# Chapter 3: Multigroup Factor Analysis

### Identification of multigroup models

The main reason for estimating a multigroup factor analysis model is typically that we wish to estimate and compare means, variances or covariances of the factors between the groups. The requirement for the identifiability of the model is then that it should be possible to uniquely identify distinct values for these parameters in the different groups.

If the measurement model has full invariance of measurement, these country-specific distributions of the factors are identified if the measurement model is such that it would be identified also for single-group factor analysis (as discussed in Chapter 2) and if the factor means and variances are fixed in one group as discussed earlier in this chapter.

The remaining question is then whether and when the multigroup model is identified if the measurement model includes some non-invariance of measurement. Here we give some conditions for this. Consider first models which have different types of partial non-invariance for all of the observed items which are used as indicators of a factor:

- Means, variances and covariances of all factors are identified separately in each group if full or scalar invariance holds for all the items.
- Variances and covariances of the factors are identified also if metric invariance holds for all the items.
- Correlations of the factors are identified even under complete non-invariance, i.e. when configural invariance holds but all measurement parameters are different across the groups. Example 2 of Chapter 2 gives an example of estimating such correlations from separate country-specific models.

In the case of models for one factor, we have the following results on partial non-invariance by item:

- Means and variances of the factor are identified if at least two observed items have full measurement invariance.
- Neither means nor variances of the factor are identified if only one item is fully invariant and all other items are fully non-invariant. All such models are equivalent to each other, whichever item is chosen to be the one invariant item, and also equivalent to an infinite number of models where all the items are fully non-invariant ([Asp14]). In other words, all such models fit the data equally well but give different conclusions about the means and variances of the latent factor across the groups.

#### References

- [Asp14] Asparouhov, T. and Muthén, B. (2014). Multiple-group factor analysis alignment. Structural Equation Modeling, 21, 495–508.