# Chapter 5: Latent variable models with categorical indicators

### One-factor model for binary items: Estimation

The latent trait model implies also a joint distribution for the *p* observed items. When all the items are categorical variables, this distribution is defined by the probabilities of the cells in the *p*-variate contingency table of the items. When the items *y*_{1}, ..., *y*_{p}, are all binary with values 0 and 1, this is a 2 x 2 x ... x 2 contingency table with a total of 2^{p} cells, where each cell corresponds to one combination of values (*k*_{1}, ..., *k*_{p}) for the items, with each *k*_{j} being either 0 or 1.

A one-factor latent trait model implies that the probabilities in this joint distribution are given by

where the integral is over all the possible values of η, and *p*(η) denotes the distribution (more precisely, the probability density function) of η (which under the identification assumption stated in the previous section is a normal distribution with mean 0 and variance 1).

Maximum likelihood (ML) estimates of the parameters of the model are those values of the parameters which imply the closest match (in terms of the likelihood function for the data) between these model-implied probabilities and the sample probabilities in the observed contingency table of the items. These estimates can be found using an iterative computational algorithm, as was also the case for factor analysis and structural equation modelling. Furthermore, and unlike in the case of those previous models, estimation of a latent trait model also requires that integrals over the distributions of the factors (of the kind shown above) need to be evaluated at each iteration using computer-intensive numerical methods of integration. Because of this numerical integration, latent trait models are much harder and slower to estimate than factor analysis and structural equation models. In practice this means that latent trait models with more than one or two latent factors may often be computationally too demanding to be easily and routinely used.