By neglecting to include any terms for interaction in their model, the difference test proposed by the authors will always be based on an error term that includes all potential sources of interaction, as well as plot effects, with correspondingly large degrees of freedom.
Later on, the model was also used to model differences in test-taking motivation in low-stakes tests in order to separate motivated from unmotivated test-taking behavior (e.g., Mittelhaëuser et al. 2015).
Gender equivalence of factor loadings was then tested by comparing the nested model (i.e., the model with factor loadings constrained across genders) to the base model (i.e., the model with factor loadings freely estimated across genders) using the nested-model chi-square difference test.
A number of fit indices were employed to provide an overall assessment of fit of the raw data to the specified model (RMSEA, GFI, CFI and NFI) and also to compare alternative models (chi square difference test) [ 50]. 791 participants were approached in the Beijing area to participate in the study, 8 people refused to take part due to time constraints or inconvenience.
However, the point estimate of the RMSEA for each model was lying outside the CI of the corresponding other model and chi-square difference test indicated a significant difference between both models favouring Model 4 (p < 0.001).
Both assortment models caused a significant decrease in model fit, but the SH model fitted the data relatively worse than the PA model (according to χ difference test, Akaike's information criterion (AIC), Bayesian information criterion (BIC), and Deviance information criterion (DIC)).
Thus, because the unitary model is nested in the correlated skill model, a chi-square difference test was performed, whose results show that the latter was slightly better than the former (χ 2 difference = 4.039, df difference = 1, p = .044).044
Allowing path coefficients and parameter estimates to vary freely between both groups did not improve model fit (chi-square difference test χ(11) = 12.7, P = 0.313), suggesting that overall model results applied equally well to both groups.
The best fitting second-degree FP was then compared against the null model using a deviance difference test with four degrees of freedom to determine whether BMI should be included in the model.
