# Multiple Imputation and its Application

## Books

A practical guide to analysing partially observed data.

Collecting, analysing and drawing inferences from data is central to research in the medical and social sciences. Unfortunately, it is rarely possible to collect all the intended data. The literature on inference from the resulting incomplete  data is now huge, and continues to grow both as methods are developed for large and complex data structures, and as increasing computer power and suitable software enable researchers to apply these methods.

This book focuses on a particular statistical method for analysing and drawing inferences from incomplete data, called Multiple Imputation (MI). MI is attractive because it is both practical and widely applicable. The authors aim is to clarify the issues raised by missing data, describing the rationale for MI, the relationship between the various imputation models and associated algorithms and its application to increasingly complex data structures.

Multiple Imputation and its Application:

• Discusses the issues raised by the analysis of partially observed data, and the assumptions on which analyses rest.
• Presents a practical guide to the issues to consider when analysing incomplete data from both observational studies and randomized trials.
• Provides a detailed discussion of the practical use of MI with real-world examples drawn from medical and social statistics.
• Explores handling non-linear relationships and interactions with multiple imputation, survival analysis, multilevel multiple imputation, sensitivity analysis via multiple imputation, using non-response weights with multiple imputation and doubly robust multiple imputation.

Multiple Imputation and its Application is aimed at quantitative researchers and students in the medical and social sciences with the aim of clarifying the issues raised by the analysis of incomplete data data, outlining the rationale for MI and describing how to consider and address the issues that arise in its application.

Preface xi

Data acknowledgements xiii

Acknowledgements xv

Glossary xvii

PART I FOUNDATIONS 1

1 Introduction 3

1.1 Reasons for missing data 4

1.2 Examples 6

1.3 Patterns of missing data 7

1.3.1 Consequences of missing data 9

1.4 Inferential framework and notation 10

1.4.1 Missing Completely At Random (MCAR) 11

1.4.2 Missing At Random (MAR) 12

1.4.3 Missing Not At Random (MNAR) 17

1.4.4 Ignorability 21

1.5 Using observed data to inform assumptions about the missingness mechanism 21

1.6 Implications of missing data mechanisms for regression analyses 24

1.6.1 Partially observed response 24

1.6.2 Missing covariates 28

1.6.3 Missing covariates and response 30

1.6.4 Subtle issues I: The odds ratio 30

1.6.5 Implication for linear regression 32

1.6.6 Subtle issues II: Subsample ignorability 33

1.6.7 Summary: When restricting to complete records is valid 34

1.7 Summary 35

2 The multiple imputation procedure and its justification 37

2.1 Introduction 37

2.2 Intuitive outline of the MI procedure 38

2.3 The generic MI procedure 44

2.4 Bayesian justification of MI 46

2.5 Frequentist inference 48

2.5.1 Large number of imputations 49

2.5.2 Small number of imputations 49

2.6 Choosing the number of imputations 54

2.7 Some simple examples 55

2.8 MI in more general settings 62

2.8.1 Survey sample settings 70

2.9 Constructing congenial imputation models 70

2.10 Practical considerations for choosing imputation models 71

2.11 Discussion 73

PART II MULTIPLE IMPUTATION FOR CROSS SECTIONAL DATA 75

3 Multiple imputation of quantitative data 77

3.1 Regression imputation with a monotone missingness pattern 77

3.1.1 MAR mechanisms consistent with a monotone pattern 79

3.1.2 Justification 81

3.2 Joint modelling 81

3.2.1 Fitting the imputation model 82

3.3 Full conditional specification 85

3.3.1 Justification 86

3.4 Full conditional specification versus joint modelling 87

3.5 Software for multivariate normal imputation 88

3.6 Discussion 88

4 Multiple imputation of binary and ordinal data 90

4.1 Sequential imputation with monotone missingness pattern 90

4.2 Joint modelling with the multivariate normal distribution 92

4.3 Modelling binary data using latent normal variables 94

4.3.1 Latent normal model for ordinal data 98

4.4 General location model 103

4.5 Full conditional specification 103

4.5.1 Justification 103

4.6 Issues with over-fitting 104

4.7 Pros and cons of the various approaches 109

4.8 Software 110

4.9 Discussion 111

5 Multiple imputation of unordered categorical data 112

5.1 Monotone missing data 112

5.2 Multivariate normal imputation for categorical data 114

5.3 Maximum indicant model 114

5.3.1 Continuous and categorical variable 117

5.3.2 Imputing missing data 119

5.3.3 More than one categorical variable 120

5.4 General location model 121

5.5 FCS with categorical data 122

5.6 Perfect prediction issues with categorical data 124

5.7 Software 126

5.8 Discussion 126

6 Nonlinear relationships 127

6.1 Passive imputation 128

6.2 No missing data in nonlinear relationships 130

6.3 Missing data in nonlinear relationships 133

6.3.1 Predictive Mean Matching (PMM) 133

6.3.2 Just Another Variable (JAV) 134

6.3.3 Joint modelling approach 135

6.3.4 Extension to more general models and missing data patterns 138

6.3.5 Metropolis-Hastings sampling 140

6.3.6 Rejection sampling 141

6.3.7 FCS approach 143

6.4 Discussion 145

7 Interactions 147

7.1 Interaction variables fully observed 147

7.2 Interactions of categorical variables 151

7.3 General nonlinear relationships 155

7.4 Software 163

7.5 Discussion 164

8 Survival data, skips and large datasets 167

8.1 Time-to-event data 167

8.1.1 Imputing missing covariate values 169

8.1.2 Survival data as categorical 173

8.1.3 Imputing censored survival times 177

8.2 Nonparametric, or ‘hot deck’ imputation 180

8.2.1 Nonparametric imputation for survival data 182

8.3 Multiple imputation for skips 184

8.4 Two-stage MI 188

8.5 Large datasets 190

8.5.1 Large datasets and joint modelling 190

8.5.2 Shrinkage by constraining parameters 192

8.5.3 Comparison of the two approaches 195

8.6 Multiple imputation and record linkage 195

8.7 Measurement error 197

8.8 Multiple imputation for aggregated scores 200

8.9 Discussion 202

9 Multilevel multiple imputation 203

9.1 Multilevel imputation model 203

9.2 MCMC algorithm for imputation model 214

9.3 Imputing level-2 covariates using FCS 220

9.4 Individual patient meta-analysis 222

9.4.1 When to apply Rubin’s rules 224

9.5 Extensions 225

9.5.1 Random level-1 covariance matrices 226

9.5.2 Model fit 228

9.6 Discussion 228

10 Sensitivity analysis: MI unleashed 229

10.1 Review of MNAR modelling 230

10.2 Framing sensitivity analysis 233

10.3 Pattern mixture modelling with MI 235

10.3.1 Missing covariates 240

10.3.2 Application to survival analysis 241

10.4 Pattern mixture approach with longitudinal data via MI 246

10.4.1 Change in slope post-deviation 247

10.5 Piecing together post-deviation distributions from other trial arms 249

10.6 Approximating a selection model by importance weighting 257

10.6.1 Algorithm for approximate sensitivity analysis by re-weighting 259

10.7 Discussion 268

11 Including survey weights 269

11.1 Using model based predictions 269

11.2 Bias in the MI variance estimator 271

11.2.1 MI with weights 274

11.2.2 Estimation in domains 276

11.3 A multilevel approach 277

11.4 Further developments 280

11.5 Discussion 281

12 Robust multiple imputation 282

12.1 Introduction 282

12.2 Theoretical background 284

12.2.1 Simple estimating equations 284

12.2.2 The Probability Of Missingness (POM) model 285

12.2.3 Augmented inverse probability weighted estimating equation 286

12.3 Robust multiple imputation 287

12.3.1 Univariate MAR missing data 287

12.3.2 Longitudinal MAR missing data 289

12.4 Simulation studies 292

12.4.1 Univariate MAR missing data 292

12.4.2 Longitudinal monotone MAR missing data 293

12.4.3 Longitudinal nonmonotone MAR missing data 293

12.4.4 Nonlongitudinal nonmonotone MAR missing data 297

12.4.5 Results and discussion 297

12.5 The RECORD study 302

12.6 Discussion 304

Appendix A Markov Chain Monte Carlo 306

Appendix B Probability distributions 310

B.1 Posterior for the multivariate normal distribution 313

Bibliography 316

Index of Authors 327

Index of Examples 332

Index 334

View all

View all