c Parameter

Why Is the c Parameter Needed?

Selected-response item formats, such as multiple-choice and true–false items, are frequently used on assessments because of their efficiency; selected-response items can sample more of the content domain that a test covers, per unit of testing time, than other item formats. However, some examinees can answer these items correctly as a result of some degree of chance, perhaps depending on how many response options the examinee can accurately eliminate. Guessing is generally not a concern with constructed-response item types. The c parameter helps IRT account for an examinee with extremely low ability correctly answering selected-response items.

Unidimensional IRT Models for Dichotomous Items

The most commonly used IRT models assume that (a) a single ability underlies an examinee’s response process and (b) items are dichotomously scored (e.g., right vs. wrong). The relationship between the probability p of a correct response to a [Page 230]dichotomously scored item i and examinee ability θ 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). Several parameters are used within IRT to control the position and shape of the ICCs. The 3PL model has the following:

• 1.
  a discrimination parameter, denoted a, that controls the slope of the ICC;
• 2.
  the difficulty parameter, denoted b, that indicates the point on the θ scale where there is an inflection (i.e., where the concavity of the curve changes) in the ICC; and
• 3.
  a pseudo-guessing parameter, denoted c, that represents the lower asymptote of the ICC.

The probability of a correct response can then be calculated as:

$$P\left(\theta \right)=c+\left(1-c\right)\frac{1}{1+e^{-a\left(\theta -b\right)}}.$$

Figure 1 shows 3PL ICCs for four hypothetical items. All 4 items have b parameters that are fairly high in value to allow the lower asymptote to be visually prominent at the lowest θ values shown in this graph. Depending on any given item’s b parameter value and the range of θs shown in the graph, the lower asymptote may not always be as distinct. As shown in Figure 1, the higher the value of c, the more likely it is that examinees with extremely low ability will answer the item correctly.

The theoretical value of the c parameter should be 1 divided by the number of response options for the item. For an item with two response options, like a true–false item, the theoretical value would be 1/2, for an item with three options, the suspected value would be 1/3, and so on. Although these theoretical expectations are reasonable, in reality, they do not hold in most cases. Very frequently, c is lower than the inverse of the number of response options. It is well-known that experienced item writers use common student misconceptions in the incorrect response options and that will make the empirical c value lower than the theoretical value. Of course, the empirical c value can also be higher than the theoretical value, perhaps when some misconceptions are not very common among students.

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