# Chapter 5: Latent variable models with categorical indicators

### One-factor model for binary items: Assessment of model fit

Methods that can be used to assess the goodness of fit of a latent variable model have been discussed for factor analysis models in Chapter 2. Most of the general ideas introduced there, and some but not all of the methods, apply also to model assessment for latent trait models:

- An overall goodness of fit test can again be defined as a likelihood ratio test between the fitted and saturated models. Both of these are now models for the cell probabilities of the contingency table of the categorical items: the probabilities from the saturated model are estimated simply by the sample proportions of the cells, and the probabilities from the fitted model are implied by its estimated parameters. As in factor analysis, this test has high power when the sample size is even moderately large, so the test often rejects most models.
- Likelihood ratio tests can be used also to compare nested pairs of non-saturated models, in the same way and for similar purposes as in factor analysis.
- AIC and BIC “information criterion” statistics can be used to compare models (even non-nested ones) in the same way as in factor analysis.
- On the other hand, most of the model fit indices that have been developed specifically for factor analysis and linear structural equation models cannot be used for latent trait models. These include statistics such as RMSEA and CFI.
- A different class of methods which can be used to examine the goodness of fit of latent trait models is based on marginal residuals for the fitted model. These are calculated for lower-order marginal tables derived from the full contingency table of the
*p*observed items. In particular, we may examine in turn all the two-way tables for each pair of two items, each aggregated over the other items. For each such table, we can calculate its cell probabilities in the table derived from the observed sample, and the probabilities implied by the fitted model. The (appropriately standardized) differences between these observed and fitted probabilities are the marginal residuals, which can be used to assess how well the model fits for each pair of items. Methods of using such residuals are discussed in more detail by [Bar08] and [Bar11]. They are, however, not yet routinely implemented by all software for fitting latent trait models.

If we consider a model with one latent factor and conclude that it does not fit well, this suggests that either some of the items could be omitted to obtain a model which fits better for the remainder of the items, or that a model with more than one factor should be considered. Models with multiple factors are discussed briefly in the next section, together with other extensions.

#### References

- [Bar08] Bartholomew, D. J., Steele, F., Moustaki, I. and Galbraith, J. G. (2008). Analysis of multivariate social science data (Second edition). Chapman & Hall/CRC.
- [Bar11] Bartholomew, D. J., Knott, M. and Moustaki, I. (2011). Latent Variable Models and Factor Analysis: a unified approach (Third edition). Wiley.