Item Response Theory

Statistical Models in the Item Response Theory Family

IRT encompasses a family of parametric statistical models that are fit to item response data to estimate scores on a latent trait variable. Many models within the family fall into one of the two types: those that are built for dichotomous item responses (i.e., the data for each item can take on only two values) and those that are built for polytomous item responses (i.e., the data for each item can take on three or more values). Fitting an IRT model to a data set of item responses produces, among other things, a series of item characteristic curves (ICCs) that relate the underlying latent trait variable to the probability of scoring in each of the item response categories. Figure 1 shows examples of ICCs for several dichotomously and polytomously scored items.

Figure 1 Item Characteristic Curves

Conceptually, the ICC is the statistically estimated answer to the measurement question: “What is the probability of an examinee providing a particular response to this test item, given that the examinee has a particular trait level?” The ICC reflects the nature of all IRT model specifications in that the latent trait variable is treated as an independent variable that has a functional relationship to the multiple dependent variables of observed item responses. For dichotomously scored items, the ICC is displayed as an S-shaped curve representing the conditional probability of scoring a 1 on the item as opposed to a 0. The curve representing the conditional probability of scoring a 0 is not shown because it is redundant information (i.e., the probability of a 0 response is one minus the probability of a 1 response). For polytomously scored items, the ICC is displayed as a series of conditional probability curves, one for each response category on the item. At any point on the latent trait variable, the sum of the probabilities of scoring in each of the possible response categories is 1.

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