Item–Response (Rasch) Models

Item–response models define a set of data reduction techniques that relate observed discrete variables to a much smaller set of unobserved latent variables. These models are primarily used in political science to relate observed votes on legislative roll calls to legislators' unobserved ideology. Originally developed in the context of educational testing, these models were designed to recover characteristics of test questions (items) and the “abilities” of student test takers. Consequently, much of the model description and interpretation retains education terminology. The data involves N test takers (indexed by i) answering M test questions (indexed by j), with an answer (response) by student i on question j (yij) equal to one if correct, zero otherwise. In legislative voting, the “test takers” are the legislators, the “items” are the roll calls, and the “ability” is the legislator's ideology. The canonical model for the answers posits the following probability model:

where xi is the student's ability and parameters βj and αj characterize the question. This is simply a logit model of a correct answer as a function of one independent variable, the ability measured by the test, with the additional complication that the independent variable is unobserved. Of course, other cumulative density functions can be used instead (e.g., the normal for a probit model). This is a latent variable model, similar to factor analysis. The test items are all observable indicators (manifest variables) of the unobservable common factor x, with βj analogous to a factor loading. In the item–response case, all the indicators are binary, whereas they are continuous in factor analysis.

The probability model is frequently reformulated as

with specific interpretations to each item parameter. The slope coefficients βj are the discrimination parameters and the values of κj are difficulty parameters. Discrimination is the extent to which a test question separates students with higher and lower levels of abilities. If βj = 0; then the test question does not measure the ability gauged by the test. Higher values of κj indicate a more difficult question. κj represents the ability level of a student who has a 50% probability of answering the question correctly and 50% answering incorrectly. Note, κj and xi are on the same scale; if xi = κj; Pr(yij = 1) = .5.

Other Item–Response Models

A special case is the one-parameter model, known as the Rasch model (after Georg Rasch):

This model assumes that every item discriminates abilities equally well (βj = 1 for all questions j). In this model, all the observed information is contained in the total score, the number of correct responses: an extremely desirable property for tests. This property is only true for the logistic one-parameter model though.

A generalization of the two-parameter model is the three-parameter model:

where γj is a “pseudoguessing” parameter. It allows for the absolute worst student (with ability approaching negative infinity) to have a positive probability of answering a question correctly; that is, a student answers a question correctly, not because of a higher level of ability but because he or she happened to mark the answer correctly. The “pseudo” aspect of guessing is that students do not necessarily guess but use reasoning processes unrelated to their ability (sometimes actually worse than pure guessing).

...