulton CORRESPONDENCE ANALYSIS IN PRACTICE, SECOND EDITION M. Greenacre DESIGN AND ANALYSIS OF QUALITY OF LIFE STUDIES IN CLINICAL TRIALS, SECOND EDITION D.L. Fairclough DYNAMICAL SEARCH L. Pronzato, H. Wynn, and A. Zhigljavsky FLEXIBLE IMPUTATION OF MISSING DATA S. van Buuren GENERALIZED LATENT VARIABLE MODELING: MULTILEVEL, LONGITUDI- NAL, AND STRUCTURAL EQUATION MODELS A. Skrondal and S. Rabe-Hesketh GRAPHICAL ANALYSIS OF MULTI-RESPONSE DATA K. Basford and J. Tukey INTRODUCTION TO COMPUTATIONAL BIOLOGY: MAPS, SEQUENCES, AND GENOMES M. Waterman MARKOV CHAIN MONTE CARLO IN PRACTICE W. Gilks, S. Richardson, and D. Spiegelhalter MEASUREMENT ERROR ANDMISCLASSIFICATION IN STATISTICS AND EPIDE- MIOLOGY: IMPACTS AND BAYESIAN ADJUSTMENTS P. Gustafson MEASUREMENT ERROR: MODELS, METHODS, AND APPLICATIONS J. P. Buonaccorsi MEASUREMENT ERROR: MODELS, METHODS, AND APPLICATIONS J. P. Buonaccorsi © 2016 Taylor & Francis Group, LLC Published titles MENDELIAN RANDOMIZATION: METHODS FOR USING GENETIC VARIANTS IN CAUSAL ESTIMATION S.Burgess and S.G. Thompson META-ANALYSIS OF BINARY DATA USINGPROFILE LIKELIHOOD D. Böhning, R. Kuhnert, and S. Rattanasiri POWER ANALYSIS OF TRIALS WITH MULTILEVEL DATA M. Moerbeek and S. Teerenstra STATISTICAL ANALYSIS OF GENE EXPRESSION MICROARRAY DATA T. Speed STATISTICAL AND COMPUTATIONAL PHARMACOGENOMICS R. Wu and M. Lin STATISTICS IN MUSICOLOGY J. Beran STATISTICS OF MEDICAL IMAGING T. Lei STATISTICAL CONCEPTS AND APPLICATIONS IN CLINICAL MEDICINE J. Aitchison, J.W. Kay, and I.J. Lauder STATISTICAL AND PROBABILISTIC METHODS IN ACTUARIAL SCIENCE P.J. Boland STATISTICAL DETECTION AND SURVEILLANCE OF GEOGRAPHIC CLUSTERS P. Rogerson and I. Yamada STATISTICS FOR ENVIRONMENTAL BIOLOGY AND TOXICOLOGY A. Bailer and W. Piegorsch STATISTICS FOR FISSION TRACK ANALYSIS R.F. Galbraith VISUALIZING DATA PATTERNS WITH MICROMAPS D.B. Carr and L.W. Pickle © 2016 Taylor & Francis Group, LLC © 2016 Taylor & Francis Group, LLC Chapman & Hall/CRC Interdisciplinary Statistics Series Power Analysis of Trials with Multilevel Data Mirjam Moerbeek Utrecht University, The Netherlands Steven Teerenstra Radboud University Medical Center, The Netherlands © 2016 Taylor & Francis Group, LLC CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2016 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Version Date: 20150417 International Standard Book Number-13: 978-1-4987-2990-1 (eBook - PDF) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. 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Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com © 2016 Taylor & Francis Group, LLC Contents List of figures xi List of tables xv Preface xvii 1 Introduction 1 1.1 Experimentation . . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.1 Problems with random assignment . . . . . . . . . . . 5 1.2 Hierarchical data structures . . . . . . . . . . . . . . . . . . . 6 1.3 Research design . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3.1 Cluster randomized trial . . . . . . . . . . . . . . . . . 10 1.3.2 Multisite trial . . . . . . . . . . . . . . . . . . . . . . . 11 1.3.3 Pseudo cluster randomized trial . . . . . . . . . . . . . 12 1.3.4 Individually randomized group treatment trial . . . . 12 1.3.5 Longitudinal intervention study . . . . . . . . . . . . . 13 1.3.6 Some guidance to design choice . . . . . . . . . . . . . 14 1.4 Power analysis for experimental research . . . . . . . . . . . 15 1.5 Aim and contents of the book . . . . . . . . . . . . . . . . . 18 1.5.1 Aim . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.5.2 Contents . . . . . . . . . . . . . . . . . . . . . . . . . 18 2 Multilevel statistical models 21 2.1 The basic two-level model . . . . . . . . . . . . . . . . . . . . 21 2.2 Estimation and hypothesis test . . . . . . . . . . . . . . . . . 26 2.3 Intraclass correlation coefficient . . . . . . . . . . . . . . . . 29 2.4 Multilevel models for dichotomous outcomes . . . . . . . . . 32 2.5 More than two levels of nesting . . . . . . . . . . . . . . . . . 35 2.6 Software for multilevel analysis . . . . . . . . . . . . . . . . . 37 3 Concepts of statistical power analysis 39 3.1 Background of power analysis . . . . . . . . . . . . . . . . . 39 3.1.1 Hypotheses testing . . . . . . . . . . . . . . . . . . . . 39 3.1.2 Power calculations for continuous outcomes . . . . . . 41 3.1.3 Power calculations for dichotomous outcomes . . . . . 45 3.1.3.1 Risk difference . . . . . . . . . . . . . . . . . 45 3.1.3.2 Odds ratio . . . . . . . . . . . . . . . . . . . 46 vii © 2016 Taylor & Francis Group, LLC viii Contents 3.2 Types of power analysis . . . . . . . . . . . . . . . . . . . . . 47 3.3 Timing of power analysis . . . . . . . . . . . . . . . . . . . . 49 3.4 Methods for power analysis . . . . . . . . . . . . . . . . . . . 50 3.5 Robustness of power and sample size calculations . . . . . . 52 3.6 Procedure for a priori power analysis . . . . . . . . . . . . . 53 3.6.1 An example . . . . . . . . . . . . . . . . . . . . . . . . 56 3.7 The optimal design of experiments . . . . . . . . . . . . . . . 57 3.7.1 An example (continued) . . . . . . . . . . . . . . . . . 59 3.8 Sample size and precision analysis . . . . . . . . . . . . . . . 59 3.9 Sample size and accuracy of parameter estimates . . . . . . . 61 4 Cluster randomized trials 63 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.2 Multilevel model . . . . . . . . . . . . . . . . . . . . . . . . . 65 4.3 Sample size calculations for continuous outcomes . . . . . . . 68 4.3.1 Factors that influence power. . . . . . . . . . . . . . . 69 4.3.2 Design effect . . . . . . . . . . . . . . . . . . . . . . . 72 4.3.3 Sample size formulae for fixed cluster size or fixed number of clusters . . . . . . . . . . . . . . . . . . . . 73 4.3.4 Including budgetary constraints . . . . . . . . . . . . . 75 4.4 Sample size calculations for dichotomous outcomes . . . . . . 78 4.4.1 Risk difference . . . . . . . . . . . . . . . . . . . . . . 79 4.4.2 Odds ratio . . . . . . . . . . . . . . . . . . . . . . . . 80 4.5 An example . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 5 Improving statistical power in cluster randomized trials 83 5.1 Inclusion of covariates . . . . . . . . . . . . . . . . . . . . . . 84 5.2 Minimization, matching, pre-stratification . . . . . . . . . . . 87 5.3 Taking repeated measurements . . . . . . . . . . . . . . . . . 90 5.4 Crossover in cluster randomized trials . . . . . . . . . . . . . 94 5.5 Stepped wedge designs . . . . . . . . . . . . . . . . . . . . . 101 6 Multisite trials 107 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 6.2 Multilevel model . . . . . . . . . . . . . . . . . . . . . . . . . 109 6.3 Sample size calculations for continuous outcomes . . . . . . . 115 6.3.1 Factors that influence power. . . . . . . . . . . . . . . 115 6.3.2 Design effect . . . . . . . . . . . . . . . . . . . . . . . 118 6.3.3 Sample size formulae for fixed cluster size or fixed number of clusters . . . . . . . . . . . . . . . . . . . . 120 6.3.4 Including budgetary constraints . . . . . . . . . . . . . 121 6.3.5 Constant treatment effect . . . . . . . . . . . . . . . . 122 6.4 Sample size calculations for dichotomous outcomes . . . . . . 124 6.4.1 Odds ratio . . . . . . . . . . . . . . . . . . . . . . . . 125 6.5 An example . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 © 2016 Taylor & Francis Group, LLC Contents ix 7 Pseudo cluster randomized trials 129 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 129 7.2 Multilevel model . . . . . . . . . . . . . . . . . . . . . . . . . 132 7.3 Sample size calculations for continuous outcomes . . . . . . . 134 7.3.1 Factors that influence power. . . . . . . . . . . . . . . 134 7.3.2 Design effect . . . . . . . . . . . . . . . . . . . . . . . 136 7.3.3 Sample size formulae for fixed cluster size or fixed number of clusters . . . . . . . . . . . . . . . . . . . . 137 7.4 Sample size calculations for binary outcomes . . . . . . . . . 138 7.5 An example . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 8 Individually randomized group treatment trials 141 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 8.2 Multilevel model . . . . . . . . . . . . . . . . . . . . . . . . . 143 8.2.1 Clustering in both treatment arms . . . . . . . . . . . 143 8.2.2 Clustering in one treatment arm . . . . . . . . . . . . 145 8.3 Sample size calculations for continuous outcomes . . . . . . . 146 8.3.1 Clustering in both treatment arms . . . . . . . . . . . 146 8.3.1.1 Factors that influence power . . . . . . . . . 146 8.3.1.2 Sample size formulae for fixed cluster sizes . 147 8.3.1.3 Including budgetary constraints . . . . . . . 148 8.3.2 Clustering in one treatment arm . . . . . . . . . . . . 150 8.3.2.1 Factors that influence power . . . . . . . . . 150 8.3.2.2 Sample size formulae for fixed cluster sizes . 151 8.3.2.3 Including budgetary constraints . . . . . . . 151 8.4 Sample size calculations for dichotomous outcomes . . . . . . 153 8.4.1 Clustering in both treatment arms . . . . . . . . . . . 153 8.4.2 Clustering in one treatment arm . . . . . . . . . . . . 154 8.5 An example . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 9 Longitudinal intervention studies 159 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 9.2 Multilevel model . . . . . . . . . . . . . . . . . . . . . . . . . 161 9.3 Sample size calculations for continuous outcomes . . . . . . . 165 9.3.1 Factors that influence power. . . . . . . . . . . . . . . 165 9.3.2 Sample size formula for fixed number of measurements 168 9.3.3 Including budgetary constraints . . . . . . . . . . . . . 169 9.4 Sample size calculations for dichotomous outcomes . . . . . . 170 9.4.1 Odds ratio . . . . . . . . . . . . . . . . . . . . . . . . 171 9.5 The effect of drop-out on statistical power . . . . . . . . . . 172 9.5.1 The effects of different drop-out patterns . . . . . . . 173 9.5.2 Including budgetary constraints . . . . . . . . . . . . . 179 9.6 An example . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 © 2016 Taylor & Francis Group, LLC x Contents 10 Extensions: three levels of nesting and factorial designs 183 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 10.2 Three-level cluster randomized trials . . . . . . . . . . . . . . 184 10.3 Multisite cluster randomized trials . . . . . . . . . . . . . . . 188 10.4 Repeated measures in cluster randomized trials and multisite trials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193 10.5 Factorial designs . . . . . . . . . . . . . . . . . . . . . . . . . 198 10.5.1 Continuous outcome . . . . . . . . . . . . . . . . . . . 198 10.5.2 Binary outcome. . . . . . . . . . . . . . . . . . . . . . 199 10.5.3 Sample size calculation for factorial designs . . . . . . 200 11 The problem of unknown intraclass correlation coefficients 203 11.1 Estimates from previous research . . . . . . . . . . . . . . . . 204 11.2 Sample size re-estimation . . . . . . . . . . . . . . . . . . . . 205 11.3 Bayesian sample size calculation . . . . . . . . . . . . . . . . 211 11.4 Maximin optimal designs . . . . . . . . . . . . . . . . . . . . 214 12 Computer software for power calculations 217 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 12.2 Computer program SPA-ML . . . . . . . . . . . . . . . . . . 218 References 229 Author Index 255 Subject Index 265 © 2016 Taylor & Francis Group, LLC