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Measures of explained variance, ?R2 and f, 2 are routinely used to evaluate the size of moderation effects. However, they suffer from several drawbacks: (a) Not all the variance components of the outcome variable Y are related to the effect of moderation, and so an effect size with the total variance of Y as the denominator cannot accurately characterize the moderation effect; (b) moderation and interaction are conflated; and (c) the assumption of homoscedasticity might be violated when moderation exists. By arguing that measures for the size of moderation effect should be based on the variance of the outcome Y via the predictor variable X (i.e., X-Y), this article develops a new conceptualization of moderation effects that leads to 2 ways of defining new measures of moderation effects size. One is by using regression models that include the moderator, the predictor, and the product term sequentially. The other is based on a variance decomposition of the outcome variable Y. These new effect size measures effectively differentiate the role of the predictor variable from that of the moderator variable. Two empirical examples are provided to contrast the new measures against the traditional? R2 and f2, and to illustrate the applications of the new ones. R code is also provided for researchers to compute the new effect size measures. [ABSTRACT FROM AUTHOR]

issing values that are missing not at random (MNAR) can result from a variety of missingness processes. However, two fundamental subtypes of MNAR values can be obtained from the definition of the MNAR mechanism itself. The distinction between them deserves consideration because they have characteristic differences in how they distort relationships in the data. This has implications for the validity of statistical results and generalizability of methodological findings that are based on data (empirical or generated) with MNAR values. However, these MNAR subtypes have largely gone unnoticed by the literature. As few studies have considered both subtypes, their relevance to methodological and substantive research has been overlooked. This article systematically introduces the two MNAR subtypes and gives them descriptive names. A case study demonstrates they are mechanically distinct from each other and from other missing-data mechanisms. Applied examples are given to help researchers conceptually identify MNAR subtypes in real data. Methods are provided to generate missing values from both subtypes in simulation studies. Simulation studies for regression and growth curve modeling contexts show MNAR subtypes consistently differ in the severity of their impact on statistical inference. This behavior is examined in light of how relationships in the data become characteristically distorted. The contents of this article are intended to provide a foundation and tools for organized consideration of MNAR subtypes. [ABSTRACT FROM AUTHOR]

Motivated by the need to effectively evaluate the quality of the mean structure in growth curve modeling (GCM), this article proposes to separately evaluate the goodness of fit of the mean structure from that of the covariance structure. Several fit indices are defined, and rationales are discussed. Particular considerations are given for polynomial and piecewise polynomial models because fit indices for them are valid regardless of the underlying population distribution of the data. Examples indicate that the newly defined fit indices remove the confounding issues with indices jointly evaluating mean and covariance structure models and provide much more reliable evaluation of the mean structure in GCM. Examples also show that pseudo R-squares and concordance correlations are unable to reflect the goodness of mean structures in GCM. Proper use of the fit indices for the purpose of model diagnostics is discussed. A window-based program, WebSEM, is also introduced for easily computing these fit indices by applied researchers. [ABSTRACT FROM AUTHOR]

Multigroup structural equation modeling (SEM) plays a key role in studying measurement invariance and in group comparison. When population covariance matrices are deemed not equal across groups, the next step to substantiate measurement invariance is to see whether the sample covariance matrices in all the groups can be adequately fitted by the same factor model, called configural invariance. After configural invariance is established, cross-group equalities of factor loadings, error variances, and factor variances-covariances are then examined in sequence. With mean structures, cross-group equalities of intercepts and factor means are also examined. The established rule is that if the statistic at the current model is not significant at the level of .05, one then moves on to testing the next more restricted model using a chi-square-difference statistic. This article argues that such an established rule is unable to control either Type I or Type II errors. Analysis, an example, and Monte Carlo results show why and how chi-square-difference tests are easily misused. The fundamental issue is that chi-square-difference tests are developed under the assumption that the base model is sufficiently close to the population, and a nonsignificant chi-square statistic tells little about how good the model is. To overcome this issue, this article further proposes that null hypothesis testing in multigroup SEM be replaced by equivalence testing, which allows researchers to effectively control the size of misspecification before moving on to testing a more restricted model. R code is also provided to facilitate the applications of equivalence testing for multigroup SEM. [ABSTRACT FROM AUTHOR]

Normal-distribution-based maximum likelihood (NML) is the most widely used method in structural equation modeling (SEM), although practical data tend to be nonnormally distributed. The effect of nonnormally distributed data or data contamination on the normal-distribution-based likelihood ratio (LR) statistic is well understood due to many analytical and empirical studies. In SEM, fit indices are used as widely as the LR statistic. In addition to NML, robust procedures have been developed for more efficient and less biased parameter estimates with practical data. This article studies the effect of outliers and leverage observations on fit indices following NML and two robust methods. Analysis and empirical results indicate that good leverage observations following NML and one of the robust methods lead most fit indices to give more support to the substantive model. While outliers tend to make a good model superficially bad according to many fit indices following NML, they have little effect on those following the two robust procedures. Implications of the results to data analysis are discussed, and recommendations are provided regarding the use of estimation methods and interpretation of fit indices. [ABSTRACT FROM AUTHOR]

This article introduces two simple scatter plots for model diagnosis in structural equation modeling. One plot contrasts a residual-based M-distance of the structural model with the M-distance for the factor score. It contains information on outliers, good leverage observations, bad leverage observations, and normal cases. The other plot contrasts the residual-based M-distance with the quantile of a chi distribution. It allows the researcher to visually identify clusters of potential outliers. The article further studies the effect of the potential outliers on the overall model evaluation when they are removed according to the order of the clusters exhibited in the plot. Suggestions are provided on determining the outlier status of outstanding cases in real data analysis. Recommendations are also made on the choice of robust methods and maximum likelihood following outlier removal. [ABSTRACT FROM AUTHOR]

This paper studies an information criterion for selecting covariance structure models using the generalized least squares (GLS) procedure. A risk assessed by the predictive GLS discrepancy function is introduced and used to determine the quality of a model. By correcting the biases in the sample GLS discrepancy function, four GLS discrepancy based information criteria are proposed. Monte Carlo results illustrate the merits of each criterion in model selection and in minimizing the risk. [ABSTRACT FROM AUTHOR]

Various fit indices exist in structural equation models. Most of these indices are related to the noncentrality parameter (NCP) of the chi-square distribution that the involved test statistic is implicitly assumed to follow. Existing literature suggests that few statistics can be well approximated by chi-square distributions. The meaning of the NCP is not clear when the behavior of the statistic cannot he described by a chi- square distribution. In this paper we define a new measure of model misfit (MMM) as the difference between the expected values of a statistic under the alternative and null hypotheses This definition does not need to assume that the population covariance matrix is in the vicinity of the proposed model, nor does it need for the test statistic to follow any distribution of a known form. The MMM does not necessarily equal the discrepancy between the model and the population covariance matrix as has been assumed in existing literature. Bootstrap approaches to estimating the MMM and a related quantity are developed. An algorithm for obtaining bootstrap confidence intervals of the MMM is constructed. Examples with practical data sets contrast several measures of model misfit. The quantile-quantile plot is used to illustrate the unrealistic nature of chi-square distribution assumptions under either the null or an alternative hypothesis in practice. [ABSTRACT FROM AUTHOR]