Abstract: Evaluation of Test Statistics for Robust Structural Equation Modeling With Nonnormal Missing Data

Traditional structural equation modeling (SEM) techniques have trouble dealing with incomplete and/or nonnormal data that are often encountered in practice. Yuan and Zhang (2011a) developed a two-stage procedure for SEM to handle nonnormal missing data and proposed four test statistics for overall model evaluation. Although these statistics have been shown to work well with complete data, their performance for incomplete data has not been investigated in the context of robust statistics.

Focusing on a linear growth curve model, a systematic simulation study is conducted to evaluate the accuracy of the parameter estimates and the performance of five test statistics including the naive statistic derived from normal distribution based maximum likelihood (ML), the Satorra-Bentler scaled chi-square statistic (RML), the mean- and variance-adjusted chi-square statistic (AML), Yuan-Bentler residual-based test statistic (CRADF), and Yuan-Bentler residual-based F statistic (RF). Data are generated and analyzed in R using the package rsem (Yuan & Zhang, 2011b).

Based on the simulation study, we can observe the following: (a) The traditional normal distribution-based method cannot yield accurate parameter estimates for nonnormal data, whereas the robust method obtains much more accurate model parameter estimates for nonnormal data and performs almost as well as the normal distribution based method for normal distributed data. (b) With the increase of sample size, or the decrease of missing rate or the number of outliers, the parameter estimates are less biased and the empirical distributions of test statistics are closer to their nominal distributions. (c) The ML test statistic does not work well for nonnormal or missing data. (d) For nonnormal complete data, CRADF and RF work relatively better than RML and AML. (e) For missing completely at random (MCAR) missing data, in almost all the cases, RML and AML work better than CRADF and RF. (f) For nonnormal missing at random (MAR) missing data, CRADF and RF work better than AML. (g) The performance of the robust method does not seem to be influenced by the symmetry of outliers.     

Estimating Latent Variable Interactions With Nonnormal Observed Data: A Comparison of Four Approaches

A Monte Carlo simulation was conducted to investigate the robustness of 4 latent variable interaction modeling approaches (Constrained Product Indicator [CPI], Generalized Appended Product Indicator [GAPI], Unconstrained Product Indicator [UPI], and Latent Moderated Structural Equations [LMS]) under high degrees of nonnormality of the observed exogenous variables. Results showed that the CPI and LMS approaches yielded biased estimates of the interaction effect when the exogenous variables were highly nonnormal. When the violation of nonnormality was not severe (normal; symmetric with excess kurtosis < 1), the LMS approach yielded the most efficient estimates of the latent interaction effect with the highest statistical power. In highly nonnormal conditions, the GAPI and UPI approaches with maximum likelihood (ML) estimation yielded unbiased latent interaction effect estimates, with acceptable actual Type I error rates for both the Wald and likelihood ratio tests of interaction effect at N ≥ 500. An empirical example illustrated the use of the 4 approaches in testing a latent variable interaction between academic self-efficacy and positive family role models in the prediction of academic performance.     

An Investigation of the Sample Performance of Two Nonnormality Corrections for RMSEA

The root mean square error of approximation (RMSEA) is a popular fit index in structural equation modeling (SEM). Typically, RMSEA is computed using the normal theory maximum likelihood (ML) fit function. Under nonnormality, the uncorrected sample estimate of the ML RMSEA tends to be inflated. Two robust corrections to the sample ML RMSEA have been proposed, but the theoretical and empirical differences between the 2 have not been explored. In this article, we investigate the behavior of these 2 corrections. We show that the virtually unknown correction due to Li and Bentler (2006) Li, L. and Bentler, P. M. 2006. “Robust statistical tests for evaluating the hypothesis of close fit of misspecified mean and covariance structural models.”. In UCLA Statistics Preprint #506. Los Angeles: University of California..  [Google Scholar], which we label the sample-corrected robust RMSEA, is a consistent estimate of the population ML RMSEA yet drastically reduces bias due to nonnormality in small samples. On the other hand, the popular correction implemented in several SEM programs, which we label the population-corrected robust RMSEA, has poor properties because it estimates a quantity that decreases with increasing nonnormality. We recommend the use of the sample-corrected RMSEA with nonnormal data and its wide implementation.     

On the Computation of the RMSEA and CFI from the Mean-And-Variance Corrected Test Statistic with Nonnormal Data in SEM

A new type of nonnormality correction to the RMSEA has recently been developed, which has several advantages over existing corrections. In particular, the new correction adjusts the sample estimate of the RMSEA for the inflation due to nonnormality, while leaving its population value unchanged, so that established cutoff criteria can still be used to judge the degree of approximate fit. A confidence interval (CI) for the new robust RMSEA based on the mean-corrected (“Satorra-Bentler”) test statistic has also been proposed. Follow up work has provided the same type of nonnormality correction for the CFI (Brosseau-Liard & Savalei, 2014). These developments have recently been implemented in lavaan. This note has three goals: a) to show how to compute the new robust RMSEA and CFI from the mean-and-variance corrected test statistic; b) to offer a new CI for the robust RMSEA based on the mean-and-variance corrected test statistic; and c) to caution that the logic of the new nonnormality corrections to RMSEA and CFI is most appropriate for the maximum likelihood (ML) estimator, and cannot easily be generalized to the most commonly used categorical data estimators.     

A Heterogeneous Growth Curve Model for Nonnormal Data

The heterogeneous growth curve model (HGM; Klein &; Muthén, 2006 Klein, A.G., &; Muthén, B.O. (2006). Modeling heterogeneity of latent growth depending on initial status. Journal of Educational and Behavioral Statistics, 31, 357–375. doi: 10.3102/10769986031004357[Crossref], [Web of Science ®] , [Google Scholar]) is a method for modeling heterogeneity of growth rates with a heteroscedastic residual structure for the slope factor. It has been developed as an extension of a conventional growth curve model and a complementary tool to growth curve mixture models. In this article, a robust version of the heterogeneous growth curve model (HGM-R) is presented that extends the original HGM with a mixture model to allow for an unbiased parameter estimation under the condition of nonnormal data. In two simulation studies, the performance of the method is examined under the condition of nonnormality and a misspecified heteroscedastic residual structure. The results of the simulation studies suggest an unbiased estimation of the heterogeneity by the HGM-R when sample size was large enough and a good approximation of the heteroscedastic residual structure even when the functional form of the heteroscedasticity was misspecified. The practical application of the approach is demonstrated for a data set from HIV-infected patients.     

Abstract: Adequacy of Semiparametric Approximations for Growth Models with Nonnormal Random Effects

Abstract

Conventional growth models assume that the random effects describing individual trajectories are conditionally normal. In practice, this assumption may often be unrealistic. As an alternative, Nagin (2005) Nagin, D. 2005. Group-based modeling of development, Cambridge: Harvard University Press. [Crossref] , [Google Scholar] suggested a semiparametric group-based approach (SPGA) which approximates an unknown, continuous distribution of individual trajectories with a mixture of group trajectories.

Prior simulations (Brame, Nagin, &; Wasserman, 2006 Brame, R., Nagin, D. and Wasserman, L. 2006. Exploring some analytical characteristics of finite mixture models.. Journal of Quantitative Criminology, 22: 31–59. [Crossref], [Web of Science ®] , [Google Scholar]; Nagin, 2005 Nagin, D. 2005. Group-based modeling of development, Cambridge: Harvard University Press. [Crossref] , [Google Scholar]) indicated that SPGA could generate nearly-unbiased estimates of means and variances of a nonnormal distribution of individual trajectories, as functions of group-trajectory estimates. However, these studies used few random effects—usually only a random intercept. Based on the analytical relationship between SPGA and adaptive quadrature, we hypothesized that SPGA's ability to approximate (a) random effect variances/covariances and (b) effects of time-invariant predictors of growth should deteriorate as the dimensionality of the random effects distribution increases. We expected this problem to be mitigated by correlations among the random effects (highly correlated random effects functioning as fewer dimensions) and sample size (larger N supporting more groups).

We tested these hypotheses via simulation, varying the number of random effects (1, 2, or 3), correlation among the random effects (0 or .6), and N (250, 500). Results indicated that, as the number of random effects increased, SPGA approximations remained acceptable for fixed effects, but became increasingly negatively biased for random effect variances. Whereas correlated random effects and larger N reduced this underestimation, correlated random effects sometimes distorted recovery of predictor effects. To illustrate this underestimation, Figure 1 depicts SPGA's approximation of the intercept variance from a three correlated random effect generating model (N < eqid1 > 500). These results suggest SPGA approximations are inadequate for the nonnormal, high-dimensional distributions of individual trajectories often seen in practice.

Abstract: Adequacy of Semiparametric Approximations for Growth Models with Nonnormal Random Effects

All authors

Sonya K. Sterba, Ruth E. Mathiowetz &; Daniel J. Bauer

https://doi.org/10.1080/00273170802640509

Published online:

18 December 2008

FIGURE 1 SPGA-approximated intercept variance from a three correlated random effect generating model. Notes. The dashed horizontal lines denote + 10% bias. The solid horizontal line denotes the population-generating parameter value; * denotes the best-BIC selected number of groups. The vertical bars denote 90% confidence intervals.     

8.

Generating Nonnormal Multivariate Data Using Copulas: Applications to SEM     

Patrick Mair  Albert Satorra  Peter M. Bentler 《Multivariate behavioral research》2013,48(4):547-565

This article develops a procedure based on copulas to simulate multivariate nonnormal data that satisfy a prespecified variance-covariance matrix. The covariance matrix used can comply with a specific moment structure form (e.g., a factor analysis or a general structural equation model). Thus, the method is particularly useful for Monte Carlo evaluation of structural equation models within the context of nonnormal data. The new procedure for nonnormal data simulation is theoretically described and also implemented in the widely used R environment. The quality of the method is assessed by Monte Carlo simulations. A 1-sample test on the observed covariance matrix based on the copula methodology is proposed. This new test for evaluating the quality of a simulation is defined through a particular structural model specification and is robust against normality violations.     

9.

The Problem with Having Two Watches: Assessment of Fit When RMSEA and CFI Disagree     

Keke Lai  Samuel B. Green 《Multivariate behavioral research》2016,51(2-3):220-239

The root mean square error of approximation (RMSEA) and the comparative fit index (CFI) are two widely applied indices to assess fit of structural equation models. Because these two indices are viewed positively by researchers, one might presume that their values would yield comparable qualitative assessments of model fit for any data set. When RMSEA and CFI offer different evaluations of model fit, we argue that researchers are likely to be confused and potentially make incorrect research conclusions. We derive the necessary as well as the sufficient conditions for inconsistent interpretations of these indices. We also study inconsistency in results for RMSEA and CFI at the sample level. Rather than indicating that the model is misspecified in a particular manner or that there are any flaws in the data, the two indices can disagree because (a) they evaluate, by design, the magnitude of the model's fit function value from different perspectives; (b) the cutoff values for these indices are arbitrary; and (c) the meaning of “good” fit and its relationship with fit indices are not well understood. In the context of inconsistent judgments of fit using RMSEA and CFI, we discuss the implications of using cutoff values to evaluate model fit in practice and to design SEM studies.     

10.

Abstract: Bayesian Inference for Growth Mixture Models With Nonignorable Missing Data     

Zhenqiu Laura Lu  Zhiyong Johnny Zhang  Gitta Lubke 《Multivariate behavioral research》2013,48(6):1028-1029

     

11.

Abstract: Evaluating Shared Parameter Mixture Models for Longitudinal Analysis With Missing Data     

Nisha C. Gottfredson 《Multivariate behavioral research》2013,48(6):1025-1026

     

12.

Promoting Forgiveness: A Comparison of Two Brief Psychoeducational Group Interventions With a Waiting-List Control     

MICHAEL E. MCCULLOUGH  EVERETT L. WORTHINGTON 《Counseling and values》1995,40(1):55-68

The authors studied the effects of 2 brief psychoeducational group interventions on participants' forgiveness for an offender and compared them with a waiting-list control. The Self-Enhancement group justified forgiveness because of its physical and psychological benefits to the forgiver. The Interpersonal group justified forgiveness because of its utility in restoring interpersonal relationships. Both groups led to decreased feelings of revenge, increased positive feelings toward the offender, and greater reports of conciliatory behavior. The Self-Enhancement group also increased affirming attributions toward the offender, decreased feelings of revenge, and increased conciliatory behavior more effectively than did the Interpersonal group.     

13.

Investigating the Effect of Missing Data on the Population CFI and RMSEA values     

Xijuan Zhang  Victoria Savalei 《Multivariate behavioral research》2018,53(1)

     

14.

A Simple Simulation Technique for Nonnormal Data with Prespecified Skewness,Kurtosis, and Covariance Matrix     

Njål Foldnes  Ulf Henning Olsson 《Multivariate behavioral research》2016,51(2-3):207-219

We present and investigate a simple way to generate nonnormal data using linear combinations of independent generator (IG) variables. The simulated data have prespecified univariate skewness and kurtosis and a given covariance matrix. In contrast to the widely used Vale-Maurelli (VM) transform, the obtained data are shown to have a non-Gaussian copula. We analytically obtain asymptotic robustness conditions for the IG distribution. We show empirically that popular test statistics in covariance analysis tend to reject true models more often under the IG transform than under the VM transform. This implies that overly optimistic evaluations of estimators and fit statistics in covariance structure analysis may be tempered by including the IG transform for nonnormal data generation. We provide an implementation of the IG transform in the R environment.     

15.

Abstract: Local Influence and Robust Methods for Mediation Models     

Jiyun Zu  Ke-Hai Yuan 《Multivariate behavioral research》2013,48(4)

Mediation analysis investigates how certain variables mediate the effect of predictors on outcome variables. Existing studies of mediation models have been limited to normal theory maximum likelihood (ML) or least squares with normally distributed data. Because real data in the social and behavioral sciences are seldom normally distributed and often contain outliers, classical methods can result in biased and inefficient estimates, which lead to inaccurate or unreliable test of the meditated effect. The authors propose two approaches for better mediation analysis. One is to identify cases that strongly affect test results of mediation using local influence methods and robust methods. The other is to use robust methods for parameter estimation, and then test the mediated effect based on the robust estimates. Analytic details of both local influence and robust methods particular for mediation models were provided and one real data example was given. We first used local influence and robust methods to identify influential cases. Then, for the original data and the data with the identified influential cases removed, the mediated effect was tested using two estimation methods: normal theory ML and the robust method, crossing two tests of mediation: the Sobel (1982) Sobel, M. E. 1982. “Asymptotic confidence intervals for indirect effects in structural equation models”. In Sociological methodology, Edited by: Leinhardt, S. 290–312. Washington, DC: American Sociological Association. [Crossref] , [Google Scholar] test using information-based standard error (z _I ) and sandwich-type standard error (z _SW ). Results show that local influence and robust methods rank the influence of cases similarly, while the robust method is more objective. The widely used z _I statistic is inflated when the distribution is heavy-tailed. Compared to normal theory ML, the robust method provides estimates with smaller standard errors and more reliable test.     

16.

Abstract: A Comparison of the Factor Mixture Model and Taxometric Procedures     

Stephen Tueller  Gitta Lubke 《Multivariate behavioral research》2013,48(4)

Taxometric procedures and the Factor Mixture Model (FMM) have a complimentary set of strengths and weaknesses. Both approaches purport to detect evidence of a latent class structure. Taxometric procedures, popular in psychiatric and psychopathology literature, make no assumptions beyond those needed to compute means and covariances. However, Taxometric procedures assume that observed items are uncorrelated within a class or taxon. This assumption is violated when there are individual differences in the trait underlying the items (i.e., severity differences within class). FMMs can model within-class covariance structures ranging from local independence to multidimensional within-class factor models and permits the specification of more than two classes. FMMs typically rely on normality assumptions for within-class factors and error terms. FMMs are highly parameterized and susceptible to misspecifications of the within-class covariance structure.

The current study compared the Taxometric procedures MAXEIG and the Base-Rate Classification Technique to the FMM in their respective abilities to (1) correctly detect the two-class structure in simulated data, and to (2) correctly assign subjects to classes. Two class data were simulated under conditions of balanced and imbalanced relative class size, high and low class separation, and 1-factor and 2-factor within-class covariance structures. For the 2-factor data, simple and cross-loaded factor loading structures, and positive and negative factor correlations were considered. For the FMM, both correct and incorrect within-class factor structures were fit to the data.

FMMs generally outperformed Taxometric procedures in terms of both class detection and in assigning subjects to classes. Imbalanced relative class size (e.g., a small minority class and a large majority class) negatively impacted both FMM and Taxometric performance while low class separation was much more problematic for Taxometric procedures than the FMM. Comparisons of alterative FMMs based on information criteria generally resulted in correct model choice but deteriorated when small class separation was combined with imbalanced relative class size.     

17.

The Performance of Methods to Test Upper-Level Mediation in the Presence of Nonnormal Data     

Keenan A. Pituch  Laura M. Stapleton 《Multivariate behavioral research》2013,48(2):237-267

A Monte Carlo study compared the statistical performance of standard and robust multilevel mediation analysis methods to test indirect effects for a cluster randomized experimental design under various departures from normality. The performance of these methods was examined for an upper-level mediation process, where the indirect effect is a fixed effect and a group-implemented treatment is hypothesized to impact a person-level outcome via a person-level mediator. Two methods—the bias-corrected parametric percentile bootstrap and the empirical-M test—had the best overall performance. Methods designed for nonnormal score distributions exhibited elevated Type I error rates and poorer confidence interval coverage under some conditions. Although preliminary, the findings suggest that new mediation analysis methods may provide for robust tests of indirect effects.     

18.

Abstract: A Comparison of Classification Techniques for Diagnosis in an Alzheimer's Study     

Jodi M. Casabianca  David V. Budescu  Denise C. Fyffe  Charles Lewis 《Multivariate behavioral research》2013,48(6)

     

19.

Abstract: Modeling Single-Case Designs With Generalized Additive Models     

Kristynn J. Sullivan  William R. Shadish 《Multivariate behavioral research》2013,48(1)

     

20.

Abstract: Study of Pupil Creativity in Two Institutional Settings     

ARUN KUMAR GUPTA 《创造性行为杂志》1978,12(3):218-218

     

Generating Nonnormal Multivariate Data Using Copulas: Applications to SEM

This article develops a procedure based on copulas to simulate multivariate nonnormal data that satisfy a prespecified variance-covariance matrix. The covariance matrix used can comply with a specific moment structure form (e.g., a factor analysis or a general structural equation model). Thus, the method is particularly useful for Monte Carlo evaluation of structural equation models within the context of nonnormal data. The new procedure for nonnormal data simulation is theoretically described and also implemented in the widely used R environment. The quality of the method is assessed by Monte Carlo simulations. A 1-sample test on the observed covariance matrix based on the copula methodology is proposed. This new test for evaluating the quality of a simulation is defined through a particular structural model specification and is robust against normality violations.     

The Problem with Having Two Watches: Assessment of Fit When RMSEA and CFI Disagree

The root mean square error of approximation (RMSEA) and the comparative fit index (CFI) are two widely applied indices to assess fit of structural equation models. Because these two indices are viewed positively by researchers, one might presume that their values would yield comparable qualitative assessments of model fit for any data set. When RMSEA and CFI offer different evaluations of model fit, we argue that researchers are likely to be confused and potentially make incorrect research conclusions. We derive the necessary as well as the sufficient conditions for inconsistent interpretations of these indices. We also study inconsistency in results for RMSEA and CFI at the sample level. Rather than indicating that the model is misspecified in a particular manner or that there are any flaws in the data, the two indices can disagree because (a) they evaluate, by design, the magnitude of the model's fit function value from different perspectives; (b) the cutoff values for these indices are arbitrary; and (c) the meaning of “good” fit and its relationship with fit indices are not well understood. In the context of inconsistent judgments of fit using RMSEA and CFI, we discuss the implications of using cutoff values to evaluate model fit in practice and to design SEM studies.     

Promoting Forgiveness: A Comparison of Two Brief Psychoeducational Group Interventions With a Waiting-List Control

The authors studied the effects of 2 brief psychoeducational group interventions on participants' forgiveness for an offender and compared them with a waiting-list control. The Self-Enhancement group justified forgiveness because of its physical and psychological benefits to the forgiver. The Interpersonal group justified forgiveness because of its utility in restoring interpersonal relationships. Both groups led to decreased feelings of revenge, increased positive feelings toward the offender, and greater reports of conciliatory behavior. The Self-Enhancement group also increased affirming attributions toward the offender, decreased feelings of revenge, and increased conciliatory behavior more effectively than did the Interpersonal group.     

A Simple Simulation Technique for Nonnormal Data with Prespecified Skewness,Kurtosis, and Covariance Matrix

We present and investigate a simple way to generate nonnormal data using linear combinations of independent generator (IG) variables. The simulated data have prespecified univariate skewness and kurtosis and a given covariance matrix. In contrast to the widely used Vale-Maurelli (VM) transform, the obtained data are shown to have a non-Gaussian copula. We analytically obtain asymptotic robustness conditions for the IG distribution. We show empirically that popular test statistics in covariance analysis tend to reject true models more often under the IG transform than under the VM transform. This implies that overly optimistic evaluations of estimators and fit statistics in covariance structure analysis may be tempered by including the IG transform for nonnormal data generation. We provide an implementation of the IG transform in the R environment.     

Abstract: Local Influence and Robust Methods for Mediation Models

Mediation analysis investigates how certain variables mediate the effect of predictors on outcome variables. Existing studies of mediation models have been limited to normal theory maximum likelihood (ML) or least squares with normally distributed data. Because real data in the social and behavioral sciences are seldom normally distributed and often contain outliers, classical methods can result in biased and inefficient estimates, which lead to inaccurate or unreliable test of the meditated effect. The authors propose two approaches for better mediation analysis. One is to identify cases that strongly affect test results of mediation using local influence methods and robust methods. The other is to use robust methods for parameter estimation, and then test the mediated effect based on the robust estimates. Analytic details of both local influence and robust methods particular for mediation models were provided and one real data example was given. We first used local influence and robust methods to identify influential cases. Then, for the original data and the data with the identified influential cases removed, the mediated effect was tested using two estimation methods: normal theory ML and the robust method, crossing two tests of mediation: the Sobel (1982) Sobel, M. E. 1982. “Asymptotic confidence intervals for indirect effects in structural equation models”. In Sociological methodology, Edited by: Leinhardt, S. 290–312. Washington, DC: American Sociological Association. [Crossref] , [Google Scholar] test using information-based standard error (z _I ) and sandwich-type standard error (z _SW ). Results show that local influence and robust methods rank the influence of cases similarly, while the robust method is more objective. The widely used z _I statistic is inflated when the distribution is heavy-tailed. Compared to normal theory ML, the robust method provides estimates with smaller standard errors and more reliable test.     

Abstract: A Comparison of the Factor Mixture Model and Taxometric Procedures

Taxometric procedures and the Factor Mixture Model (FMM) have a complimentary set of strengths and weaknesses. Both approaches purport to detect evidence of a latent class structure. Taxometric procedures, popular in psychiatric and psychopathology literature, make no assumptions beyond those needed to compute means and covariances. However, Taxometric procedures assume that observed items are uncorrelated within a class or taxon. This assumption is violated when there are individual differences in the trait underlying the items (i.e., severity differences within class). FMMs can model within-class covariance structures ranging from local independence to multidimensional within-class factor models and permits the specification of more than two classes. FMMs typically rely on normality assumptions for within-class factors and error terms. FMMs are highly parameterized and susceptible to misspecifications of the within-class covariance structure.

The current study compared the Taxometric procedures MAXEIG and the Base-Rate Classification Technique to the FMM in their respective abilities to (1) correctly detect the two-class structure in simulated data, and to (2) correctly assign subjects to classes. Two class data were simulated under conditions of balanced and imbalanced relative class size, high and low class separation, and 1-factor and 2-factor within-class covariance structures. For the 2-factor data, simple and cross-loaded factor loading structures, and positive and negative factor correlations were considered. For the FMM, both correct and incorrect within-class factor structures were fit to the data.

FMMs generally outperformed Taxometric procedures in terms of both class detection and in assigning subjects to classes. Imbalanced relative class size (e.g., a small minority class and a large majority class) negatively impacted both FMM and Taxometric performance while low class separation was much more problematic for Taxometric procedures than the FMM. Comparisons of alterative FMMs based on information criteria generally resulted in correct model choice but deteriorated when small class separation was combined with imbalanced relative class size.     

The Performance of Methods to Test Upper-Level Mediation in the Presence of Nonnormal Data

A Monte Carlo study compared the statistical performance of standard and robust multilevel mediation analysis methods to test indirect effects for a cluster randomized experimental design under various departures from normality. The performance of these methods was examined for an upper-level mediation process, where the indirect effect is a fixed effect and a group-implemented treatment is hypothesized to impact a person-level outcome via a person-level mediator. Two methods—the bias-corrected parametric percentile bootstrap and the empirical-M test—had the best overall performance. Methods designed for nonnormal score distributions exhibited elevated Type I error rates and poorer confidence interval coverage under some conditions. Although preliminary, the findings suggest that new mediation analysis methods may provide for robust tests of indirect effects.