that affirm x or more of the questions, then, assuming no DIF, equation (17) is modified, as in equation (16), that is,

(18)

Equations (17) and (18) express the probabilities, P{A ≥ x} and P{Ax | G = g}, as the average of the conditional probability P{Ax | } in a standard way. For any plausible IRT model, P{Ax | } is an increasing function of , ranging from a small value for low values of to nearly 1 for large values of .

If, for some value of x, P{Ax |} were a step function, that was zero to the left of * a nd 1 to the right of *, then the prevalence rate in equation (15) and the affirmation rate in equation (17) would be equal. However, for any value of x, P{Ax | } is far from a step function, due to the inherent measurement error between the latent and manifest variables. It is possible that, for an appropriate choice of x, the parts of t he function, P{Ax | }, above and below the cut point, *, would “balance,” but this would have to be investigated in each case and for equation (18) could depend on the value of g. The difference between equations (15) and (17) is the bias arising from the use of a cut point based on the manifest variables and the use of one defined on the latent scale. This bias was addressed in a way in Nord (1999). How well he was able to investigate this bias is not clear to us due to the complexity of the task.

The form assumed for the latent distribution, f(), can make a difference in the estimated prevalence of food insecurity. This can be studied to some extent by trying out different assumptions and seeing what effect they have. As the number of manifest variables increases, the effects of different assumptions about the latent distribution grow less, but in the case of food insecurity the number of manifest variables is too small for this to be assumed.

Johnson (2005) describes several approaches to estimating prevalence rates that are defined by cut points along the -scale and of the form in equation (15) rather than equation (17). These methods avoid the biases mentioned above and apply to either overall or subgroup-specific prevalence rates. An example of the bias in prevalence estimates that arises from the failure to condition appropriately on the subgroup is given in Mislevy et al. (1992) for an educational testing application.

The Consequences of Measurement Error

As discussed earlier, the latent posterior distribution in equation (11), f( | X1 = x1, …, Xp = xp), summarizes all that is known about the latent



The National Academies | 500 Fifth St. N.W. | Washington, D.C. 20001
Copyright © National Academy of Sciences. All rights reserved.
Terms of Use and Privacy Statement