Contents

Preface

PART I OVERVIEW AND BASIC APPROACHES

1. Introduction

1.1. The Problem of Missing Data

1.2. Missing-Data Patterns

1.3. Mechanisms That Lead to Missing Data

1.4. A Taxonomy of Missing-Data Methods

2. Missing Data in Experiments

2.1. Introduction

2.2. The Exact Least Squares Solution with Complete Data

2.3. The Correct Least Squares Analysis with Missing Data

2.4. Filling in Least Squares Estimates

2.5. Bartlett’s ANCOVA Method

2.6. Least Squares Estimates of Missing Values by ANCOVA Using Only Complete-Data Methods

2.7. Correct Least Squares Estimates of Standard Errors and One Degree of Freedom Sums of Squares

2.8. Correct Least Squares Sums of Squares with More Than One Degree of Freedom

3. Complete-Case and Available-Case Analysis, Including Weighting Methods

3.1. Introduction

3.2. Complete-Case Analysis

3.3. Weighted Complete-Case Analysis

3.4. Available-Case Analysis

4. Single Imputation Methods

4.1. Introduction

4.2. Imputing Means from a Predictive Distribution

4.3. Imputing Draws from a Predictive Distribution

4.4. Conclusions

5. Estimation of Imputation Uncertainty

5.1. Introduction

5.2. Imputation Methods that Provide Valid Standard Errors from a Single Filled-in Data Set

5.3. Standard Errors for Imputed Data by Resampling

5.4. Introduction to Multiple Imputation

5.5. Comparison of Resampling Methods and Multiple Imputation

PART II LIKELIHOOD-BASED APPROACHES TO THE ANALYSIS OF MISSING DATA

6. Theory of Inference Based on the Likelihood Function

6.1. Review of Likelihood-Based Estimation for Complete Data

6.2. Likelihood-Based Inference with Incomplete Data

6.3. A Generally Flawed Alternative to Maximum Likelihood: Maximizing Over the Parameters and the Missing Data

6.4. Likelihood Theory for Coarsened Data

7. Factored Likelihood Methods, Ignoring the Missing-Data Mechanism

7.1. Introduction

7.2. Bivariate Normal Data with One Variable Subject to Nonresponse: ML Estimation

7.3. Bivariate Normal Monotone Data: Small-Sample Inference

7.4. Monotone Data With More Than Two Variables

7.5. Factorizations for Special Nonmonotone Patterns

8. Maximum Likelihood for General Patterns of Missing Data: Introduction and Theory with Ignorable Nonresponse

8.1. Alternative Computational Strategies

8.2. Introduction to the EM Algorithm

8.3. The E and M Steps of EM

8.4. Theory of the EM Algorithm

8.5. Extensions of EM

8.6. Hybrid Maximization Methods

9. Large-Sample Inference Based on Maximum Likelihood Estimates

9.1. Standard Errors Based on the Information Matrix

9.2. Standard Errors via Methods that do not Require Computing and Inverting an Estimate of the Observed Information Matrix

10. Bayes and Multiple Imputation

10.1. Bayesian Iterative Simulation Methods

10.2. Multiple Imputation

PART III LIKELIHOOD-BASED APPROACHES TO THE ANALYSIS OF INCOMPLETE DATA: SOME EXAMPLES

11. Multivariate Normal Examples, Ignoring the Missing-Data Mechanism

11.1. Introduction

11.2. Inference for a Mean Vector and Covariance Matrix with Missing Data Under Normality

11.3. Estimation with a Restricted Covariance Matrix

11.4. Multiple Linear Regression

11.5. A General Repeated-Measures Model with Missing Data

11.6. Time Series Models

12. Robust Estimation

12.1. Introduction

12.2. Robust Estimation for a Univariate Sample

12.3. Robust Estimation of the Mean and Covariance Matrix

12.4. Further Extensions of the t Model

13. Models for Partially Classified Contingency Tables, Ignoring the Missing-Data Mechanism

13.1. Introduction

13.2. Factored Likelihoods for Monotone Multinomial Data

13.3. ML and Bayes Estimation for Multinomial Samples with General Patterns of Missing Data

13.4. Loglinear Models for Partially Classified Contingency Tables

14. Mixed Normal and Non-normal Data with Missing Values, Ignoring the Missing-Data Mechanism

14.1. Introduction

14.2. The General Location Model

14.3. The General Location Model with Parameter Constraints

14.4. Regression Problems Involving Mixtures of Continuous and Categorical Variables

14.5. Further Extensions of the General Location Model

15. Nonignorable Missing-Data Models

15.1. Introduction

15.2. Likelihood Theory for Nonignorable Models

15.3. Models with Known Nonignorable Missing-Data Mechanisms: Grouped and Rounded Data

15.4. Normal Selection Models

15.5. Normal Pattern-Mixture Models

15.6. Nonignorable Models for Normal Repeated-Measures Data

15.7. Nonignorable Models for Categorical Data

References

Author Index

Subject Index

WILEY SERIES IN PROBABILITY AND STATISTICS

Established by WALTER A. SHEWHART and SAMUEL S. WILKS

Editors: David J. Balding, Peter Bloomfield, Noel A. C. Cressie, Nicholas I. Fisher, Iain M. Johnstone, J. B. Kadane, Louis M. Ryan, David W Scott, Adrian F. M. Smith, Jozef L. Teugels Editors Emeriti: Vic Barnet, J. Stuart Hunder, David G. Kendall

A complete list of the titles in this series appears at the end of this volume.

Copyright © 2002 by John Wiley & Sons, Inc. All rights reserved.

Published by John Wiley & Sons, Inc., Hoboken, New Jersey.
Published simultaneously in Canada

No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except as permitted under Sections 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400, fax 978750-4470, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008, e-mail: permcoordinator@wiley.com.

Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of marchantability or fitness for a aprticular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other danages.

For general information on our other products and services please contact our Customer Care Department within the U.S. at 877-762-2974, outside the U.S. at 317-572-3993 or fax 317-572-4002.

Wiley also publishes its books in a variety of electronic formats. Some content that appears in print, however, may not be available in electronic format.

Library of Congress Cataloging-in-Publication Data

Little, Roderick J. A.

      Statistical analysis with missing data / Roderick J Little, Donald B. Rubin. -- 2nd ed.
        p. cm. - - (Wiley series in probability and statistics)
      “A Wiley-Interscience publication.”
      Includes bibliographical references and index.
      ISBN 0-471-18386-5 (acid-free paper)
        1. Mathematical statistics. 2. Missing observations (Statistics) I Rubin, Donald B. II. Title. III. Series

      QA276 .L57 2002

      519.5--dc21                                                             2002027006

ISBN 0-471-18386-5

20 19 18 17 16 15 14

Preface

The literature on the statistical analysis of data with missing values has flourished since the early 1970s, spurred by advances in computer technology that made previously laborious numerical calculations a simple matter. This book aims to survey current methodology for handling missing-data problems and present a likelihood-based theory for analysis with missing data that systematizes these methods and provides a basis for future advances. Part I of the book discusses historical approaches to missing-value problems in three important areas of statistics: analysis of variance of planned experiments, survey sampling, and multivariate analysis. These methods, although not without value, tend to have an ad hoc character, often being solutions worked out by practitioners with limited research into theoretical properties. Part II presents a systematic approach to the analysis of data with missing values, where inferences are based on likelihoods derived from formal statistical models for the data-generating and missing-data mechanisms. Part III presents applications of the approach in a variety of contexts, including ones involving regression, factor analysis, contingency table analysis, time series, and sample survey inference. Many of the historical methods in Part I can be derived as examples (or approximations) of this likelihood-based approach.

The book is intended for the applied statistician and hence emphasizes examples over the precise statement of regularity conditions or proofs of theorems. Nevertheless, readers are expected to be familiar with basic principles of inference based on likelihoods, briefly reviewed in Section 6.1. The book also assumes an understanding of standard models of complete-data analysis—the normal linear model, multinomial models for counted data—and the properties of standard statistical distributions, especially the multivariate normal distribution. Some chapters assume familiarity in particular areas of statistical activity—analysis of variance for experimental designs (Chapter 2), survey sampling (Chapters 3, 4, and 5), or loglinear models for contingency tables (Chapter 13). Specific examples also introduce other statistical topics, such as factor analysis or time series (Chapter 11). The discussion of these examples is self-contained and does not require specialized knowledge, but such knowledge will, of course, enhance the reader’s appreciation of the main statistical issues. We have managed to cover about three-quarters of the material in the book in a 40-hour graduate statistics course.

When the first edition of this book was written in the mid-l980s, a weakness in the literature was that missing-data methods were mainly confined to the derivation of point estimates of parameters and approximate standard errors, with interval estimation and testing based on large-sample theory. Since that time, Bayesian methods for simulating posterior distributions have received extensive development, and these developments are reflected in the second edition. The closely related technique of multiple imputation also receives greater emphasis than in the first edition, in recognition of its increasing role in the theory and practice of handling missing data, including commercial software. The first part of the book has been reorganized to improve the flow of the material. Part II includes extensions of the EM algorithm, not available at the time of the first edition, and more Bayesian theory and computation, which have become standard tools in many areas of statistics. Applications of the likelihood approach have been assembled in a new Part III. Work on diagnostic tests of model assumptions when data are incomplete remains somewhat sketchy.

Because the second edition has some major additions and revisions, we provide a map showing where to locate the material originally appearing in Edition 1.

First Edition

1. Introduction

2. Missing Data in Experiments

3.2. Complete-Case Analysis

3.3. Available-Case Analysis

3.4. Filling in the Missing Values

4.2., 4.3. Randomization Inference with and without Missing Data

4.4. Weighting Methods

4.5. Imputation Procedures

4.6. Estimation of Sampling Variance with Nonresponse

5. Theory of Inference Based on the Likelihood Function

6. Factored Likelihood Methods

7. Maximum Likelihood for General Patterns of Missing Data

8. ML for Normal Examples

9. Partially Classified Contingency Tables

10.2 The General Location Model

10.3., 10.4. Extensions

10.5. Robust Estimation

11. Nonignorable Models

12.1., 12.2. Survey Nonresponse

12.3. Ignorable Nonresponse Models

12.4. Multiple Imputation

12.5. Nonignorable Nonresponse

Second Edition

1. Introduction

2. Missing Data in Experiments

3.2. Complete-Case Analysis

3.4. Available-Case Analysis

4.2. Imputing Means from a Predictive Distribution

Omitted

3.3. Weighted Complete-Case Analysis

4. Imputation

5. Estimation of Imputation Uncertainty

6. Theory of Inference Based on the Likelihood Function

7. Factored Likelihood Methods, Ignoring the Missing-Data Mechanism

8. Maximum Likelihood for General Patterns

9.1. Standard Errors Based on the Information Matrix.

11. Multivariate Normal Examples

13. Partially Classified Contingency Tables

14. Mixed Normal and Categorical Data with Missing Values

12. Models for Robust Estimation

15. Nonignorable Models

3.3. Weighted Complete-Case Analysis

4. Imputation

5.4. Introduction to Multiple Imputation

10. Bayes and Multiple Imputation

15.5. Normal Pattern-Mixture Models

The statistical literature on missing data has expanded greatly since the first edition, in terms of scope of applications and methodological developments. Thus, we have not found it possible to survey all the statistical work and still keep the book of tolerable length. We have tended to confine discussion to applications in our own range of experience, and we have focused methodologically on Bayesian and likelihood-based methods, which we believe provide a strong theoretical foundation for applications. We leave it to others to describe other approaches, such as that based on generalized estimating equations.

Many individuals are due thanks for their help in producing this book. NSF and NIMH (through grants NSF-SES-83-11428, NSF-SES-84-11804, NIMH-MH- 37188, DMS-9803720, and NSF-0106914) helped support some aspects of the research reported here. For the first edition, Mark Schluchter helped with computations, Leisa Weld and T. E. Raghunathan carefully read the final manuscript and made helpful suggestions, and our students in Biomathematics M232 at UCLA and Statistics 220r at Harvard University also made helpful suggestions. Judy Siesen typed and retyped our many drafts, and Bea Shube provided kind support and encouragement. For the second edition, we particularly thank Chuanhai Liu for help with computation, and Mingyao Li, Fang Liu, and Ying Yuan for help with examples. Many readers have helped by finding typographical and other errors, and we particularly thank Adi Andrei, Samantha Cook, Shane Jensen, Elizabeth Stuart, and Daohai Yu for their help on this aspect.

In closing, we continue to find that many statistical problems can be usefully viewed as missing-value problems even when the data set is fully recorded, and moreover, that missing-data research can be an excellent springboard for learning about statistics in general. We hope our readers will agree with us and find the book stimulating.

Ann Arbor, Michigan
Cambridge, Massachusetts

R. J. A. LITTLE
D. B. RUBIN

PART I

Overview and Basic Approaches

CHAPTER 1

Introduction

1.1. THE PROBLEM OF MISSING DATA

Standard statistical methods have been developed to analyze rectangular data sets. Traditionally, the rows of the data matrix represent units, also called cases, observations, or subjects depending on context, and the columns represent variables measured for each unit. The entries in the data matrix are nearly always real numbers, either representing the values of essentially continuous variables, such as age and income, or representing categories of response, which may be ordered (e.g., level of education) or unordered (e.g., race, sex). This book concerns the analysis of such a data matrix when some of the entries in the matrix are not observed. For example, respondents in a household survey may refuse to report income. In an industrial experiment some results are missing because of mechanical breakdowns unrelated to the experimental process. In an opinion survey some individuals may be unable to express a preference for one candidate over another. In the first two examples it is natural to treat the values that are not observed as missing, in the sense that there are actual underlying values that would have been observed if survey techniques had been better or the industrial equipment had been better maintained. In the third example, however, it is less clear that a well-defined candidate preference has been masked by the nonresponse; thus it is less natural to treat the unobserved values as missing. Instead the lack of a response is essentially an additional point in the sample space of the variable being measured, which identifies a “no preference” or “don’t know” stratum of the population.

Most statistical software packages allow the identification of nonrespondents by creating one or more special codes for those entries of the data matrix that are not observed. More than one code might be used to identify particular types of non-response, such as “don’t know,” or “refuse to answer,” or “out of legitimate range.” Some statistical packages typically exclude units that have missing value codes for any of the variables involved in an analysis. This strategy, which we term a “complete-case analysis,” is generally inappropriate, since the investigator is usually interested in making inferences about the entire target population, rather than the portion of the target population that would provide responses on all relevant variables in the analysis. Our aim is to describe a collection of techniques that are more generally appropriate than complete-case analysis when missing entries in the data set mask underlying values.

1.2. MISSING-DATA PATTERNS

We find it useful to distinguish the missing-data pattern, which describes which values are observed in the data matrix and which values are missing, and the missing-data mechanism (or mechanisms), which concerns the relationship between missingness and the values of variables in the data matrix. Some methods of analysis, such as those described in Chapter 7, are intended for particular patterns of missing data and use only standard complete-data analyses. Other methods, such as those described in Chapters 8–10, are applicable to more general missing-data patterns, but usually involve more computing than methods designed for special patterns. Thus it is beneficial to sort rows and columns of the data according to the pattern of missing data to see if an orderly pattern emerges. In this section we discuss some important patterns, and in the next section we formalize the idea of missing-data mechanisms.

Let Y = (yij) denote an (n × K) rectangular data set without missing values, with ith row yi = (yi1, … , yiK) where yij is the value of variable Yj for subject i. With missing data, define the missing-data indicator matrix M = (mij), such that mij = 1 if yij is missing and mij = 0 if yij is present. The matrix M then defines the pattern of missing data. Figure 1.1 shows some examples of missing-data patterns. Some methods for handling missing data apply to any pattern of missing data, whereas other methods are restricted to a special pattern.

EXAMPLE 1.1. Univariate Missing Data. Figure 1.1a illustrates univariate missing data, where missingness is confined to a single variable. The first incomplete-data problem to receive systematic attention in the statistics literature has the pattern of Figure 1.1a, namely, the problem of missing data in designed experiments. In the context of agricultural trials this situation is often called the missing-plot problem. Interest is in the relationship between a dependent variable YK, such as yield of crop, on a set of factors Y₁, … , YK-1, such as variety, type of fertilizer, and temperature, all of which are intended to be fully observed. (In the figure, K = 5.) Often a balanced experimental design is chosen that yields orthogonal factors and hence a simple analysis. However, sometimes the outcomes for some of the experimental units are missing (for example because of lack of germination of a seed, or because the data were incorrectly recorded). The result is the pattern with YK incomplete and Y₁, … , YK-1 fully observed. Missing-data techniques fill in the missing values of YK in order to retain the balance in the original experimental design. Historically important methods, reviewed in Chapter 2, were motivated by computational simplicity and hence are less important in our era of high-speed computers, but they can still be useful in high-dimensional problems.

Figure 1.1. Examples of missing-data patterns. Rows correspond to observations, columns to variables.

EXAMPLE 1.2. Unit and Item Nonresponse in Surveys. Another common pattern is obtained when the single incomplete variable YK in Figure 1.1a is replaced by a set of variables YJ+1, … , YK, all observed or missing on the same set of cases (see Figure 1.1, where = 5 and = 2). An example of this pattern is unit nonresponse in sample surveys, where a questionnaire is administered and a subset of sampled individuals do not complete the questionnaire because of noncontact, refusal, or some other reason. In that case the survey items are the incomplete variables, and the fully observed variables consist of survey design variables measured for respondents and nonrespondents, such as household location or characteristics measured in a listing operation prior to the survey. Common techniques for addressing unit nonresponse in surveys are discussed in Chapter 3. Survey practitioners call missing values on particular items in the questionnaire . These missing values typically have a haphazard pattern, such as that in . Item nonresponse in surveys is typically handled by imputation methods as discussed in Chapter 4, although the methods discussed in Part II of the book are also appropriate and relevant. For other discussions of missing data in the survey context, see Madow and Olkin (1983), Madow, Nisselson, and Olkin (1983), Madow, Olkin, and Rubin (1983), Rubin (1987a) and Groves et al. (2002).

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!

Lesen Sie weiter in der vollständigen Ausgabe!