This was a fun paper to write. Most people who use structural equation modeling don’t realize that a model’s parameterization can influence the standard errors and significance tests. The effect can be quite large and can make significant effects nonsignificant or nonsignificant effects significant. People assumed that parameterization can be ignored because the overall goodness of fit is unaffected. We point out the problem and present a simple way to test parameters in structural equation modeling that is unaffected by parameterization.

Gonzalez, R. & Griffin, D. (2001). Testing parameters in structural equation modeling: Every “one” matters. Psychological Methods, 6, 258-269. doi:10.1037/1082- 989X.6.3.258 PMid:11570231 (PDF)

Abstract

A problem with standard errors estimated by many structural equation modeling programs is described. In such programs, a parameter’s standard error is sensitive to how the model is identified (i.e., how scale is set). Alternative but equivalent ways to identify a model may yield different standard errors, and hence different Z tests for a parameter, even though the identifications produce the same overall model fit. This lack of invariance due to model identification creates the possibility that different analysts may reach different conclusions about a parameter’s significance level even though they test equivalent models on the same data. The authors suggest that parameters be tested for statistical significance through the likelihood ratio test, which is invariant to the identification choice.