Chapter 3: Multigroup Factor Analysis
Example 2 on Multigroup factor analysis: Assessing noninvariance of measurement
Consider the data on the questions D18D20 which are treated as measures of the factor "obligation to obey the police", for data from Denmark, Norway and Sweden. Fit multigroup models with one factor, and compare models with different specifications of measurement invariance and noninvariance in the items. How well do these different models fit the data, and how do they affect conclusions about crossnational comparisons of the mean of the factor?
Here we consider only three countries and one factor, to keep the command file in Stata relatively short (the commands for all 27 countries and more factors and items would be an obvious extension of these).
Results for the fitted models are summarized in Table 3.2. Here we consider for illustration seven different specifications for the measurement models across the three countries: The model where full invariance holds, scalar invariance for all of the three items, scalar invariance, metric invariance and complete noninvariance for item D18, and complete noninvariance for items D19 and D20. Note that models with metric invariance in all the items or models with noninvariance in two or more items are not included, because such models would not allow the identification of distinct estimates of factor means for the different countries.
Considering first the goodness of fit of the models, likelihood ratio tests indicate that allowing for noninvariance in any one item would improve the fit compared to the full invariance model. The tests of partial invariance models shown for item D18 suggest that for this item at least the measurement intercepts in particular are significantly different between the countries. The AIC and BIC statistics indicate that the model preferred by each of them includes noninvariance of measurement. So in these data, even with just three items and three culturally and linguistically fairly similar countries, a multigroup analysis suggests significant deviations from exact invariance of measurement.
What matters most for substantive interpretation, however, is whether comparative conclusions about the constructs being measured are affected by different choices for the measurement models. Here they are not. Table 3.2 also shows that all the models considered here yield very similar estimates for the estimated means of the factor. According to all of them, the average level of felt obligation to obey the police is around 0.52 in Norway and around 0.35 in Sweden, on a scale where the mean in Denmark is fixed at 0, and the standard deviation in Denmark fixed at 1. The estimated standard errors of these estimates are around 0.04, so all the differences between the country means are statistically significant. Since all the models give similar results about the factors, here we could without difficulty focus on the simplest model which assumes invariance of measurement.
Estimated mean (and standard error) of the factor [Obligation to obey the police] 


Measurement model  LRtest againt full invariance model: PValue  AIC  BIC  Denmark (Constrained) 
Norway  Sweden 
Full invariance  57417  57501  0  0.52 (0.04) 
0.36 (0.04) 

Scalar invariance (error variances free) 
<0.001  57394  57516  0  0.51 (0.04) 
0.36 (0.04) 
Scalar invariance for item D18 
0.10  57417  57513  0  0.52 (0.04) 
0.36 (0.04) 
Metric invariance for item D18 (error variance and intercept free) 
<0.001  57384  57493  0  0.50 (0.04) 
0.37 (0.04) 
Complete noninvariance for item D18  <0.001  57381  57503  0  0.50 (0.04) 
0.37 (0.04) 
Complete noninvariance for item D19  <0.001  57397  57519  0  0.57 (0.05) 
0.34 (0.04) 
Complete noninvariance for item D20  <0.001  57396  57518  0  0.52 (0.04) 
0.37 (0.04) 