Multiple Imputation and Its Application

ISBN-13: 9781119756088 / Angielski / Twarda / 2023 / 416 str.

James R. Carpenter; Jonathan Bartlett; Tim Morris

Spis treści

PrefaceData acknowledgmentsGlossaryI Foundations 11 Introduction 21.1 Reasons for missing data . . . . . . . . . . . . . . . . . . . . . 51.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3 Patterns of missing data . . . . . . . . . . . . . . . . . . . . . 81.3.1 Consequences of missing data . . . . . . . . . . . . . . . 101.4 Inferential framework and notation . . . . . . . . . . . . . . . . 131.4.1 Missing Completely At Random (MCAR) . . . . . . . . 151.4.2 Missing At Random (MAR) . . . . . . . . . . . . . . . . 161.4.3 Missing Not At Random (MNAR) . . . . . . . . . . . . 221.4.4 Ignorability . . . . . . . . . . . . . . . . . . . . . . . . . 271.5 Using observed data to inform assumptions about the missingness mechanism . .. . . . . . . 281.6 Implications of missing data mechanisms for regression analyses 321.6.1 Partially observed response . . . . . . . . . . . . . . . . 331.6.2 Missing covariates . . . . . . . . . . . . . . . . . . . . . 371.6.3 Missing covariates and response . . . . . . . . . . . . . . 401.6.4 Subtle issues I: the odds ratio . . . . . . . . . . . . . . . 401.6.5 Implication for linear regression . . . . . . . . . . . . . . 431.6.6 Subtle issues II: sub sample ignorability . . . . . . . . . 441.6.7 Summary: when restricting to complete records is valid 451.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472 The Multiple Imputation Procedure and Its Justification 522.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522.2 Intuitive outline of the MI procedure . . . . . . . . . . . . . . 542.3 The generic MI Procedure . . . . . . . . . . . . . . . . . . . . . 612.4 Bayesian justification of MI . . . . . . . . . . . . . . . . . . . . 642.5 Frequentist Inference . . . . . . . . . . . . . . . . . . . . . . . 662.6 Choosing the number of imputations . . . . . . . . . . . . . . . 732.7 Some simple examples . . . . . . . . . . . . . . . . . . . . . . . 752.8 MI in More General Settings . . . . . . . . . . . . . . . . . . . 842.8.1 Proper imputation . . . . . . . . . . . . . . . . . . . . . 842.8.2 Congenial imputation and substantive model . . . . . . 852.8.3 Uncongenial imputation and substantive models . . . . 872.8.4 Survey Sample Settings . . . . . . . . . . . . . . . . . . 942.9 Constructing congenial imputation models . . . . . . . . . . . . 952.10 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 962.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 971.6.3 Missing covariates and response . . . . . . . . . . . . . . 401.6.4 Subtle issues I: the odds ratio . . . . . . . . . . . . . . . 401.6.5 Implication for linear regression . . . . . . . . . . . . . . 431.6.6 Subtle issues II: sub sample ignorability . . . . . . . . . 441.6.7 Summary: when restricting to complete records is valid 451.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472 The Multiple Imputation Procedure and Its Justification 522.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522.2 Intuitive outline of the MI procedure . . . . . . . . . . . . . . 542.3 The generic MI Procedure . . . . . . . . . . . . . . . . . . . . . 612.4 Bayesian justification of MI . . . . . . . . . . . . . . . . . . . . 642.5 Frequentist Inference . . . . . . . . . . . . . . . . . . . . . . . 662.6 Choosing the number of imputations . . . . . . . . . . . . . . . 732.7 Some simple examples . . . . . . . . . . . . . . . . . . . . . . . 752.8 MI in More General Settings . . . . . . . . . . . . . . . . . . . 842.8.1 Proper imputation . . . . . . . . . . . . . . . . . . . . . 842.8.2 Congenial imputation and substantive model . . . . . . 852.8.3 Uncongenial imputation and substantive models . . . . 872.8.4 Survey Sample Settings . . . . . . . . . . . . . . . . . . 942.9 Constructing congenial imputation models . . . . . . . . . . . . 952.10 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 962.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97II Multiple imputation for simple data structures 1043 Multiple imputation of quantitative data 1053.1 Regression imputation with a monotone missingness pattern . . 1053.1.1 MAR mechanisms consistent with a monotone pattern . 1073.1.2 Justification . . . . . . . . . . . . . . . . . . . . . . . . 1093.2 Joint modelling . . . . . . . . . . . . . . . . . . . . . . . . . . . 1103.2.1 Fitting the imputation model . . . . . . . . . . . . . . 1113.2.2 Adding covariates . . . . . . . . . . . . . . . . . . . . . 1153.3 Full conditional specification . . . . . . . . . . . . . . . . . . . 1183.3.1 Justification . . . . . . . . . . . . . . . . . . . . . . . . . 1193.4 Full conditional specification versus joint modelling . . . . . . . 1213.5 Software for multivariate normal imputation . . . . . . . . . . . 1213.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1223.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1234 Multiple imputation of binary and ordinal data 1254.1 Sequential imputation with monotone missingness pattern . . 1254.2 Joint modelling with the multivariate normal distribution . . . 1274.3 Modelling binary data using latent normal variables . . . . . . 1304.3.1 Latent normal model for ordinal data . . . . . . . . . . 1374.4 General location model . . . . . . . . . . . . . . . . . . . . . . 1414.5 Full conditional specification . . . . . . . . . . . . . . . . . . . 1424.5.1 Justification . . . . . . . . . . . . . . . . . . . . . . . . . 1434.6 Issues with over-fitting . . . . . . . . . . . . . . . . . . . . . . 1444.7 Pros and cons of the various approaches . . . . . . . . . . . . . 1504.8 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1524.9 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1534.10 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1535 Imputation of unordered categorical data 1565.1 Monotone missing data . . . . . . . . . . . . . . . . . . . . . . 1575.2 Multivariate normal imputation for categorical data . . . . . . 1585.3 Maximum indicant model . . . . . . . . . . . . . . . . . . . . . 1595.3.1 Continuous and categorical variable . . . . . . . . . . . 1625.3.2 Imputing missing data . . . . . . . . . . . . . . . . . . . 1645.4 General location model . . . . . . . . . . . . . . . . . . . . . . 1655.5 FCS with categorical data . . . . . . . . . . . . . . . . . . . . 1695.6 Perfect prediction issues with categorical data . . . . . . . . . . 1705.7 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1715.8 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1725.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173III Multiple imputation in practice 1756 Non-linear relationships, interactions, and other derived variables 1766.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1776.1.1 Interactions . . . . . . . . . . . . . . . . . . . . . . . . . 1786.1.2 Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . 1796.1.3 Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1806.1.4 Sum scores . . . . . . . . . . . . . . . . . . . . . . . . . 1816.1.5 Composite endpoints . . . . . . . . . . . . . . . . . . . . 1826.2 No missing data in derived variables . . . . . . . . . . . . . . . 1846.3 Simple methods . . . . . . . . . . . . . . . . . . . . . . . . . . 1866.3.1 Impute then transform . . . . . . . . . . . . . . . . . . . 1876.3.2 Transform then impute / just another variable . . . . . 1876.3.3 Adapting standard imputation models and passive imputation .. . . . . . . . . . . . . . . . . . . . . . 1906.3.4 Predictive mean matching . . . . . . . . . . . . . . . . . 1916.3.5 Imputation separately by groups for interactions . . . . 1956.4 Substantive-model-compatible imputation . . . . . . . . . . . . 2006.4.1 The basic idea . . . . . . . . . . . . . . . . . . . . . . . 2006.4.2 Latent-normal joint model SMC imputation . . . . . . . 2076.4.3 Factorised conditional model SMC imputation . . . . . 2096.4.4 Substantive model compatible fully conditional specification . . . . . . . . . . . . . . . . . . . . . . . . . 2126.4.5 Auxiliary variables . . . . . . . . . . . . . . . . . . . . . 2136.4.6 Missing outcome values . . . . . . . . . . . . . . . . . . 2146.4.7 Congeniality vs. compatibility . . . . . . . . . . . . . . . 2146.4.8 Discussion of SMC . . . . . . . . . . . . . . . . . . . . . 2166.5 Returning to the problems . . . . . . . . . . . . . . . . . . . . . 2176.5.1 Ratios . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2176.5.2 Splines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2186.5.3 Fractional polynomials . . . . . . . . . . . . . . . . . . . 2186.5.4 Multiple imputation with conditional questions or 'skips'2236.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2287 Survival data 2317.1 Missing covariates in time to event data . . . . . . . . . . . . . 2317.1.1 Approximately compatible approaches . . . . . . . . . . 2327.1.2 Substantive model compatible approaches . . . . . . . . 2417.2 Imputing censored survival times . . . . . . . . . . . . . . . . . 2457.3 Non-parametric, or 'hot deck' imputation . . . . . . . . . . . . 2487.3.1 Non-parametric imputation for survival data . . . . . . 2517.4 Case-cohort designs . . . . . . . . . . . . . . . . . . . . . . . . 2547.4.1 Standard analysis of case-cohort studies . . . . . . . . . 2547.4.2 Multiple imputation for case-cohort studies . . . . . . . 2557.4.3 Full-cohort . . . . . . . . . . . . . . . . . . . . . . . . . 2567.4.4 Intermediate approaches . . . . . . . . . . . . . . . . . . 2577.4.5 Substudy approach . . . . . . . . . . . . . . . . . . . . . 2577.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2617.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2618 Prognostic models, missing data and multiple imputation 2658.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2658.2 Motivating example . . . . . . . . . . . . . . . . . . . . . . . . 2668.3 Missing data at model implementation . . . . . . . . . . . . . 2678.4 Multiple imputation for prognostic modelling . . . . . . . . . . 2688.5 Model building . . . . . . . . . . . . . . . . . . . . . . . . . . . 2688.5.1 Model building with missing data . . . . . . . . . . . . . 2688.5.2 Imputing predictors when model building is to be performed . . . . . . . . . . . . . . . . . . . . . . . . . 2708.6 Model performance . . . . . . . . . . . . . . . . . . . . . . . . 2718.6.1 How should we pool MI results for estimation of performance? . . . . . . . . . . . . . . . . . . . . . . . 2718.6.2 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . 2728.6.3 Discrimination . . . . . . . . . . . . . . . . . . . . . . . 2738.6.4 Model performance measures with clinical interpretability2738.7 Model validation . . . . . . . . . . . . . . . . . . . . . . . . . . 2748.7.1 Internal model validation . . . . . . . . . . . . . . . . . 2748.7.2 External model validation . . . . . . . . . . . . . . . . . 2758.8 Incomplete data at implementation . . . . . . . . . . . . . . . 2768.8.1 MI for incomplete data at implementation . . . . . . . . 2768.8.2 Alternatives to multiple imputation . . . . . . . . . . . 2788.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2819 Multilevel multiple imputation 2839.1 Multilevel imputation model . . . . . . . . . . . . . . . . . . . 2849.1.1 Imputation of level 1 variables . . . . . . . . . . . . . . 2879.1.2 Imputation of level 2 variables . . . . . . . . . . . . . . 2919.1.3 Accommodating the substantive model . . . . . . . . . . 2969.2 MCMC algorithm for imputation model . . . . . . . . . . . . . 2979.2.1 Checking model convergence . . . . . . . . . . . . . . . 3059.3 Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3079.3.1 Cross-classification and 3-level data . . . . . . . . . . . 3079.3.2 Random level 1 covariance matrices . . . . . . . . . . . 3089.3.3 Model fit . . . . . . . . . . . . . . . . . . . . . . . . . . 3109.4 Other imputation methods . . . . . . . . . . . . . . . . . . . . 3119.4.1 1-step and 2-step FCS . . . . . . . . . . . . . . . . . . . 3129.4.2 Substantive model compatible imputation . . . . . . . . 3139.4.3 Non-parametric methods . . . . . . . . . . . . . . . . . 3149.4.4 Comparisons of different methods . . . . . . . . . . . . 3149.5 Individual participant data meta-analysis . . . . . . . . . . . . 3159.5.1 When to apply Rubin's rules . . . . . . . . . . . . . . . 3189.5.2 Homoscedastic vs heteroscedastic imputation model . . 3209.6 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3209.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3219.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32210 Sensitivity analysis: MI unleashed 32610.1 Review of MNAR modelling . . . . . . . . . . . . . . . . . . . 32810.2 Framing sensitivity analysis: Estimands . . . . . . . . . . . . . 33110.3 Pattern mixture modelling with MI . . . . . . . . . . . . . . . 33510.3.1 Missing covariates . . . . . . . . . . . . . . . . . . . . . 34110.3.2 Sensitivity with multiple variables: the NAR FCS procedure . . . . . . .. . . . . . . . . . . . . . . . . . . 34410.3.3 Application to survival analysis . . . . . . . . . . . . . . 34610.4 Pattern mixture approach with longitudinal data via MI . . . . 35110.4.1 Change in slope post-deviation . . . . . . . . . . . . . . 35310.5 Reference based imputation . . . . . . . . . . . . . . . . . . . . 35610.5.1 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . 36110.5.2 Information Anchoring . . . . . . . . . . . . . . . . . . 36810.6 Approximating a selection model by importance weighting . . 37210.6.1 Weighting the imputations . . . . . . . . . . . . . . . . 37510.6.2 Stacking the imputations and applying the weights . . . 37610.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38610.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38711 Multiple imputation for measurement error and misclassification 39211.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39311.2 Multiple imputation with validation data . . . . . . . . . . . . 39411.2.1 Measurement error . . . . . . . . . . . . . . . . . . . . . 39611.2.2 Misclassification . . . . . . . . . . . . . . . . . . . . . . 39711.2.3 Imputing assuming error is non-differential . . . . . . . 39911.2.4 Non-linear outcome models . . . . . . . . . . . . . . . . 40011.3 Multiple imputation with replication data . . . . . . . . . . . . 40111.3.1 Measurement error . . . . . . . . . . . . . . . . . . . . . 40311.3.2 Misclassification . . . . . . . . . . . . . . . . . . . . . . 40811.4 External information on the measurement process . . . . . . . 40911.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41111.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41312 Multiple imputation with weights 41612.1 Using model based predictions in strata . . . . . . . . . . . . . 41712.2 Bias in the MI Variance Estimator . . . . . . . . . . . . . . . . 41812.3 MI with weights . . . . . . . . . . . . . . . . . . . . . . . . . . 42212.3.1 Conditions for consistency of thetabMI . . . . . . . . . . . . 42212.3.2 Conditions for the consistency of Vb MI . . . . . . . . . . 42412.4 A multilevel approach . . . . . . . . . . . . . . . . . . . . . . . 42612.4.1 Evaluation of the multilevel multiple imputation approach for handling survey weights . . . 42912.4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 43412.5 Further topics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43712.5.1 Estimation in Domains . . . . . . . . . . . . . . . . . . 43712.5.2 Two-stage analysis . . . . . . . . . . . . . . . . . . . . 43712.5.3 Missing values in the weight model . . . . . . . . . . . . 43812.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43812.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43913 Multiple imputation for causal inference 44313.1 Multiple imputation for causal inference in point exposure studies44413.1.1 Randomised trials . . . . . . . . . . . . . . . . . . . . . 44513.1.2 Observational studies . . . . . . . . . . . . . . . . . . . 44613.2 Multiple imputation and propensity scores . . . . . . . . . . . . 45013.2.1 Propensity scores for confounder adjustment . . . . . . 45013.2.2 Multiple imputation of confounders . . . . . . . . . . . . 45213.2.3 Imputation model specification . . . . . . . . . . . . . . 45613.3 Principal stratification via multiple imputation . . . . . . . . . 45713.3.1 Principal strata effects . . . . . . . . . . . . . . . . . . 45813.3.2 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . 45913.4 Multiple imputation for instrumental variable analysis . . . . . 46113.4.1 Instrumental variable analysis for non-adherence . . . . 46113.4.2 Instrumental variable analysis via multiple imputation . 46413.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46713.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46814 Using multiple imputation in practice 47214.1 A general approach . . . . . . . . . . . . . . . . . . . . . . . . 47314.2 Objections to multiple imputation . . . . . . . . . . . . . . . . 47714.3 Reporting of analyses with incomplete data . . . . . . . . . . . 48214.4 Presenting incomplete baseline data . . . . . . . . . . . . . . . 48314.5 Model diagnostics . . . . . . . . . . . . . . . . . . . . . . . . . 48614.6 How many imputations? . . . . . . . . . . . . . . . . . . . . . . 48714.6.1 Using the jack-knife estimate of the Monte-Carlo standard error . . . . . . . . . . . . . . . . . . . . 49014.7 Multiple imputation for each substantive model, project ordataset? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49214.8 Large datasets . . . . . . . . . . . . . . . . . . . . . . . . . . . 49314.8.1 Large datasets and joint modelling . . . . . . . . . . . 49414.8.2 Shrinkage by constraining parameters . . . . . . . . . . 49614.8.3 Comparison of the two approaches . . . . . . . . . . . . 49914.9 Multiple Imputation and record linkage . . . . . . . . . . . . . 50014.10Setting random number seeds for multiple imputation analyses 50214.11Simulation studies including multiple imputation . . . . . . . . 50314.11.1Random number seeds for simulation studies includingmultiple imputation . . . . . . . . . . . . . . . . . . . . 50314.11.2Repeated simulation of all data or only the missingnessmechanism? . . . . . . . . . . . . . . . . . . . . . . . . 50414.11.3How many imputations for simulation studies? . . . . . 50514.11.4Multiple imputation for data simulation . . . . . . . . . 50714.12Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 508A Markov Chain Monte Carlo 512B Probability distributions 517B.1 Posterior for the multivariate normal distribution . . . . . . . 521C Overview of multiple imputation in R, Stata 524C.1 Basic multiple imputation using R . . . . . . . . . . . . . . . . 524C.2 Basic MI using Stata . . . . . . . . . . . . . . . . . . . . . . . . 526Bibliography 530Index 555

O autorze

JAMES R. CARPENTER is Professor of Medical Statistics at the London School of Hygiene & Tropical Medicine and Programme Leader in Methodology at the MRC Clinical Trials Unit at UCL, UK.JONATHAN W. BARTLETT is a Professor of Medical Statistics at the London School of Hygiene & Tropical Medicine, UK.TIM P. MORRIS is Principal Research Fellow in Medical Statistics at the MRC Clinical Trials Unit at UCL, UK.ANGELA M. WOOD is Professor of Health Data Science in the Department of Public Health and Primary Care, University of Cambridge, UK.MATTEO QUARTAGNO is Senior Research Fellow in Medical Statistics at the MRC Clinical Trials Unit at UCL, UK.MICHAEL G. KENWARD retired in 2016 after sixteen years as GlaxoSmithKline Professor of Biostatistics at the London School of Hygiene & Tropical Medicine, UK.