a Parameter

Defining the a Parameter

Using a mathematical formula, the IRT theory defines the probability of an examinee’s correct response to an item as a function of the latent ability of the examinee and that item’s properties. This function, the item characteristic curve (ICC), is also referred to as the item response function. An ICC defines a smooth nonlinear relationship between latent trait constructs (θ) and probability of a correct response. If assumptions are met, the ICCs can be stable over groups of examinees, and the θ scale also can be stable even when the test includes different items. A graphical representation of an ICC is given in Figure 1.

Figure 1 Example of an ICC

The generic mathematics function for Figure 1 is shown in Equation 1.

$$P(x_i=1|\theta )=c_i+(1-c_i)\frac{\exp (a_i(\theta -b_i))}{1+\exp (a_i(\theta -b_i))},$$

where θ is the examinee ability parameter, ci is often referred to as the pseudo-guessing or lower asymptote parameter with a value typically between 0 and 0.25, bi is the location or difficulty [Page 2]parameter, and ai is the discrimination or slope parameter. The parameter ai indicates the steepness of ICC at θ = b, where probability of correctly answering an item changes most rapidly. The logistic function presented in Equation 1 is called the three-parameter logistic (3PL) model, which was presented by Allan Birnbaum in a pioneering work by Frederic Lord and Melvin Novick in 1968. The two-parameter logistic (2PL; Equation 2) and one-parameter logistic (1PL) model (Equation 3) can be considered as special cases of the 3PL IRT model. As indicated in the corresponding formula, the 2PL model only has the a and the b parameter, whereas the 1PL model only has the b parameter and the a parameter is fixed.

$$P(x_i=1|\theta )=\frac{\exp (a_i(\theta -b_i))}{1+\exp (a_i(\theta -b_i))},$$

$$P(x_i=1|\theta )=\frac{\exp (\theta -b_i)}{1+\exp (\theta -b_i)}.$$

Interpreting the a Parameter

Figure 2 represents ICCs for three dichotomous items under the unidimensional 3PL IRT model. Among these 3 items, all b’s = 0 and all c’s = 0.2, but a values differ: a1 = 0.5, a2 = 1, and a3 = 1.5.

Figure 2 ICCs for three example items

As shown in Figure 2, the slope of Item 1, at the location where examinees’ abilities are about the same as the item’s difficulty (θ = b = 0), is the flattest among the three items, whereas the slope of Item 3 at the same location is the steepest. Therefore, Item 1 has the lowest discrimination value among the three and Item 3 has the highest. If identifying two examinees of whom one has an ability larger than zero and one smaller than zero on the ability axis, the difference between the probabilities of the two students answering Item 3 correctly will be greater than Item 1 or 2. It is therefore easier to discriminate between the two examinees using Item 3, compared to Item 1 or 2. With all else equal, Item 3 can be concluded as more desirable because it can effectively distinguish among examinees differing in ability.

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