Exemplar Theoretical Item Response Function (IRF) for the Three-Parameter Logistic Item Response Theory (IRT) Model

This figure contains a plot of the theoretical item response function (IRF) for a dichotomous item. The item is a multiple-choice item with one correct response and three alternative responses. In the plot, the horizontal axis represents the theta () scale, while the vertical axis represents the probability of a correct response. The solid curve is the theoretical IRF based on the item parameter estimates and the equation for the three-parameter logistic IRT model. In this case the scale has a mean of zero and a standard deviation of one.

Example Plot

SOURCE: U.S. Department of Education, Institute of Education Sciences, National Center for Education Statistics, National Assessment of Educational Progress (NAEP), 2000 and 2001 Assessment.

For items of this type, if students randomly select one of the four responses they would be expected to choose the correct response about 25 percent of the time. Because this is so, students low on the scale are expected to have a chance to respond correctly to this item. In fact, for this particular item, exactly 25 percent of the students low on the scale will respond correctly to the item (c = 0.25). In practice, some students may be able to rule out one or more of the alternative responses or one of the alternatives might be very attractive to some students, so the chance that students low on the scale would select the correct response exactly 25 percent of the time for an actual multiple-choice item with 4 response choices is not likely. As can be expected, students high on the scale are more likely to respond correctly to this item.

The b-parameter indicates the place on the scale where the curve is most steep. The value of the b-parameter is at the point that students have a probability of responding correctly to the item that is halfway between the c-parameter value and 1.0. In this case, that happens when the probability is 0.625 (0.25 + (1.0 - 0.25)/2). For this item b = 1.0. If the c-parameter value is 0.25, then items with b-parameter values greater than 1.0 are more difficult than this item and items with b-parameter values less than 1.0 are less difficult than this item.

The a-parameter indicates the maximum slope (steepness) of the curve. This happens at the point on this curve that corresponds to the b-parameter value 1.0. The a-parameter value for this item is 1.2. If the c-parameter is 0.25, then items with a-parameter values greater than 1.2 are more discriminating than this item and items with a-parameter values less than 1.2 are less discriminating. When an item is more discriminating the curve is steeper, allowing a more exact prediction on the basis of their responses to this item of whether students are above or below 1.0 on the scale.

Last updated 18 July 2008 (KL)

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