# Chapter 4: Structural Equation Models

### Example 3 on Structural equation modelling: A multigroup model

Fit the structural equation model considered in the Examples 2 and 3, but now as a multigroup model fitted to the data from Denmark, Norway and Sweden (we limit the analysis to these countries to keep the example simple). Using likelihood ratio tests, examine if the regression coefficients in the structural model vary significantly between these countries.

The command and output files for Stata and R show examples of how these multigroup models can be fitted and likelihood ratio tests can be used to test cross-country constraints of equality for the regression coefficients. In all of these models we specify full cross-national invariance of measurement for all of the indicators of the factors. In the structural models, we allow free cross-national variation in the parameters of the distributions of the exogenous factors (Effectiveness and Procedural fairness) and in all the intercept terms and the residual variances and residual covariances of the models for the endogenous factors.

Consider for example the comparison between the most flexible and the most restrictive structural models here, that is the model where all of the regression coefficients vary between the countries, and the model where none of them do. The likelihood ratio test between these models has a p-value of p=0.56, which is not significant at any conventional level of significance. This means that we do not reject the null hypothesis that each of the coefficients are equal across the countries, i.e. that the associations between the latent factors in this model are of equal strengths in each of Denmark, Norway and Sweden. The common estimated coefficients from this model are shown in Figure 4.3. They are all statistically significant and all positive in sign, except that the coefficient of Effectiveness in the model for Co-operation is negative, as it was also for several other countries in Example 2 of this chapter.

*Figure 4.3: Estimated regression coefficients in the structural model in Example 3, fitted to data on respondents from Denmark, Norway and Sweden, and with these coefficients constrained to be equal across the countries.*