The statistical properties of molecular events in concurrent variable-interval schedules

Abstract

Choice between concurrent variable-interval variable-interval schedules of reinforcement is examined from a molecular perspective. Molecular measures of performance include the duration of time spent responding before a changeover and the number of responses emitted before a changeover. A molecular-ratio measure of preference was derived as the logarithm of the ratio of consecutive inter-changeover responding-times. Molecular-ratio measures of performance may be profitably used to test aspects of mathematically-oriented theories of choice not usually encountered directly in the experimental analysis of behaviour. The Bradley-Terry-Luce (Choice Axiom) and Thurstone choice models were reviewed as possible candidates to summarise the statistical and distributional characteristics of the molecular-ratio measures of choice. The assumptions and predictions of the Choice Axiom closely resemble the details of the strict matching law which states that the proportion of responses emitted on an alternative equals the proportion of reinforcements obtained. As predicted by the Choice Axiom, the logistic distribution is shown to account for the stable-state ln responding-time ratio data. It is also proposed that a transformation of the variance of the logistic distribution of ln responding-time ratios from a single experimental session gives an acceptable approximation to the slope of the molar-level response-reinforcer relation. The logistic distribution was derived as a description of the distribution of the ratio of reinforcements. It is shown theoretically how the variance in the distribution of reinforcements in time is affected by whether exponential or arithmetic schedules are arranged, and how this variance affects the slope of the relation in the generalised matching law which states that the logarithm of the ratio of responses is a linear function of the logarithm of the ratio of reinforcements obtained. The theory predicts that concurrent (exponential) schedules will provide a slope of 1.0 (conforming to the matching law), but concurrent (arithmetic) schedules will provide a slope of approximately.78. A reanalysis of 18 published experimental reports confirmed these predictions for response-allocation data. The predictions were also confirmed for time allocation on arithmetic schedules. However, time allocation, as measured in concurrent schedules, may not always be an unbiased measure. Some statistical distributions are proposed and examined as descriptive accounts of molecular-level responding. In general, the empirical distributions of interchangeover responding-times at an alternative were not well fitted by the exponential distribution. The model of response allocation between two alternatives advanced by Heyman (1979a) was shown to be inadequate. Two procedures were used to assess response-by-response dependence; one was based on a Bayesian modification of Akaike’s Information Criterion, the other was based on modified geometric distributions of response run lengths conditioned on first-, second- and third-order dependence. For concurrent (exponential) variable-interval schedules, programmed without a changeover delay, models based on first- and second-order dependence provided acceptable fits for most of the data. Response run lengths on concurrent (arithmetic) schedules, arranged with a changeover delay, could not be fitted to models consistent with less than third-order dependence.