This means that an item's information and thus its precision of measurement is highest for those persons whose ability complies with the item's difficulty and who consequently have a probability of P i = .50 of solving that item (see section "The Rasch model").
For an item to fall into the T category, one of the following two criteria must be met: Either the item is designed in such a way that some theorem presumably known to the student significantly simplifies the task of solving the item, or the item is such that one would expect use of some theorem from the student.
Finally, since items referred to contextual stimuli, which were presented in textual formats, reading ability can be regarded as another, shared demand of solving the items.
The expectancy of solving an item successfully and the personal value of taking the test are considered antecedents of test-taking engagement.
However, the perceived expectancy of being capable of solving an item may vary considerably across test-takers (Asseburg and Frey 2013; Cole et al. 2008).
The probability of solving an item depends only on the respondent's ability, or proficiency, and two item parameters characterizing the properties of the item (item difficulty and item discrimination).
These two parameters relate negatively (θi – δi), because the probability of solving an item increases with the person's ability but decreases with greater item difficulty (Wright & Stone, 1979).
Traditional approaches to measurement scales rely on averages or a simple summation of the test scores; IRT models instead reflect the assumption that the probability of solving an item depends on the test taker's latent trait or ability (i.e., θi = person parameter), combined with the item difficulty (i.e., δi = item parameter).
Thus, a basic model of test-taking motivation and engagement should include the "expectancy" of solving the test item as well as the "value" that the test-taker attaches to solving the test item (see Fig. 1).
A specific person's probability P ni of solving item i (right side of Eq. 1) is dependent upon the person's ability B n and the item's difficulty D i (left side of Eq.1).
