Multiple Imputation and Its Application 2e

Multiple Imputation and Its Application 2e（Statistics in Practice）

多重插补及其应用第2版
生物医学工程学

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作者

James Carpenter

出版社

Wiley

出版时间

2023年07月19日

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ＩＳＢＮ

9781119756088

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页码

464

开本

16.83 x 24.45 cm.

语种

英文

版次

2nd ed.

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Multiple imputation remains the most widely used methodology for missing data. Since the publication of the first edition, both MI methodology and the range of applications has continued to expand and develop. Methodological advances include extended MI methodologies for multilevel data and causal models, alongside important practical developments in sensitivity analysis. Key practical applications are clinical trials, prognostic modelling and causal modelling. Following on from the first edition, the authors here present the concepts in an intuitive way, setting out the issues raised by missing data, describing the rationale for MI, and show how it can be applied in increasingly complex settings with a range of examples. Also available for the first time are theoretical and computer-based exercises using Stata and R to help the instructor. Multiple Imputation and its Application, Second Edition is aimed at quantitative medical and social researchers by presenting the concepts in an intuitive way, illustrating with a range of examples. Alongside this, inclusion of key mathematical details, and theoretical and computer-based exercises will make the text suitable for graduate teaching and short courses.

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 skips223 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 Rubins 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 bMI . . . . . . . . . . . . 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 imputa

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