Chapter 5: Latent variable models with categorical indicators

Example 2 on Latent trait models for binary items: Multigroup analysis for cross-national comparison of the factor means

Note: The discussion of this example focuses on somewhat more advanced topics than most of the rest of this module.

Using the same three binary indicators of trust in the procedural fairness of the police, estimate and compare the averages of this factor between the countries in the ESS.

This analysis expands that of Example 1 by adding a structural model where the country of the respondent is used as an explanatory variable for the factor. This is thus a multigroup analysis with country as the group and assuming cross-national equivalence of measurement for all of the observed items. Unlike for the sem command for factor analysis and structural equation modelling, the gsem command in Stata does not have separate command syntax for multigroup analysis. Instead, the group (here country) is simply specified as a categorical explanatory variable for the factor, and entered in the form of dummy variables for the groups (omitting the dummy variable for one reference group, which is here taken to be Belgium).

Stata commands:

// A one-factor latent trait model for binary items:
// Models which involve country as explanatory variable

// ** Version 1: Fitting measurement and structural models together:
// Initial fit with smaller number of integration points
// (faster but less accurate):
gsem (ProcFair -> respect fair explain, logit) ///
(ProcFair <- ///
, var(e.ProcFair@1) from(b,skip) intmethod(ghermite) intpoints(3)
matrix b2=e(b)
// Final fit with more integration points, with starting values from
// the first fit above:
gsem (ProcFair -> respect fair explain, logit) ///
(ProcFair <- ///
, var(e.ProcFair@1) from(b2) intmethod(ghermite)

// ** Version 2: A three-step model where the factor score from Example 1
// is used as the response variable
reg procFairScore

Stata output and notes

The Stata commands included above show two ways of carrying out this analysis:

These approaches have slightly different characteristics:

Two further characteristics are common to both Methods 2 and 3:

The differences between these different ways of fitting the model matter ultimately only if they lead to meaningful differences in the main conclusions that we aim to draw from the analysis. In this example these questions of interest are the comparisons of average levels of the factor (trust in the procedural fairness of the police) between the countries. Figure 5.2 shows these country averages estimated using Methods 1 and 2. It is clear that both sets of estimates are very similar, in that both give essentially the same relative differences and rankings of the countries. Many of the differences between the countries are clearly statistically significant (the standard errors of the means from Method 1 are around 0.05). The ordering of the countries shows fairly consistent geographic regularities, with levels of trust in procedural fairness of the police mostly highest in the North and West of Europe and lower in the South and East.

Figure 5.3: Averages of the factor “Procedural fairness of the police” in the countries in the ESS, as estimated in Example 2. The plot shows two estimates of these averages, from a joint model (“Method 1” as discussed in the text, on the horizontal axis), and from a linear model for factor scores derived from the measurement model fitted in Example 1 (“Method 2”, on the vertical axis). The main conclusion from this comparison is that both sets of estimates give very similar results about comparisons between the countries.