Testing the linearity and independence assumptions of the generalized matching law for reinforcer magnitude: a residual meta-analysis

We conducted a residual meta-analysis to test the assumptions of the generalized matching law that effects of relative reinforcer magnitude on response allocation in concurrent schedules can be described by a power function and are independent from the effects of relative reinforcer rate. We identified five studies which varied magnitude ratios over at least four levels and six studies in which reinforcer rate and magnitude ratios were varied factorially. The generalized matching law provided a reasonably good description of the data, accounting for 77.1% and 90.1% of the variance in the two sets of studies. Results of polynomial regressions showed that there were no systematic patterns in pooled residuals as a function of predicted log response ratios for data sets in which relative magnitude was varied. For data sets in which relative rate and magnitude were varied factorially, there was a significant negative cubic pattern in the pooled residuals, suggesting that obtained response allocation was less extreme than predicted for conditions with extreme predicted values. However, subsequent analyses showed that this result was associated with results from conditions in one study in which the product of the rate and magnitude ratios was 63:1, and in which response allocation may have been attenuated by a ceiling effect. When data from these conditions were omitted, there were no significant components in the residuals. Although the number of available studies was small, results provide tentative support for the assumptions of the generalized matching law that effects of reinforcer magnitude ratios on choice can be described by a power function and are independent from reinforcer rate ratios.