b Parameter

Parameter within Different Item Response Theory Models

The precise interpretation of the b parameter is dependent on the specific IRT model within which it is considered, the most common being the one-parameter logistic (1PL) or Rasch model, the two-parameter logistic (2PL) model, and three-parameter logistic (3PL) model. Under the 1PL model, the b parameter is the single item feature by which items are distinguished in characterizing the likelihood of a correct response. Specifically, the probability of correct response (Xij = 1) by examinee i to item j is given by

where θj represents an ability-level (or trait-level) parameter of the examinee. An interpretation of the b parameter follows from its being attached to the same metric as that assigned to θ.

Usually this metric is continuous and unbounded; the indeterminacy of the metric is often handled by assigning either the mean of θ (across examinees) or b (across items) to 0. Commonly b parameters will assume values between − 3 and 3, with more extreme positive values representing more difficult (or infrequently endorsed) items, and more extreme negative values representing easy (or frequently endorsed) items.

The 2PL and 3PL models include additional item parameters that interact with the b parameter in determining the probability of correct response. The 2PL model adds an item discrimination parameter (aj), so the probability of correct response is

and the 3PL model adds a lower asymptote (“pseudoguessing”) parameter, resulting in

While the same general interpretation of the b parameter as a difficulty parameter still applies under the 2PL and 3PL models, the discrimination and lower asymptote parameters also contribute to the likelihood of a correct response at a given ability level.

Interpretation of the B Parameter

Figure 1 provides an illustration of the b parameter with respect to the 1PL, 2PL, and 3PL models. In [Page 110]this figure, item characteristic curves (ICCs) for three example items are shown with respect to each model. Each curve represents the probability of a correct response as a function of the latent ability level of the examinee. Across all three models, it can be generally seen that as the b parameter increases, the ICC tends to decrease, implying a lower probability of correct response.

In the 1PL and 2PL models, the b parameter has the interpretation of representing the level of the ability or trait at which the respondent has a .50 probability of answering correctly (endorsing the item). For each of the models, the b parameter also identifies the ability level that corresponds to the inflection point of the ICC, and thus the b parameter can be viewed as determining the ability level at which the item is maximally informative. Consequently, the b parameter is a critical element in determining where along the ability continuum an item provides its most effective estimation of ability, and thus the parameter has a strong influence on how items are selected when administered adaptively, such as in a computerized adaptive testing environment.

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