Response-reinforcer contingency and spatially defined operants: testing an invariance property of phi.
Phi coefficient invariance breaks down under one diagonal-cell permutation, so don’t assume it always captures local contingency processes.
01Research in Context
What this study did
Fetterman et al. (1989) tested whether the phi coefficient stays the same when you flip the layout of response and reinforcer cells. They ran four permutations of a spatial operant task with percentile schedules.
The goal was to see if phi, a common measure of response-reinforcer contingency, is truly invariant under different spatial mappings.
What they found
Phi held steady for three of the four layouts. In the fourth, a diagonal-cell flip, the prediction broke down.
This means phi invariance has a boundary condition. It does not always capture local contingency processes when the spatial mapping changes.
How this fits with other research
Roper (1978) had already argued that reinforcement works through learned predictive relations, not invisible response strengthening. Fetterman et al. (1989) give lab data showing one metric of those relations can fail.
Davison et al. (1984) warned that matching on concurrent VI VR might be an artifact of the schedule itself. Fetterman et al. (1989) echo that caution: another beloved measure, phi, can also mislead if the setup shifts.
Together, these papers tell us to test our tools, not just trust them.
Why it matters
When you design teaching arrangements, do not assume that a high phi value always equals a strong contingency. If you change where responses happen or how reinforcers are mapped in space, rerun your analysis. Check that the relation still holds before you stake your intervention on it.
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02At a glance
03Original abstract
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.
Journal of the experimental analysis of behavior, 1989 · doi:10.1901/jeab.1989.51-145