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It is shown that the maximum likelihood estimator of the widely used omega coefficient for reliability of multicomponent measuring instruments converges almost surely to the population reliability coefficient for normal congeneric measures with uncorrelated errors as sample size increases indefinitely. This strong consistency implies convergence in probability (consistency) as well as in distribution for the omega estimator. Strong consistency is also demonstrated for the maximal reliability estimator associated with the optimal linear combination of the instrument components. The findings of this note add (i) to the recommendation to use in the general normality case the omega estimator in empirical research, (ii) to the critical literature on the popular coefficient alpha then, and (iii) to the literature on the properties of the optimal linear combination of observed measures and the maximal reliability estimator. [ABSTRACT FROM AUTHOR]

In many applications of multilevel modeling, group-level (L2) variables for assessing group-level effects are generated by aggregating variables from a lower level (L1). However, the observed group mean might not be a reliable measure of the unobserved true group mean. In this article, we propose a Bayesian approach for estimating a multilevel latent contextual model that corrects for measurement error and sampling error (i.e., sampling only a small number of L1 units from a L2 unit) when estimating group-level effects of aggregated L1 variables. Two simulation studies were conducted to compare the Bayesian approach with the maximum likelihood approach implemented in Mplus. The Bayesian approach showed fewer estimation problems (e.g., inadmissible solutions) and more accurate estimates of the group-level effect than the maximum likelihood approach under problematic conditions (i.e., small number of groups, predictor variable with a small intraclass correlation). An application from educational psychology is used to illustrate the different estimation approaches. [ABSTRACT FROM PUBLISHER]

Multiple ratings are becoming increasingly popular for the assessment of a wide range of behaviors. Teaching evaluation designs and 360-degree feedback often use multiple raters alongside self-ratings. Eid and colleagues (2008) proposed a multilevel structural equation model for the analysis of data stemming from such designs that assumes that raters stem from independent populations of raters. However, it is quite common for raters to rate multiple targets, thus implying a cross-classified data structure. A simulation study was conducted to assess the effects of this rater nonindependence on parameter and standard error estimates in multilevel structural equation models. Results show parameter estimation biases to be small, whereas standard errors pertaining to the Level 1 covariance matrices as well as the mean structure on Level 2 are underestimated. [ABSTRACT FROM AUTHOR]

Nonrecursive structural equation models generally take the form of feedback loops, involving 2 latent variables that are connected by 2 unidirectional paths, 1 starting with each variable and terminating in the other variable. Nonrecursive models belong to a larger class of path models that require the use of instrumental variables (IVs) to achieve model identification. Prior research has focused on SEM parameter estimation with IVs when indicators were continuous and normally distributed. Much less is known about how estimators function in the presence of categorical indicators, which are commonly used in the social sciences, such as with cognitive and affective instruments. In this study, there was specific interest in comparing the 2-stage least squares (2SLS) estimator and its categorical variant to other recommended estimators. This study compares the performance of several estimation approaches for fitting structural equation models with categorical indicator variables when IVs are necessary to obtain proper model estimates. Across conditions, 1 extension of the nonlinear 2SLS (N2SLS) approach, the nonlinear 3-stage least squares (N3SLS), which accounts for correlated errors among regressors within each model (as does the N2SLS), as well as correlations of errors across models, which N2SLS does not, appears to work the best among methods compared. [ABSTRACT FROM AUTHOR]

Determining an appropriate sample size for use in latent variable modeling techniques has presented ongoing challenges to researchers. In particular, small sample sizes are known to present concerns over sampling error for the variances and covariances on which model estimation is based, as well as for fit indexes and convergence failures. The literature on the topic has focused on conducting power analyses as well as identifying rules of thumb for deciding an appropriate sample size. Often the advice involves an assumption that sample size requirement is moderated by aspects of the model in question. In this study, an effort was undertaken to extend the findings of Gagné and Hancock (2006) on measurement model quality and solution propriety to a broader set of confirmatory factor analysis models. As well, we examined whether Herzog, Boomsma, and Reinecke's (2007) findings for the Swain correction to the χ2statistic for large models would generalize to models in our study. Our findings suggest that Gagné and Hancock's approach extends to large models with surprisingly little increase in sample size requirements and that the Swain correction to χ2performs fairly well. We argue that likely rejection or model fit should be taken into account when determining sample size requirements and therefore, provide an updated table of minimum sample size that incorporates Gagné and Hancock's method and model fit. [ABSTRACT FROM AUTHOR]