Richard L. Shull

Mathematical description of operant behavior : an introduction

Publisher Summary This chapter discusses mathematical description of operant behavior. The language of mathematics encourages descriptions that are precise, succinct, and general. An equation, for example, describes precisely and succinctly how the terms are related to one another. The term on the left specifies the dependent variable, and the term on the right specifies the independent variable. A common strategy for generating ideas for mathematical models is to conceptualize the behavioral phenomenon as analogous to some other phenomenon for which effective mathematical descriptions have already been developed. Then, with appropriate adjustments and translations, the mathematical description can be applied to the new case. Under many learning procedures, the probability of a response increases as a function of trials. But the size of the increase progressively decreases with each successive trial.

Bouts, Changeovers, and Units of Operant Behavior

Responding of various types occurs as bouts of activity separated by pauses. Deprivation and reinforcement affect the tendency to initiate and persist engaging in bouts but have little effect on performance once engaged. Such a two-state conception encourages time-based measurement even when the response is recorded as a brief, discrete event like a lever press. The two-state conception also emphasizes that response rate, as commonly calculated, is a mixture of different units controlled by different sets of variables. Focusing on the components of the two states--especially bout-initiations (or changeovers)--might help solve some puzzles presented by total response-rate data and might help clarify how schedules of reinforcement affect response rate.

Schedule-Induced Running and Chamber Size

Running-wheel behavior was examined as a function of the floor area of the experimental chamber in three food-deprived rats when a food pellet was delivered each minute and when it was omitted. Running-wheel behavior when food was omitted was unsystematically related to the floor area. When food was scheduled, three measures of running-wheel behavior were found to be decreasing functions of floor area: percentage of the session time spent in the wheel, percentage of the interfood intervals with a wheel entry, and the mean stay time per interfood interval with a wheel entry. Wheel revolutions per session varied unsystematically, and the local rate of running was an increasing function of floor area when food was scheduled. These results explicate some inconsistent findings in the literature, and provide support for the notion that wheel-running is not a schedule-induced behavior.

Postreinforcement pause duration on fixed-interval and-fixed time schedules of food reinforcement

Three pigeons were studied under three different schedules that arranged equivalent constant minimum-time intervals between food presentations. In the first, a fixed-interval schedule, reinforcement depended on a response after 2 min had elapsed. In the second, a conjunctive fixed-time fixed-ratio 1 schedule, reinforcement depended on an elapsed 2-min interval and a response without regard to order. In the third, a fixed-time schedule, food was presented every 2 min without regard to the bird’s responding. Response rates were much lower under the latter two schedules than under the fixed-interval schedule. Postreinforcement pauses were about equal under all three conditions, provided the duration of the postreinforcement pause for the fixed-time schedule was based only on intervals containing at least one response.

Statistical inference in behavior analysis: Discussant's remarks.

A collection of essays on the roles of inferential statistics in behavior-analytic research prompted consideration of five issues: (a) the acceptance of research that focuses on the behavior of individual organisms; (b) the need to apply methods thoughtfully; (c) the heuristic value of statistical description; (d) the treatment of aberrant data in the search for general principles; and (e) the role of derived measures in the search for invariances.

Effects of changeover delay in concurrent schedules of reinforcement of key pecking by pigeons

Examination of pigeons’ key-pecking performance, reinforced on concurrent variable-interval 1-min variable-interval 3-min schedules, as a function of the changeover delay revealed the following relationships. As the changeover delay was increased from 0 through 20 sec (1) relative response frequency on the key color associated with the 1-min schedule increased, (2) relative time spent in the 1-min schedule increased, (3) obtained relative frequency of reinforcement in the 1-min schedule increased, and (4) changeover rate decreased.

Latencies on response-initiated fixed-interval schedules: Effects of signaling food availability

Three pigeons were trained on a response-initiated fixed-interval schedule of food reinforcement where the first keypeck in the presence of one key color changed the key color and started a fixed time interval, after which a keypeck produced food. The latency to begin the fixed-interval component was an increasing function of the fixed-interval duration, which was either 40 or 80 sec. At both interval durations, a condition was also studied where the key color changed again at the end of the fixed interval, signaling the availability of food reinforcement. Adding the signal reduced the rate of keypecking during the fixed-interval component but did not systematically affect the initial latency to start the fixed-interval component.

BRIEF STIMULI PAIRED WITH FOOD: COMPARISON OF TWO SCHEDULE ARRANGEMENTS~

Pecks by pigeons produced food according to a fixed-interval schedule. Pecks also produced a brief blackout according to a small fixed-ratio schedule. Each food delivery was immediately preceded by a brief blackout, but not all blackouts were followed by food. The schedule of food and the schedule of blackouts were combined two ways. On one, a second-order schedule, each fixed-ratio completed during the fixed-interval produced the blackout. The first fixed-ratio completed after the fixed-interval elapsed produced the blackout-food compound. The second, a conjoint schedule, was identical except that the first peck after the fixed-interval elapsed produced the blackout-food compound regardless of the number of responses since the last blackout. Although the blackout was paired with food and was produced on a small fixed-ratio schedule under both arrangements, there was evidence of fixed-ratio-like response patterns only on the second-order schedule.