Gaussian distribution for the case of food insecurity because the full set of food insecurity questions is asked only for those households that are likely to have large values of .

The latent distribution, f(), along with the item response functions, P{X_i = x_i | }, may be combined using equation (5) to specify the joint distribution of the manifest variables, i.e.,

(9)

for continuous latent variables; a sum replaces the integral in equation (9) for discrete latent variables.

The parameters of the joint distribution of the X’s in equation (9) include both the item parameters from the item response functions and possibly other parameters from the latent distribution. It is this joint distribution for the manifest variables that allows these parameters to be estimated and for the latent variable model to be tested against data.

It often happens that important subgroups of respondents need to be studied separately. For example, households with children are asked questions that are not appropriate for households without children. It is possible for the latent distribution to vary with the subgroup. When there are large differences in these latent distributions, it may be important to include them in the model for the manifest variables.

To denote this situation, let G be a variable that distinguishes between different subgroups of respondents. For example G = 0 could indicate a household without children, while G = 1 indicates a household with children. In this setting, equation (9) can be expanded to

(10)

where G = g denotes one of the subgroups of interest. In many IRT applications, the latent distributions, f(| G = g), are assumed to be Gaussian with means and variances that vary with g.

In order for equation (10) to be a correct formula, has to “explain” (in the sense of conditional independence mentioned earlier) any dependence between subgroup membership and responses to the questions. In