Response-reinforcer contingency and spatially defined operants: testing an invariance property of phi.
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A chamber containing 72 response keys defining the circumference of a circle 1 m in diameter was used to examine the relation between differentiation of response location and a measure of response-reinforcer contingency known as the phi coefficient. A different target key was specified in each successive phase, and response location was differentiated with respect to the target. Criterional and noncriterional responses (i.e., responses "near" and "far" from the target) were defined using targeted percentile schedules to control the overall probability of each response class. By manipulating criterional (and, hence, noncriterional) response probability and the reinforcement probabilities conditional on each, a mathematical invariance property peculiar to phi in contingency analysis was examined. Specifically, diagonally interchanging cell frequencies in a 2 x 2 table relating criterional/noncriterional responses to reinforcement/nonreinforcement leaves phi unchanged. Hence, the degree of response differentiation predicted by phi remains unchanged under the four permutations implied by the various diagonal interchanges. This predicted invariance was examined under values of phi equal to .33, .58, and .82. Increasing phi generally increased the stereotypy of response location. Three of the permutations generated almost interchangeable performance at different phi values. The remaining permutation, however, generated functions relating response concentration to phi with slopes shallower than those obtained under the other permutations. This resulted from relatively higher levels of differentiation, compared to the other permutations, at low phi values. These data strongly suggest boundary conditions on the ability of phi to reflect completely the local processes that are indexed by phi at a molar level.