# Chapter 3: Multigroup Factor Analysis

### Introduction

A key aim of many social surveys is to measure the same constructs in different groups in order to make cross-group comparisons of the distributions of the constructs. This is clearly the case in cross-national surveys such as the ESS, where the populations of individuals in the different countries are the key groups of interest. A defining purpose of a cross-national survey is to provide data for making comparisons between countries, often in terms of the distributions of latent constructs which are measured by multiple indicators.

Since the latent variables in factor analysis are assumed to follow a multivariate normal distribution, such cross-national comparisons focus on the means, variances and covariances of the factors, which together fully define the multivariate normal distribution. *Multigroup factor analysis* can be used for such comparisons. It extends the standard (single-group) factor analysis model by allowing some parameters of the model to vary across the groups.

In order for cross-group comparisons to be meaningful, the variable of interest should be measured in the same way and on the same scale across the groups. In the case of a latent variable this requirement amounts to the condition that at least a sufficiently large part of the measurement model of the variable should have the same form and identical parameter values in all of the groups. If this is the case, we can say that those parts of the measurement model are invariant (or equivalent) across the groups, and that cross-group *measurement invariance* (or *measurement equivalence*) holds for them.

In the rest of this chapter we first describe multigroup factor analysis models where measurement invariance is assumed to hold for the entire measurement model, before discussing how and when this condition may be checked and perhaps partially relaxed by allowing partial non-invariance of measurement.