other words, given , both Xi and G must be conditionally independent for any item.
It is possible for this type of conditional independence to fail. When Xi and G are not conditionally independent given , the item indicated by Xi is said to exhibit differential item functioning (DIF), that is, to function differently for some or all of the subgroups indicated by G. For example, some of the FSS questions regarding adults could be easier to affirm for adults in households with children than for those without children. This might occur if the available household resources were used to feed the children first while the adults went without food for some period. In this hypothetical example, the item would function differently in terms of for households with and without children. Differential item functioning is discussed extensively in Holland and Wainer (1993). If an item is found to exhibit DIF in two groups, it may be appropriate to allow different item parameters for it that depend on which subgroup the respondent is in rather than the usual no-DIF assumption that they are the same for all subgroups. USDA should evaluate the amount and consequences of DIF in the FSS.
It is evident that what can be deduced about the latent variable for a respondent depends on the values of the manifest variables for that respondent and the features assumed for the latent variable model. In particular, the latent posterior distribution, denoted here by f( | X1 = x1, …, Xp = xp), of given the values of the manifest variables, X1 = x1, …, Xp = xp, summarizes everything that is known about the value of for any respondent with a given pattern of values of the manifest variables, X1 = x1, …, Xp = xp. When the household’s membership in a subgroup, indicated by G = g, is also considered to be important, then what is known about f can be summarized by the latent posterior distribution, f( | X1 = x1, …, Xp = xp, G = g), where the conditioning is now on both the pattern of values of the manifest variables and the group membership indicated by G.
It is well known that the latent posterior distribution is given by the expression
The numerator of equation (11) is the product of the item response functions and the latent distribution, which can be estimated once the item parameters are estimated. The denominator of equation (11) is the joint distribution of the manifest variables given in equation (9).