| Abstract | I refer to a competence test, Tm, consisting of m binary items. The items are represented by variables, X1, . . . , Xm, having values 1 (if there is a correct answer) or 0 (otherwise). Values of these variables for the members of a population (or sample) P are given by vectors xi= (xi1, . . . , xim), the sum score is denoted si:= Σjxij; i identifies members of P. A standard approach to the estimation of individual abilities w.r.t. Tm uses the Rasch model. This model postulates item parameters δ=(δ1, . . . , δm), and for each person i a parameter θi, which together deter-mine probabilities πRij:= Pr(Xj= 1|θi, δj) :=L(θi−δj) where L(x) := exp(x)/(1 exp(x)), for person i’s correctly answering to item j. A problem with this approach concerns the interpretation of these probabilities. How to understand, for example, that a person can correctly solve a mathematical task with a probability 0.2, or 0.4, or 0.6? In this paper I consider an alternative approach which defines response probabilities πij by a reference to a distinction between ‘knowing’ and ‘not knowing’ (and possibly guessing) the correct answer to an item. By introducing interval-valued response probabilities, this approach also allows one to express the idea that a person’s ability to correctly solving items is, to some degree, a vague notion. In Section 2 I introduce the approach for tests containing items which cannot be solved by guessing. In Section 3 I discuss multiple-choice (MC) items, and in Section 4 I compare the approach with the Rasch model. (Orig.). |
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