## Experimental Evidence and the Quantitative Law of Effect

The quantitative law of effect (Equation 9.6) has been analyzed in laboratory experiments. In an early investigation, Catania and Reynolds (1968) conducted an exhaustive study of six pigeons that pecked a key for food on different variable-interval (VI) schedules. The rate of reinforcement ranged from 8 to 300 food presentations each hour. Herrnstein

FIG. 9.13. Rate of response as a function of rate of food reinforcement for six pigeons on single VI schedules. The k and Revalues for each fitted curve are shown. Reprinted from Fig. 8 of "On the Law of Effect," by R. J. Herrnstein, 1970, Journal of the Experimental Analysis of Behavior, 21, pp. 243-266; which in turn is based on data from Catania and Reynolds, 1968; copyright 1970 by the Society for the Experimental Analysis of Behavior, Inc.

FIG. 9.13. Rate of response as a function of rate of food reinforcement for six pigeons on single VI schedules. The k and Revalues for each fitted curve are shown. Reprinted from Fig. 8 of "On the Law of Effect," by R. J. Herrnstein, 1970, Journal of the Experimental Analysis of Behavior, 21, pp. 243-266; which in turn is based on data from Catania and Reynolds, 1968; copyright 1970 by the Society for the Experimental Analysis of Behavior, Inc.

(1970), in his classic article on the law of effect, replotted the data from the Catania and Reynolds experiment on X,Y coordinates. Figure 9.13 shows the plots for the six birds, with reinforcements per hour on the X-axis and responses per minute on the Y-axis.

Herrnstein used a statistical procedure to fit his equation to the data of each pigeon. Figure 9.13 presents the curves that best fit these results. Notice that all of the birds produce rates of response that are described as a hyperbolic function of rate of reinforcement. Some of the curves fit the data almost perfectly (e.g., pigeon 281), whereas others are less satisfactory (e.g., pigeon 129). Overall, Herrnstein's quantitative law of effect is well supported by these findings.

The quantitative law of effect has been extended to magnitude of food reinforcement, brain stimulation, quality of reinforcement, delay of positive reinforcement, rate of negative reinforcement, magnitude or intensity of negative reinforcement, and delay of negative reinforcement (see de Villiers, 1977, for a thorough review). In a summary of the evidence, Peter de Villiers (1977) stated:

The remarkable generality of Herrnstein's equation is apparent from this survey. The behavior of rats, pigeons, monkeys and. .. people is equally well accounted for, whether the behavior is lever pressing, key pecking, running speed, or response latency in a variety of experimental settings. The reinforcers can be as different as food, sugar water, escape from shock or loud noise or cold water, electrical stimulation of a variety of brain loci, or turning a comedy record back on. Out of 53 tests of Equation [9.6] on group data the least-squares fit of the equation accounts for over 90% of the variance in 42 cases and for over 80% in another six cases. Out of 45 tests on individual data, the equation accounts for over 90% of the variance in 32 cases and for over 80% in another seven cases. The literature appears to contain no evidence for a substantially different equation than Equation [9.6]... . This equation therefore provides a powerful but simple framework for the quantification of the relation between response strength and both positive and negative reinforcement. (p. 262)