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Mediation analysis plays an important role in understanding causal processes in social and behavioral sciences. While path analysis with composite scores was criticized to yield biased parameter estimates when variables contain measurement errors, recent literature has pointed out that the population values of parameters of latent-variable models are determined by the subjectively assigned scales of the latent variables. Thus, conclusions in existing studies comparing structural equation modeling (SEM) and path analysis with weighted composites (PAWC) on the accuracy and precision of the estimates of the indirect effect in mediation analysis have little validity. Instead of comparing the size on estimates of the indirect effect between SEM and PAWC, this article compares parameter estimates by signal-to-noise ratio (SNR), which does not depend on the metrics of the latent variables once the anchors of the latent variables are determined. Results show that PAWC yields greater SNR than SEM in estimating and testing the indirect effect even when measurement errors exist. In particular, path analysis via factor scores almost always yields greater SNRs than SEM. Mediation analysis with equally weighted composites (EWCs) also more likely yields greater SNRs than SEM. Consequently, PAWC is statistically more efficient and more powerful than SEM in conducting mediation analysis in empirical research. The article also further studies conditions that cause SEM to have smaller SNRs, and results indicate that the advantage of PAWC becomes more obvious when there is a strong relationship between the predictor and the mediator, whereas the size of the prediction error in the mediator adversely affects the performance of the PAWC methodology. Results of a real-data example also support the conclusions. [ABSTRACT FROM AUTHOR]

This study applies equivalence testing methods on a representative sample of published empirical structural equation modeling studies in the psychological sciences and assesses the extent to which reported fit results are replicated when compared to those obtained via traditional methods. Results of 382 models from a sample of 242 articles published in 5 top ranked journals from the general field of Developmental Psychology were reported. The results indicated that a sizeable number of models designated in the original studies as 'good' were in fact not reproducible when examined through an equivalence testing lens. These questionable models displayed substantial discrepancy with the data and should not have qualified for further consideration because they were simply not plausible models. Implications of the results and suggestions for best modeling practices within the psychological sciences are discussed. [ABSTRACT FROM AUTHOR]

Structural equation modeling (SEM) has been deemed as a proper method when variables contain measurement errors. In contrast, path analysis with composite scores is preferred for prediction and diagnosis of individuals. While path analysis with composite scores has been criticized for yielding biased parameter estimates, recent literature pointed out that the population values of parameters in a latent-variable model depend on artificially assigned scales. Consequently, bias in parameter estimates is not a well-grounded concept for models involving latent constructs. This article compares path analysis with composite scores against SEM with respect to effect size and statistical power in testing the significance of the path coefficients, via the z- or t-statistics. The data come from many sources with various models that are substantively determined. Results show that SEM is not as powerful as path analysis even with equally weighted composites. However, path analysis with Bartlett-factor scores and the partial least-squares approach to SEM perform the best with respect to effect size and power. [ABSTRACT FROM AUTHOR]

This article proposes a two-level moderated mediation (2moME) model with single level data, and develops measures to quantify the moderated mediation (moME) effect sizes for both the conventional moME model and the 2moME model. A Bayesian approach is developed to estimate and test moME effects and the corresponding effect sizes (ES). Monte Carlo results indicate that (1) the 2moME model yields more accurate estimates of the parameters than the conventional moME model; (2) the 95% credibility interval following the 2moME model covers the moME effects and the ESs more accurately than that following the conventional moME model; and (3) statistical tests for the existence of the moME effects with the 2moME model are more reliable in controlling type I errors than those with the conventional moME model, especially under heteroscedasticity conditions. In addition, the developed measures of ES are more interpretable, and directly answer the questions regarding the extent to which a moderator can account for the change of the mediation effect between the predictor and the outcome variable through the mediator variable. An empirical example illustrates the application of the 2moME model and the ES measures. [ABSTRACT FROM AUTHOR]

Differential item functioning (DIF) analysis is an important step in establishing the validity of measurements. Most traditional methods for DIF analysis use an item-by-item strategy via anchor items that are assumed DIF-free. If anchor items are flawed, these methods will yield misleading results due to biased scales. In this article, based on the fact that the item's relative change of difficulty difference (RCD) does not depend on the mean ability of individual groups, a new DIF detection method (RCD-DIF) is proposed by comparing the observed differences against those with simulated data that are known DIF-free. The RCD-DIF method consists of a D-QQ (quantile quantile) plot that permits the identification of internal references points (similar to anchor items), a RCD-QQ plot that facilitates visual examination of DIF, and a RCD graphical test that synchronizes DIF analysis at the test level with that at the item level via confidence intervals on individual items. The RCD procedure visually reveals the overall pattern of DIF in the test and the size of DIF for each item and is expected to work properly even when the majority of the items possess DIF and the DIF pattern is unbalanced. Results of two simulation studies indicate that the RCD graphical test has Type I error rate comparable to those of existing methods but with greater power. [ABSTRACT FROM AUTHOR]

Unless data are missing completely at random (MCAR), proper methodology is crucial for the analysis of incomplete data. Consequently, methods for effectively testing the MCAR mechanism become important, and procedures were developed via testing the homogeneity of means and variances-covariances across the observed patterns (e.g., Kim & Bentler in Psychometrika 67:609-624, 2002; Little in J Am Stat Assoc 83:1198-1202, 1988). The current article shows that the population counterparts of the sample means and covariances of a given pattern of the observed data depend on the underlying structure that generates the data, and the normal-distribution-based maximum likelihood estimates for different patterns of the observed sample can converge to the same values even when data are missing at random or missing not at random, although the values may not equal those of the underlying population distribution. The results imply that statistics developed for testing the homogeneity of means and covariances cannot be safely used for testing the MCAR mechanism even when the population distribution is multivariate normal. [ABSTRACT FROM AUTHOR]

In structural equation modeling (SEM), parameter estimates are typically computed by the Fisher-scoring algorithm, which often has difficulty in obtaining converged solutions. Even for simulated data with a correctly specified model, non-converged replications have been repeatedly reported in the literature. In particular, in Monte Carlo studies it has been found that larger factor loadings or smaller error variances in a confirmatory factor model correspond to a higher rate of convergence. However, studies of a ridge method in SEM indicate that adding a diagonal matrix to the sample covariance matrix also increases the rate of convergence for the Fisher-scoring algorithm. This article addresses these two seemingly contradictory phenomena. Using statistical and numerical analyses, the article clarifies why both approaches increase the rate of convergence in SEM. Monte Carlo results confirm the analytical results. Recommendations are provided on how to increase both the speed and rate of convergence in parameter estimation. [ABSTRACT FROM AUTHOR]

The conventional setup for multi-group structural equation modeling requires a stringent condition of cross-group equality of intercepts before mean comparison with latent variables can be conducted. This article proposes a new setup that allows mean comparison without the need to estimate any mean structural model. By projecting the observed sample means onto the space of the common scores and the space orthogonal to that of the common scores, the new setup allows identifying and estimating the means of the common and specific factors, although, without replicate measures, variances of specific factors cannot be distinguished from those of measurement errors. Under the new setup, testing cross-group mean differences of the common scores is done independently from that of the specific factors. Such independent testing eliminates the requirement for cross-group equality of intercepts by the conventional setup in order to test cross-group equality of means of latent variables using chi-square-difference statistics. The most appealing piece of the new setup is a validity index for mean differences, defined as the percentage of the sum of the squared observed mean differences that is due to that of the mean differences of the common scores. By analyzing real data with two groups, the new setup is shown to offer more information than what is obtained under the conventional setup. [ABSTRACT FROM AUTHOR]

Means and covariance/dispersion matrix are the building blocks for many statistical analyses. By naturally extending the score functions based on a multivariate $$t$$ -distribution to estimating equations, this article defines a class of M-estimators of means and dispersion matrix for samples with missing data. An expectation-robust (ER) algorithm solving the estimating equations is obtained. The obtained relationship between the ER algorithm and the corresponding estimating equations allows us to obtain consistent standard errors when robust means and dispersion matrix are further analyzed. Estimating equations corresponding to existing ER algorithms for computing M- and S-estimators are also identified. Monte Carlo results show that robust methods outperform the normal-distribution-based maximum likelihood when the population distribution has heavy tails or when data are contaminated. Applications of the results to robust analysis of linear regression and growth curve models are discussed. [ABSTRACT FROM AUTHOR]

Survey data typically contain many variables. Structural equation modeling (SEM) is commonly used in analyzing such data. The most widely used statistic for evaluating the adequacy of a SEM model is T, a slight modification to the likelihood ratio statistic. Under normality assumption, T approximately follows a chi-square distribution when the number of observations ( N) is large and the number of items or variables ( p) is small. However, in practice, p can be rather large while N is always limited due to not having enough participants. Even with a relatively large N, empirical results show that T rejects the correct model too often when p is not too small. Various corrections to T have been proposed, but they are mostly heuristic. Following the principle of the Bartlett correction, this paper proposes an empirical approach to correct T so that the mean of the resulting statistic approximately equals the degrees of freedom of the nominal chi-square distribution. Results show that empirically corrected statistics follow the nominal chi-square distribution much more closely than previously proposed corrections to T, and they control type I errors reasonably well whenever N≥max(50,2 p). The formulations of the empirically corrected statistics are further used to predict type I errors of T as reported in the literature, and they perform well. [ABSTRACT FROM AUTHOR]

Moderation analysis is widely used in social and behavioral research. The most commonly used model for moderation analysis is moderated multiple regression (MMR) in which the explanatory variables of the regression model include product terms, and the model is typically estimated by least squares (LS). This paper argues for a two-level regression model in which the regression coefficients of a criterion variable on predictors are further regressed on moderator variables. An algorithm for estimating the parameters of the two-level model by normal-distribution-based maximum likelihood (NML) is developed. Formulas for the standard errors (SEs) of the parameter estimates are provided and studied. Results indicate that, when heteroscedasticity exists, NML with the two-level model gives more efficient and more accurate parameter estimates than the LS analysis of the MMR model. When error variances are homoscedastic, NML with the two-level model leads to essentially the same results as LS with the MMR model. Most importantly, the two-level regression model permits estimating the percentage of variance of each regression coefficient that is due to moderator variables. When applied to data from General Social Surveys 1991, NML with the two-level model identified a significant moderation effect of race on the regression of job prestige on years of education while LS with the MMR model did not. An R package is also developed and documented to facilitate the application of the two-level model. [ABSTRACT FROM AUTHOR]

The paper clarifies the relationship among several information matrices for the maximum likelihood estimates (MLEs) of item parameters. It shows that the process of calculating the observed information matrix also generates a related matrix that is the middle piece of a sandwich-type covariance matrix. Monte Carlo results indicate that standard errors (SEs) based on the observed information matrix are robust to many, but not all, conditions of model/distribution misspecifications. SEs based on the sandwich-type covariance matrix perform most consistently across conditions. Results also suggest that SEs based on other matrices are either not consistent or perform not as robust as those based on the sandwich-type covariance matrix or the observed information matrix. [ABSTRACT FROM AUTHOR]

The paper develops a two-stage robust procedure for structural equation modeling (SEM) and an R package rsem to facilitate the use of the procedure by applied researchers. In the first stage, M-estimates of the saturated mean vector and covariance matrix of all variables are obtained. Those corresponding to the substantive variables are then fitted to the structural model in the second stage. A sandwich-type covariance matrix is used to obtain consistent standard errors (SE) of the structural parameter estimates. Rescaled, adjusted as well as corrected and F-statistics are proposed for overall model evaluation. Using R and EQS, the R package rsem combines the two stages and generates all the test statistics and consistent SEs. Following the robust analysis, multiple model fit indices and standardized solutions are provided in the corresponding output of EQS. An example with open/closed book examination data illustrates the proper use of the package. The method is further applied to the analysis of a data set from the National Longitudinal Survey of Youth 1997 cohort, and results show that the developed procedure not only gives a better endorsement of the substantive models but also yields estimates with uniformly smaller standard errors than the normal-distribution-based maximum likelihood. [ABSTRACT FROM AUTHOR]

The paper obtains consistent standard errors (SE) and biases of order O(1/ n) for the sample standardized regression coefficients with both random and given predictors. Analytical results indicate that the formulas for SEs given in popular text books are consistent only when the population value of the regression coefficient is zero. The sample standardized regression coefficients are also biased in general, although it should not be a concern in practice when the sample size is not too small. Monte Carlo results imply that, for both standardized and unstandardized sample regression coefficients, SE estimates based on asymptotics tend to under-predict the empirical ones at smaller sample sizes. [ABSTRACT FROM AUTHOR]

Indefinite symmetric matrices that are estimates of positive-definite population matrices occur in a variety of contexts such as correlation matrices computed from pairwise present missing data and multinormal based methods for discretized variables. This note describes a methodology for scaling selected off-diagonal rows and columns of such a matrix to achieve positive definiteness. As a contrast to recently developed ridge procedures, the proposed method does not need variables to contain measurement errors. When minimum trace factor analysis is used to implement the theory, only correlations that are associated with Heywood cases are shrunk. [ABSTRACT FROM AUTHOR]

In a variety of statistical problems the estimate Θn of a parameter Θ is defined as the root of a generalized estimating equation Gn(Θnγn)=0 where γn is an estimate of a nuisance parameter γ. We give sufficient conditions for the asymptotic normality of #x0398;n defined in this way and derive their asymptotic distribution. A circumstance under which the asymptotic distribution of #x0398;n will not be influenced by that of γn) is noted. As an example, we consider a covariance structure analysis in which both the population mean and the population fourth-order moment are nuisance parameters. Applications to pseudo maximum likelihood, generalized least squares with estimated weights, and M-estimation with an estimated scale parameter are discussed briefly. [ABSTRACT FROM AUTHOR]

Robust schemes in regression are adapted to mean and covariance structure analysis, providing an iteratively reweighted least squares approach to robust structural equation modeling. Each case is properly weighted according to its distance, based on first and second order moments, from the structural model. A simple weighting function is adopted because of its flexibility with changing dimensions. The weight matrix is obtained from an adaptive way of using residuals. Test statistic and standard error estimators are given, based on iteratively reweighted least squares. The method reduces to a standard distribution-free methodology if all cases are equally weighted. Examples demonstrate the value of the robust procedure. [ABSTRACT FROM AUTHOR]

Current practice in factor analysis typically involves analysis of correlation rather than covariance matrices. We study whether the standard z-statistic that evaluates whether a factor loading is statistically necessary is correctly applied in such situations and more generally when the variables being analyzed are arbitrarily rescaled. Effects of rescaling on estimated standard errors of factor loading estimates, and the consequent effect on z-statistics, are studied in three variants of the classical exploratory factor model under canonical, raw varimax, and normal varimax solutions. For models with analytical solutions we find that some of the standard errors as well as their estimates are scale equivariant, while others are invariant. For a model in which an analytical solution does not exist, we use an example to illustrate that neither the factor loading estimates nor the standard error estimates possess scale equivariance or invariance, implying that different conclusions could be obtained with different scalings. Together with the prior findings on parameter estimates, these results provide new guidance for a key statistical aspect of factor analysis. [ABSTRACT FROM AUTHOR]