Structural Equation Modelling

Abstract

Structural equation modeling (SEM) is a family of models where multivariate techniques are used to examine simultaneously complex relationships among variables. The goal of SEM is to evaluate the extent to which proposed relationships reflect the actual pattern of relationships present in the data. SEM users employ specialized software to develop a model, which then generates a model-implied covariance matrix. The model-implied covariance matrix is based on the user-defined theoretical model and represents the user’s beliefs about relationships among the variables. Guided by the user’s predefined constraints, SEM software employs a combination of factor analysis and regression to generate a set of parameters (often through maximum likelihood [ML] estimation) to create the model-implied covariance matrix, which represents the relationships between variables included in the model. Structural equation modeling capitalizes on the benefits of both factor analysis and path analytic techniques to address complex research questions. Structural equation modeling consists of six basic steps: model specification; identification; estimation; evaluation of model fit; model modification; and reporting of results. Conducting SEM analyses requires certain data considerations as data-related problems are often the reason for software failures. These considerations include sample size, data screening for multivariate normality, examining outliers and multicollinearity, and assessing missing data. Furthermore, three notable issues SEM users might encounter include common method variance, subjectivity and transparency, and alternative model testing. First, analyzing common method variance includes recognition of three types of variance: common variance (variance shared with the factor); specific variance (reliable variance not explained by common factors); and error variance (unreliable and inexplicable variation in the variable). Second, SEM still lacks clear guidelines for the modeling process which threatens replicability. Decisions are often subjective and based on the researcher’s preferences and knowledge of what is most appropriate for achieving the best overall model. Finally, reporting alternatives to the hypothesized model is another issue that SEM users should consider when analyzing structural equation models. When testing a hypothesized model, SEM users should consider alternative (nested) models derived from constraining or eliminating one or more paths in the hypothesized model. Alternative models offer several benefits; however, they should be driven and supported by existing theory. It is important for the researcher to clearly report and provide findings on the alternative model(s) tested. Common model-specific issues are often experienced by users of SEM. Heywood cases, nonidentification, and nonpositive definite matrices are among the most common issues. Heywood cases arise when negative variances or squared multiple correlations greater than 1.0 are found in the results. The researcher could resolve this by considering a small plausible value that could be used to constrain the residual. Non-positive definite matrices result from linear dependencies and/or correlations greater than 1.0. To address this, researchers can attempt to ensure all indicator variables are independent, inspect output manually for negative residual variances, evaluate if sample size is appropriate, or re-specify the proposed model. When used properly, structural equation modeling is a powerful tool that allows for the simultaneous testing of complex models.

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“…Model fit indices are essential tools in structural equation modeling (SEM) for evaluating how well a hypothesized model corresponds to the observed data. Among the common fit indices, the Comparative Fit Index (CFI) compares the fit of a user-specified model to a baseline model, with values above 0.90 indicating a good fit [41,42]. The Tucker-Lewis Index (TLI) operates similarly to the CFI but includes a penalty for model complexity, again suggesting a good fit for values above 0.90 [41,42].…”

“…Among the common fit indices, the Comparative Fit Index (CFI) compares the fit of a user-specified model to a baseline model, with values above 0.90 indicating a good fit [41,42]. The Tucker-Lewis Index (TLI) operates similarly to the CFI but includes a penalty for model complexity, again suggesting a good fit for values above 0.90 [41,42]. The Bentler-Bonett Non-normed Fit Index (NNFI) adjusts for both model complexity and sample size, with a threshold of 0.90 for an acceptable fit [41,42].…”

“…The Tucker-Lewis Index (TLI) operates similarly to the CFI but includes a penalty for model complexity, again suggesting a good fit for values above 0.90 [41,42]. The Bentler-Bonett Non-normed Fit Index (NNFI) adjusts for both model complexity and sample size, with a threshold of 0.90 for an acceptable fit [41,42]. The Relative Noncentrality Index (RNI) considers noncentrality parameters to measure model fit, with values above 0.90 denoting a good fit [41,42].…”

“…The Bentler-Bonett Non-normed Fit Index (NNFI) adjusts for both model complexity and sample size, with a threshold of 0.90 for an acceptable fit [41,42]. The Relative Noncentrality Index (RNI) considers noncentrality parameters to measure model fit, with values above 0.90 denoting a good fit [41,42]. The Bentler-Bonett Normed Fit Index (NFI) compares the chi-square value of the user-specified model to that of a null model, where values exceeding 0.90 are desirable [41,42].…”

“…The Relative Noncentrality Index (RNI) considers noncentrality parameters to measure model fit, with values above 0.90 denoting a good fit [41,42]. The Bentler-Bonett Normed Fit Index (NFI) compares the chi-square value of the user-specified model to that of a null model, where values exceeding 0.90 are desirable [41,42]. Bollen's Relative Fit Index (RFI), similar to the NFI, adjusts for degrees of freedom, and values above 0.90 are considered indicative of a good fit [41,42].…”