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.