Abstract
We put forward a new item response model which is an extension of the binomial error model first introduced by Keats and Lord. Like the binomial error model, the basic latent variable can be interpreted as a probability of responding in a certain way to an arbitrarily specified item. For a set of dichotomous items, this model gives predictions that are similar to other single parameter IRT models (such as the Rasch model) but has certain advantages in more complex cases. The first is that in specifying a flexible two-parameter Beta distribution for the latent variable, it is easy to formulate models for randomized experiments in which there is no reason to believe that either the latent variable or its distribution vary over randomly composed experimental groups. Second, the elementary response function is such that extensions to more complex cases (e.g., polychotomous responses, unfolding scales) are straightforward. Third, the probability metric of the latent trait allows tractable extensions to cover a wide variety of stochastic response processes.
Highlights
In this paper we introduce a class of item response models for the analysis of response distributions derived from survey data
In contrast to a Rasch model, our estimation of item parameters is not separable from the estimation of person parameters; the advantage of specifying a flexible family of distributions like the Beta comes in the capacity to analyze factorial experiments in which either the trait, or item hardnesses, or both may be affected by treatments, as we show below
We find a correlation > .99 between the natural log of the extended binomial error model’s item parameters and those of the Rasch model
Summary
In this paper we introduce a class of item response models for the analysis of response distributions derived from survey data. The simplest item response function in this class is a generalization of the binomial error model for ability testing of Keats and Lord [1]. Similar item response functions were considered briefly by Lazarsfeld [2] and Coleman [3] but not implemented as models for response data. The distinguishing characteristic of the approach taken here is that the probability of an item response is written as a function of a latent variable that can be interpreted as a probability. The choice of a probability metric suggests the Beta density as a natural choice to model the distribution of the latent variable. A response function formulated in this way is easy to modify to accommodate variations in the nature and complexity of response tasks
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