Multidimensional Item Response Theory

Unidimensional IRT Models

Unidimensional IRT refers to a set of statistical models designed for use with responses to items on tests, questionnaires, and other such instruments in order to obtain estimates of individuals’ levels on the construct measured by the scale as a whole. For example, a commonly used model is the three-parameter logistic (3PL) model, which contains a parameter specific to respondents [Page 1103](person parameter) and three parameters specific to the items measuring the construct of interest. The person parameter is the estimate of the latent trait being measured by the scale (e.g., reading ability and depression) and is referred to as θ. The item parameters include (a) location on the latent trait scale, (b) the item’s ability to differentiate among individuals with different levels of the construct, and (c) the likelihood that an individual will endorse the item due solely to chance. The construct itself is a latent variable that is measured by the set of items, which serve the role of indicators much as do the observed variables in factor analysis models. Indeed, there is a direct relationship between IRT and factor analysis models and their parameters, so that one can be easily converted to the other. IRT models can be divided into two broad families based on whether they model dichotomous item responses with two categories or polytomous items with three or more categories.

The simplest unidimensional IRT model is the Rasch model, which is expressed as

$$P{x}_j=1|\theta ,b_j=e\theta ,-b_{j}1+e\theta ,-b_j.$$

In Equation 1, xj is the response to item j with 1 being correct in the context of an achievement test and 0 being incorrect. An individual’s level of the latent trait being measured by the set of items is represented by θ, and the item location is bj. For a math test, θ corresponds to an examinee’s math ability while bj is the difficulty of item j. For a depression inventory, θ would be the patient’s level of depression and bj the likelihood of an individual endorsing the behavior measured by item j. An important strength of all IRT models is that item difficulty and examinee ability are placed on the same scale. Therefore, it is possible to directly compare where an individual lies on the latent trait scale with the location of any item on the instrument. In addition, b and θ are both centered at 0, which represents average or typical location for both.

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