b Parameter

Context

IRT models the probability that an examinee will make a particular response to an item based on examinee’s standing on the trait that a test measures. For a mathematics achievement test, IRT can model the probability that an examinee will earn a particular score on an item based on the examinee’s achievement in math. For an instrument measuring extroversion, IRT can model the probability that an examinee will select the response very accurate for the statement I make friends easily based on the examinee’s degree of extroversion. Virtually any type of instrument (e.g., ability and achievement tests, personality and attitude assessments, and questionnaires and surveys) and any type of item (e.g., true or false, multiple choice, Likert) can be analyzed using IRT.

Importance of b Parameter

There are many practical benefits to the b parameter relative to classical measures of item difficulty. First, b parameters are not group dependent (provided the IRT model fits the data). Another advantage is that item difficulty, b, and examinee ability, θ, are on the same scale. This, combined with the fact that an item’s maximum information occurs at or near the b parameter, means that inspection of the b parameters is helpful in the test construction process.

Unidimensional IRT Models for Dichotomous Items

The most common IRT models assume that (a) a single ability underlies the examinees’ response processes and (b) items are dichotomously scored (e.g., right vs. wrong). The relationship between the probability of a correct response and examinee ability has a monotonically increasing ICC that is roughly s shaped, although it can be compressed and stretched at various points along the ability scale (see Figure 1).

Figure 1 Item characteristic curves for five hypothetical items

The b parameter describes the ICC’s location on the θ scale. Specifically, it shows where there is an inflection point on the θ scale (i.e., where the concavity of the curve changes) in the ICC.

[Page 162]The three-parameter logistic (3PL) model has:

• 1.
  a discrimination parameter, denoted a, that controls the slope of the ICC;
• 2.
  a pseudo-guessing parameter, denoted c, that represents the lower asymptote of the ICC; and
• 3.
  a difficulty parameter, denoted b.

The two-parameter logistic (2PL) model does not have the pseudo-guessing parameter, c (or equivalently, one can consider c = 0). The one-parameter logistic (1PL) model also does not have the pseudo-guessing parameter, c, and all items are assigned a common slope, meaning items are modeled with only the b parameter.

...