Probabilistic Models for Some Intelligence and Attainment Tests

Often in social science research, data are collected from a number of examinees that are each scored on their performance on items of a test. Here, a “test” broadly refers to an assessment of an examinee's level of ability in a particular domain such as math or reading, or a survey of an examinee's behaviors or attitudes toward something. In Probabilistic Models for Some Intelligence and Attainment Tests,George Rasch proposed a model for analyzing such test data. The model, known as the Rasch model, and its extensions are perhaps among the most known models for the analysis of test data. In the analysis of data with a Rasch model, the aim is to measure each examinee's level of a latent trait (e.g., math ability, attitude toward capital punishment) that underlies his or her scores on items of a test. This entry takes a closer look at the Rasch model proposed in Probabilistic Models for Some Intelligence and Attainment Testsand its extensions.

Suppose that items of a test are each scored on two levels, say 0 = Incorrect and 1 = Correct, or 0 = False and 1 = True. The Rasch model can be represented by

where Xijis a random variable for the score of examinee i{i= 1,…, n)on the jth item of the test (j= 1,…, J), θiis the latent trait parameter of the jth examinee, and δjis the difficulty parameter of the jth test item. Also, G(η)is a cumulative distribution function (c.d.f.), usually assumed to be the c.d.f. of the standard logistic distribution defined by

although another popular choice, G,is the standard normal c.d.f., with G(η) = Normal(η|0,1). A Rasch model that assumes G to be a standard logistic distribution or a standard normal distribution is called a logistic Rasch model or a normalogive Rasch model, respectively.

Many tests contain items that are scored on more than two levels. Suppose that the /th item of [Page 1103]a test has possible scores denoted by k= 0, 1,…, J. for all test items indexed by / = 1,…, /. The Rasch partial credit model provides an approach to analyze test items scored on multiple levels, given by

for examinees i= 1, …, nand test items indexed by j= 1, …, j, where δjkrepresents the difficulty of attaining the kth score level in item /jfixing G(θi- δi0) = 1, for all / = 1, …, J]. The Rasch rating scale model is jspecial case of the partial credit model, where it is assumed that all items of a test are scored on the same number of levels, with the levels having difficulties that are the same across items, that is, mi= m(> 1) and δjk= τkfor all test items j= 1, …, Jand all score levels k= 0, 1, …, m. Note also that the Rasch model for two-level items, presented in Equation 1, is also a special case of the partial credit model, where mj= 1 and δjk= δjfor all test items j= 1, …, J. Henceforth, to maintain simplicity in discussion, the item parameters of a Rasch model are represented by δjk(for all k= 1, …, mjand all j= 1, …, J), with the understanding that δjkhas a specific form for either the dichotomous, partial credit model, or rating scale Rasch model, as mentioned.

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