Item response theory (IRT; see Baker, 1992; Hamble-ton & Swaminathan, 1989) usually is considered a class of statistical models for categorical data to which the Rasch model (RM) and the two-parameter logistic model belong, but not latent class analysis (LCA). However, there are at least two reasons why LCA can and should be seen as an IRT model. Whereas the Rasch model tries to measure a latent trait of the persons, LCA tries to identify latent classes of persons. Both measure a latent variable, which is a quantitative variable in the case of the RM and a categorical variable in the case of LCA. There are lots of good examples where test or questionnaire data have been analyzed using LCA. The second reason is that the RM can be generalized to a mixture distribution model, which is at the same time a generalized latent class model.
The Rasch Model
The Rasch model refers to a data matrix that is set up by the two factors persons and items (see Table 18.1).
The entries of such a data matrix are the responses xyj of a set of persons on a set of items. In the simplest case, these item responses are dichoto-mous, meaning that they only distinguish a correct or "yes" response (xvj =1) from an incorrect or "no" response (xy.= 0). To calculate the probability of an item response, each person is characterized by an
TABLF: 18.1
The Data Structure of IRT Models
_Factor: items_
1 ri
3 r3
Factor: persons r
"ability" parameter that is not necessarily an ability but any kind of latent trait, and each item is characterized by an item parameter. The Rasch model (Rasch, 1960, 1980) assumes only one parameter per item, that is, a difficulty parameter, whereas other IRT models have a second parameter that is defined as a discrimination parameter. The former can be formalized by the following equation.
In this equation, the probability for person v to respond to item i is a logistic function of the person parameter 0v and the item parameter a.. For one item this relationship is represented by the so-called item characteristic curve (ICC; see Figure 18.1), which is defined as the probability of an item response as a function of the person parameter value. The item parameter a. is defined as the x-coordinate of the turning point of the logistic function. Because the x-axis represents the latent dimension, the item parameter is defined on the same scale and has the same metric as the person parameters.
The reason why the present chapter only deals with Rasch-type models has to do with some advantageous statistical properties of this "simple" one-parameter logistic model (Rost, 2001). Rasch-type models are the only IRT models where the unweighted sum of correct item responses (the "number correct") is a sufficient statistic for the estimation of the trait parameters. Therefore, the
FIGURE 18.1. Nonintersecting item characteristic curves as assumed by the Rasch model.
Rasch model provides a formal framework for what is done anyway in naive analyses of test data: counting the number of items that were answered by a respondent in a certain direction. This property of Rasch models is called sufficiency and refers to the fact that the sum of correct responses (xvj = 1) of a person contains all the information necessary to estimate the parameter of this person. The same is true for the item parameters: The sum of correct responses to an item contains all information necessary to estimate the item parameter (see following section).
The model Equation (1) shows that both types of parameters—person and item parameters—are combined additively, which is called the property of latent additivity. It can be seen from this equation that the item parameter is defined as a difficulty parameter because the item parameter is subtracted from the person parameter.
The Rasch model does not provide a second item parameter defining the discrimination of an item, which implies that all ICCs of a Rasch homogeneous test are parallel, so they do not intersect (see Figure 18.1). Parallel ICCs and constant item discriminations also mean that all items measure the same latent trait equally well (property of item homogeneity). Item homogeneity is a necessary condition for measuring the persons independently of the distribution of measures of the items (property of specific objectivity). Fischer (1995a) showed that the family of Rasch models is the only model type that fulfills this condition and that specific objectivity is related to the property of latent additivity: Whenever specific objectivity holds for a set of items and a set of persons, then a representation of the model exists where the person and the item parameter are connected by addition or subtraction as is the case for the Rasch model (Model 1).
Because of these properties, the likelihood of the data—the probability of the observed data given the assumed model—is just a function of the marginal sums of the data matrix nj5 the number of persons that solved item i, and rv, the number of items that were solved by person v, that is, the number correct scores.
FIGURE 18.1. Nonintersecting item characteristic curves as assumed by the Rasch model.
Because the likelihood function L is used for estimating the parameters, it follows that the marginal sums are sufficient for estimating the parameters. The pattern of responses of a person does not contribute anything to the estimation of the person parameter that cannot be drawn from the simple sum score.
Parameter estimation procedures that are based on the preceding likelihood function are called joint maximum likelihood procedures because both types of parameters are jointly estimated. With Rasch models it is also possible to estimate the item parameters without estimating the person parameters simultaneously or knowing them beforehand. This is done by conditioning the likelihood on the sufficient statistics of the person parameters, that is, on the rv scores. The consequence is a conditional likelihood function that does not contain the person parameters. Estimation procedures based on this likelihood are called conditional maximum likelihood methods (see following section).
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