Journal of Behavioral Education, VoL 2, No. 2, 1992, pp. 205-209
Commentary
Applied Quantitative Behavior Analysis: A View From the Laboratory Michael Davison, Ph.D. 1,2
Accepted: August 23, 1991. Action Editor: Nirbhay N. Singh
My bookshelf tells me that I ceased reading in Applied Behavior Analysis at the end of 1976. The first and second paragraphs of Martens' (1992) paper correctly diagnosed the reason for my extinction: I saw only the application of an outmoded and limited behavioral technology, with no interest in the way that the basic science, on which behavioral technology was supposedly built, was changing and developing. I became bored. But, one rainy Sunday afternoon in about 1980, I wrote a theoretical paper on how matching theory could be used effectively in a classroom situation because I was concerned about the technology gap. I submitted it to JABA and had it roundly rejected - - I suppose it was clear that I know little about the nuances of classrooms, and the anti-quantitative biases of practitioners (does this derive from the anti-statistical bias?). As a result of this history, I appreciate and approve of Martens' (1992) attempts to bring the technology up to speed, or to moderate speed at least. What Martens is not doing, in his paper, is to bring the application of quantitative theories up to date, for the "theories" that he discusses are those firmly rooted in 1970 with Herrnstein's (1970) paper. At the risk of turning everyone off again, things have changed a great deal since 1970. Many of us now are not at all happy with strict matching (Martens' Equation 1). Baum (1974) replaced strict matching with generalized matching, which added some useful and informative parameters to the strict-matching relation, and better described the empirical data. But since that time, many 1Professor of Psychology, University of Auckland, New Zealand. ZCorrespondence should be directed to Michael Davison, Department of Psychology, University of Auckland, Private Bag, Auckland, New Zealand. 207 1053-0819/92/0600-0207506.50/0 9 1992Human SciencesPress,Inc.
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of us have fallen out of love with the generalized matching "law" because of some empirical failures (see, for example, Logue and Chavarro, 1987, and Davison, 1988), and there have been a number of attempts to replace it with something better (e.g., Davison & Jenkins, 1985). Few, also, remain happy with Herrnstein's hyperbola (Martens' Equation 2). For instance, McDowell and Wood (1984, 1985) have found some empirical problems, and Warren-Boulton, Silberberg, Gray, and Ollom (1985) have discussed some analytic and theoretical problems. But, despite these important quibbles, both the strict matching law and Herrnstein's hyperbola remain good, if not entirely accurate, empirical descriptions of behavior. They are useful and effective, and will not lead a technologist far wrong in most situations. Technologists, though, should be aware of the pitfalls and the methods for circumventing them for the odd case in which predictions are entirely wrong. My approach in 1980 was rather different from that taken by Martens (1992) and, so far as I know, by any other researchers. Like Martens' approach, it was firmly rooted in the premise (at least, initially) of strict matching. But there was a second, important, starting point, known as the principle of indifference from irrelevant alternatives (or, the constant-ratio rule; see Luce, 1959; Davison & McCarthy, 1988). This rule, which has been found correct for a number of concurrent-schedule procedures, states that the preference between two constant alternatives is unaffected by the presence, absence, or rate of reinforcers for, other alternatives. My analysis was premised on the suggestion that a cost-effective modification of behavior could be done in the following way: Assume that you have a limited container of resources for the modification, and that your goal is to change behavior by an amount that can be exactly s p e c i f i e d - from, for example, spending 75 % of classroom time off task, to spending 20% off task. It is not possible to predict, from the strict matching law, what rate of resources contingent on on-task behavior you will need to do this for a number of reasons: (1), you cannot a priori know the value of your resources as reinforcers; (2), you do not know the value of Ron, the value of on-task reinforcers, and (3), equally, you do not know the value of Roff, the rate of off-task reinforcers. These latter values may be very high, in which case you will need to make your resources contingent on the on-task behavior at a high rate to get any substantial change in performance. But the on-task and off-task values may be small, in which case you may need only to use a small amount of your resources to get a large change. Without knowing these values, and the value of your resources, it is too easy either to undershoot the required change (and either terminate the modification, or beg more resources), or to use up a greater amount of your resources than you need (and put other modification projects in jeopardy, and maybe get
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the behavior in question trapped into an extreme and possibly unproductive level, such as 100% on task). The procedure must commence, then, with a baseline measurement phase, and this is followed by a "resources measurement" phase. In the latter, you arrange a third (for example) task for the subject, one which is incompatible with the behaviors of interest and takes time from them. You reinforce this behavior using your resources, and measure time allocation to on-task, off-task, and "resources-task" behaviors. When the time-allocation measures have stabilized, these measures can be used directly in the strict matching law to measure the rates of the on-task and off-task reinforcers in terms of the rate of resources reinforcers. If Ri is the rate of reinforcers for incompatible responses that you chose to use, and knowing Ton, Woff, Ti, and Ri, then applying the strict matching relation to firstly on-task versus incompatible responses, and then to off-task versus incompatible responses, the unknown values of Ron and Roff can be obtained algebraically. It is a simple matter, then, to calculate the rate of reinforcers (call it Rm) that must be made additionally contingent to on-task behavior to change the on-/off-task ratio to whatever is required. The equation used is: Ton Ron + Rm Ton + Toff
Ron + R m § Rof f
where Ton/(Ton + Toff) is the required proportion of time spent on task. Obviously, the first thing that should be done is to test this suggestion. If the more recent developments in matching research (which imply less proportional change than is predicted by strict matching) are correct, this should underpredict the amount required a little. This amount will be proportional to the amount of the required change away from equal time allocation. A more up-to-date analysis would take this underprediction into account and use the generalized matching law. This is written in logarithmic terms as: = a log~,~-~,rf) * log c where a is called sensitivity to reinforcement, and is often less than the value of 1.0 required by strict matching (see Baum, 1979). Log c is called bias, and measures any constant proportional preference for one alternative over the other. This relation can be used in exactly the same way as the strict matching law, above, but the finding of the values of a and log c requires more measurement to be done in the "resources measurement" phase. At least five different levels of R i need to be arranged so that two linear regressions (for on versus incompatible, and off versus incompatible, tasks) can be done to find the values of a and log c. This procedure would allow much more accurate prediction of the resources required for a par-
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ticular quantitative behavior change, but it would take longer before remediation could commence, and more of the resources would be used in the measurement phase. Experience will tell whether this is necessary - - it may be sufficient to apply a rule-of-thumb correction to the predictions of the matching relation rather than waste time and resources. As I wrote in my 1980 paper, these procedures, which could be applied in a very wide range of situations, have very important benefits. First, they allow a very efficient use of resources to produce defined behavior changes speedily (without having to experiment on reinforcer levels during remediation), accurately, and above all economically. Further, after a "resources measurement" phase, the procedures allow a therapist to be informed of the resources required, and to be able to contract with either the client or with an organization, for a particular quantitative change for a particular remuneration. Such contracts would benefit both those providing therapy and, more importantly, those receiving it. I would not like to pretend that such procedures are easy, and a great deal of research would need to be done to understand the many pitfalls in such applications, though the research mentioned by Martens (1992) is a strong step toward this. In writing this commentary, I hope that readers will not simply dismiss these suggestions as coming from another pejorative "pigeon person" with no understanding of the real world (though there is some truth in this!). Rather, my commentary, I hope, might motivate some researchers to give these procedures a try.
REFERENCES Baum, W. M. (1974). On two types of deviation from the matching law: Bias and undermatching. Journal of the Experimental Analysis of Behavior, 22, 231-242. Baum, W. M. (1979). Matching, undermatching, and overmatching in studies of choice. Journal of the Experimental Analysis of Behavior, 32, 269-281. Davison, M. (1988). Concurrent schedules: Interaction of reinforcer frequency and reinforcer duration. Journal of the Experimental Analysis of Behavior, 49, 339-349. Davison, M., & Jenkins, P. E. (1985). Stimulus discriminability, contingency discriminability, and schedule performance. Animal Learning & Behavior, 13, 77-84. Davison, M., & McCarthy, D. (1988). The matching law: A research rev&w. NY: Erlbaum. Herrnstein, R. J. (1970). On the law of effect. Journal of the Experimental Analysis of Behavior, 13, 243-266. Logue, A. W., & Chavarro, A. (1987). Effect on choice of absolute and relative values of reinforcer delay, amount, and frequency. Journal of Experimental Psychology- Animal Behavior Processes, 13, 280-291. Luce, R. D. (1959). Individual choice behavior: A theoretical analysis. NY: Wiley. Martens, B. K. (1992). Contingency and choice: The importance of matching theory for classroom instruction. Journal of Behavioral Education, _9, 121-137. McDowell, J. J, & Wood, H. M. (1985). Confirmation of linear system theory prediction: Changes in Herrnstein's k as a function of changes in reinforcer magnitude. Journal of the Experimental Analysis of Behavior, 41, 183-192.
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McDowell, J. J, & Wood, H. M. (1984). Confirmation of linear system theory prediction: Changes in Herrnstein's k as a function of response-force requirements. Journal of the Experimental Analysis of Behavior, 43, 61-73. Warren-Boulton, F. R., Silberberg, A., Gray, M., & Ollom, R. (1985). Reanalysis of the equation for simple action. Journal of the Experimental Analysis of Behavior, 43, 265-277.
