Writing about Structural Equation Models R.H. Hoyle Abigail T. Panter

The Conceptual Model Introduce model with a diagram. Provide a written explanation in the text for each relation or path.

The Statistical Model Include a path diagram. Describe the statistical model in the Methods section.

Details about the Data Include a correlation matrix with standard deviations of the variables. Explain any assumptions about measurement levels of the raw data. Provide information about the distributions of the individual variables and the multivariate distribution. Include statistics on univariate kurtosis and multivariate normality.

Describing the Results Report results from ML estimation. If necessary report results of alternative estimation procedures. Include the Indicators of Fit recommended in Table 9.1.

Recommended Indexes of Overall Model Fit Stand-Alone/Absolute Indexes Reference Description Chi-Square Scaled Chi-square Bollen (1989b) Sattora & Bentler (1994) Statistical test of the lack of fit resulting from overidentifying restrictions placed on a model Goodness-of-fit index Joreskog & Sorbom (1981) Indexes the relative amount of the observed variances and covariances accounted for by a model Analogous to R2

Recommended Indexes of Overall Model Fit Type-2 Indexes Reference Description Tucker-Lewis Index (TLI)/Nonnormed Fit Index (NNFI) Bentler & Bonett (1980) Tucker & Lewis (1973) Compares the lack of fit of a target model to the lack of fit of a baseline model, usually the independence model. Value estimates the relative improvement per df of the target model over a baseline model. Not recommended for very small samples (<150) or with GLS estimation.

Recommended Indexes of Overall Model Fit Type 2 Indexes Reference Description Incremental fit index (IFI) Bollen (1989a) Same interpretation as TLI/NNFI. Less variable than TLI/NNFI in small samples and more consistent across estimators than TLI/NNFI.

Recommended Indexes of Overall Model Fit Type-3 Indexes Reference Description Comparative fit index (CFI) Bentler (1989, 1990) Indexes the relative reduction in lack of fit as estimated by the noncentral chi square of a target model versus a baseline model. Varies between 0 and 1. Fit index (FI)/Relative noncentrality index (RNI) McDonald & Marsh (1990) Equivalent to CFI but can exceed 1 and 1 because of sampling error or over-fitting.

Parameter Estimates Provide all parameter estimates, error variances, and variances of latent variables. Include standard error of estimates, critical ratios and p-values. Indicate parameters fixed at specific values.

Alternative Models The strongest SEM analysis compares competing theoretical models. Include substantive reasons for post-hoc modifications. Report any equivalent models. If possible, cross-validate the final model.

Interpretation Directionality is established by logic, manipulation, or strong theoretical arguments. Associations in SEMs are necessary but not sufficient evidence of causal relations. Include additional limitations regarding the interpretation of the SEM results in the discussion.

Reference R.H. Hoyle and A. T. Panter, “Writing about Structural Equation models,” In: R. H. Hoyle (ed.), Structural Equation Modeling; Concepts, Issues and Applications. Sage Publications, 1995, pp. 158-176.