Ineligibility Constraints in .NET

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Ineligibility Constraints
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A natural way to implement item-eligibility decisions is through the shadowtest approach in the preceding section The decision of item i being ineligible for the current examinee implies the addition of the following simple constraint in the model in (23) (29): xi 0: 2:11
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If item i remains eligible, no ineligibility constraint is added to the model The idea is to make these constraints probabilistic, that is, impose them with a certain probability for each item in the pool The probabilities are chosen to be adaptive; that is, as we will show below, they are updated to a larger value if an item tends to be exposed too frequently and to a lower value otherwise For a real-life CAT program, the number of items that need control is invariably small It is only the items with the largest value of their information function over a certain range of abilities that tend to be overexposed The probability of ineligibility will therefore be negligible for the majority of the items Nevertheless, in principle, it is possible that too many ineligibility constraints are imposed for an occasional examinee and the model becomes infeasible, that is, has no solution left In the empirical study reported below, this never happened, but for the sake of completeness we will consider the possibility of infeasibility in our derivation of the probabilities of eligibility Our assumption is that if infeasibility does happen, all ineligibility constraints are removed from the model for the examinee, and the shadow test is calculated for this relaxed model The adaptive nature of the probabilistic experiment our model automatically corrects for an occasional extra exposure of an item due to this measure later in the testing process
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Derivation of Probabilities of Item Eligibility
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The process of item administration for a given pool of items and population of examinees is determined by the probabilities of four different events for a random examinee The events are (1) event Ei of item i being eligible for the
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examinee, (2) event F of the shadow test remaining feasible after all eligibility constraints have been added to the model for the examinee, (3) event Si of item i being selected for the examinee, and (4) event Ai of item i being administered to the examinee We will use a bar over the symbols of these events to denote a complementary event For example, the event of infeasibility, (and hence of removing the ineligibility constraints from the model) is denoted as F For these four events it holds generally that an item can be administered only if it is selected Likewise, it can be selected only if it is eligible or all ineligibility constraints have been removed from the model because of infeasibility In set-theory notation, it thus holds that Ai & Si & fEi [ Fg: For the probabilities of the four events it follows that P Ai P Ai j Si P Si j Ei [ F P Ei [ F : However, the rst two factors in the right side simplify as follows: P Ai j Si P Si j Ei [ F P Ai j Ei [ F : Therefore, the probability of administering item i can be written as P Ai P Ai j Ei [ F P Ei [ F : 2:14 2:13 2:12
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Substituting this expression for P Ai into the one with the upper bound in (210) shows that this upper bound is always met if P Ei [ F rmax : P Ai j Ei [ F 2:15
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This inequality does not impose any direct constraint on the probabilities of item eligibility P Ei , but making a very mild assumption on the (unlikely) event of infeasibility and following the derivation by van der Linden and Veldkamp (2004) it can be shown that the following constraint should be imposed: P Ei 1 1 rmax P Ei [ F ; P F P Ai P F P Ai > 0; P F > 0; 2:16
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with P j Ai > 0 and P j F > 0: Thus, if we want to keep the probabilities of item administration P Ai below a maximum rmax , all we have to do is keep the probabilities of item eligibility below the upper bound in (216)
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