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More About This Title Multiple Imputation and its Application
English
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.
English
James Carpenter, Medical Statistics Unit, London School of Hygiene and Tropical Medicine, UK.
Michael G. Kenward, Medical Statistics Unit, London School of Hygiene and Tropical Medicine, UK
Amongst other areas Professor Kenward has worked in pre-clinical and clinical medicine and epidemiology for over twenty years, holding a number of international positions. He has also been a statistical consultant for over twenty years, predominantly in medical research. He has taught over 80 short courses in biostatistics throughout the world, and is the author of the book Analysis of Repeated Measurements.
Both authors act as consultants in missing data problems in biostatistics for several major pharmaceutical companies. They have been funded since 2002 by the UK Economic and Social Research Council to develop multiple imputation software for multilevel data, and to provide training for research scientists in the handling of missing data from observational studies.
English
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
PART III ADVANCED TOPICS 165
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
