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Library of Congress Cataloging-in-Publication Data

Names: Little, Roderick J.A., author. | Rubin, Donald B., author.

Title: Statistical analysis with missing data / Roderick J.A. Little, Donald B. Rubin.

Description: Third edition | Hoboken, NJ : Wiley, 2020. | Series: Wiley series in probability and statistics | Includes index. |

Identifiers: LCCN 2018058860 (print) | LCCN 2018061330 (ebook) | ISBN 9781118596012 (Adobe PDF) | ISBN 9781118595695 (ePub) | ISBN 9780470526798 (hardcover)

Subjects: LCSH: Mathematical statistics. | Mathematical statistics--Problems, exercises, etc. | Missing observations (Statistics) | Missing observations (Statistics)--Problems, exercises, etc.

Classification: LCC QA276 (ebook) | LCC QA276 .L57 2019 (print) | DDC 519.5--dc23

LC record available at https://lccn.loc.gov/2018058860

Cover image: Wiley

Cover design by Wiley

CONTENTS

Cover

Preface to the Third Edition

Part I Overview and Basic Approaches

1 Introduction
1. 1.1 The Problem of Missing Data
2. 1.2 Missingness Patterns and Mechanisms
3. 1.3 Mechanisms That Lead to Missing Data
4. 1.4 A Taxonomy of Missing Data Methods
5. Problems
6. Note
2 Missing Data in Experiments
1. 2.1 Introduction
2. 2.2 The Exact Least Squares Solution with Complete Data
3. 2.3 The Correct Least Squares Analysis with Missing Data
4. 2.4 Filling in Least Squares Estimates
5. 2.5 Bartlett's ANCOVA Method
6. 2.6 Least Squares Estimates of Missing Values by ANCOVA Using Only Complete-Data Methods
7. 2.7 Correct Least Squares Estimates of Standard Errors and One Degree of Freedom Sums of Squares
8. 2.8 Correct Least-Squares Sums of Squares with More Than One Degree of Freedom
9. Problems
3 Complete-Case and Available-Case Analysis, Including Weighting Methods
1. 3.1 Introduction
2. 3.2 Complete-Case Analysis
3. 3.3 Weighted Complete-Case Analysis
4. 3.4 Available-Case Analysis
5. Problems
4 Single Imputation Methods
1. 4.1 Introduction
2. 4.2 Imputing Means from a Predictive Distribution
3. 4.3 Imputing Draws from a Predictive Distribution
4. 4.4 Conclusion
5. Problems
5 Accounting for Uncertainty from Missing Data
1. 5.1 Introduction
2. 5.2 Imputation Methods that Provide Valid Standard Errors from a Single Filled-in Data Set
3. 5.3 Standard Errors for Imputed Data by Resampling
4. 5.4 Introduction to Multiple Imputation
5. 5.5 Comparison of Resampling Methods and Multiple Imputation
6. Problems

Part II Likelihood-Based Approaches to the Analysis of Data with Missing Values

6 Theory of Inference Based on the Likelihood Function
1. 6.1 Review of Likelihood-Based Estimation for Complete Data
2. 6.2 Likelihood-Based Inference with Incomplete Data
3. 6.3 A Generally Flawed Alternative to Maximum Likelihood: Maximizing over the Parameters and the Missing Data
4. 6.4 Likelihood Theory for Coarsened Data
5. Problems
6. Notes
7 Factored Likelihood Methods When the Missingness Mechanism Is Ignorable
1. 7.1 Introduction
2. 7.2 Bivariate Normal Data with One Variable Subject to Missingness: ML Estimation
3. 7.3 Bivariate Normal Monotone Data: Small-Sample Inference
4. 7.4 Monotone Missingness with More Than Two Variables
5. 7.5 Factored Likelihoods for Special Nonmonotone Patterns
6. Problems
8 Maximum Likelihood for General Patterns of Missing Data: Introduction and Theory with Ignorable Nonresponse
1. 8.1 Alternative Computational Strategies
2. 8.2 Introduction to the EM Algorithm
3. 8.3 The E Step and The M Step of EM
4. 8.4 Theory of the EM Algorithm
5. 8.5 Extensions of EM
6. 8.6 Hybrid Maximization Methods
7. Problems
9 Large-Sample Inference Based on Maximum Likelihood Estimates
1. 9.1 Standard Errors Based on The Information Matrix
2. 9.2 Standard Errors via Other Methods
3. Problems
10 Bayes and Multiple Imputation
1. 10.1 Bayesian Iterative Simulation Methods
2. 10.2 Multiple Imputation
3. Problems
4. Notes

Part III Likelihood-Based Approaches to the Analysis of Incomplete Data: Some Examples

11 Multivariate Normal Examples, Ignoring the Missingness Mechanism
1. 11.1 Introduction
2. 11.2 Inference for a Mean Vector and Covariance Matrix with Missing Data Under Normality
3. 11.3 The Normal Model with a Restricted Covariance Matrix
4. 11.4 Multiple Linear Regression
5. 11.5 A General Repeated-Measures Model with Missing Data
6. 11.6 Time Series Models
7. 11.7 Measurement Error Formulated as Missing Data
8. Problems
12 Models for Robust Estimation
1. 12.1 Introduction
2. 12.2 Reducing the Influence of Outliers by Replacing the Normal Distribution by a Longer-Tailed Distribution
3. 12.3 Penalized Spline of Propensity Prediction
4. Problems
5. Notes
13 Models for Partially Classified Contingency Tables, Ignoring the Missingness Mechanism
1. 13.1 Introduction
2. 13.2 Factored Likelihoods for Monotone Multinomial Data
3. 13.3 ML and Bayes Estimation for Multinomial Samples with General Patterns of Missingness
4. 13.4 Loglinear Models for Partially Classified Contingency Tables
5. Problems
14 Mixed Normal and Nonnormal Data with Missing Values, Ignoring the Missingness Mechanism
1. 14.1 Introduction
2. 14.2 The General Location Model
3. 14.3 The General Location Model with Parameter Constraints
4. 14.4 Regression Problems Involving Mixtures of Continuous and Categorical Variables
5. 14.5 Further Extensions of the General Location Model
6. Problems
15 Missing Not at Random Models
1. 15.1 Introduction
2. 15.2 Models with Known MNAR Missingness Mechanisms: Grouped and Rounded Data
3. 15.3 Normal Models for MNAR Missing Data
4. 15.4 Other Models and Methods for MNAR Missing Data
5. Problems

References

Author Index

Subject Index

End User License Agreement

List of Tables

Chapter 1

Table 1.1
Table 1.2
Table 1.3

Chapter 2

Table 2.1
Table 2.2

Chapter 3

Table 3.1

Chapter 4

Table 4.1

Chapter 7

Table 7.1
Table 7.2
Table 7.3
Table 7.4

Chapter 8

Table 8.1

Chapter 9

Table 9.1

Chapter 10

Table 10.1

Chapter 11

Table 11.1
Table 11.2
Table 11.3
Table 11.4
Table 11.5
Table 11.6

Chapter 12

Table 12.1
Table 12.2

Chapter 13

Table 13.1
Table 13.2
Table 13.3
Table 13.4
Table 13.5
Table 13.6
Table 13.7
Table 13.8
Table 13.9
Table 13.10
Table 13.11

Chapter 14

Table 14.1
Table 14.2
Table 14.3

Chapter 15

Table 15.1
Table 15.2
Table 15.3
Table 15.4
Table 15.5
Table 15.6
Table 15.7
Table 15.8
Table 15.9
Table 15.10
Table 15.11
Table 15.12
Table 15.13
Table 15.14

List of Illustrations

Chapter 1

Figure 1.1 Examples of missingness patterns; rows correspond to units and columns to varia...
Figure 1.2 Example 1.10: stem and leaf displays of distribution of standard normal sample with...
Figure 1.3 Example 1.12: observed and population distributions of historical heights. Popu...
Figure 1.4 Example 1.15: missingness pattern corresponding to main sample data where a var...

Chapter 4

Figure 4.1 Example 4.2: regression imputation for K = 2 variables.

Chapter 7

Figure 7.2 Example 7.11: data structure with three new tests: 0 = score observed, 1 = scor...

Chapter 8

Figure 8.1 The missingness pattern for Example 8.3.

Chapter 10

Figure 10.1 Posterior distribution of θ (Jeffreys' prior).

Chapter 11

Figure 11.1 Example 11.1, posterior distributions of mean differences μ_low − μ_med/high, St....
Figure 11.2 Example 11.6, posterior distributions of mean differences μ_low − μ_med/high, St....
Figure 11.3 Missingness pattern for Example 11.10, with shaded values observed. X = true co...

Chapter 12

Figure 12.1 Example 12.2, distributions of weights from robust ML methods, applied to multi...
Figure 12.2 Example 12.3, posterior distributions of mean differences μ_low − μ_med/high, St....

Chapter 13

Figure 13.1 The data in Example 13.1.
Figure 13.2 Example 13.2, plots of posterior distributions of the cell probabilities.

Chapter 14

Figure 14.1A Example 14.2, (a) Sequences of draws from the posterior distributions of the me...
Figure 14.1B Example 14.2, (a) Sequences of Gibbs' sampler draws from the Fisher-Z transform...
Figure 14.2 Pattern of missing data leading to simple ML estimates. 0, Observed; 1, missing...

Chapter 15

Figure 15.1 Example 15.15, 95% intervals for generated data with (a) ρ = 0.8 and (b) ρ = 0....
Figure 15.2 Missing data structure for subsample ignorable likelihood methods.
Figure 15.3 Example 15.19, empirical densities of the observed income values in the first q...
Figure 15.4 Example 15.20, hazard ratio and 95% intervals for combined rivaroxaban vs. plac...
Figure 15.5 Example 15.20, hazard ratio and 95% intervals for combined rivaroxaban vs. plac...
Figure 15.6 Example 15.21, an enhanced tipping-point display from Liublinska and Rubin (201...

Preface to the Third Edition

There has been tremendous growth in the literature on statistical methods for handling missing data, and associated software, since the publication of the second edition of “Statistical Analysis with Missing Data” in 2002. Attempting to cover this literature comprehensively would add excessively to the length of the book and also change its character. Therefore, our additions have focused mainly on work with which we have been associated and we can write about with some authority. The main changes from the second edition are as follows:

Concerning theory, we have changed the “obs” and “mis” notation for observed and missing data, which, though intuitive, caused some confusion because subscripting data by “obs” was not intended to imply conditioning on the pattern of observed values. We now use subscript (0) to denote observed values and subscript (1) to denote missing values, which is in fact similar to the notation employed by Rubin's original (1976a) paper. We have also been more specific about assumptions for ignoring the missing data mechanism for likelihood-based/Bayesian analyses and asymptotic frequentist analysis; the latter involves changing missing data patterns in repeated analysis. These changes reflect material in Mealli and Rubin (2015). A definition of “partially missing at random” and ignorability for parameter subsets has been added, based on Little et al. (2016a).
Data previously termed “not missing at random” are now called “missing not at random,” which we think is clearer.
Applications place greater emphasis on multiple imputation rather than direct computation of the posterior distribution of parameters. This new emphasis reflects the expansion of flexible software for multiple imputation, which makes the method attractive to applied statisticians.
We have added a number of additional missing data applications to measurement error, disclosure limitation, robust inference, and clinical trial data.
Chapter 15, on missing not at random data, has been completely revamped, including a number of new applications to subsample regression and sensitivity analysis
A number of minor errors in the previous edition have been corrected, although (as in all books), some probably remain and other new ones may have crept in – for which we apologize.

The ideal of using a consistent notation across all chapters, avoiding the use of the same symbol to mean different concepts, proved too hard given the range of topics covered. However, we have tried to maintain a consistent notation within chapters, and defined new uses of common letters as they arise. We hope different uses of the same symbol across chapters is not too confusing, and welcome suggestions for improvements.

3rd Edition

3^rd Edition