Testing Structural Equation Models or Detection of Misspecifications?

Abstract

Assessing the correctness of a structural equation model is essential to avoid drawing incorrect conclusions from empirical research. In the past, the chi-square test was recommended for assessing the correctness of the model but this test has been criticized because of its sensitivity to sample size. As a reaction, an abundance of fit indexes have been developed. The result of these developments is that structural equation modeling packages are now producing a large list of fit measures. One would think that this progression has led to a clear understanding of evaluating models with respect to model misspecifications. In this article we question the validity of approaches for model evaluation based on overall goodness-of-fit indexes. The argument against such usage is that they do not provide an adequate indication of the “size” of the model's misspecification. That is, they vary dramatically with the values of incidental parameters that are unrelated with the misspecification in the model. This is illustrated using simple but fundamental models. As an alternative method of model evaluation, we suggest using the expected parameter change in combination with the modification index (MI) and the power of the MI test.

Notes

Research of the second author was supported by grants SEJ2006-13537 and PR2007-0221 from the Spanish Ministry of Science and Technology.

¹The effect (β₂₁) could also be underestimated if the correlation between the disturbance term is negative.

²Note that power = 1 – β.

^aThe chi-square is equal to the noncentrality parameter in this case, because population data are analyzed (Saris & Satorra, 1985).

^bThe power values were computed with α = .05 and 4 df (because the model has 4 df). To obtain the correct power value, we used tables, such as can be found in Saris and Stronkhorst (1984), that relate the power, the degrees of freedom of the test, and the noncentrality parameter.

³The power of the test is estimated on the basis of the noncentrality parameters obtained by analyzing population data (Satorra & Saris 1985). The noncentrality parameter is equal to the chi-square statistic in this case.

^aThe noncentrality parameter (ncp) is equal to the chi-square in this case because population data are analyzed (Saris & Satorra, 1985).

^bThe power values were computed with α = .05 and 2 df (because the model has 2 df). To obtain the correct power value, we used tables, such as can be found in Saris and Stronkhorst (1984), that relate the power, the degrees of freedom of the test, and the noncentrality parameter.

⁴In general, this is not the case. It is due to a specific character of this model.

⁵A computer program, JRule, that produces statistics for all the restricted parameters, based on the output of LISREL, has been developed by us. The program can be requested by sending an e-mail to [email protected] and putting JRule in the subject line.