Generating Synthetic Missing Data: A Review by Missing Mechanism <s> II. PRELIMINARY NOTATION <s> Preface.PART I: OVERVIEW AND BASIC APPROACHES.Introduction.Missing Data in Experiments.Complete-Case and Available-Case Analysis, Including Weighting Methods.Single Imputation Methods.Estimation of Imputation Uncertainty.PART II: LIKELIHOOD-BASED APPROACHES TO THE ANALYSIS OF MISSING DATA.Theory of Inference Based on the Likelihood Function.Methods Based on Factoring the Likelihood, Ignoring the Missing-Data Mechanism.Maximum Likelihood for General Patterns of Missing Data: Introduction and Theory with Ignorable Nonresponse.Large-Sample Inference Based on Maximum Likelihood Estimates.Bayes and Multiple Imputation.PART III: LIKELIHOOD-BASED APPROACHES TO THE ANALYSIS OF MISSING DATA: APPLICATIONS TO SOME COMMON MODELS.Multivariate Normal Examples, Ignoring the Missing-Data Mechanism.Models for Robust Estimation.Models for Partially Classified Contingency Tables, Ignoring the Missing-Data Mechanism.Mixed Normal and Nonnormal Data with Missing Values, Ignoring the Missing-Data Mechanism.Nonignorable Missing-Data Models.References.Author Index.Subject Index. <s> BIB001 </s> Generating Synthetic Missing Data: A Review by Missing Mechanism <s> II. PRELIMINARY NOTATION <s> Part 1. An Introduction to Missing Data. 1.1 Introduction. 1.2 Chapter Overview. 1.3 Missing Data Patterns. 1.4 A Conceptual Overview of Missing Data heory. 1.5 A More Formal Description of Missing Data Theory. 1.6 Why Is the Missing Data Mechanism Important? 1.7 How Plausible Is the Missing at Random Mechanism? 1.8 An Inclusive Analysis Strategy. 1.9 Testing the Missing Completely at Random Mechanism. 1.10 Planned Missing Data Designs. 1.11 The Three-Form Design. 1.12 Planned Missing Data for Longitudinal Designs. 1.13 Conducting Power Analyses for Planned Missing Data Designs. 1.14 Data Analysis Example. 1.15 Summary. 1.16 Recommended Readings. Part 2. Traditional Methods for Dealing with Missing Data. 2.1 Chapter Overview. 2.2 An Overview of Deletion Methods. 2.3 Listwise Deletion. 2.4 Pairwise Deletion. 2.5 An Overview of Single Imputation Techniques. 2.6 Arithmetic Mean Imputation. 2.7 Regression Imputation. 2.8 Stochastic Regression Imputation. 2.9 Hot-Deck Imputation. 2.10 Similar Response Pattern Imputation. 2.11 Averaging the Available Items. 2.12 Last Observation Carried Forward. 2.13 An Illustrative Simulation Study. 2.14 Summary. 2.15 Recommended Readings. Part 3. An Introduction to Maximum Likelihood Estimation. 3.1 Chapter Overview. 3.2 The Univariate Normal Distribution. 3.3 The Sample Likelihood. 3.4 The Log-Likelihood. 3.5 Estimating Unknown Parameters. 3.6 The Role of First Derivatives. 3.7 Estimating Standard Errors. 3.8 Maximum Likelihood Estimation with Multivariate Normal Data. 3.9 A Bivariate Analysis Example. 3.10 Iterative Optimization Algorithms. 3.11 Significance Testing Using the Wald Statistic. 3.12 The Likelihood Ratio Test Statistic. 3.13 Should I Use the Wald Test or the Likelihood Ratio Statistic? 3.14 Data Analysis Example 1. 3.15 Data Analysis Example 2. 3.16 Summary. 3.17 Recommended Readings. Part 4. Maximum Likelihood Missing Data Handling. 4.1 Chapter Overview. 4.2 The Missing Data Log-Likelihood. 4.3 How Do the Incomplete Data Records Improve Estimation? 4.4 An Illustrative Computer Simulation Study. 4.5 Estimating Standard Errors with Missing Data. 4.6 Observed Versus Expected Information. 4.7 A Bivariate Analysis Example. 4.8 An Illustrative Computer Simulation Study. 4.9 An Overview of the EM Algorithm. 4.10 A Detailed Description of the EM Algorithm. 4.11 A Bivariate Analysis Example. 4.12 Extending EM to Multivariate Data. 4.13 Maximum Likelihood Software Options. 4.14 Data Analysis Example 1. 4.15 Data Analysis Example 2. 4.16 Data Analysis Example 3. 4.17 Data Analysis Example 4. 4.18 Data Analysis Example 5. 4.19 Summary. 4.20 Recommended Readings. Part 5. Improving the Accuracy of Maximum Likelihood Analyses. 5.1 Chapter Overview. 5.2 The Rationale for an Inclusive Analysis Strategy. 5.3 An Illustrative Computer Simulation Study. 5.4 Identifying a Set of Auxiliary Variables. 5.5 Incorporating Auxiliary Variables Into a Maximum Likelihood Analysis. 5.6 The Saturated Correlates Model. 5.7 The Impact of Non-Normal Data. 5.8 Robust Standard Errors. 5.9 Bootstrap Standard Errors. 5.10 The Rescaled Likelihood Ratio Test. 5.11 Bootstrapping the Likelihood Ratio Statistic. 5.12 Data Analysis Example 1. 5.13 Data Analysis Example 2. 5.14 Data Analysis Example 3. 5.15 Summary. 5.16 Recommended Readings. Part 6. An Introduction to Bayesian Estimation. 6.1 Chapter Overview. 6.2 What Makes Bayesian Statistics Different? 6.3 A Conceptual Overview of Bayesian Estimation. 6.4 Bayes' Theorem. 6.5 An Analysis Example. 6.6 How Does Bayesian Estimation Apply to Multiple Imputation? 6.7 The Posterior Distribution of the Mean. 6.8 The Posterior Distribution of the Variance. 6.9 The Posterior Distribution of a Covariance Matrix. 6.10 Summary. 6.11 Recommended Readings. Part 7. The Imputation Phase of Multiple Imputation. 7.1 Chapter Overview. 7.2 A Conceptual Description of the Imputation Phase. 7.3 A Bayesian Description of the Imputation Phase. 7.4 A Bivariate Analysis Example. 7.5 Data Augmentation with Multivariate Data. 7.6 Selecting Variables for Imputation. 7.7 The Meaning of Convergence. 7.8 Convergence Diagnostics. 7.9 Time-Series Plots. 7.10 Autocorrelation Function Plots. 7.11 Assessing Convergence from Alternate Starting Values. 7.12 Convergence Problems. 7.13 Generating the Final Set of Imputations. 7.14 How Many Data Sets Are Needed? 7.15 Summary. 7.16 Recommended Readings. Part 8. The Analysis and Pooling Phases of Multiple Imputation. 8.1 Chapter Overview. 8.2 The Analysis Phase. 8.3 Combining Parameter Estimates in the Pooling Phase. 8.4 Transforming Parameter Estimates Prior to Combining. 8.5 Pooling Standard Errors. 8.6 The Fraction of Missing Information and the Relative Increase in Variance. 8.7 When Is Multiple Imputation Comparable to Maximum Likelihood? 8.8 An Illustrative Computer Simulation Study. 8.9 Significance Testing Using the t Statistic. 8.10 An Overview of Multiparameter Significance Tests. 8.11 Testing Multiple Parameters Using the D1 Statistic. 8.12 Testing Multiple Parameters by Combining Wald Tests. 8.13 Testing Multiple Parameters by Combining Likelihood Ratio Statistics. 8.14 Data Analysis Example 1. 8.15 Data Analysis Example 2. 8.16 Data Analysis Example 3. 8.17 Summary. 8.18 Recommended Readings. Part 9. Practical Issues in Multiple Imputation. 9.1 Chapter Overview. 9.2 Dealing with Convergence Problems. 9.3 Dealing with Non-Normal Data. 9.4 To Round or Not to Round? 9.5 Preserving Interaction Effects. 9.6 Imputing Multiple-Item Questionnaires. 9.7 Alternate Imputation Algorithms. 9.8 Multiple Imputation Software Options. 9.9 Data Analysis Example 1. 9.10 Data Analysis Example 2. 9.11 Summary. 9.12 Recommended Readings. Part 10. Models for Missing Not at Random Data. 10.1 Chapter Overview. 10.2 An Ad Hoc Approach to Dealing with MNAR Data. 10.3 The Theoretical Rationale for MNAR Models. 10.4 The Classic Selection Model. 10.5 Estimating the Selection Model. 10.6 Limitations of the Selection Model. 10.7 An Illustrative Analysis. 10.8 The Pattern Mixture Model. 10.9 Limitations of the Pattern Mixture Model. 10.10 An Overview of the Longitudinal Growth Model. 10.11 A Longitudinal Selection Model. 10.12 Random Coefficient Selection Models. 10.13 Pattern Mixture Models for Longitudinal Analyses. 10.14 Identification Strategies for Longitudinal Pattern Mixture Models. 10.15 Delta Method Standard Errors. 10.16 Overview of the Data Analysis Examples. 10.17 Data Analysis Example 1. 10.18 Data Analysis Example 2. 10.19 Data Analysis Example 3. 10.20 Data Analysis Example 4. 10.21 Summary. 10.22 Recommended Readings. Part 11. Wrapping Things Up: Some Final Practical Considerations. 11.1 Chapter Overview. 11.2 Maximum Likelihood Software Options. 11.3 Multiple Imputation Software Options. 11.4 Choosing between Maximum Likelihood and Multiple Imputation. 11.5 Reporting the Results from a Missing Data Analysis. 11.6 Final Thoughts. 11.7 Recommended Readings. <s> BIB002
In order to provide a formal description of the missing data mechanisms, it is first necessary to establish some basic notation and terminology. Let us assume a dataset X represented by a n × p matrix, where i = 1, · · · , n patterns and j = 1, · · · , p features. The elements of X are denoted by x i,j , each individual feature in X is denoted by x j and each pattern is referred to as In classification and missing theory domains, each pattern is also assigned a target class t i ∈ {C 1 , C 2 , · · · , C c } and a missing indicator , which indicates the features that are missing for each pattern x i . We can now define a missing data indicator M as a n × p binary matrix, defined as follows  : M indicates the locations of the missing values in the dataset and X may be divided into two components, X = (X obs , X miss ). X obs and X miss represent, respectively, the observed and missing values in X, i.e., X obs contains all elements x i,j where m i,j = 0 while X miss contains all elements x i,j where m i,j = 1. Rubin's missing data theory  , BIB001 establishes that the probability distribution of M may depend on X = (X obs , X miss ), and that this relationship describes the missing data mechanisms, p(M | X, ξ ), whose parameters are herein denoted by ξ BIB002 ,  . In practice, ξ cannot be determined with certainty; however, it is not important to know these parameters in detail, it is only necessary to understand whether there is or there is not a relation between M and X components: X obs and X miss . A dataset X can suffer from different percentages of missing data, which are referred to as missing rates (MRs) and they can be defined for each feature individually or for the entire dataset. Consider Table 1 , which illustrates the concepts presented above. Table 1a represents the matrix of data X, where the number of patterns is n = 20 (20 records/lines in the table), and the number of features is p = 2 (''Age'' and ''Number of Cigarettes''). Only feature x 2 (''Number of Cigarettes'') has missing values, denoted by ''⊗'', but there are several patterns that contain missing values, {x 2 , x 3 , x 8 , x 10 , x 13 , x 15 , x 17 , x 18 }. Table 1b represents the missing data indicator matrix M, where positions x ij of Table 1a are coded as 0/1 values according to their presence/absence. As an example, M 2,1 = 0 since ''Age'' is observed in pattern x 2 , while M 2,2 = 1 since ''Number of Cigarettes'' is missing in x 2 . Regarding the MR, feature x 1 has a MR of 0% (there are no missing values in ''Age''), and feature x 2 has a MR of 40% (out of 20 values, 8 are missing in ''Number of Cigarettes'', 8 20 = 40%). We may also define the MR considering the entire dataset, that is, the total of x i,j elements that are missing. In this case, there are a total of patterns × features elements (20 × 2 = 40 elements), and 8 of them are missing, thus giving a MR of 8 40 = 20%, if the entire dataset is considered.