Compositions and their application to the analysis of choice

Descriptions of steady-state patterns of choice allocation under concurrent schedules of reinforcement have long relied on the "generalized matching law" (Baum, 1974), a log-odds power function. Although a powerful model in some contexts, a series of conflicting empirical results have cast its generality in doubt. The relevance and analytic relevance of matching models can be greatly expanded by considering them in terms of compositions (Aitchison, 1986). A composition encodes a set of ratios (e.g., 5:3:2) as a vector with a constant sum, and this constraint (called closure) restricts the data to a nonstandard sample space. By exploiting this sample space, unbiased estimates of model parameters can be obtained to predict behavior given any number of choice alternatives. Additionally, the compositional analysis of choice provides tools that can accommodate both violations of scale invariance and unequal discriminability of stimuli signaling schedules of reinforcement. In order to demonstrate how choice data can be analyzed using the compositional approach, data from three previously published studies are reanalyzed. Additionally, new data is reported comparing matching behavior given four, six, and eight response alternatives.