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CFAs of multidimensional constructs often fail to meet standards of good measurement (e.g., goodness-of-fit, measurement invariance, and well-differentiated factors). Exploratory structural equation modeling (ESEM) represents a compromise between exploratory factor analysis' (EFA) flexibility, and CFA/SEM's rigor and parsimony, but lacks parsimony (particularly in large models) and might confound constructs that need to be kept separate. In Set-ESEM, two or more a priori sets of constructs are modeled within a single model such that cross-loadings are permissible within the same set of factors (as in Full-ESEM) but are constrained to be zero for factors in different sets (as in CFA). The different sets can reflect the same set of constructs on multiple occasions, and/or different constructs measured within the same wave. Hence, Set-ESEM that represents a middle-ground between the flexibility of traditional-ESEM (hereafter referred to as Full-ESEM) and the rigor and parsimony of CFA/SEM. Thus, the purposes of this article are to provide an overview tutorial on Set-ESEM, juxtapose it with Full-ESEM, and to illustrate its application with simulated data and diverse "real" data applications with accessible, heuristic explanations of best practice.

Scalar invariance is an unachievable ideal that in practice can only be approximated; often using potentially questionable approaches such as partial invariance based on a stepwise selection of parameter estimates with large modification indices. Study 1 demonstrates an extension of the power and flexibility of the alignment approach for comparing latent factor means in large-scale studies (30 OECD countries, 8 factors, 44 items, N = 249,840), for which scalar invariance is typically not supported in the traditional confirmatory factor analysis approach to measurement invariance (CFA-MI). Importantly, we introduce an alignment-within-CFA (AwC) approach, transforming alignment from a largely exploratory tool into a confirmatory tool, and enabling analyses that previously have not been possible with alignment (testing the invariance of uniquenesses and factor variances/covariances; multiple-group MIMIC models; contrasts on latent means) and structural equation models more generally. Specifically, it also allowed a comparison of gender differences in a 30-country MIMIC AwC (i.e., a SEM with gender as a covariate) and a 60-group AwC CFA (i.e., 30 countries × 2 genders) analysis. Study 2, a simulation study following up issues raised in Study 1, showed that latent means were more accurately estimated with alignment than with the scalar CFA-MI, and particularly with partial invariance scalar models based on the heavily criticized stepwise selection strategy. In summary, alignment augmented by AwC provides applied researchers from diverse disciplines considerable flexibility to address substantively important issues when the traditional CFA-MI scalar model does not fit the data. (PsycINFO Database Record, ((c) 2018 APA, all rights reserved).)

The factor structure of the Values in Action Inventory of Strengths (VIA-IS; Peterson & Seligman, 2004) has not been well established as a result of methodological challenges primarily attributable to a global positivity factor, item cross-loading across character strengths, and questions concerning the unidimensionality of the scales assessing character strengths. We sought to overcome these methodological challenges by applying exploratory structural equation modeling (ESEM) at the item level using a bifactor analytic approach to a large sample of 447,573 participants who completed the VIA-IS with all 240 character strengths items and a reduced set of 107 unidimensional character strength items. It was found that a 6-factor bifactor structure generally held for the reduced set of unidimensional character strength items; these dimensions were justice, temperance, courage, wisdom, transcendence, humanity, and an overarching general factor that is best described as dispositional positivity. (PsycINFO Database Record, ((c) 2017 APA, all rights reserved).)

Confirmatory factor analyses (CFAs) typically fail to support the a priori 5-factor structure of Big Five self-report instruments, due in part to the overly restrictive CFA assumptions. We show that exploratory structural equation modeling (ESEM), an integration of CFA and exploratory factor analysis, overcomes these problems in relation to responses to the 44-item Big Five Inventory (BFI) administered to a large Italian community sample. ESEM fitted the data better and resulted in less correlated factors than CFA, although ESEM and CFA factor scores correlated at near unity with observed raw scores. Tests of gender invariance with a 13-model taxonomy of full measurement invariance showed that the factor structure of the BFI is gender-invariant and that women score higher on Neuroticism, Agreeableness, Extraversion, and Conscientiousness. Through ESEM one could address substantively important issues about BFI psychometric properties that could not be appropriately addressed through traditional approaches.

In multilevel modeling, group-level variables (L2) for assessing contextual effects are frequently generated by aggregating variables from a lower level (L1). A major problem of contextual analyses in the social sciences is that there is no error-free measurement of constructs. In the present article, 2 types of error occurring in multilevel data when estimating contextual effects are distinguished: unreliability that is due to measurement error and unreliability that is due to sampling error. The fact that studies may or may not correct for these 2 types of error can be translated into a 2 × 2 taxonomy of multilevel latent contextual models comprising 4 approaches: an uncorrected approach, partial correction approaches correcting for either measurement or sampling error (but not both), and a full correction approach that adjusts for both sources of error. It is shown mathematically and with simulated data that the uncorrected and partial correction approaches can result in substantially biased estimates of contextual effects, depending on the number of L1 individuals per group, the number of groups, the intraclass correlation, the number of indicators, and the size of the factor loadings. However, the simulation study also shows that partial correction approaches can outperform full correction approaches when the data provide only limited information in terms of the L2 construct (i.e., small number of groups, low intraclass correlation). A real-data application from educational psychology is used to illustrate the different approaches.

Interactions between (multiple indicator) latent variables are rarely used because of implementation complexity and competing strategies. Based on 4 simulation studies, the traditional constrained approach performed more poorly than did 3 new approaches--unconstrained, generalized appended product indicator, and quasi-maximum-likelihood (QML). The authors' new unconstrained approach was easiest to apply. All 4 approaches were relatively unbiased for normally distributed indicators, but the constrained and QML approaches were more biased for nonnormal data; the size and direction of the bias varied with the distribution but not with the sample size. QML had more power, but this advantage was qualified by consistently higher Type I error rates. The authors also compared general strategies for defining product indicators to represent the latent interaction factor.