Tests of the " Beta Model

over the set of r alternatives available to an organism. This vector gives the probability that each alternative will be chosen on trial n. A transition operator T is postulated such that ( i ) T does not depend upon n (independence of path); (ii) T depends upon the choice made on trial n and on the outcome; and (iii) pn+, = Tpn. For the most part linear (matrix) operators have been studied-partly because their mathematical properties are comparatively simple, partly because of Estes' stimulus-sampling rationale [3], and partly because the combining-ofclasses condition ([I], [ Z j ) leads to a particular type of linear operator. Nonetheless, it is still an open question whether linearity is a tenable assumption or whether one of the possible nonlinear operators will be better able to describe data. The problem, of course, is how to select among all the possible nonlinear operators. The purpose of this paper is to study some properties of a nonlinear model, called the beta model, and to apply it to three published experiments (Chapter 14 and [6]). The linear model, called the alpha model, and the beta model can both be arrived at from the same general considerations. For this reason, comparisons between these two models are made.