+ V/a rather than to . The item parameter, a, determines how much the measurement error changes . The parameter a is the previously mentioned item discrimination parameter from equation (7) in a different guise, just as the threshold b is the item difficulty parameter mentioned earlier. A large a-value results in little measurement error, and a small a-value indicates a large amount of measurement error. The stochastic quantity V is taken to be independent of and to have a symmetric distribution with mean 0. Hence, the value of + V/a fluctuates around in a random way.
The probability of affirming the item, the item response function, is given by:
where, in equation (8), FV(t) denotes the cumulative distribution function (cdf) of V. For the Rasch and 2PL models, FV(t) is assumed to be the logistic cdf, while for the Normal Ogive model it is the Gaussian cdf.
Finally, the conditional independence assumption in equation (5) corresponds to the assumption that the measurement errors for different items, Vi, are statistically independent. Threshold models provide a convenient way to fit many types of latent variable models into a common framework.
Only two of the items on the Household Food Security Survey Module (HFSSM) are actually dichotomous or binary, requesting a yes or no response. The other questions are either trichotomous or are two-part questions that, when considered together, have four possible ordered responses. The current use by USDA is to reduce the nondichotomous item response to binary responses by collapsing the response options to two possibilities that are regarded as either affirming or not affirming the question. A later section briefly considers more general item response functions that are directly applicable to the case of polytomous ordered responses to the food insecurity questions.
In order to be able to specify the joint distribution of the manifest variables, X1, X2, … Xp, it is necessary to integrate out or marginalize over the latent distribution, f(). Depending on the continuous or discrete nature of , f() is assumed to be either a probability density or a discrete probability function. The latent distribution reflects the heterogeneity of across the population of respondents at a relevant point in time. A common assumption for IRT models is that f() is the Gaussian distribution with mean 0 and variance 1. However, the latent distribution may have parameters that describe both the location and the degree of variation in over a particular population of respondents as well. Johnson (2005) suggested a left-truncated