# Chapter 3: Multigroup Factor Analysis

### Multigroup models under full measurement invariance

Consider again a model for one or more *η*_{j}, which are measured by multiple indicators in a way described by a factor analysis measurement model. Suppose now that we have data on respondents from *G* known groups such as countries. In this section we assume that complete measurement invariance across the groups holds for the measurement model, so that this model is exactly the same in all the groups. There is then nothing new to say about the measurement model, which is defined and interpreted in the same ways as in the single-group situation discussed before. We can thus focus on the changes in how the distribution of the latent factors is specified.

For concreteness of notation, suppose that there are two factors *η*_{1} and *η*_{2}. We now assume that among individuals in each group *g* = 1, ..., *G*, the factors are jointly normally distributed with means

*E( η_{1}) = κ_{1}^{(g)}* and

*E(*

*η*_{2}) =*κ*_{2}^{(g)}variances

*var( η_{1}) = φ_{1}^{(g)}* and

*var(*

*η*_{2}) =*φ*_{2}^{(g)}and covariance

*cov( η_{1}, η_{2}) = φ_{12}^{(g)}*

In other words, this allows all of the parameters which describe the distribution of the factors to be different in different groups.

It is still necessary to impose some constraints on these parameters in order to identify the scales of the latent factors. This, however, now needs to be done only in one group (which can be chosen freely), leaving all the parameters free to be estimated in all the other groups. For example, we may choose group 1 as the reference group and fix the factor means *κ*_{1}^{(1)} and *κ*_{2}^{(1)} to be 0 and the factor variances *φ*_{1}^{(1)} and *φ*_{2}^{(1)} to be 1 in that group (the factor covariance *φ*_{12}^{(g)} can be freely estimated in all groups, including group 1). When this is done, the mean of 0 in group 1 becomes a benchmark against which the means *κ*_{1}^{(g)} and *κ*_{2}^{(g)} in the other groups, and values of the factors for individuals in all groups, can be compared, and similarly the value 1 in group 1 becomes a benchmark for the factor variances *φ*_{1}^{(g)} and *φ*_{2}^{(g)}.

Such models can be easily fitted in standard software, and estimates from them allow us to compare distributions of factors across countries. This is illustrated by Example 1 later in this chapter.