Statistical Analysis with Missing Data

Roderick J. A. Little, PhD., is Richard D. Remington Distinguished University Professor of Biostatistics, Professor of Statistics, and Research Professor, Institute for Social Research, at the University of Michigan. Donald B. Rubin, PhD., is Professor, Yau Mathematical Sciences Center, Tsinghua University; Murray Shusterman Senior Research Fellow, Department of Statistical Science, Fox School of Business at Temple University; and Professor Emeritus, Harvard University.

• Preface to the Third Edition xiPart I Overview and Basic Approaches 11 Introduction 31.1 The Problem of Missing Data 31.2 Missingness Patterns and Mechanisms 81.3 Mechanisms That Lead to Missing Data 131.4 A Taxonomy of Missing Data Methods 232 Missing Data in Experiments 292.1 Introduction 292.2 The Exact Least Squares Solution with Complete Data 302.3 The Correct Least Squares Analysis with Missing Data 322.4 Filling in Least Squares Estimates 332.4.1 Yates’s Method 332.4.2 Using a Formula for the Missing Values 342.4.3 Iterating to Find the Missing Values 342.4.4 ANCOVA with Missing Value Covariates 352.5 Bartlett’s ANCOVA Method 352.5.1 Useful Properties of Bartlett’s Method 352.5.2 Notation 362.5.3 The ANCOVA Estimates of Parameters and Missing Y-Values 362.5.4 ANCOVA Estimates of the Residual Sums of Squares and the Covariance Matrix of 𝛽̂ 372.6 Least Squares Estimates of Missing Values by ANCOVA Using Only Complete-Data Methods 382.7 Correct Least Squares Estimates of Standard Errors and One Degree of Freedom Sums of Squares 402.8 Correct Least-Squares Sums of Squares with More Than One Degree of Freedom 423 Complete-Case and Available-Case Analysis, Including Weighting Methods 473.1 Introduction 473.2 Complete-Case Analysis 473.3 Weighted Complete-Case Analysis 503.3.1 Weighting Adjustments 503.3.2 Poststratification and Raking to Known Margins 583.3.3 Inference from Weighted Data 603.3.4 Summary of Weighting Methods 613.4 Available-Case Analysis 614 Single Imputation Methods 674.1 Introduction 674.2 Imputing Means from a Predictive Distribution 694.2.1 Unconditional Mean Imputation 694.2.2 Conditional Mean Imputation 704.3 Imputing Draws from a Predictive Distribution 734.3.1 Draws Based on Explicit Models 734.3.2 Draws Based on Implicit Models – Hot Deck Methods 764.4 Conclusion 815 Accounting for Uncertainty from Missing Data 855.1 Introduction 855.2 Imputation Methods that Provide Valid Standard Errors from a Single Filled-in Data Set 865.3 Standard Errors for Imputed Data by Resampling 905.3.1 Bootstrap Standard Errors 905.3.2 Jackknife Standard Errors 925.4 Introduction to Multiple Imputation 955.5 Comparison of Resampling Methods and Multiple Imputation 100Part II Likelihood-Based Approaches to the Analysis of Data with Missing Values 1076 Theory of Inference Based on the Likelihood Function 1096.1 Review of Likelihood-Based Estimation for Complete Data 1096.1.1 Maximum Likelihood Estimation 1096.1.2 Inference Based on the Likelihood 1186.1.3 Large Sample Maximum Likelihood and Bayes Inference 1196.1.4 Bayes Inference Based on the Full Posterior Distribution 1266.1.5 Simulating Posterior Distributions 1306.2 Likelihood-Based Inference with Incomplete Data 1326.3 A Generally Flawed Alternative to Maximum Likelihood: Maximizing over the Parameters and the Missing Data 1416.3.1 The Method 1416.3.2 Background 1426.3.3 Examples 1436.4 Likelihood Theory for Coarsened Data 1457 Factored Likelihood Methods When the Missingness Mechanism Is Ignorable 1517.1 Introduction 1517.2 Bivariate Normal Data with One Variable Subject to Missingness: ML Estimation 1537.2.1 ML Estimates 1537.2.2 Large-Sample Covariance Matrix 1577.3 Bivariate Normal Monotone Data: Small-Sample Inference 1587.4 Monotone Missingness with More Than Two Variables 1617.4.1 Multivariate Data with One Normal Variable Subject to Missingness 1617.4.2 The Factored Likelihood for a General Monotone Pattern 1627.4.3 ML Computation for Monotone Normal Data via the Sweep Operator 1667.4.4 Bayes Computation forMonotone Normal Data via the Sweep Operator 1747.5 Factored Likelihoods for Special Nonmonotone Patterns 1758 Maximum Likelihood for General Patterns of Missing Data: Introduction and Theory with Ignorable Nonresponse 1858.1 Alternative Computational Strategies 1858.2 Introduction to the EM Algorithm 1878.3 The E Step and The M Step of EM 1888.4 Theory of the EM Algorithm 1938.4.1 Convergence Properties of EM 1938.4.2 EM for Exponential Families 1968.4.3 Rate of Convergence of EM 1988.5 Extensions of EM 2008.5.1 The ECM Algorithm 2008.5.2 The ECME and AECM Algorithms 2058.5.3 The PX-EM Algorithm 2068.6 Hybrid Maximization Methods 2089 Large-Sample Inference Based on Maximum Likelihood Estimates 2139.1 Standard Errors Based on The Information Matrix 2139.2 Standard Errors via Other Methods 2149.2.1 The Supplemented EM Algorithm 2149.2.2 Bootstrapping the Observed Data 2199.2.3 Other Large-Sample Methods 2209.2.4 Posterior Standard Errors from Bayesian Methods 22110 Bayes and Multiple Imputation 22310.1 Bayesian Iterative Simulation Methods 22310.1.1 Data Augmentation 22310.1.2 The Gibbs’ Sampler 22610.1.3 Assessing Convergence of Iterative Simulations 23010.1.4 Some Other Simulation Methods 23110.2 Multiple Imputation 23210.2.1 Large-Sample Bayesian Approximations of the Posterior Mean and Variance Based on a Small Number of Draws 23210.2.2 Approximations Using Test Statistics or p-Values 23510.2.3 Other Methods for Creating Multiple Imputations 23810.2.4 Chained-Equation Multiple Imputation 24110.2.5 Using Different Models for Imputation and Analysis 243Part III Likelihood-Based Approaches to the Analysis of Incomplete Data: Some Examples 24711 Multivariate Normal Examples, Ignoring the Missingness Mechanism 24911.1 Introduction 24911.2 Inference for a Mean Vector and Covariance Matrix with Missing Data Under Normality 24911.2.1 The EM Algorithm for Incomplete Multivariate Normal Samples 25011.2.2 Estimated Asymptotic Covariance Matrix of (𝜃 − ) 25211.2.3 Bayes Inference and Multiple Imputation for the Normal Model 25311.3 The Normal Model with a Restricted Covariance Matrix 25711.4 Multiple Linear Regression 26411.4.1 Linear Regression with Missingness Confined to the Dependent Variable 26411.4.2 More General Linear Regression Problems with Missing Data 26611.5 A General Repeated-Measures Model with Missing Data 26911.6 Time Series Models 27311.6.1 Introduction 27311.6.2 Autoregressive Models for Univariate Time Series with Missing Values 27311.6.3 Kalman Filter Models 27611.7 Measurement Error Formulated as Missing Data 27712 Models for Robust Estimation 28512.1 Introduction 28512.2 Reducing the Influence of Outliers by Replacing the Normal Distribution by a Longer-Tailed Distribution 28612.2.1 Estimation for a Univariate Sample 28612.2.2 Robust Estimation of the Mean and Covariance Matrix with Complete Data 28812.2.3 Robust Estimation of the Mean and Covariance Matrix from Data with Missing Values 29012.2.4 Adaptive Robust Multivariate Estimation 29112.2.5 Bayes Inference for the t Model 29212.2.6 Further Extensions of the t Model 29412.3 Penalized Spline of Propensity Prediction 29813 Models for Partially Classified Contingency Tables, Ignoring the Missingness Mechanism 30113.1 Introduction 30113.2 Factored Likelihoods for Monotone Multinomial Data 30213.2.1 Introduction 30213.2.2 ML and Bayes for Monotone Patterns 30313.2.3 Precision of Estimation 31213.3 ML and Bayes Estimation for Multinomial Samples with General Patterns of Missingness 31313.4 Loglinear Models for Partially Classified Contingency Tables 31713.4.1 The Complete-Data Case 31713.4.2 Loglinear Models for Partially Classified Tables 32013.4.3 Goodness-of-Fit Tests for Partially Classified Data 32614 Mixed Normal and Nonnormal Data with Missing Values, Ignoring the Missingness Mechanism 32914.1 Introduction 32914.2 The General Location Model 32914.2.1 The Complete-DataModel and Parameter Estimates 32914.2.2 ML Estimation with Missing Values 33114.2.3 Details of the E Step Calculations 33414.2.4 Bayes’ Computation for the Unrestricted General Location Model 33514.3 The General Location Model with Parameter Constraints 33714.3.1 Introduction 33714.3.2 Restricted Models for the Cell Means 34014.3.3 LoglinearModels for the Cell Probabilities 34014.3.4 Modifications to the Algorithms of Previous Sections to Accommodate Parameter Restrictions 34014.3.5 SimplificationsWhen Categorical Variables are More Observed than Continuous Variables 34314.4 Regression Problems InvolvingMixtures of Continuous and Categorical Variables 34414.4.1 Normal Linear Regression with Missing Continuous or Categorical Covariates 34414.4.2 Logistic Regression with Missing Continuous or Categorical Covariates 34614.5 Further Extensions of the General Location Model 34715 Missing Not at RandomModels 35115.1 Introduction 35115.2 Models with Known MNAR Missingness Mechanisms: Grouped and Rounded Data 35515.3 Normal Models for MNAR Missing Data 36215.3.1 Normal Selection and Pattern-Mixture Models for Univariate Missingness 36215.3.2 Following up a Subsample of Nonrespondents 36415.3.3 The Bayesian Approach 36615.3.4 Imposing Restrictions on Model Parameters 36915.3.5 Sensitivity Analysis 37615.3.6 Subsample Ignorable Likelihood for Regression with Missing Data 37915.4 Other Models and Methods for MNAR Missing Data 38215.4.1 MNAR Models for Repeated-Measures Data 38215.4.2 MNAR Models for Categorical Data 38515.4.3 Sensitivity Analyses for Chained-Equation Multiple Imputations 39115.4.4 Sensitivity Analyses in Pharmaceutical Applications 396References 405Author Index 429Subject Index 437

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