Guessing Parameter

In item response theory (IRT), the guessing parameter is a term informally used for the lower asymptote parameter in a three-parameter-logistic (3PL) model. Among examinees who demonstrate very low levels of the trait or ability measured by the test, the value of the guessing parameter is the expected proportion that will answer the item correctly or endorse the item in the scored direction. This can be understood more easily by examining the 3PL model:

where θ is the value of the trait or ability; P(θ) is the probability of correct response or item endorsement, conditional on θ; ai is the slope or discrimination for item i; bi is the difficulty or threshold for item i; and ci is the lower asymptote or guessing parameter for item i. Sometimes, the symbol g is used instead of c. In Equation 1, as θ decreases relative to b, the second term approaches zero and thus the probability approaches c. If it is reasonable to assume that the proportion of examinees with very low θ who know the correct answer is virtually zero, it is reasonable to assume that those who respond correctly do so by guessing. Hence, the lower asymptote is often labeled the guessing parameter. Figure 1 shows the probabilities from Equation 1 plotted across the range of θ, for a = 1.5, b = 0.5, and c = 0.2. The range of θ is infinite, but the range chosen for the plot was −3 to +3 because most examinees or respondents would fall within this range if the metric were set such that θ had a mean of 0 and standard deviation of 1 (a common, though arbitrary, way of defining the measurement metric in IRT). This function is called an item characteristic curve (ICC) or item response function (IRF). The value of the lower asymptote or guessing parameter in Figure 1 is 0.2, so as θ becomes infinitely low, the probability of a correct response approaches 0.2.

Figure 1 Item Characteristic Curve

Guessing does not necessarily mean random guessing. If the distractors function effectively, the correct answer should be less appealing than the distractors and thus would be selected less than would be expected by random chance. The lower asymptote parameter would then be less than 1/number of options. Frederic Lord suggested that this is often the case empirically for large-scale tests. Such tests tend to be well-developed, with items that perform poorly in pilot testing discarded before the final forms are assembled. In more typical classroom test forms, one or more of the distractors may be implausible or otherwise not function well. Low-ability examinees may guess randomly from a subset of plausible distractors, yielding a lower-asymptote greater than 1/number of options. This could also happen when there is a clue to the right answer, such as the option length. The same effect would occur if examinees can reach the correct answer even by using faulty reasoning or knowledge. Another factor is that examinees who guess tend to choose middle response options; if the correct answer is B or C, the probability of a correct response by guessing would be higher than if the correct answer were A or D.

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