MICHAEL J. CAMPBELL and STEPHEN J. WALTERS, Medical Statistics Group, School of Health and Related Research, University of Sheffield, UK

“Overall, the reviewers are enthusiastic about the book. The authors have covered all important areas of cRCTs, using a practical and pragmatic approach to the topic. The code is helpful for the practical implementation of the examples. The material is simple to understand, which will appeal to applied researchers, not only to biostatisticians. As such, we clearly recommend this book to all researchers interested in cRCTs. For biostatisticians involved in cRCTs and investigators of cRCTs, it is a must-have on the bookshelf.”  (Biometrical Journal, 1 May 2015)

• Preface xiiiAcronyms and abbreviations xv1 Introduction 11.1 Randomised controlled trials 11.1.1 A-Allocation at random 11.1.2 B-Blindness 21.1.3 C-Control 21.2 Complex interventions 31.3 History of cluster randomised trials 41.4 Cohort and field trials 41.5 The field/community trial 51.5.1 The REACT trial 51.5.2 The Informed Choice leaflets trial 61.5.3 The Mwanza trial 71.5.4 The paramedics practitioner trial 71.6 The cohort trial 81.6.1 The PoNDER trial 81.6.2 The DESMOND trial 91.6.3 The Diabetes Care from Diagnosis trial 101.6.4 The REPOSE trial 111.6.5 Other examples of cohort cluster trials 111.7 Field versus cohort designs 111.8 Reasons for cluster trials 121.9 Between- and within-cluster variation 141.10 Random-effects models for continuous outcomes 151.10.1 The model 151.10.2 The intracluster correlation coefficient 161.10.3 Estimating the intracluster correlation (ICC) coefficient 161.10.4 Link between the Pearson correlation coefficient and the intraclass correlation coefficient 171.11 Random-effects models for binary outcomes 181.11.1 The model 181.11.2 The ICC for binary data 191.11.3 The coefficient of variation 191.11.4 Relationship between cvc and 𝜌 for binary data 201.12 The design effect 201.13 Commonly asked questions 211.14 Websources 21Exercise 22Appendix 1.A 222 Design issues 272.1 Introduction 272.2 Issues for a simple intervention 282.2.1 Phases of a trial 282.2.2 ‘Pragmatic’ and ‘explanatory’ trials 292.2.3 Intention-to-treat and per-protocol analyses 292.2.4 Non-inferiority and equivalence trials 302.3 Complex interventions 302.3.1 Design of complex interventions 302.3.2 Phase I modelling/qualitative designs 322.3.3 Pilot or feasibility studies 332.3.4 Example of pilot/feasibility studies in cluster trials 332.4 Recruitment bias 342.5 Matched-pair trials 342.5.1 Design of matched-pair studies 342.5.2 Limitations of matched-pairs designs 362.5.3 Example of matched-pair design: The Family Heart Study 362.6 Other types of designs 372.6.1 Cluster factorial designs 372.6.2 Example cluster factorial trial 382.6.3 Cluster crossover trials 382.6.4 Example of a cluster crossover trial 392.6.5 Stepped wedge 392.6.6 Pseudorandomised trials 402.7 Other design issues 412.8 Strategies for improving precision 412.9 Randomisation 422.9.1 Reasons for randomisation 422.9.2 Simple randomisation 432.9.3 Stratified randomisation 432.9.4 Restricted randomisation 432.9.5 Minimisation 44Exercise 45Appendix 2.A 483 Sample size: How many subjects/clusters do I need for my cluster randomised controlled trial? 503.1 Introduction 513.1.1 Justification of the requirement for a sample size 513.1.2 Significance tests, P-values and power 513.1.3 Sample size and cluster trials 533.2 Sample size for continuous data – comparing two means 533.2.1 Basic formulae 533.2.2 The design effect (DE) in cluster RCTs 543.2.3 Example from general practice 553.3 Sample size for binary data – comparing two proportions 563.3.1 Sample size formula 563.3.2 Example calculations 573.3.3 Example: The Informed Choice leaflets study 583.4 Sample size for ordered categorical (ordinal) data 593.4.1 Sample size formula 593.4.2 Example calculations 603.5 Sample size for rates 623.5.1 Formulae 623.5.2 Example comparing rates 633.6 Sample size for survival 633.6.1 Formulae 633.6.2 Example of sample size for survival 643.7 Equivalence/non-inferiority studies 643.7.1 Equivalence/non-inferiority versus superiority 643.7.2 Continuous data – comparing the equivalence of two means 653.7.3 Example calculations for continuous data 653.7.4 Binary data – comparing the equivalence of two proportions 663.8 Unknown standard deviation and effect size 663.9 Practical problems 673.9.1 Tips on getting the SD 673.9.2 Non-response 673.9.3 Unequal groups 673.10 Number of clusters fixed 683.10.1 Number of clusters and number of subjects per cluster 683.10.2 Example with number of clusters fixed 693.10.3 Increasing the number of clusters or number of patients per cluster? 693.11 Values of the ICC 693.12 Allowing for imprecision in the ICC 703.13 Allowing for varying cluster sizes 703.13.1 Formulae 703.13.2 Example of effect of variable cluster size 713.14 Sample size re-estimation 713.14.1 Adjusting for covariates 723.15 Matched-pair studies 723.15.1 Sample sizes for matched designs 723.15.2 Example of a sample size calculation for a matched study 723.16 Multiple outcomes/endpoints 733.17 Three or more groups 743.18 Crossover trials 743.18.1 Formulae 753.18.2 Example of a sample size formula in a crossover trial 753.19 Post hoc sample size calculations 753.20 Conclusion: Usefulness of sample size calculations 763.21 Commonly asked questions 76Exercise 77Appendix 3.A 784 Simple analysis of cRCT outcomes using aggregate cluster-level summaries 834.1 Introduction 834.1.1 Methods of analysing cluster randomised trials 834.1.2 Choosing the statistical method 844.2 Aggregate cluster-level analysis – carried out at the cluster level, using aggregate summary data 844.3 Statistical methods for continuous outcomes 864.3.1 Two independent-samples t-test 864.3.2 Example 884.4 Mann–Whitney U test 914.5 Statistical methods for binary outcomes 944.6 Analysis of a matched design 954.7 Discussion 984.8 Commonly asked question 98Exercise 99Appendix 4.A 995 Regression methods of analysis for continuous outcomes using individual person-level data 1025.1 Introduction 1025.2 Incorrect models 1045.2.1 The simple (independence) model 1045.2.2 Fixed effects 1045.3 Linear regression with robust standard errors 1055.3.1 Robust standard errors 1055.3.2 Example of use of robust standard errors 1075.3.3 Cluster-specific versus population-averaged models 1075.4 Random-effects general linear models in a cohort study 1085.4.1 General models 1085.4.2 Fitting a random-effects model 1095.4.3 Example of a random-effects model from the PoNDER study 1105.4.4 Checking the assumptions 1105.5 Marginal general linear model with coefficients estimated by generalised estimating equations (GEE) 1125.5.1 Generalised estimating equations 1125.5.2 Example of a marginal model from the PoNDER study 1135.6 Summary of methods 1145.7 Adjusting for individual-level covariates in cohort studies 1155.8 Adjusting for cluster-level covariates in cohort studies 1185.9 Models for cross-sectional designs 1195.10 Discussion of model fitting 120Exercise 122Appendix 5.A 1236 Regression methods of analysis for binary, count and time-to-event outcomes for a cluster randomised controlled trial 1266.1 Introduction 1266.2 Difference between a cluster-specific model and a population-averaged or marginal model for binary data 1276.3 Analysis of binary data using logistic regression 1296.4 Review of past simulations to determine efficiency of different methods for binary data 1306.5 Analysis using summary measures 1316.6 Analysis using logistic regression (ignoring clustering) 1326.7 Random-effects logistic regression 1346.8 Marginal models using generalised estimating equations 1356.9 Analysis of count data 1356.10 Survival analysis with cluster trials 1376.11 Missing data 1396.12 Discussion 139Exercise 139Appendix 6.A 1407 The protocol 1437.1 Introduction 1437.2 Abstract 1447.3 Protocol background 1477.4 Research objectives 1477.5 Outcome measures 1477.6 Design 1477.7 Intervention details 1487.8 Eligibility 1487.9 Randomisation 1497.10 Assessment and data collection 1497.11 Statistical considerations 1507.11.1 Sample size 1507.11.2 Statistical analysis 1517.11.3 Interim analyses 1527.12 Ethics 1537.12.1 Declaration of Helsinki 1537.12.2 Informed consent 1547.13 Organisation 1557.13.1 The team 1557.13.2 Trial forms 1557.13.3 Data management 1557.13.4 Protocol amendments 1567.14 Further reading 156Exercise 1568 Reporting of cRCTs 1598.1 Introduction: Extended CONSORT guidelines for reporting and presenting the results from cRCTs 1598.2 Patient flow diagram 1608.3 Comparison of entry characteristics 1608.4 Incomplete data 1678.5 Reporting the main outcome 1718.6 Subgroup analysis and analysis of secondary outcomes/endpoints 1748.7 Estimates of between-cluster variability 1758.7.1 Example of reporting the ICC: The PoNDER cRCT 1758.8 Further reading 175Exercise 1769 Practical issues 1789.1 Preventing bias in cluster randomised controlled trials 1789.1.1 Problems with identifying and recruiting patients to cluster trials 1789.1.2 Preventing biased recruitment 1799.2 Developing complex interventions 1819.3 Choice of method of analysis 1829.4 Missing data 1859.5 Example sensitivity analysis: Imputation of missing 6-month EPDS data for at-risk women from the PoNDER cRCT 1889.6 Multiplicity of outcomes 1929.6.1 Limiting the number of confirmatory tests 1929.6.2 Summary measures and statistics 1939.6.3 Global tests and multiple comparison procedures 1939.6.4 Which multiple comparison procedure to use? 19410 Computing software 19510.1 R 19510.1.1 History 19510.1.2 Installing R 19610.1.3 Simple use of R 19710.1.4 An example of an R program 19810.2 Stata (version 12) 19910.2.1 Introduction to Stata 19910.2.2 Aggregate cluster-level analysis – carried out at the cluster level, using aggregate summary data 20110.2.3 Random-effects models – continuous outcomes 20210.2.4 Random-effects models – binary outcomes 20510.2.5 Random-effects models – count outcomes 20610.2.6 Marginal models – continuous outcomes 20810.2.7 Marginal models – binary outcomes 20910.2.8 Marginal models – count outcomes 21010.3 SPSS (version 19) 21210.3.1 Introduction to SPSS 21210.3.2 Comparing cluster means using aggregate cluster-level analysis – carried out at the cluster level, using aggregate summary data 21310.3.3 Marginal models 21510.3.4 Random-effects models 22710.4 Conclusion and further reading 232References 234Index 243

MICHAEL J. CAMPBELL and STEPHEN J. WALTERS, Medical Statistics Group, School of Health and Related Research, University of Sheffield, UK

“Overall, the reviewers are enthusiastic about the book. The authors have covered all important areas of cRCTs, using a practical and pragmatic approach to the topic. The code is helpful for the practical implementation of the examples. The material is simple to understand, which will appeal to applied researchers, not only to biostatisticians. As such, we clearly recommend this book to all researchers interested in cRCTs. For biostatisticians involved in cRCTs and investigators of cRCTs, it is a must-have on the bookshelf.”  (Biometrical Journal, 1 May 2015)

• Preface xiiiAcronyms and abbreviations xv1 Introduction 11.1 Randomised controlled trials 11.1.1 A-Allocation at random 11.1.2 B-Blindness 21.1.3 C-Control 21.2 Complex interventions 31.3 History of cluster randomised trials 41.4 Cohort and field trials 41.5 The field/community trial 51.5.1 The REACT trial 51.5.2 The Informed Choice leaflets trial 61.5.3 The Mwanza trial 71.5.4 The paramedics practitioner trial 71.6 The cohort trial 81.6.1 The PoNDER trial 81.6.2 The DESMOND trial 91.6.3 The Diabetes Care from Diagnosis trial 101.6.4 The REPOSE trial 111.6.5 Other examples of cohort cluster trials 111.7 Field versus cohort designs 111.8 Reasons for cluster trials 121.9 Between- and within-cluster variation 141.10 Random-effects models for continuous outcomes 151.10.1 The model 151.10.2 The intracluster correlation coefficient 161.10.3 Estimating the intracluster correlation (ICC) coefficient 161.10.4 Link between the Pearson correlation coefficient and the intraclass correlation coefficient 171.11 Random-effects models for binary outcomes 181.11.1 The model 181.11.2 The ICC for binary data 191.11.3 The coefficient of variation 191.11.4 Relationship between cvc and 𝜌 for binary data 201.12 The design effect 201.13 Commonly asked questions 211.14 Websources 21Exercise 22Appendix 1.A 222 Design issues 272.1 Introduction 272.2 Issues for a simple intervention 282.2.1 Phases of a trial 282.2.2 ‘Pragmatic’ and ‘explanatory’ trials 292.2.3 Intention-to-treat and per-protocol analyses 292.2.4 Non-inferiority and equivalence trials 302.3 Complex interventions 302.3.1 Design of complex interventions 302.3.2 Phase I modelling/qualitative designs 322.3.3 Pilot or feasibility studies 332.3.4 Example of pilot/feasibility studies in cluster trials 332.4 Recruitment bias 342.5 Matched-pair trials 342.5.1 Design of matched-pair studies 342.5.2 Limitations of matched-pairs designs 362.5.3 Example of matched-pair design: The Family Heart Study 362.6 Other types of designs 372.6.1 Cluster factorial designs 372.6.2 Example cluster factorial trial 382.6.3 Cluster crossover trials 382.6.4 Example of a cluster crossover trial 392.6.5 Stepped wedge 392.6.6 Pseudorandomised trials 402.7 Other design issues 412.8 Strategies for improving precision 412.9 Randomisation 422.9.1 Reasons for randomisation 422.9.2 Simple randomisation 432.9.3 Stratified randomisation 432.9.4 Restricted randomisation 432.9.5 Minimisation 44Exercise 45Appendix 2.A 483 Sample size: How many subjects/clusters do I need for my cluster randomised controlled trial? 503.1 Introduction 513.1.1 Justification of the requirement for a sample size 513.1.2 Significance tests, P-values and power 513.1.3 Sample size and cluster trials 533.2 Sample size for continuous data – comparing two means 533.2.1 Basic formulae 533.2.2 The design effect (DE) in cluster RCTs 543.2.3 Example from general practice 553.3 Sample size for binary data – comparing two proportions 563.3.1 Sample size formula 563.3.2 Example calculations 573.3.3 Example: The Informed Choice leaflets study 583.4 Sample size for ordered categorical (ordinal) data 593.4.1 Sample size formula 593.4.2 Example calculations 603.5 Sample size for rates 623.5.1 Formulae 623.5.2 Example comparing rates 633.6 Sample size for survival 633.6.1 Formulae 633.6.2 Example of sample size for survival 643.7 Equivalence/non-inferiority studies 643.7.1 Equivalence/non-inferiority versus superiority 643.7.2 Continuous data – comparing the equivalence of two means 653.7.3 Example calculations for continuous data 653.7.4 Binary data – comparing the equivalence of two proportions 663.8 Unknown standard deviation and effect size 663.9 Practical problems 673.9.1 Tips on getting the SD 673.9.2 Non-response 673.9.3 Unequal groups 673.10 Number of clusters fixed 683.10.1 Number of clusters and number of subjects per cluster 683.10.2 Example with number of clusters fixed 693.10.3 Increasing the number of clusters or number of patients per cluster? 693.11 Values of the ICC 693.12 Allowing for imprecision in the ICC 703.13 Allowing for varying cluster sizes 703.13.1 Formulae 703.13.2 Example of effect of variable cluster size 713.14 Sample size re-estimation 713.14.1 Adjusting for covariates 723.15 Matched-pair studies 723.15.1 Sample sizes for matched designs 723.15.2 Example of a sample size calculation for a matched study 723.16 Multiple outcomes/endpoints 733.17 Three or more groups 743.18 Crossover trials 743.18.1 Formulae 753.18.2 Example of a sample size formula in a crossover trial 753.19 Post hoc sample size calculations 753.20 Conclusion: Usefulness of sample size calculations 763.21 Commonly asked questions 76Exercise 77Appendix 3.A 784 Simple analysis of cRCT outcomes using aggregate cluster-level summaries 834.1 Introduction 834.1.1 Methods of analysing cluster randomised trials 834.1.2 Choosing the statistical method 844.2 Aggregate cluster-level analysis – carried out at the cluster level, using aggregate summary data 844.3 Statistical methods for continuous outcomes 864.3.1 Two independent-samples t-test 864.3.2 Example 884.4 Mann–Whitney U test 914.5 Statistical methods for binary outcomes 944.6 Analysis of a matched design 954.7 Discussion 984.8 Commonly asked question 98Exercise 99Appendix 4.A 995 Regression methods of analysis for continuous outcomes using individual person-level data 1025.1 Introduction 1025.2 Incorrect models 1045.2.1 The simple (independence) model 1045.2.2 Fixed effects 1045.3 Linear regression with robust standard errors 1055.3.1 Robust standard errors 1055.3.2 Example of use of robust standard errors 1075.3.3 Cluster-specific versus population-averaged models 1075.4 Random-effects general linear models in a cohort study 1085.4.1 General models 1085.4.2 Fitting a random-effects model 1095.4.3 Example of a random-effects model from the PoNDER study 1105.4.4 Checking the assumptions 1105.5 Marginal general linear model with coefficients estimated by generalised estimating equations (GEE) 1125.5.1 Generalised estimating equations 1125.5.2 Example of a marginal model from the PoNDER study 1135.6 Summary of methods 1145.7 Adjusting for individual-level covariates in cohort studies 1155.8 Adjusting for cluster-level covariates in cohort studies 1185.9 Models for cross-sectional designs 1195.10 Discussion of model fitting 120Exercise 122Appendix 5.A 1236 Regression methods of analysis for binary, count and time-to-event outcomes for a cluster randomised controlled trial 1266.1 Introduction 1266.2 Difference between a cluster-specific model and a population-averaged or marginal model for binary data 1276.3 Analysis of binary data using logistic regression 1296.4 Review of past simulations to determine efficiency of different methods for binary data 1306.5 Analysis using summary measures 1316.6 Analysis using logistic regression (ignoring clustering) 1326.7 Random-effects logistic regression 1346.8 Marginal models using generalised estimating equations 1356.9 Analysis of count data 1356.10 Survival analysis with cluster trials 1376.11 Missing data 1396.12 Discussion 139Exercise 139Appendix 6.A 1407 The protocol 1437.1 Introduction 1437.2 Abstract 1447.3 Protocol background 1477.4 Research objectives 1477.5 Outcome measures 1477.6 Design 1477.7 Intervention details 1487.8 Eligibility 1487.9 Randomisation 1497.10 Assessment and data collection 1497.11 Statistical considerations 1507.11.1 Sample size 1507.11.2 Statistical analysis 1517.11.3 Interim analyses 1527.12 Ethics 1537.12.1 Declaration of Helsinki 1537.12.2 Informed consent 1547.13 Organisation 1557.13.1 The team 1557.13.2 Trial forms 1557.13.3 Data management 1557.13.4 Protocol amendments 1567.14 Further reading 156Exercise 1568 Reporting of cRCTs 1598.1 Introduction: Extended CONSORT guidelines for reporting and presenting the results from cRCTs 1598.2 Patient flow diagram 1608.3 Comparison of entry characteristics 1608.4 Incomplete data 1678.5 Reporting the main outcome 1718.6 Subgroup analysis and analysis of secondary outcomes/endpoints 1748.7 Estimates of between-cluster variability 1758.7.1 Example of reporting the ICC: The PoNDER cRCT 1758.8 Further reading 175Exercise 1769 Practical issues 1789.1 Preventing bias in cluster randomised controlled trials 1789.1.1 Problems with identifying and recruiting patients to cluster trials 1789.1.2 Preventing biased recruitment 1799.2 Developing complex interventions 1819.3 Choice of method of analysis 1829.4 Missing data 1859.5 Example sensitivity analysis: Imputation of missing 6-month EPDS data for at-risk women from the PoNDER cRCT 1889.6 Multiplicity of outcomes 1929.6.1 Limiting the number of confirmatory tests 1929.6.2 Summary measures and statistics 1939.6.3 Global tests and multiple comparison procedures 1939.6.4 Which multiple comparison procedure to use? 19410 Computing software 19510.1 R 19510.1.1 History 19510.1.2 Installing R 19610.1.3 Simple use of R 19710.1.4 An example of an R program 19810.2 Stata (version 12) 19910.2.1 Introduction to Stata 19910.2.2 Aggregate cluster-level analysis – carried out at the cluster level, using aggregate summary data 20110.2.3 Random-effects models – continuous outcomes 20210.2.4 Random-effects models – binary outcomes 20510.2.5 Random-effects models – count outcomes 20610.2.6 Marginal models – continuous outcomes 20810.2.7 Marginal models – binary outcomes 20910.2.8 Marginal models – count outcomes 21010.3 SPSS (version 19) 21210.3.1 Introduction to SPSS 21210.3.2 Comparing cluster means using aggregate cluster-level analysis – carried out at the cluster level, using aggregate summary data 21310.3.3 Marginal models 21510.3.4 Random-effects models 22710.4 Conclusion and further reading 232References 234Index 243

MICHAEL J. CAMPBELL and STEPHEN J. WALTERS, Medical Statistics Group, School of Health and Related Research, University of Sheffield, UK

“Overall, the reviewers are enthusiastic about the book. The authors have covered all important areas of cRCTs, using a practical and pragmatic approach to the topic. The code is helpful for the practical implementation of the examples. The material is simple to understand, which will appeal to applied researchers, not only to biostatisticians. As such, we clearly recommend this book to all researchers interested in cRCTs. For biostatisticians involved in cRCTs and investigators of cRCTs, it is a must-have on the bookshelf.”  (Biometrical Journal, 1 May 2015)

• Preface xiiiAcronyms and abbreviations xv1 Introduction 11.1 Randomised controlled trials 11.1.1 A-Allocation at random 11.1.2 B-Blindness 21.1.3 C-Control 21.2 Complex interventions 31.3 History of cluster randomised trials 41.4 Cohort and field trials 41.5 The field/community trial 51.5.1 The REACT trial 51.5.2 The Informed Choice leaflets trial 61.5.3 The Mwanza trial 71.5.4 The paramedics practitioner trial 71.6 The cohort trial 81.6.1 The PoNDER trial 81.6.2 The DESMOND trial 91.6.3 The Diabetes Care from Diagnosis trial 101.6.4 The REPOSE trial 111.6.5 Other examples of cohort cluster trials 111.7 Field versus cohort designs 111.8 Reasons for cluster trials 121.9 Between- and within-cluster variation 141.10 Random-effects models for continuous outcomes 151.10.1 The model 151.10.2 The intracluster correlation coefficient 161.10.3 Estimating the intracluster correlation (ICC) coefficient 161.10.4 Link between the Pearson correlation coefficient and the intraclass correlation coefficient 171.11 Random-effects models for binary outcomes 181.11.1 The model 181.11.2 The ICC for binary data 191.11.3 The coefficient of variation 191.11.4 Relationship between cvc and 𝜌 for binary data 201.12 The design effect 201.13 Commonly asked questions 211.14 Websources 21Exercise 22Appendix 1.A 222 Design issues 272.1 Introduction 272.2 Issues for a simple intervention 282.2.1 Phases of a trial 282.2.2 ‘Pragmatic’ and ‘explanatory’ trials 292.2.3 Intention-to-treat and per-protocol analyses 292.2.4 Non-inferiority and equivalence trials 302.3 Complex interventions 302.3.1 Design of complex interventions 302.3.2 Phase I modelling/qualitative designs 322.3.3 Pilot or feasibility studies 332.3.4 Example of pilot/feasibility studies in cluster trials 332.4 Recruitment bias 342.5 Matched-pair trials 342.5.1 Design of matched-pair studies 342.5.2 Limitations of matched-pairs designs 362.5.3 Example of matched-pair design: The Family Heart Study 362.6 Other types of designs 372.6.1 Cluster factorial designs 372.6.2 Example cluster factorial trial 382.6.3 Cluster crossover trials 382.6.4 Example of a cluster crossover trial 392.6.5 Stepped wedge 392.6.6 Pseudorandomised trials 402.7 Other design issues 412.8 Strategies for improving precision 412.9 Randomisation 422.9.1 Reasons for randomisation 422.9.2 Simple randomisation 432.9.3 Stratified randomisation 432.9.4 Restricted randomisation 432.9.5 Minimisation 44Exercise 45Appendix 2.A 483 Sample size: How many subjects/clusters do I need for my cluster randomised controlled trial? 503.1 Introduction 513.1.1 Justification of the requirement for a sample size 513.1.2 Significance tests, P-values and power 513.1.3 Sample size and cluster trials 533.2 Sample size for continuous data – comparing two means 533.2.1 Basic formulae 533.2.2 The design effect (DE) in cluster RCTs 543.2.3 Example from general practice 553.3 Sample size for binary data – comparing two proportions 563.3.1 Sample size formula 563.3.2 Example calculations 573.3.3 Example: The Informed Choice leaflets study 583.4 Sample size for ordered categorical (ordinal) data 593.4.1 Sample size formula 593.4.2 Example calculations 603.5 Sample size for rates 623.5.1 Formulae 623.5.2 Example comparing rates 633.6 Sample size for survival 633.6.1 Formulae 633.6.2 Example of sample size for survival 643.7 Equivalence/non-inferiority studies 643.7.1 Equivalence/non-inferiority versus superiority 643.7.2 Continuous data – comparing the equivalence of two means 653.7.3 Example calculations for continuous data 653.7.4 Binary data – comparing the equivalence of two proportions 663.8 Unknown standard deviation and effect size 663.9 Practical problems 673.9.1 Tips on getting the SD 673.9.2 Non-response 673.9.3 Unequal groups 673.10 Number of clusters fixed 683.10.1 Number of clusters and number of subjects per cluster 683.10.2 Example with number of clusters fixed 693.10.3 Increasing the number of clusters or number of patients per cluster? 693.11 Values of the ICC 693.12 Allowing for imprecision in the ICC 703.13 Allowing for varying cluster sizes 703.13.1 Formulae 703.13.2 Example of effect of variable cluster size 713.14 Sample size re-estimation 713.14.1 Adjusting for covariates 723.15 Matched-pair studies 723.15.1 Sample sizes for matched designs 723.15.2 Example of a sample size calculation for a matched study 723.16 Multiple outcomes/endpoints 733.17 Three or more groups 743.18 Crossover trials 743.18.1 Formulae 753.18.2 Example of a sample size formula in a crossover trial 753.19 Post hoc sample size calculations 753.20 Conclusion: Usefulness of sample size calculations 763.21 Commonly asked questions 76Exercise 77Appendix 3.A 784 Simple analysis of cRCT outcomes using aggregate cluster-level summaries 834.1 Introduction 834.1.1 Methods of analysing cluster randomised trials 834.1.2 Choosing the statistical method 844.2 Aggregate cluster-level analysis – carried out at the cluster level, using aggregate summary data 844.3 Statistical methods for continuous outcomes 864.3.1 Two independent-samples t-test 864.3.2 Example 884.4 Mann–Whitney U test 914.5 Statistical methods for binary outcomes 944.6 Analysis of a matched design 954.7 Discussion 984.8 Commonly asked question 98Exercise 99Appendix 4.A 995 Regression methods of analysis for continuous outcomes using individual person-level data 1025.1 Introduction 1025.2 Incorrect models 1045.2.1 The simple (independence) model 1045.2.2 Fixed effects 1045.3 Linear regression with robust standard errors 1055.3.1 Robust standard errors 1055.3.2 Example of use of robust standard errors 1075.3.3 Cluster-specific versus population-averaged models 1075.4 Random-effects general linear models in a cohort study 1085.4.1 General models 1085.4.2 Fitting a random-effects model 1095.4.3 Example of a random-effects model from the PoNDER study 1105.4.4 Checking the assumptions 1105.5 Marginal general linear model with coefficients estimated by generalised estimating equations (GEE) 1125.5.1 Generalised estimating equations 1125.5.2 Example of a marginal model from the PoNDER study 1135.6 Summary of methods 1145.7 Adjusting for individual-level covariates in cohort studies 1155.8 Adjusting for cluster-level covariates in cohort studies 1185.9 Models for cross-sectional designs 1195.10 Discussion of model fitting 120Exercise 122Appendix 5.A 1236 Regression methods of analysis for binary, count and time-to-event outcomes for a cluster randomised controlled trial 1266.1 Introduction 1266.2 Difference between a cluster-specific model and a population-averaged or marginal model for binary data 1276.3 Analysis of binary data using logistic regression 1296.4 Review of past simulations to determine efficiency of different methods for binary data 1306.5 Analysis using summary measures 1316.6 Analysis using logistic regression (ignoring clustering) 1326.7 Random-effects logistic regression 1346.8 Marginal models using generalised estimating equations 1356.9 Analysis of count data 1356.10 Survival analysis with cluster trials 1376.11 Missing data 1396.12 Discussion 139Exercise 139Appendix 6.A 1407 The protocol 1437.1 Introduction 1437.2 Abstract 1447.3 Protocol background 1477.4 Research objectives 1477.5 Outcome measures 1477.6 Design 1477.7 Intervention details 1487.8 Eligibility 1487.9 Randomisation 1497.10 Assessment and data collection 1497.11 Statistical considerations 1507.11.1 Sample size 1507.11.2 Statistical analysis 1517.11.3 Interim analyses 1527.12 Ethics 1537.12.1 Declaration of Helsinki 1537.12.2 Informed consent 1547.13 Organisation 1557.13.1 The team 1557.13.2 Trial forms 1557.13.3 Data management 1557.13.4 Protocol amendments 1567.14 Further reading 156Exercise 1568 Reporting of cRCTs 1598.1 Introduction: Extended CONSORT guidelines for reporting and presenting the results from cRCTs 1598.2 Patient flow diagram 1608.3 Comparison of entry characteristics 1608.4 Incomplete data 1678.5 Reporting the main outcome 1718.6 Subgroup analysis and analysis of secondary outcomes/endpoints 1748.7 Estimates of between-cluster variability 1758.7.1 Example of reporting the ICC: The PoNDER cRCT 1758.8 Further reading 175Exercise 1769 Practical issues 1789.1 Preventing bias in cluster randomised controlled trials 1789.1.1 Problems with identifying and recruiting patients to cluster trials 1789.1.2 Preventing biased recruitment 1799.2 Developing complex interventions 1819.3 Choice of method of analysis 1829.4 Missing data 1859.5 Example sensitivity analysis: Imputation of missing 6-month EPDS data for at-risk women from the PoNDER cRCT 1889.6 Multiplicity of outcomes 1929.6.1 Limiting the number of confirmatory tests 1929.6.2 Summary measures and statistics 1939.6.3 Global tests and multiple comparison procedures 1939.6.4 Which multiple comparison procedure to use? 19410 Computing software 19510.1 R 19510.1.1 History 19510.1.2 Installing R 19610.1.3 Simple use of R 19710.1.4 An example of an R program 19810.2 Stata (version 12) 19910.2.1 Introduction to Stata 19910.2.2 Aggregate cluster-level analysis – carried out at the cluster level, using aggregate summary data 20110.2.3 Random-effects models – continuous outcomes 20210.2.4 Random-effects models – binary outcomes 20510.2.5 Random-effects models – count outcomes 20610.2.6 Marginal models – continuous outcomes 20810.2.7 Marginal models – binary outcomes 20910.2.8 Marginal models – count outcomes 21010.3 SPSS (version 19) 21210.3.1 Introduction to SPSS 21210.3.2 Comparing cluster means using aggregate cluster-level analysis – carried out at the cluster level, using aggregate summary data 21310.3.3 Marginal models 21510.3.4 Random-effects models 22710.4 Conclusion and further reading 232References 234Index 243