Imputation is the substitution of some value for a missing data point or a missing component of a data point. Once all missing values have been imputed, the dataset can then be analysed using standard techniques for complete data. This book is written with three main aims; to provide a thorough introduction to the general MI methods, to provide a detailed discussion of the practical use of the MI method and to present real-world examples drawn from the field of biostatistics. Illustrated throughout, using different issues that arise in the use of MI in observational and clinical trial settings. Relevant computer code and data will be provided for the examples used throughout the book and will include SAS, Stata, WinBUGS, MLwiN and R.

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.

I Foundations

1 Introduction

1.1 Reasons for missing data

1.1.1 Patterns of missing data

1.1.2 Consequences of missing data

1.2 Inferential framework and notation

1.2.1 Missing Completely At Random (MCAR)

1.2.2 Missing At Random (MAR)

1.2.3 Missing Not At Random (MNAR)

1.2.4 Ignorability

1.3 Using observed data to inform assumptions about the missingness mechanism

1.4 Implications of missing data mechanisms for regression analyses

1.4.1 Partially observed response

1.4.2 Missing covariates

1.4.3 Missing covariates and response

1.4.4 Subtle issues I: the odds ratio

1.4.5 Implication for linear regression

1.4.6 Subtle issues II: sub sample ignorability

1.4.7 Summary: when restricting to complete records is valid

1.5 Summary

2 The Multiple Imputation Procedure and Its Justification

2.1 Introduction

2.2 Intuitive outline of the MI procedure

2.3 The generic MI Procedure

2.4 Bayesian justification of MI

2.5 Frequentist Inference

2.6 Choosing the number of imputations

2.7 Some simple examples

2.8 MI in More General Settings

2.8.1 Survey Sample Settings

2.9 Practical considerations for choosing imputation models

2.10 Discussion

II Multiple imputation for cross sectional data

3 Multiple imputation of quantitative data

3.1 Regression imputation with a monotone missingness pattern

3.1.1 MAR mechanisms consistent with a monotone pattern

3.1.2 Justification

3.2 Joint modelling

3.2.1 Fitting the imputation model

3.3 Full conditional specification

3.3.1 Justification

3.4 Full conditional specification versus joint modelling

3.5 Software for multivariate normal imputation

3.6 Discussion

4 Multiple imputation of binary and ordinal data

4.1 Sequential imputation with monotone missingness pattern

4.2 Joint modelling with the multivariate normal distribution

4.3 Modelling binary data using latent normal variables

4.3.1 Latent normal model for ordinal data

4.4 General location model

4.5 Full conditional specification

4.5.1 Justification

4.6 Issues with over-fitting

4.7 Pros and cons of the various approaches

4.8 Software

4.9 Discussion

5 Imputation of unordered categorical data

5.1 Monotone missing data

5.2 Multivariate normal imputation for categorical data

5.3 Maximum indicant model

5.3.1 Continuous and categorical variable

5.3.2 Imputing missing data

5.3.3 More than one categorical variable

5.4 General location model

5.5 FCS with categorical data

5.6 Perfect prediction issues with categorical data
5.7 Software

5.8 Discussion

6 Non-linear relationships

6.1 Passive imputation

6.2 No missing data in non-linear relationships

6.3 Missing data in non-linear relationships

6.3.1 Predictive Mean Matching (PMM)

6.3.2 Just Another Variable (JAV)

6.3.3 Joint modelling approach

6.3.4 Extension to more general models and missing data pattern

6.3.5 Metropolis Hastings sampling

6.3.6 Rejection sampling

6.3.7 FCS approach

6.4 Discussion

7 Interactions

7.1 Interaction variables fully observed

7.2 Interactions of categorical variables

7.3 General non-linear relationships

7.4 Software

7.5 Discussion

III Advanced Topics

8 Survival data, skips and large datasets

8.1 Time to event data

8.1.1 Imputing missing covariate values

8.1.2 Survival data as categorical

8.1.3 Imputing censored survival times

8.2 Non-parametric, or `hot deck' imputation

8.2.1 Non-parametric imputation for survival data

8.3 Multiple imputation for skips

8.4 Two-stage MI

8.5 Large datasets

8.5.1 Large datasets and joint modelling

8.5.2 Shrinkage by constraining parameters

8.5.3 Comparison of the two approaches

8.6 Multiple Imputation and record linkage

8.7 Measurement error

8.8 Multiple imputation for aggregated scores

8.9 Discussion

9 Multilevel multiple imputation

9.1 Multilevel imputation model

9.2 MCMC algorithm for imputation model

9.3 Imputing level 2 covariates using FCS

9.4 Individual patient meta-analysis

9.4.1 When to apply Rubin's rules

9.5 Extensions

9.5.1 Random level-1 covariance matrices

9.5.2 Model_t

9.6 Discussion

10 Sensitivity analysis: MI unleashed

10.1 Review of MNAR modelling

10.2 Framing sensitivity analysis

10.3 Pattern mixture modelling with MI

10.3.1 Missing covariates

10.3.2 Application to survival analysis

10.4 Pattern mixture approach with longitudinal
data via MI

10.4.1 Change in slope post-deviation

10.5 Piecing together post-deviation distributions from other trial arms

10.6 Approximating a selection model by importance weighting

10.6.1 Algorithm for approximate sensitivity analysis by reweighting

10.7 Discussion

11 Including survey weights

11.1 Using model based predictions

11.2 Bias in the MI Variance Estimator

11.2.1 MI with weights

11.2.2 Estimation in Domains

11.3 A multilevel approach

11.4 Further developments

11.5 Discussion

12 Robust Multiple Imputation

12.1 Introduction

12.2 Theoretical background

12.2.1 Simple Estimating equations

12.2.2 The probability of missingness (POM) model

12.2.3 Augmented inverse probability weighted
estimating equation

12.3 Robust Multiple Imputation

12.3.1 Univariate MAR missing data

12.3.2 Longitudinal MAR missing data

12.4 Simulation studies

12.4.1 Univariate MAR missing data

12.4.2 Longitudinal monotone MAR missing data

12.4.3 Longitudinal non-monotone MAR missing data

12.4.4 Non-longitudinal non-monotone MAR missing data

12.4.5 Conclusions

12.5 The RECORD study

12.6 Discussion

Appendix A Markov Chain Monte Carlo

Appendix B Probability distributions

B.1 Posterior for the multivariate normal distribution

Bibliography

Index

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