Multiple Imputation and its Application

The most up-to-date edition of a bestselling guide to analyzing partially observed data

In this comprehensively revised Second Edition of Multiple Imputation and its Application, a team of distinguished statisticians delivers an overview of the issues raised by missing data, the rationale for multiple imputation as a solution, and the practicalities of applying it in a multitude of settings.

With an accessible and carefully structured presentation aimed at quantitative researchers, Multiple Imputation and its Application is illustrated with a range of examples and offers key mathematical details. The book includes a wide range of theoretical and computer-based exercises, tested in the classroom, which are especially useful for users of R or Stata. Readers will find:

• A comprehensive overview of one of the most effective and popular methodologies for dealing with incomplete data sets
• Careful discussion of key concepts
• A range of examples illustrating the key ideas
• Practical advice on using multiple imputation
• Exercises and examples designed for use in the classroom and/or private study

Written for applied researchers looking to use multiple imputation with confidence, and for methods researchers seeking an accessible overview of the topic, Multiple Imputation and its Application will also earn a place in the libraries of graduate students undertaking quantitative analyses.

James R. Carpenter is Professor of Medical Statistics at the London School of Hygiene & Tropical Medicine and Programme Leader in Methodology at the MRC Clinical Trials Unit at UCL, UK.

Jonathan W. Bartlett is a Professor of Medical Statistics at the London School of Hygiene & Tropical Medicine, UK.

Tim P. Morris is Principal Research Fellow in Medical Statistics at the MRC Clinical Trials Unit at UCL, UK.

Angela M. Wood is Professor of Health Data Science in the Department of Public Health and Primary Care, University of Cambridge, UK.

Matteo Quartagno is Senior Research Fellow in Medical Statistics at the MRC Clinical Trials Unit at UCL, UK.

Michael G. Kenward retired in 2016 after sixteen years as GlaxoSmithKline Professor of Biostatistics at the London School of Hygiene & Tropical Medicine, UK.

Preface

Data acknowledgments

Glossary

I Foundations 1

1 Introduction 2

1.1 Reasons for missing data . . . . . . . . . . . . . . . . . . . . . 5

1.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.3 Patterns of missing data . . . . . . . . . . . . . . . . . . . . . 8

1.3.1 Consequences of missing data . . . . . . . . . . . . . . . 10

1.4 Inferential framework and notation . . . . . . . . . . . . . . . . 13

1.4.1 Missing Completely At Random (MCAR) . . . . . . . . 15

1.4.2 Missing At Random (MAR) . . . . . . . . . . . . . . . . 16

1.4.3 Missing Not At Random (MNAR) . . . . . . . . . . . . 22

1.4.4 Ignorability . . . . . . . . . . . . . . . . . . . . . . . . . 27

1.5 Using observed data to inform assumptions about the missingness mechanism . .. . . . . . . 28

1.6 Implications of missing data mechanisms for regression analyses 32

1.6.1 Partially observed response . . . . . . . . . . . . . . . . 33

1.6.2 Missing covariates . . . . . . . . . . . . . . . . . . . . . 37

1.6.3 Missing covariates and response . . . . . . . . . . . . . . 40

1.6.4 Subtle issues I: the odds ratio . . . . . . . . . . . . . . . 40

1.6.5 Implication for linear regression . . . . . . . . . . . . . . 43

1.6.6 Subtle issues II: sub sample ignorability . . . . . . . . . 44

1.6.7 Summary: when restricting to complete records is valid 45

1.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

1.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2 The Multiple Imputation Procedure and Its Justification 52

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

2.2 Intuitive outline of the MI procedure . . . . . . . . . . . . . . 54

2.3 The generic MI Procedure . . . . . . . . . . . . . . . . . . . . . 61

2.4 Bayesian justification of MI . . . . . . . . . . . . . . . . . . . . 64

2.5 Frequentist Inference . . . . . . . . . . . . . . . . . . . . . . . 66

2.6 Choosing the number of imputations . . . . . . . . . . . . . . . 73

2.7 Some simple examples . . . . . . . . . . . . . . . . . . . . . . . 75

2.8 MI in More General Settings . . . . . . . . . . . . . . . . . . . 84

2.8.1 Proper imputation . . . . . . . . . . . . . . . . . . . . . 84

2.8.2 Congenial imputation and substantive model . . . . . . 85

2.8.3 Uncongenial imputation and substantive models . . . . 87

2.8.4 Survey Sample Settings . . . . . . . . . . . . . . . . . . 94

2.9 Constructing congenial imputation models . . . . . . . . . . . . 95

2.10 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

2.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

1.6.3 Missing covariates and response . . . . . . . . . . . . . . 40

1.6.4 Subtle issues I: the odds ratio . . . . . . . . . . . . . . . 40

1.6.5 Implication for linear regression . . . . . . . . . . . . . . 43

1.6.6 Subtle issues II: sub sample ignorability . . . . . . . . . 44

1.6.7 Summary: when restricting to complete records is valid 45

1.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

1.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2 The Multiple Imputation Procedure and Its Justification 52

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

2.2 Intuitive outline of the MI procedure . . . . . . . . . . . . . . 54

2.3 The generic MI Procedure . . . . . . . . . . . . . . . . . . . . . 61

2.4 Bayesian justification of MI . . . . . . . . . . . . . . . . . . . . 64

2.5 Frequentist Inference . . . . . . . . . . . . . . . . . . . . . . . 66

2.6 Choosing the number of imputations . . . . . . . . . . . . . . . 73

2.7 Some simple examples . . . . . . . . . . . . . . . . . . . . . . . 75

2.8 MI in More General Settings . . . . . . . . . . . . . . . . . . . 84

2.8.1 Proper imputation . . . . . . . . . . . . . . . . . . . . . 84

2.8.2 Congenial imputation and substantive model . . . . . . 85

2.8.3 Uncongenial imputation and substantive models . . . . 87

2.8.4 Survey Sample Settings . . . . . . . . . . . . . . . . . . 94

2.9 Constructing congenial imputation models . . . . . . . . . . . . 95

2.10 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

2.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

II Multiple imputation for simple data structures 104

3 Multiple imputation of quantitative data 105

3.1 Regression imputation with a monotone missingness pattern . . 105

3.1.1 MAR mechanisms consistent with a monotone pattern . 107

3.1.2 Justification . . . . . . . . . . . . . . . . . . . . . . . . 109

3.2 Joint modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

3.2.1 Fitting the imputation model . . . . . . . . . . . . . . 111

3.2.2 Adding covariates . . . . . . . . . . . . . . . . . . . . . 115

3.3 Full conditional specification . . . . . . . . . . . . . . . . . . . 118

3.3.1 Justification . . . . . . . . . . . . . . . . . . . . . . . . . 119

3.4 Full conditional specification versus joint modelling . . . . . . . 121

3.5 Software for multivariate normal imputation . . . . . . . . . . . 121

3.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

3.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

4 Multiple imputation of binary and ordinal data 125

4.1 Sequential imputation with monotone missingness pattern . . 125

4.2 Joint modelling with the multivariate normal distribution . . . 127

4.3 Modelling binary data using latent normal variables . . . . . . 130

4.3.1 Latent normal model for ordinal data . . . . . . . . . . 137

4.4 General location model . . . . . . . . . . . . . . . . . . . . . . 141

4.5 Full conditional specification . . . . . . . . . . . . . . . . . . . 142

4.5.1 Justification . . . . . . . . . . . . . . . . . . . . . . . . . 143

4.6 Issues with over-fitting . . . . . . . . . . . . . . . . . . . . . . 144

4.7 Pros and cons of the various approaches . . . . . . . . . . . . . 150

4.8 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

4.9 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

4.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153

5 Imputation of unordered categorical data 156

5.1 Monotone missing data . . . . . . . . . . . . . . . . . . . . . . 157

5.2 Multivariate normal imputation for categorical data . . . . . . 158

5.3 Maximum indicant model . . . . . . . . . . . . . . . . . . . . . 159

5.3.1 Continuous and categorical variable . . . . . . . . . . . 162

5.3.2 Imputing missing data . . . . . . . . . . . . . . . . . . . 164

5.4 General location model . . . . . . . . . . . . . . . . . . . . . . 165

5.5 FCS with categorical data . . . . . . . . . . . . . . . . . . . . 169

5.6 Perfect prediction issues with categorical data . . . . . . . . . . 170

5.7 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

5.8 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

5.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173

III Multiple imputation in practice 175

6 Non-linear relationships, interactions, and other derived variables 176

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

6.1.1 Interactions . . . . . . . . . . . . . . . . . . . . . . . . . 178

6.1.2 Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

6.1.3 Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

6.1.4 Sum scores . . . . . . . . . . . . . . . . . . . . . . . . . 181

6.1.5 Composite endpoints . . . . . . . . . . . . . . . . . . . . 182

6.2 No missing data in derived variables . . . . . . . . . . . . . . . 184

6.3 Simple methods . . . . . . . . . . . . . . . . . . . . . . . . . . 186

6.3.1 Impute then transform . . . . . . . . . . . . . . . . . . . 187

6.3.2 Transform then impute / just another variable . . . . . 187

6.3.3 Adapting standard imputation models and passive imputation .. . . . . . . . . . . . . . . . . . . . . . 190

6.3.4 Predictive mean matching . . . . . . . . . . . . . . . . . 191

6.3.5 Imputation separately by groups for interactions . . . . 195

6.4 Substantive-model-compatible imputation . . . . . . . . . . . . 200

6.4.1 The basic idea . . . . . . . . . . . . . . . . . . . . . . . 200

6.4.2 Latent-normal joint model SMC imputation . . . . . . . 207

6.4.3 Factorised conditional model SMC imputation . . . . . 209

6.4.4 Substantive model compatible fully conditional specification . . . . . . . . . . . . . . . . . . . . . . . . . 212

6.4.5 Auxiliary variables . . . . . . . . . . . . . . . . . . . . . 213

6.4.6 Missing outcome values . . . . . . . . . . . . . . . . . . 214

6.4.7 Congeniality vs. compatibility . . . . . . . . . . . . . . . 214

6.4.8 Discussion of SMC . . . . . . . . . . . . . . . . . . . . . 216

6.5 Returning to the problems . . . . . . . . . . . . . . . . . . . . . 217

6.5.1 Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

6.5.2 Splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

6.5.3 Fractional polynomials . . . . . . . . . . . . . . . . . . . 218

6.5.4 Multiple imputation with conditional questions or ‘skips’223

6.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228

7 Survival data 231

7.1 Missing covariates in time to event data . . . . . . . . . . . . . 231

7.1.1 Approximately compatible approaches . . . . . . . . . . 232

7.1.2 Substantive model compatible approaches . . . . . . . . 241

7.2 Imputing censored survival times . . . . . . . . . . . . . . . . . 245

7.3 Non-parametric, or ‘hot deck’ imputation . . . . . . . . . . . . 248

7.3.1 Non-parametric imputation for survival data . . . . . . 251

7.4 Case–cohort designs . . . . . . . . . . . . . . . . . . . . . . . . 254

7.4.1 Standard analysis of case–cohort studies . . . . . . . . . 254

7.4.2 Multiple imputation for case-cohort studies . . . . . . . 255

7.4.3 Full-cohort . . . . . . . . . . . . . . . . . . . . . . . . . 256

7.4.4 Intermediate approaches . . . . . . . . . . . . . . . . . . 257

7.4.5 Substudy approach . . . . . . . . . . . . . . . . . . . . . 257

7.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

7.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261

8 Prognostic models, missing data and multiple imputation 265

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

8.2 Motivating example . . . . . . . . . . . . . . . . . . . . . . . . 266

8.3 Missing data at model implementation . . . . . . . . . . . . . 267

8.4 Multiple imputation for prognostic modelling . . . . . . . . . . 268

8.5 Model building . . . . . . . . . . . . . . . . . . . . . . . . . . . 268

8.5.1 Model building with missing data . . . . . . . . . . . . . 268

8.5.2 Imputing predictors when model building is to be performed . . .  . . . . . . . . . . . . . . . . . . . . . . 270

8.6 Model performance . . . . . . . . . . . . . . . . . . . . . . . . 271

8.6.1 How should we pool MI results for estimation of performance? . . . . . . . . . . . . . . . . . . . . . . . 271

8.6.2 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . 272

8.6.3 Discrimination . . . . . . . . . . . . . . . . . . . . . . . 273

8.6.4 Model performance measures with clinical interpretability273

8.7 Model validation . . . . . . . . . . . . . . . . . . . . . . . . . . 274

8.7.1 Internal model validation . . . . . . . . . . . . . . . . . 274

8.7.2 External model validation . . . . . . . . . . . . . . . . . 275

8.8 Incomplete data at implementation . . . . . . . . . . . . . . . 276

8.8.1 MI for incomplete data at implementation . . . . . . . . 276

8.8.2 Alternatives to multiple imputation . . . . . . . . . . . 278

8.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

9 Multilevel multiple imputation 283

9.1 Multilevel imputation model . . . . . . . . . . . . . . . . . . . 284

9.1.1 Imputation of level 1 variables . . . . . . . . . . . . . . 287

9.1.2 Imputation of level 2 variables . . . . . . . . . . . . . . 291

9.1.3 Accommodating the substantive model . . . . . . . . . . 296

9.2 MCMC algorithm for imputation model . . . . . . . . . . . . . 297

9.2.1 Checking model convergence . . . . . . . . . . . . . . . 305

9.3 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 307

9.3.1 Cross-classification and 3-level data . . . . . . . . . . . 307

9.3.2 Random level 1 covariance matrices . . . . . . . . . . . 308

9.3.3 Model fit . . . . . . . . . . . . . . . . . . . . . . . . . . 310

9.4 Other imputation methods . . . . . . . . . . . . . . . . . . . . 311

9.4.1 1-step and 2-step FCS . . . . . . . . . . . . . . . . . . . 312

9.4.2 Substantive model compatible imputation . . . . . . . . 313

9.4.3 Non-parametric methods . . . . . . . . . . . . . . . . . 314

9.4.4 Comparisons of different methods . . . . . . . . . . . . 314

9.5 Individual participant data meta-analysis . . . . . . . . . . . . 315

9.5.1 When to apply Rubin’s rules . . . . . . . . . . . . . . . 318

9.5.2 Homoscedastic vs heteroscedastic imputation model . . 320

9.6 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320

9.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321

9.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322

10 Sensitivity analysis: MI unleashed 326

10.1 Review of MNAR modelling . . . . . . . . . . . . . . . . . . . 328

10.2 Framing sensitivity analysis: Estimands . . . . . . . . . . . . . 331

10.3 Pattern mixture modelling with MI . . . . . . . . . . . . . . . 335

10.3.1 Missing covariates . . . . . . . . . . . . . . . . . . . . . 341

10.3.2 Sensitivity with multiple variables: the NAR FCS procedure . . . . . . .. . . . . . . . . . . . . . . . . . . 344

10.3.3 Application to survival analysis . . . . . . . . . . . . . . 346

10.4 Pattern mixture approach with longitudinal data via MI . . . . 351

10.4.1 Change in slope post-deviation . . . . . . . . . . . . . . 353

10.5 Reference based imputation . . . . . . . . . . . . . . . . . . . . 356

10.5.1 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . 361

10.5.2 Information Anchoring . . . . . . . . . . . . . . . . . . 368

10.6 Approximating a selection model by importance weighting . . 372

10.6.1 Weighting the imputations . . . . . . . . . . . . . . . . 375

10.6.2 Stacking the imputations and applying the weights . . . 376

10.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386

10.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 387

11 Multiple imputation for measurement error and misclassification 392

11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393

11.2 Multiple imputation with validation data . . . . . . . . . . . . 394

11.2.1 Measurement error . . . . . . . . . . . . . . . . . . . . . 396

11.2.2 Misclassification . . . . . . . . . . . . . . . . . . . . . . 397

11.2.3 Imputing assuming error is non-differential . . . . . . . 399

11.2.4 Non-linear outcome models . . . . . . . . . . . . . . . . 400

11.3 Multiple imputation with replication data . . . . . . . . . . . . 401

11.3.1 Measurement error . . . . . . . . . . . . . . . . . . . . . 403

11.3.2 Misclassification . . . . . . . . . . . . . . . . . . . . . . 408

11.4 External information on the measurement process . . . . . . . 409

11.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411

11.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413

12 Multiple imputation with weights 416

12.1 Using model based predictions in strata . . . . . . . . . . . . . 417

12.2 Bias in the MI Variance Estimator . . . . . . . . . . . . . . . . 418

12.3 MI with weights . . . . . . . . . . . . . . . . . . . . . . . . . . 422

12.3.1 Conditions for consistency of θbMI . . . . . . . . . . . . 422

12.3.2 Conditions for the consistency of Vb MI . . . . . . . . . . 424

12.4 A multilevel approach . . . . . . . . . . . . . . . . . . . . . . . 426

12.4.1 Evaluation of the multilevel multiple imputation approach for handling survey weights . . . 429

12.4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 434

12.5 Further topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 437

12.5.1 Estimation in Domains . . . . . . . . . . . . . . . . . . 437

12.5.2 Two-stage analysis . . . . . . . . . . . . . . . . . . . . 437

12.5.3 Missing values in the weight model . . . . . . . . . . . . 438

12.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 438

12.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439

13 Multiple imputation for causal inference 443

13.1 Multiple imputation for causal inference in point exposure studies444

13.1.1 Randomised trials . . . . . . . . . . . . . . . . . . . . . 445

13.1.2 Observational studies . . . . . . . . . . . . . . . . . . . 446

13.2 Multiple imputation and propensity scores . . . . . . . . . . . . 450

13.2.1 Propensity scores for confounder adjustment . . . . . . 450

13.2.2 Multiple imputation of confounders . . . . . . . . . . . . 452

13.2.3 Imputation model specification . . . . . . . . . . . . . . 456

13.3 Principal stratification via multiple imputation . . . . . . . . . 457

13.3.1 Principal strata effects . . . . . . . . . . . . . . . . . . 458

13.3.2 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . 459

13.4 Multiple imputation for instrumental variable analysis . . . . . 461

13.4.1 Instrumental variable analysis for non-adherence . . . . 461

13.4.2 Instrumental variable analysis via multiple imputation . 464

13.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 467

13.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468

14 Using multiple imputation in practice 472

14.1 A general approach . . . . . . . . . . . . . . . . . . . . . . . . 473

14.2 Objections to multiple imputation . . . . . . . . . . . . . . . . 477

14.3 Reporting of analyses with incomplete data . . . . . . . . . . . 482

14.4 Presenting incomplete baseline data . . . . . . . . . . . . . . . 483

14.5 Model diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . 486

14.6 How many imputations? . . . . . . . . . . . . . . . . . . . . . . 487

14.6.1 Using the jack-knife estimate of the Monte-Carlo standard error . . . . . .  . . . . . . . . . . . . . . 490

14.7 Multiple imputation for each substantive model, project or

dataset? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 492

14.8 Large datasets . . . . . . . . . . . . . . . . . . . . . . . . . . . 493

14.8.1 Large datasets and joint modelling . . . . . . . . . . . 494

14.8.2 Shrinkage by constraining parameters . . . . . . . . . . 496

14.8.3 Comparison of the two approaches . . . . . . . . . . . . 499

14.9 Multiple Imputation and record linkage . . . . . . . . . . . . . 500

14.10Setting random number seeds for multiple imputation analyses 502

14.11Simulation studies including multiple imputation . . . . . . . . 503

14.11.1Random number seeds for simulation studies including

multiple imputation . . . . . . . . . . . . . . . . . . . . 503

14.11.2Repeated simulation of all data or only the missingness

mechanism? . . . . . . . . . . . . . . . . . . . . . . . . 504

14.11.3How many imputations for simulation studies? . . . . . 505

14.11.4Multiple imputation for data simulation . . . . . . . . . 507

14.12Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508

A Markov Chain Monte Carlo 512

B Probability distributions 517

B.1 Posterior for the multivariate normal distribution . . . . . . . 521

C Overview of multiple imputation in R, Stata 524

C.1 Basic multiple imputation using R . . . . . . . . . . . . . . . . 524

C.2 Basic MI using Stata . . . . . . . . . . . . . . . . . . . . . . . . 526

Bibliography 530

Index 555

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