Craig K. Enders

The Relative Performance of Full Information Maximum Likelihood Estimation for Missing Data in Structural Equation Models.

A Monte Carlo simulation examined the performance of 4 missing data methods in structural equation models: full information maximum likelihood (FIML), listwise deletion, pairwise deletion, and similar response pattern imputation. The effects of 3 independent variables were examined (factor loading magnitude, sample size, and missing data rate) on 4 outcome measures: convergence failures, parameter estimate bias, parameter estimate efficiency, and model goodness of fit. Results indicated that FIML estimation was superior across all conditions of the design. Under ignorable missing data conditions (missing completely at random and missing at random), FIML estimates were unbiased and more efficient than the other methods. In addition, FIML yielded the lowest proportion of convergence failures and provided near-optimal Type 1 error rates across both simulations.

Centering Predictor Variables in Cross-Sectional Multilevel Models: A New Look at an Old Issue.

Appropriately centering Level 1 predictors is vital to the interpretation of intercept and slope parameters in multilevel models (MLMs). The issue of centering has been discussed in the literature, but it is still widely misunderstood. The purpose of this article is to provide a detailed overview of grand mean centering and group mean centering in the context of 2-level MLMs. The authors begin with a basic overview of centering and explore the differences between grand and group mean centering in the context of some prototypical research questions. Empirical analyses of artificial data sets are used to illustrate key points throughout. The article provides a number of practical recommendations designed to facilitate centering decisions in MLM applications.

A Primer on Maximum Likelihood Algorithms Available for Use With Missing Data

Maximum likelihood algorithms for use with missing data are becoming commonplace in microcomputer packages. Specifically, 3 maximum likelihood algorithms are currently available in existing software packages: the multiple-group approach, full information maximum likelihood estimation, and the EM algorithm. Although they belong to the same family of estimator, confusion appears to exist over the differences among the 3 algorithms. This article provides a comprehensive, nontechnical overview of the 3 maximum likelihood algorithms. Multiple imputation, which is frequently used in conjunction with the EM algorithm, is also discussed.

Missing Data in Educational Research: A Review of Reporting Practices and Suggestions for Improvement

Missing data analyses have received considerable recent attention in the methodological literature, and two “modern” methods, multiple imputation and maximum likelihood estimation, are recommended. The goals of this article are to (a) provide an overview of missing-data theory, maximum likelihood estimation, and multiple imputation; (b) conduct a methodological review of missing-data reporting practices in 23 applied research journals; and (c) provide a demonstration of multiple imputation and maximum likelihood estimation using the Longitudinal Study of American Youth data. The results indicated that explicit discussions of missing data increased substantially between 1999 and 2003, but the use of maximum likelihood estimation or multiple imputation was rare; the studies relied almost exclusively on listwise and pairwise deletion.

An introduction to modern missing data analyses

A great deal of recent methodological research has focused on two modern missing data analysis methods: maximum likelihood and multiple imputation. These approaches are advantageous to traditional techniques (e.g. deletion and mean imputation techniques) because they require less stringent assumptions and mitigate the pitfalls of traditional techniques. This article explains the theoretical underpinnings of missing data analyses, gives an overview of traditional missing data techniques, and provides accessible descriptions of maximum likelihood and multiple imputation. In particular, this article focuses on maximum likelihood estimation and presents two analysis examples from the Longitudinal Study of American Youth data. One of these examples includes a description of the use of auxiliary variables. Finally, the paper illustrates ways that researchers can use intentional, or planned, missing data to enhance their research designs.

The impact of nonnormality on full information maximum-likelihood estimation for structural equation models with missing data.

A Monte Carlo simulation examined full information maximum-likelihood estimation (FIML) in structural equation models with nonnormal indicator variables. The impacts of 4 independent variables were examined (missing data algorithm, missing data rate, sample size, and distribution shape) on 4 outcome measures (parameter estimate bias, parameter estimate efficiency, standard error coverage, and model rejection rates). Across missing completely at random and missing at random patterns, FIML parameter estimates involved less bias and were generally more efficient than those of ad hoc missing data techniques. However, similar to complete-data maximum-likelihood estimation in structural equation modeling, standard errors were negatively biased and model rejection rates were inflated. Simulation results suggest that recently developed correctives for missing data (e.g., rescaled statistics and the bootstrap) can mitigate problems that stem from nonnormal data.

The Performance of the Full Information Maximum Likelihood Estimator in Multiple Regression Models with Missing Data

A Monte Carlo simulation examined the performance of a recently available full information maximum likelihood (FIML) estimator in a multiple regression model with missing data. The effects of four independent variables were examined (missing data technique, missing data rate, sample size, and correlation magnitude) on three outcome measures: regression coefficient bias, R 2 bias, and regression coefficient sampling variability. Three missing data patterns were examined based on Rubin’s missing data theory: missing completely at random, missing at random, and a nonrandom pattern. Results indicated that FIML estimation was superior to the three ad hoc techniques (listwise deletion, pairwise deletion, and mean imputation) across the conditions studied. FIML parameter estimates generally had less bias and less sampling variability than the three ad hoc methods.

Using the SPSS Mixed Procedure to Fit Cross-Sectional and Longitudinal Multilevel Models

Beginning with Version 11, SPSS implemented the MIXED procedure, which is capable of performing many common hierarchical linear model analyses. The purpose of this article was to provide a tutorial for performing cross-sectional and longitudinal analyses using this popular software platform. In doing so, the authors borrowed heavily from Singer’s overview of SAS PROC MIXED, duplicating her analyses using the SPSS MIXED procedure.

Missing Not at Random Models for Latent Growth Curve Analyses

The past decade has seen a noticeable shift in missing data handling techniques that assume a missing at random (MAR) mechanism, where the propensity for missing data on an outcome is related to other analysis variables. Although MAR is often reasonable, there are situations where this assumption is unlikely to hold, leading to biased parameter estimates. One such example is a longitudinal study of substance use where participants with the highest frequency of use also have the highest likelihood of attrition, even after controlling for other correlates of missingness. There is a large body of literature on missing not at random (MNAR) analysis models for longitudinal data, particularly in the field of biostatistics. Because these methods allow for a relationship between the outcome variable and the propensity for missing data, they require a weaker assumption about the missing data mechanism. This article describes 2 classic MNAR modeling approaches for longitudinal data: the selection model and the pattern mixture model. To date, these models have been slow to migrate to the social sciences, in part because they required complicated custom computer programs. These models are now quite easy to estimate in popular structural equation modeling programs, particularly Mplus. The purpose of this article is to describe these MNAR modeling frameworks and to illustrate their application on a real data set. Despite their potential advantages, MNAR-based analyses are not without problems and also rely on untestable assumptions. This article offers practical advice for implementing and choosing among different longitudinal models.

The Varieties of Religious Development in Adulthood: A Longitudinal Investigation of Religion and Rational Choice

The authors used growth mixture models to study religious development during adulthood (ages 27-80) in a sample of individuals who were identified during childhood as intellectually gifted. The authors identified 3 discrete trajectories of religious development: (a) 40% of participants belonged to a trajectory class characterized by increases in religiousness until midlife and declines in later adulthood; (b) 41% of participants belonged to a trajectory class characterized by very low religiousness in early adulthood and age-related decline; and (c) 19% of participants belonged to a trajectory class characterized by high religiousness in early adulthood and age-related increases. Gender, strength of religious upbringing, number of children, marrying, and agreeableness predicted membership in the trajectory classes. Results were largely consistent with the rational choice theory of religious involvement.