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Contents
1 • An Introduction to Missing Data
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
1.10
1.11
1.12
1.13
1.14
1.15
1.16
1
Introduction
1
Chapter Overview
2
Missing Data Patterns
2
A Conceptual Overview of Missing Data Theory
5
A More Formal Description of Missing Data Theory
9
Why Is the Missing Data Mechanism Important?
13
How Plausible Is the Missing at Random Mechanism?
14
An Inclusive Analysis Strategy
16
Testing the Missing Completely at Random Mechanism
17
Planned Missing Data Designs
21
The Three-Form Design
23
Planned Missing Data for Longitudinal Designs
28
Conducting Power Analyses for Planned Missing Data Designs
Data Analysis Example
32
Summary
35
Recommended Readings
36
30
2 • Traditional Methods for Dealing with Missing Data
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
2.10
2.11
2.12
2.13
Chapter Overview
37
An Overview of Deletion Methods
39
Listwise Deletion
39
Pairwise Deletion
40
An Overview of Single Imputation Methods
Arithmetic Mean Imputation
42
Regression Imputation
44
Stochastic Regression Imputation
46
Hot-Deck Imputation
49
Similar Response Pattern Imputation
49
Averaging the Available Items
50
Last Observation Carried Forward
51
An Illustrative Computer Simulation Study
37
42
52
xi
xii
Contents
2.14 Summary
54
2.15 Recommended Readings
55
3 • An Introduction to Maximum Likelihood Estimation
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
3.10
3.11
3.12
3.13
3.14
3.15
3.16
3.17
4 • Maximum Likelihood Missing Data Handling
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10
4.11
4.12
4.13
4.14
4.15
4.16
4.17
4.18
4.19
4.20
Chapter Overview
86
The Missing Data Log-Likelihood
88
How Do the Incomplete Data Records Improve Estimation?
An Illustrative Computer Simulation Study
95
Estimating Standard Errors with Missing Data
97
Observed versus Expected Information
98
A Bivariate Analysis Example
99
An Illustrative Computer Simulation Study
102
An Overview of the EM Algorithm
103
A Detailed Description of the EM Algorithm
105
A Bivariate Analysis Example
106
Extending EM to Multivariate Data
110
Maximum Likelihood Estimation Software Options
112
Data Analysis Example 1
113
Data Analysis Example 2
115
Data Analysis Example 3
118
Data Analysis Example 4
119
Data Analysis Example 5
122
Summary
125
Recommended Readings
126
5 • Improving the Accuracy of
Maximum Likelihood Analyses
5.1
5.2
5.3
5.4
56
Chapter Overview
56
The Univariate Normal Distribution
56
The Sample Likelihood
59
The Log-Likelihood
60
Estimating Unknown Parameters
60
The Role of First Derivatives
63
Estimating Standard Errors
65
Maximum Likelihood Estimation with Multivariate Normal Data
69
A Bivariate Analysis Example
73
Iterative Optimization Algorithms
75
Significance Testing Using the Wald Statistic
77
The Likelihood Ratio Test Statistic
78
Should I Use the Wald Test or the Likelihood Ratio Statistic?
79
Data Analysis Example 1
80
Data Analysis Example 2
81
Summary
83
Recommended Readings
85
Chapter Overview
127
The Rationale for an Inclusive Analysis Strategy
127
An Illustrative Computer Simulation Study
129
Identifying a Set of Auxiliary Variables
131
86
92
127
Contents
5.5
5.6
5.7
5.8
5.9
5.10
5.11
5.12
5.13
5.14
5.15
5.16
Incorporating Auxiliary Variables into a Maximum Likelihood Analysis
The Saturated Correlates Model
134
The Impact of Non-Normal Data
140
Robust Standard Errors
141
Bootstrap Standard Errors
145
The Rescaled Likelihood Ratio Test
148
Bootstrapping the Likelihood Ratio Statistic
150
Data Analysis Example 1
154
Data Analysis Example 2
155
Data Analysis Example 3
157
Summary
161
Recommended Readings
163
6 • An Introduction to Bayesian Estimation
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
6.9
6.10
6.11
Chapter Overview
164
What Makes Bayesian Statistics Different?
165
A Conceptual Overview of Bayesian Estimation
165
Bayes’ Theorem
170
An Analysis Example
171
How Does Bayesian Estimation Apply to Multiple Imputation?
The Posterior Distribution of the Mean
176
The Posterior Distribution of the Variance
179
The Posterior Distribution of a Covariance Matrix
183
Summary
185
Recommended Readings
186
175
187
Chapter Overview
187
A Conceptual Description of the Imputation Phase
190
A Bayesian Description of the Imputation Phase
191
A Bivariate Analysis Example
194
Data Augmentation with Multivariate Data
199
Selecting Variables for Imputation
201
The Meaning of Convergence
202
Convergence Diagnostics
203
Time-Series Plots
204
Autocorrelation Function Plots
207
Assessing Convergence from Alternate Starting Values
209
Convergence Problems
210
Generating the Final Set of Imputations
211
How Many Data Sets Are Needed?
212
Summary
214
Recommended Readings
216
8 • The Analysis and Pooling
Phases of Multiple Imputation
8.1
8.2
8.3
8.4
133
164
7 • The Imputation Phase of Multiple Imputation
7.1
7.2
7.3
7.4
7.5
7.6
7.7
7.8
7.9
7.10
7.11
7.12
7.13
7.14
7.15
7.16
xiii
Chapter Overview
217
The Analysis Phase
218
Combining Parameter Estimates in the Pooling Phase
Transforming Parameter Estimates Prior to Combining
217
219
220
xiv
Contents
8.5
8.6
8.7
8.8
8.9
8.10
8.11
8.12
8.13
8.14
8.15
8.16
8.17
8.18
Pooling Standard Errors
221
The Fraction of Missing Information and the Relative Increase in Variance
224
When Is Multiple Imputation Comparable to Maximum Likelihood?
227
An Illustrative Computer Simulation Study
229
Significance Testing Using the t Statistic
230
An Overview of Multiparameter Significance Tests
233
233
Testing Multiple Parameters Using the D1 Statistic
Testing Multiple Parameters by Combining Wald Tests
239
Testing Multiple Parameters by Combining Likelihood Ratio Statistics
240
Data Analysis Example 1
242
Data Analysis Example 2
245
Data Analysis Example 3
247
Summary
252
Recommended Readings
252
9 • Practical Issues in Multiple Imputation
9.1
9.2
9.3
9.4
9.5
9.6
9.7
9.8
9.9
9.10
9.11
9.12
254
Chapter Overview
254
Dealing with Convergence Problems
254
Dealing with Non-Normal Data
259
To Round or Not to Round?
261
Preserving Interaction Effects
265
Imputing Multiple-Item Questionnaires
269
Alternate Imputation Algorithms
272
Multiple-Imputation Software Options
278
Data Analysis Example 1
279
Data Analysis Example 2
281
Summary
283
Recommended Readings
286
10 • Models for Missing Not at Random Data
10.1
10.2
10.3
10.4
10.5
10.6
10.7
10.8
10.9
10.10
10.11
10.12
10.13
10.14
10.15
10.16
10.17
10.18
10.19
10.20
10.21
10.22
Chapter Overview
287
An Ad Hoc Approach to Dealing with MNAR Data
289
The Theoretical Rationale for MNAR Models
290
The Classic Selection Model
291
Estimating the Selection Model
295
Limitations of the Selection Model
296
An Illustrative Analysis
297
The Pattern Mixture Model
298
Limitations of the Pattern Mixture Model
300
An Overview of the Longitudinal Growth Model
301
A Longitudinal Selection Model
303
Random Coefficient Selection Models
305
Pattern Mixture Models for Longitudinal Analyses
306
Identification Strategies for Longitudinal Pattern Mixture Models
Delta Method Standard Errors
309
Overview of the Data Analysis Examples
312
Data Analysis Example 1
314
Data Analysis Example 2
315
Data Analysis Example 3
317
Data Analysis Example 4
321
Summary
326
Recommended Readings
328
287
307
Contents
11 • Wrapping Things Up:
Some Final Practical Considerations
11.1
11.2
11.3
11.4
11.5
11.6
11.7
Chapter Overview
329
Maximum Likelihood Software Options
329
Multiple-Imputation Software Options
333
Choosing between Maximum Likelihood and Multiple Imputation
Reporting the Results from a Missing Data Analysis
340
Final Thoughts
343
Recommended Readings
344
xv
329
336
References
347
Author Index
359
Subject Index
365
About the Author
377
The companion website (www.appliedmissingdata.com) includes data
files and syntax for the examples in the book, as well as up-to-date
information on software.