McDowell's expertise in mathematics and behavior modification spurred him to apply Her-rnstein's matching equation for a single operant to a clinically relevant problem. Carr and McDowell (1980) had been involved in the treatment of a 10-year-old boy who repeatedly and severely scratched himself (Fig. 9.15). Before treatment the boy had a large number of open sores on his scalp, face, back, arms, and legs. In addition, the boy's body was covered with scabs, scars, and skin discoloration, where new wounds could be produced. In their 1980 paper, Carr and McDowell demonstrated that the boy's scratching was operant
FIG. 9.15. Rate of social reinforcement and self-injurious scratching of a young boy. The data were fitted by Herrnstein's single-operant equation (Equation 9.6). Values of k and Re and percentage variance accounted for by the curve fit are shown. Adapted from Quantification of Steady-State Operant Behavior (pp. 311-324), by J. J. McDowell, 1981, Amsterdam: Elsevier/North-Holland.
behavior. Careful observation showed that the scratching occurred predominantly when he and other family members were in the living room watching television. This suggested that the self-injurious behavior was under stimulus control. In other words, the family and setting made scratching more likely to occur.
Next, Carr and McDowell (1980) looked for potential reinforcing consequences maintaining the boy's self-injurious behavior. The researchers suspected that the consequences were social, because scratching appeared to be under the stimulus control of family members. In any family interaction there are many social exchanges, and the task was to identify those consequences that reliably followed the boy's scratching. Observation showed that family members reliably reprimanded the boy when he engaged in self-injury. Reprimands are seemingly negative events, but the literature makes it clear that both approval and disapproval may serve as reinforcement. Although social reinforcement by reprimands was a good guess, it was still necessary to show that these consequences in fact functioned as reinforcement. The first step was to take baseline measures of the rate of scratching and the rate of reprimands. Following this, the family members were required to ignore the boy's behavior. That is, the presumed reinforcer was withdrawn (i.e., extinction), and the researchers continued to monitor the rate of scratching. Next, the potential reinforcer was reinstated by having the family members again reprimand the boy for his misconduct. Relative to baseline, the scratching decreased when reprimands were withdrawn and increased when they were reinstated. This test identified the reprimands as positive reinforcement for scratching. Once the reinforcement for scratching was identified, behavior modification was used to eliminate the self-injurious behavior.
In a subsequent report, McDowell (1981) analyzed the boy's baseline data in terms of the quantitative law of effect. He plotted the reprimands per hour on the X-axis and scratches per hour on the Y-axis. McDowell then fit the matching equation for a single schedule of reinforcement (Equation 9.6) to the points on the graph. Figure 9.15 shows the plot and the curve of best fit. The matching equation provides an excellent description of the boy's behavior. You will notice that most of the points are on, or very close to, the hyperbolic curve. McDowell has indicated the significance of this demonstration. He states:
As shown in the figure [9.15] the single-alternative hyperbola accounted for nearly all the variance in the data. This is especially noteworthy because the behavior occurred in an uncontrolled environment where other factors that might have influenced the behavior had ample opportunity to do so. It may be worth emphasizing that the rates of reprimanding ... occurred naturally; that is, they were not experimentally arranged Thus, the data .. . demonstrate the relevance of matching theory to the natural ecology of human behavior. (McDowell, 1988, pp. 103-104)
Overall, the quantitative law of effect or Herrnstein's hyperbolic equation has been an important contribution to the understanding of human behavior and to the modification of human behavior in applied settings (see Martens, Lochner, & Kelly, 1992, for further evidence; Fisher & Mazur, 1997, for a review).
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