# Network Meta-Analysis for Decision-Making

Statistics in Practice (Series Nr. 1)

1. Edition March 2018

488 Pages, Hardcover*Wiley & Sons Ltd*

**978-1-118-64750-9**

A practical guide to network meta-analysis with examples and code

In the evaluation of healthcare, rigorous methods of quantitative assessment are necessary to establish which interventions are effective and cost-effective. Often a single study will not provide the answers and it is desirable to synthesise evidence from multiple sources, usually randomised controlled trials. This book takes an approach to evidence synthesis that is specifically intended for decision making when there are two or more treatment alternatives being evaluated, and assumes that the purpose of every synthesis is to answer the question "for this pre-identified population of patients, which treatment is 'best'?"

A comprehensive, coherent framework for network meta-analysis (mixed treatment comparisons) is adopted and estimated using Bayesian Markov Chain Monte Carlo methods implemented in the freely available software WinBUGS. Each chapter contains worked examples, exercises, solutions and code that may be adapted by readers to apply to their own analyses.

This book can be used as an introduction to evidence synthesis and network meta-analysis, its key properties and policy implications. Examples and advanced methods are also presented for the more experienced reader.

* Methods used throughout this book can be applied consistently: model critique and checking for evidence consistency are emphasised.

* Methods are based on technical support documents produced for NICE Decision Support Unit, which support the NICE Methods of Technology Appraisal.

* Code presented is also the basis for the code used by the ISPOR Task Force on Indirect Comparisons.

* Includes extensive carefully worked examples, with thorough explanations of how to set out data for use in WinBUGS and how to interpret the output.

Network Meta-Analysis for Decision Making will be of interest to decision makers, medical statisticians, health economists, and anyone involved in Health Technology Assessment including the pharmaceutical industry.

List of Abbreviations xxi

About the Companion Website xxv

1 Introduction to Evidence Synthesis 1

1.1 Introduction 1

1.2 Why Indirect Comparisons and Network Meta?-Analysis? 2

1.3 Some Simple Methods 4

1.4 An Example of a Network Meta?-Analysis 6

1.5 Assumptions Made by Indirect Comparisons and Network Meta?-Analysis 9

1.6 Which Trials to Include in a Network 12

1.6.1 The Need for a Unique Set of Trials 12

1.7 The Definition of Treatments and Outcomes: Network Connectivity 14

1.7.1 Lumping and Splitting 14

1.7.2 Relationships Between Multiple Outcomes 15

1.7.3 How Large Should a Network Be? 15

1.8 Summary 16

1.9 Exercises 16

2 The Core Model 19

2.1 Bayesian Meta?-Analysis 19

2.2 Development of the Core Models 20

2.2.1 Worked Example: Meta?-Analysis of Binomial Data 21

2.2.1.1 Model Specification: Two Treatments 21

2.2.1.2 WinBUGS Implementation: Two Treatments 25

2.2.2 Extension to Indirect Comparisons and Network Meta?-Analysis 32

2.2.2.1 Incorporating Multi?-Arm Trials 35

2.2.3 Worked Example: Network Meta?-Analysis 36

2.2.3.1 WinBUGS Implementation 37

2.3 Technical Issues in Network Meta?-Analysis 50

2.3.1 Choice of Reference Treatment 50

2.3.2 Choice of Prior Distributions 51

2.3.3 Choice of Scale 53

2.3.4 Connected Networks 54

2.4 Advantages of a Bayesian Approach 55

2.5 Summary of Key Points and Further Reading 56

2.6 Exercises 57

3 Model Fit, Model Comparison and Outlier Detection 59

3.1 Introduction 59

3.2 Assessing Model Fit 60

3.2.1 Deviance 60

3.2.2 Residual Deviance 61

3.2.3 Zero Counts* 62

3.2.4 Worked Example: Full Thrombolytic Treatments Network 62

3.2.4.1 Posterior Mean Deviance, Dmodel 62

3.2.4.2 Posterior Mean Residual Deviance, Dres 64

3.3 Model Comparison 66

3.3.1 Effective Number of Parameters, pD 68

3.3.2 Deviance Information Criterion (DIC) 69

3.3.2.1 *Leverage Plots 70

3.3.3 Worked Example: Full Thrombolytic Treatments Network 70

3.4 Outlier Detection in Network Meta?-Analysis 75

3.4.1 Outlier Detection in Pairwise Meta?-Analysis 75

3.4.2 Predictive Cross?-Validation for Network Meta?-Analysis 79

3.4.3 Note on Multi?-Arm Trials 85

3.4.4 WinBUGS Code: Predictive Cross?-Validation for Network Meta?-Analysis 86

3.5 Summary and Further Reading 89

3.6 Exercises 90

4 Generalised Linear Models 93

4.1 A Unified Framework for Evidence Synthesis 93

4.2 The Generic Network Meta?-Analysis Models 94

4.3 Univariate Arm?-Based Likelihoods 99

4.3.1 Rate Data: Poisson Likelihood and Log Link 99

4.3.1.1 WinBUGS Implementation 100

4.3.1.2 Example: Dietary Fat 101

4.3.1.3 Results: Dietary Fat 104

4.3.2 Rate Data: Binomial Likelihood and Cloglog Link 105

4.3.2.1 WinBUGS Implementation 107

4.3.2.2 Example: Diabetes 109

4.3.2.3 Results: Diabetes 112

4.3.3 Continuous Data: Normal Likelihood and Identity Link 114

4.3.3.1 Before/After Studies: Change from Baseline Measures 115

4.3.3.2 Standardised Mean Differences 115

4.3.3.3 WinBUGS Implementation 116

4.3.3.4 Example: Parkinson's 117

4.3.3.5 Results: Parkinson's 119

4.4 Contrast?-Based Likelihoods 120

4.4.1 Continuous Data: Treatment Differences 121

4.4.1.1 Multi?-Arm Trials with Treatment Differences (Trial?-Based Summaries) 122

4.4.1.2 *WinBUGS Implementation 123

4.4.1.3 Example: Parkinson's (Treatment Differences as Data) 125

4.4.1.4 Results: Parkinson's (Treatment Differences as Data) 127

4.5 *Multinomial Likelihoods 127

4.5.1 Ordered Categorical Data: Multinomial Likelihood and Probit Link 128

4.5.1.1 WinBUGS Implementation 132

4.5.1.2 Example: Psoriasis 133

4.5.1.3 Results: Psoriasis 137

4.5.2 Competing Risks: Multinomial Likelihood and Log Link 138

4.5.2.1 WinBUGS Implementation 140

4.5.2.2 Example: Schizophrenia 141

4.5.2.3 Results: Schizophrenia 143

4.6 *Shared Parameter Models 146

4.6.1 Example: Parkinson's (Mixed Treatment Difference and Arm?-Level Data) 147

4.6.2 Results: Parkinson's (Mixed Treatment Difference and Arm?-Level Data) 148

4.7 Choice of Prior Distributions 149

4.8 Zero Cells 149

4.9 Summary of Key Points and Further Reading 150

4.10 Exercises 151

5 Network Meta?-Analysis Within Cost?-Effectiveness Analysis 155

5.1 Introduction 155

5.2 Sources of Evidence for Relative Treatment Effects and the Baseline Model 156

5.3 The Baseline Model 158

5.3.1 Estimating the Baseline Model in WinBUGS 158

5.3.2 Alternative Computation Methods for the Baseline Model 162

5.3.3 *Arm?-Based Meta?-Analytic Models 162

5.3.4 Baseline Models with Covariates 164

5.3.4.1 Using Aggregate Data 164

5.3.4.2 Risk Equations for the Baseline Model Basedon Individual Patient Data 165

5.4 The Natural History Model 165

5.5 Model Validation and Calibration Through Multi?-Parameter Synthesis 167

5.6 Generating the Outputs Required for Cost?-Effectiveness Analysis 169

5.6.1 Generating a CEA 169

5.6.2 Heterogeneity in the Context of Decision?-Making 170

5.7 Strategies to Implement Cost?-Effectiveness Analyses 173

5.7.1 Bayesian Posterior Simulation: One?-Stage Approach 174

5.7.2 Bayesian Posterior Simulation: Two?-Stage Approach 174

5.7.3 Multiple Software Platforms and Automation of Network Meta?-Analysis 175

5.8 Summary and Further Reading 177

5.9 Exercises 178

6 Adverse Events and Other Sparse Outcome Data 179

6.1 Introduction 179

6.2 Challenges Regarding the Analysis of Sparse Data in Pairwise and Network Meta?-Analysis 180

6.2.1 Network Structure and Connectivity 182

6.2.2 Assessing Convergence and Model Fit 182

6.3 Strategies to Improve the Robustness of Estimation of Effects from Sparse Data in Network Meta?-Analysis 183

6.3.1 Specifying Informative Prior Distributions for Response in Trial Reference Groups 183

6.3.2 Specifying an Informative Prior Distribution for the Between Study Variance Parameters 184

6.3.3 Specifying Reference Group Responses as Exchangeable with Random Effects 184

6.3.4 Situational Modelling Extensions 185

6.3.5 Specification of Informative Prior Distributions Versus Use of Continuity Corrections 186

6.4 Summary and Further Reading 186

6.5 Exercises 187

7 Checking for Inconsistency 189

7.1 Introduction 189

7.2 Network Structure 190

7.2.1 Inconsistency Degrees of Freedom 191

7.2.2 Defining Inconsistency in the Presence of Multi?-Arm Trials 192

7.3 Loop Specific Tests for Inconsistency 195

7.3.1 Networks with Independent Tests for Inconsistency 195

7.3.1.1 Bucher Method for Single Loops of Evidence 195

7.3.1.2 Example: HIV 196

7.3.1.3 Extension of Bucher Method to Networks with Multiple Loops: Enuresis Example 197

7.3.1.4 Obtaining the 'Direct' Estimates of Inconsistency 199

7.3.2 Methods for General Networks 200

7.3.2.1 Repeat Application of the Bucher Method 201

7.3.2.2 A Back?-Calculation Method 202

7.3.2.3 *Variance Measures of Inconsistency 202

7.3.2.4 *Node?-Splitting 203

7.4 A Global Test for Loop Inconsistency 205

7.4.1 Inconsistency Model with Unrelated Mean Relative Effects 206

7.4.2 Example: Full Thrombolytic Treatments Network 210

7.4.2.1 Adjusted Standard Errors for Multi?-Arm Trials 214

7.4.3 Example: Parkinson's 215

7.4.4 Example: Diabetes 218

7.5 Response to Inconsistency 219

7.6 The Relationship between Heterogeneity and Inconsistency 221

7.7 Summary and Further Reading 223

7.8 Exercises 225

8 Meta?-Regression for Relative Treatment Effects 227

8.1 Introduction 227

8.2 Basic Concepts 229

8.2.1 Types of Covariate 229

8.3 Heterogeneity, Meta?-Regression and Predictive Distributions 232

8.3.1 Worked Example: BCG Vaccine 233

8.3.2 Implications of Heterogeneity in Decision Making 236

8.4 Meta?-Regression Models for Network Meta?-Analysis 238

8.4.1 Baseline Risk 241

8.4.2 WinBUGS Implementation 242

8.4.3 Meta?-Regression with a Continuous Covariate 245

8.4.3.1 BCG Vaccine Example: Pairwise Meta?-Regression with a Continuous Covariate 245

8.4.3.2 Certolizumab Example: Network Meta?-Regression with Continuous Covariate 247

8.4.3.3 Certolizumab Example: Network Meta?-Regression on Baseline Risk 252

8.4.4 Subgroup Effects 255

8.4.4.1 Statins Example: Pairwise Meta?-Analysis with Subgroups 256

8.5 Individual Patient Data in Meta?-Regression 257

8.6 Models with Treatment?-Level Covariates 261

8.6.1 Accounting for Dose 261

8.6.2 Class Effects Models 263

8.6.3 Treatment Combination Models 264

8.7 Implications of Meta?-Regression for Decision Making 266

8.8 Summary and Further Reading 268

8.9 Exercises 269

9 Bias Adjustment Methods 273

9.1 Introduction 273

9.2 Adjustment for Bias Based on Meta?-Epidemiological Data 275

9.3 Estimation and Adjustment for Bias in Networks of Trials 278

9.3.1 Worked Example: Fluoride Therapies for the Prevention of Caries in Children 279

9.3.2 Extensions 285

9.3.3 Novel Agent Effects 286

9.3.4 Small?-Study Effects 287

9.3.5 Industry Sponsor Effects 287

9.3.6 Accounting for Missing Data 288

9.4 Elicitation of Internal and External Bias Distributions from Experts 289

9.5 Summary and Further Reading 290

9.6 Exercises 291

10 *Network Meta?-Analysis of Survival Outcomes 293

10.1 Introduction 293

10.2 Time?-to?-Event Data 294

10.2.1 Individual Patient Data 294

10.2.2 Reported Summary Data 295

10.2.3 Kaplan-Meier Estimate of the Survival Function 295

10.3 Parametric Survival Functions 296

10.4 The Relative Treatment Effect 298

10.5 Network Meta?-Analysis of a Single Effect Measure per Study 300

10.5.1 Proportion Alive, Median Survival and Hazard Ratio as Reported Treatment Effects 300

10.5.2 Network Meta?-Analysis of Parametric Survival Curves: Single Treatment Effect 300

10.5.3 Shared Parameter Models 301

10.5.4 Limitations 302

10.6 Network Meta?-Analysis with Multivariate Treatment Effects 302

10.6.1 Multidimensional Network Meta?-Analysis Model 302

10.6.1.1 Weibull 302

10.6.1.2 Gompertz 303

10.6.1.3 Log?-Logistic and Log?-Normal 303

10.6.1.4 Fractional Polynomial 304

10.6.1.5 Splines 304

10.6.2 Evaluation of Consistency 304

10.6.3 Meta?-Regression 305

10.7 Data and Likelihood 305

10.7.1 Likelihood with Individual Patient Data 305

10.7.2 Discrete or Piecewise Constant Hazards as Approximate Likelihood 306

10.7.3 Conditional Survival Probabilities as Approximate Likelihood 307

10.7.4 Reconstructing Kaplan-Meier Data 307

10.7.5 Constructing Interval Data 308

10.8 Model Choice 308

10.9 Presentation of Results 309

10.10 Illustrative Example 310

10.11 Network Meta?-Analysis of Survival Outcomes for Cost?-Effectiveness Evaluations 319

10.12 Summary and Further Reading 320

10.13 Exercises 322

11 *Multiple Outcomes 323

11.1 Introduction 323

11.2 Multivariate Random Effects Meta?-Analysis 324

11.3 Multinomial Likelihoods and Extensions of Univariate Methods 327

11.4 Chains of Evidence 328

11.4.1 A Decision Tree Structure: Coronary Patency 328

11.4.2 Chain of Evidence with Relative Risks: Neonatal Early Onset Group B Strep 330

11.5 Follow?-Up to Multiple Time Points: Gastro?-Esophageal Reflux Disease 332

11.6 Multiple Outcomes Reported in Different Ways: Influenza 335

11.7 Simultaneous Mapping and Synthesis 337

11.8 Related Outcomes Reported in Different Ways: Advanced Breast Cancer 342

11.9 Repeat Observations for Continuous Outcomes: Fractional Polynomials 344

11.10 Synthesis for Markov Models 345

11.11 Summary and Further Reading 347

11.12 Exercises 349

12 Validity of Network Meta?-Analysis 351

12.1 Introduction 351

12.2 What Are the Assumptions of Network Meta?-Analysis? 352

12.2.1 Exchangeability 352

12.2.2 Other Terminologies and Their Relation to Exchangeability 353

12.3 Direct and Indirect Comparisons: Some Thought Experiments 355

12.3.1 Direct Comparisons 356

12.3.2 Indirect Comparisons 359

12.3.3 Under What Conditions Is Evidence Synthesis Likely to Be Valid? 362

12.4 Empirical Studies of the Consistency Assumption 363

12.5 Quality of Evidence Versus Reliability of Recommendation 365

12.5.1 Theoretical Treatment of Validity of Network Meta?-Analysis 365

12.5.2 GRADE Assessment of Quality of Evidence from a Network Meta?-Analyses 366

12.5.3 Reliability of Recommendations Versus Quality of Evidence: The Role of Sensitivity Analysis 368

12.6 Summary and Further Reading 369

12.7 Exercises 373

Solutions to Exercises 375

Appendices 401

References 409

Index 447

A.E. ADES, University of Bristol, UK

NICKY J. WELTON, University of Bristol, UK

JEROEN P. JANSEN, Precision Health Economics, USA

ALEXANDER J. SUTTON, University of Leicester, UK