variable from the values of the manifest variables. The fact that there is measurement error in the connection between the manifest variables and the latent variable in any latent variable model results in the latent posterior distributions being spread over a range of values along the -scale, rather than being concentrated on a single point along this scale. In the special circumstances in which many manifest variables are all strongly connected to the latent variable, the estimated posterior distribution is strongly peaked over a single value, so that it then makes sense to “estimate ” by, for example, the unconditional maximum likelihood approach (Haberman, 1977; Holland, 1990). This situation often happens in educational testing applications of IRT models, in which the tests may comprise 40 to 100 test items.
However, in the case of the dichotomized HFSSM questions on the CPS, there are relatively few manifest variables on which to base our knowledge of for a given individual—at most 10 for households without children and 18 for those with children. In this circumstance, the estimated posterior distributions are not highly peaked over a single value of and spread over a range of values.
Johnson (2004, p. 23) gives a graph of two estimated posterior distributions that correspond to two different patterns of responses to the food insecurity questions. Johnson’s graphs indicate that the estimated posterior distributions have substantial standard deviations, as one would expect, because of the small number of items. In addition, the two posterior distributions almost completely overlapped. Thus, measurement error is a significant aspect of the measurement of food insecurity by USDA.
An important consequence of this type of measurement error concerns the intuitively plausible use of the distribution of estimated values of the latent variable as a proxy of the latent distribution. These estimated values of the latent variable are a side benefit of the unconditional maximum likelihood method of estimating the item parameters. However, when the effect of measurement error is large, as it is in the case of food insecurity measurement, the distribution of the estimated values of across the sampled households does not form an unbiased estimate of the latent distribution.
As mentioned earlier, an additional issue relevant to measurement error is that when the number of manifest variables is relatively small, the form assumed for the latent distribution can affect the estimated latent posterior distribution and through that estimates of prevalence. Thus, even the form of the latent posterior distribution is somewhat uncertain when the number of items is small.