Multidimensional Item Response Theory

Item response theory (IRT) models are stochastic models for responses of persons to items, where the influences of items and persons on the responses are modelled by disjunctive sets of parameters. In the framework of educational and psychological measurement, the person parameter can usually be labelled ability or proficiency; in the sequel, the term ability will be used. The definition of separate parameters for persons and items supports a comprehensive framework for many important issues in educational and psychological measurement, such as test scoring, validity, local reliability, test equating, calibration of item banks using incomplete designs, differential item functioning, optimal test construction and computerized adaptive testing (see, for instance, Lord, 1980, or Hambleton & Swaminathan, 1985).

In many instances, it suffices to assume that ability is unidimensional. However, in other instances, it may be a priori clear that multiple abilities are involved in producing the manifest responses, or the dimensionality of the ability structure might not be clear at all. In such cases, multidimensional IRT (MIRT) models can serve confirmatory and explorative purposes, respectively. As this terminology suggests, many MIRT models are closely related to factor analytic models; in fact, Takane and de Leeuw (1987) have identified a class of MIRT models that is equivalent to a factor analysis model for categorical data. This class of models will be treated first. Then attention will be given to a second class of MIRT models, defined by the existence of minimal sufficient statistics and closely related to loglinear models for the analysis of discrete data. This entry will be concluded with some remarks about the choice between the two models.

In the first class of models, MIRT models for dichotomously scored items were first presented by McDonald (1967) and Lord and Novick [Page 599](1968). These authors use a normal ogive to describe the probability of a correct response. The idea of this approach is that the dichotomous response of person i to item j is determined by an unobservable continuous random variable. This random variable has a standard normal distribution and the probability of a correct response is equal to the probability mass below some cut-off point ηij. That is, the robability of a correct response is given by

where Φ(.) is the cumulative standard normal distribution, θiq, q = 1, …, Q, are the Q ability parameters (or factor scores) of person i, βj is the difficulty of item j, and αjq, q = 1, …, Q, are Q factor loadings expressing the relative importance of the Q ability dimensions for giving a correct response to item j. Further, it is assumed that the ability parameters θiq, q = 1, …, Q, have a Q-variate normal distribution with a mean-vector μ with the elements μq, q = l, …, Q, and a covariance matrix σ. So it is assumed that Q ability dimensions play a role in test response behaviour, the relative importance of these ability dimensions in the responses to specific items is modelled by item-specific loadings αjq, and the relation between the ability dimensions in some populations of respondents is modelled by the correlation between the ability dimensions.

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