Julia is a 9-year-old student at Blackmore Elementary. You have been asked by your professor to help her teacher reduce the frequency of Julia’s hand raising in class using a differen

The Matching Law: A Tutorial for Practitioners Derek D Reed and Brent A Kaplan University of Kansas

ABSTRACT

The application of the matching law has historically been limited to use as a quantitative measurement tool in the experimental analysis of behavior to describe temporally extended patterns of behavior-environment relations. In recent years, however, applications of the matching law have been translated to clinical settings and populations to gain a better understanding of how naturally-occurring events affect socially important behaviors. This tutorial provides a brief background of the conceptual foundations of matching, an overview of the various matching equations that have been used in research, and a description of how to interpret the data derived from these equations in the context of numerous examples of matching analyses conducted with socially important behavior. An appendix of resources is provided to direct readers to primary sources, as well as useful articles and books on the topic. Keywords: choice, equations, matching law, molar analysis, tutorial

Behavior analysts have been in- terested in the environmental determinants of why behaviors are

allocated to particular choice alternatives for some time. Behaviorally speaking, choice is regarded as the distribution of behavior to reinforcement alternatives (see Fisher & Mazur, 1997). In this sense, a “choice” is nothing more than the emission of a particular response in lieu of others. Every instance of operant re- sponding represents the choice to engage in that given behavior at that moment in time, whether due to positive or negative reinforcement (Herrnstein, 1970).

As behavior analysts observe the relative distribution of behavior to reinforcement alternatives, preference may be derived by the proportion of responses allocated to each. Within this conceptual framework, more respond- ing to one alternative indicates a relative preference for that alternative. By simply recording how a client distributes their responses, we can identify preference. For example, one can—with some degree of accuracy—simply observe the behavior of children on a playground to infer their preferences with respect to games, play pals, jungle gym activities, and so on. If we aggregate responses (behaviors

emitted on the playground) over time, we can compare these aggregated re- sponses to other possible responses in the environment to determine relative preference. In more colloquial examples, consider how a teacher makes decisions as to what examples to use with her stu- dents or how a child chooses which care- giver to approach to request attention. In both cases, history of reinforcement can help explain the present choice. A teacher may use one teaching example over another because, in the past, it evoked more student interest, resulted in better student scores, was easier to explain, etc. Likewise, a child may ap- proach a particular caregiver because that caregiver provides more enthusiastic attention, delivers higher rates of atten- tion, responds to requests more quickly, etc. Each of these examples highlight the importance of understanding the broader, temporally extended, pattern of decision making within the context of reinforcer dimensions associated with each choice alternative (see Neef, Shade, & Miller, 1994).

What Is the Matching Law?

Since the early 1960s (Herrnstein, 1961), behavior analysts have theorized

that choice (i.e., relative preference) may be understood—and accurately predict- ed—by examining relative rates of rein- forcement associated with each option (e.g., pecking one of two keys, choosing one worksheet over another, emitting appropriate or problem behavior). In this conceptual framework, relatively dense sources of reinforcement will feature relatively higher rates of behavior (i.e., organisms demonstrate preference for the most reinforcing events/settings). Put simply, behavior matches reinforcement. Herrnstein (1961) formally conceptual- ized the matching law during a study assessing pigeons’ preference for sources of reinforcement. In this study, pigeons could peck one of two response keys in an operant chamber, each of which was on a variable interval (VI) reinforcement schedule. Within this preparation, these two VI schedules were concurrently available and independent of one an- other; that is, pecking on one key did not affect the schedule of reinforcement on the other. When Herrnstein plotted the relative rates of behavior against rela- tive rates of reinforcement, he found a positive relation between the two resem- bling a nearly perfect correlation (i.e., a one unit increase in reinforcement was

Behavior Analysis in Practice, 4(2), 15–24 MATCHING LAW TUTORIAL FOR PRACTITIONERS 15

associated with a one unit increase in behavior). This nearly perfect correlation of matching is depicted in Figure 1. Figure 1 illustrates that as reinforcer deliveries increase along the x-axis, proportional increases in behavior are depicted along the y-axis. This correlation is visually apparent in each data point, as each data point represents a perfect correspondence between relative rates of reinforcement and behavior. The line represents the strength of this correlation. In Figure 1, the best ft line features a slope of 1 and each x-axis value (relative rates of reinforcement) perfectly predict y-axis values (relative rates of behaviors) with strict correspondence to the matching law (i.e., all data points fall directly on the line). This observation that relative rates of behavior may be predicted by relative rates of reinforcements resulted in Herrnstein’s formulation of the matching law, which states that:

where B denotes rate of responses (e.g., responses per minute) at either alternative (denoted by subscripts 1 and 2) and R denotes rate of reinforcement (e.g., reinforcers per minute) at said alternatives. For example, if response B

1 resulted in twice

as many reinforcer deliveries relative to B 2

(i.e., R 1

is double the size of R

2 ), the matching law predicts twice as many B

1

responses. Note that proportions of time spent engaging in behaviors or consuming reinforcers may be used in lieu of rate (see Baum & Rachlin, 1969), such that time spent engaging in behaviors matches reinforcement. When investigating time/ duration-based measures of matching, B may be replaced with

Figure 1. Hypothetical plot of the relative rate of responding changing in perfect proportion to relative rates of reinforcement, that is, “matching.” The left side of Equation 1 captures relative rate of responding (y-axis), whereas relative reinforcement rate is captured by the right side of Equation 1 (x-axis).

T (time spent engaging in the response), and durations of reinforcer delivery may use as R in Equation 1.

How does matching occur in the “real world” where people are not in operant chambers with responses limited to two keys? In one example, Borrero et al. (2010) hypothesized and experimentally demonstrated that an account of matching could be found in the distribution of problem and appropriate behaviors emitted by children with developmental disabilities. These researchers proposed that children distribute either ap- propriate or inappropriate behaviors as a function of relative rates of reinforcement. Borrero and colleagues conceptualized problem behavior as B1

and appropriate behaviors as B 2 , while

experimentally manipulating rates of reinforcement for each response. According to the matching law, relative rates of problem and appropriate behavior should “match” the relative amount of reinforcement associated with each response class.

Suppose that, using the same procedures employed by Borrero et al. (2010), you review a client’s data set obtained from mand training sessions. To complete your analysis, you determine that a particular target mand (i.e., a mand you are training) shares the same function as aggression (for this ex- ample, we will use attention). Coding your data in this manner results in rates of aggression and appropriate manding as B1

and B

2 , respectively, with R

1 and R

2 representing attention delivered

contingent upon aggression and mands. If you were to plot your data in a typical matching plot where relative behavior (B

1 /[B

1 + B

2 ]) is plotted on the y-axis as a function of relative

reinforcement (R 1 /[R

1 + R

2 ]) on the x-axis, we would expect

the data points to fall along a diagonal line with a slope of

Figure 2. A hypothetical matching law plot of a client’s aggres- sive behavior and manding. The dashed diagonal line depicts perfect matching. The left side of Equation 1 captures relative rate of responding (y-axis), whereas relative reinforcement rate is captured by the right side of Equation 1 (x-axis).

MATCHING LAW TUTORIAL FOR PRACTITIONERS 16

1 (assuming perfect matching wherein one unit change in reinforcement results in one unit change of behavior). Thus, in Figure 2, theoretically “perfect” matching is denoted by the dashed diagonal line. As the hypothetical data in Figure 2 indicate, your client’s data fall almost perfectly along this line, suggesting that behavior did indeed conform to the matching law. That is, relative rates of behavior were predicted by relative rates of reinforcement for each response type.

The Generalized Matching Equation

As one might expect, however, researchers are not always successful in identifying or producing “perfect” matching. Many times, the slope of the line through the data points does not correspond to a perfect proportional change in behavior as a function of reinforcement (i.e., the slope does not equal 1). In other cases, there may be a slope of 1, but the line is shifted upward or downward such that the line does not originate (i.e., the y-intercept) from the origin of the graph (i.e., does not pass through coordinate 0,0), meaning that some preexisting bias is impacting responding. For example, if a right-handed child is expected to sort picture cards from piles on either the right or left side of a table, the child may demonstrate a bias for the right given her handedness. To account for such deviations, Baum (1974) proposed the generalized matching equation (GME),

which is algebraically equivalent to Equation 1, with the addi- tion of logarithmic transformations and free parameters s and b. The logarithmic transformation of the ratios ensures that the resulting regression line is straight, rather than curvilinear. Having linear regression lines permits an analysis that is more easily interpretable (see Baum, 1974; Shull, 1990). The GME states that:

where s represents the slope of the best ft line, and b represents the y-intercept. That s and b are free parameters implies that they are not known until linear regression has been applied to the data set (see Reed, 2009). Parameter b (bias) represents how much preference the organism has for either behavior that cannot be accounted for by reinforcement alone. Because the best-ft line allows the behavior analyst to model operant responding at any relative rate of reinforcement, we can exam- ine what responding would look like when there are exactly equal rates of reinforcement for B

1 and B

2 . In other words,

when there is no difference in the amount of reinforcement that is produced on each alternative, one would expect to see equal responding across each alternative, all else being equal. Figure 3 depicts the log ratios of reinforcement and behavior

Figure 3. The top panel depicts possible variations in bias using the GME, whereas the bottom panel depicts possible variations in sensitivity to reinforcement. The left side of Equation 2 captures relative rate of responding (y-axis), whereas relative reinforcement rate is captured by the right side of Equation 2 (x-axis).

MATCHING LAW TUTORIAL FOR PRACTITIONERS 17

along the x- and y-axis, respectively. In the top left panel of Figure 3, the log transformation (with a base of 10) of the ratio 1/1 equals zero. Thus, the behavior analyst can examine behavior when log (R

1 /R

2 ) equals zero (that is, reinforcement

rate is equal across the responses) to determine the value of the y-intercept—this would occur at the zero value on the x-axis. If the y-intercept (b) is greater than zero, there is a bias for B

1 that

is unrelated to reinforcement rate; this is because B 1

is in the numerator of the ratio, and if B

1 is greater than B

2 , the log ratio

would be positive. An example of positive bias is depicted in the top middle panel of Figure 3, where the y-intercept is above coordinates 0,0. Likewise, if the y-intercept is negative, there is a bias for B

2 (see top right panel of Figure 3). Deviations in

bias (away from zero) can result from a host of factors, such as physical characteristics of the organism or environment that unintentionally affect the ability to respond (Baum, 1974; e.g., handedness, color bias, quality of caregivers’ attention).

Of particular interest to practitioners may be the differential effects that reinforcer dimensions may have in producing the biased responding described in the previous paragraph. In one demonstration of this concept in education, Neef, Mace, Shea, and Shade (1992) offered the choice between two stacks of math problems to students receiving special education services (emotional disturbance and behavior disorders). In particular, the researchers arranged concurrent VI schedules across the two stacks in equal- and unequal-quality reinforcement phases. During equal-quality phases, the two stacks of math problems concurrently featured the same reinforcers (nickels or program money used as conditioned reinforcers in the classroom’s token economy). Results suggested that students responded across the two stacks in accordance with the matching law (i.e., rela- tive rates of math problem completion were predicted by the programmed rates of reinforcement on each stack). In the al- ternate unequal-quality phase, one stack featured nickels while the other featured program money. In this unequal-quality phase, all three students allocated relatively more responding to the stack featuring the nickel reinforcers than what was pre- dicted by the relative reinforcement rates. Thus, these students exhibited a bias for the nickels that could not be explained by rate of reinforcement alone. In a follow-up study, Neef et al. (1994) conducted analyses of students’ academic response distributions across two stacks of math problems under differ- ing reinforcer dimension comparisons. Dimensions consisted of (a) rate (i.e., the concurrent VI schedule in place for each stack of math problems), (b) quality (i.e., relative preference for reinforcers available for each stack of math problems), (c) delay (i.e., time between point delivery and exchange for backup reinforcer), and (d) effort (i.e., diffculty of the math problems). Using highly controlled comparisons wherein target dimensions varied during a session across the math problem stacks while holding other dimensions equal, Neef et al. dem- onstrated that students have idiosyncratic biases for reinforcer dimensions. For example, some students may differentially prefer sooner rewards over higher-quality ones, whereas other students may prefer less effortful contingencies that result in

delayed access to rewards over more effortful contingencies that result in immediate ones. These data highlight the notion that practitioners can engineer the environment to favor appropriate responses by arranging contingencies that make it less effortful for the learner to obtain high rates of immediately available, high quality rewards for the desired behavior, relative to those associated with undesirable behaviors.

Understanding idiosyncratic preferences for reinforcers in applied setting via matching analyses suggests the bias parameter may have utility in quantifying the degree to which reinforcers are substitutable (i.e., serve similar functions and maintain re- sponding at similar levels) in treatment scenarios. If a matching law analysis indicates no bias for two responses associated with differing dimensions of reinforcement, these reinforcers may be considered substitutable. However, if a bias is detected via reinforcer parameter manipulations, the practitioner can isolate the preferred dimension and program reinforcers accordingly; this approach may be useful in contexts that prohibit the ability to arrange all appetitive dimensions of reinforcement (e.g., spe- cifc classroom demands associated with the target response do not permit frequent rates of reinforcer delivery, but may permit more immediate or higher quality reinforcers). For example, Reed and Martens (2008) used procedures similar to those described by Neef et al. (1992, 1994) with students receiving standard educational services (i.e., not special education), to demonstrate the utility of matching to academic performance. In Experiment 1 of Reed and Martens’ study, the diffculty of the problems in each stack was matched to students’ current instructional level (i.e., in a previous assessment, the students demonstrated the ability to complete these problems accurately and fuently). Under this “symmetrical” arrangement, students’ responding was in accord with the GME (Equation 2) with little to no bias for either stack (assessed quantitatively using the b parameter from Equation 2). In Experiment 2, one stack of math problems featured diffcult (i.e., accurate but nonfuent) problems, whereas the other stack remained at the students’ instructional levels. Under this “asymmetrical” arrangement, substantial increases in bias (b) were observed toward the math problems that were at the students’ instructional level (i.e., the “easy” problems). These results suggest that, when educating students on material outside of their present instructional level—that is, outside of content in which they are accurate and fuent—students will prefer to engage in activities that are less effortful even when reinforcement favors the more effortful re- sponse. From an education standpoint, this is concerning. And as such, simply providing relatively more reinforcement for the more effortful task may not be enough to increase preference for the task. More importantly, such arrangements could result in the student choosing to engage in off-task behaviors—that are presumably less effortful and more reinforcing—that may be undesirable and problematic in classroom contexts (e.g., talk- ing to their neighbor, attending to nonacademic stimuli). We return to this discussion and provide some solutions (grounded in matching theory) for such scenarios in the Additional Implications for Practice section later in this article.

MATCHING LAW TUTORIAL FOR PRACTITIONERS 18

Figure 4. A generalized matching law plot and parameters of a hypothetical client’s aggression/manding data set. The solid line depicts the best-ft line, whereas the dashed diagonal line depicts perfect matching. The left side of Equation 2 captures relative rate of responding (y-axis), whereas relative reinforcement rate is captured by the right side of Equation 2 (x-axis).

In the GME, s, or sensitivity to reinforcement, represents the amount of change in behavior associated with each change in reinforcement. When s is close to 1, a unit change in relative rates of reinforcement features an equal unit change in relative rates of behavior. An example of strict matching is illustrated in the bottom left panel of Figure 3. In this case, increases in relative rates of reinforcement (along the x-axis) are identical to increases in relative rates of behavior (along the y-axis). For example, if the rate of reinforcement on one response alternative doubled, we would expect to see exactly twice as much responding on that alternative. If s is greater than 1, the organism is considered to be overmatching (i.e., the organism is emitting relatively more responses than what is necessary to obtain reinforcement; specifcally, the organism is disproportionally emitting more responses toward the richer reinforcement alternative). As the bottom middle panel of Figure 3 depicts, overmatching is observed when the relative rates of behavior increase more quickly along the y-axis than the change in relative rates of behavior increase along the x-axis (i.e., the slope is greater than 1). For example, if the rate of reinforcement on one response alternative doubles, we may see more than twice as much responding (e.g., 3 times as much) on that alternative. Undermatching implies that fewer responses are emitted based on the available reinforcers for one alterna- tive, and thus s is less than 1 (see bottom right panel of Figure 3). In this case, as relative rates of reinforcement increase along the y-axis, the increase in behavior is less than predicted, such that the slope is less than 1. Undermatching is also indica- tive of responding disproportionally more toward the leaner

reinforcement alternative. Extreme undermatching (s close to 0) is considered to be representative of indifference or insensi- tivity to reinforcement. Matters of sensitivity to reinforcement are important to consider in clinical or educational settings. If clients demonstrate overmatching, they are not contacting programmed reinforcers associated with the behavior on the relatively leaner schedule of reinforcement. For example, a cli- ent receives attention from Staff Member A 3 times as often as she does from Staff Member B. If the client spends 90% of her time near Staff Member A—that is, demonstrating high sensitivity to reinforcement or overmatching—she will miss out on many of the pleasant interactions Staff Member B could provide. Likewise, if the client does not change her interactions between Staff Members as they begin to alter their patterns of attending, the client’s sensitivity to reinforcement would be low (i.e., behavior is not changing as a function of reinforcement), representing undermatching.

A fnal consideration when understanding behavior- environment relations using the GME is the degree to which this theoretical model of matching accounts for variation in the data. That is, the GME is useful for describing various deviations from perfect matching, implying that the data do indeed vary. Because the GME relies on regression to ft the line to the data points (for a more detailed discussion on linear regression, see Motulsky & Christopoulos, 2004), the regression analysis can provide a quantitative account of how well the GME describes the data pattern. This account is termed variance accounted for (and is typically denoted as R2, VAC, or VAF in journal articles) because it informs the analyst to the percentage/proportion of variance in the data that is explained by the GME. If the GME perfectly describes the data (even with deviations in matching with respect to sensitivity to reinforcement or bias), one could reliably predict every data point with 100% accuracy. In this case, the percentage of variance accounted for by the GME would be 100% (or 1.0, as a proportion). This is rarely the case in matching law studies, but analysts typically hope to observe R2 as close to 1.0 as possible.

Having explained the GME (Equation 2), we will return to the hypothetical client data and reexamine matching with respect to sensitivity to reinforcement and bias using the GME (Equation 2). As Figure 4 illustrates, the slope of the line (sensitivity to reinforcement, or s) was .87. That is, when there is a one-unit change increase in reinforcement for aggression, there is slightly less than a one-unit change (.87) in aggression. In other words, the client exhibits undermatching. Moreover, with a y-intercept (bias, or b) of -.009, there was virtually no bias. That is, if we model responding when reinforcement rates for aggression and appropriate mands are equal, the relative ratio of behaviors is approximately equal; the preference for behavior appears to be based on reinforcement rates alone, and not because of any particular characteristic of response form or quality of attention for each response form. Finally, examining the R2 value of the best-ft line, it is evident that the GME provides an excellent account of the behavior pattern, with 98% of the variance being accounted for by Equation

MATCHING LAW TUTORIAL FOR PRACTITIONERS 19

2. In other words, the GME can explain 98% of the variance in relative rates of behavior, given the known relative rates of reinforcement.

Herrnstein’s Hyperbola

Behavior analysts cannot always—if ever—accurately identify all of the sources of reinforcement that may govern organisms’ choice. Both the “real world” and tightly controlled experimental settings consist of countless reinforcement alter- natives for any organism at any given time (McDowell, 1988). Put another way, it may be shortsighted to simply assume that choice is limited to two options, as is dictated by Equations 1 and 2, and in cases where only one target behavior is concerned, a single-alternative matching theory is necessary.

To account for a single-alternative conceptualization of choice, Herrnstein (1970) proposed a modification to the matching law to account for all possible responses and sources of reinforcement (Be

and Re, respectively), such that:

Further derivation of this form of matching collapses the sum of all rates of responses (sum of all Bs) to parameter k, and col- lapses R

2 into R

e . To understand what these terms describe, con-

sider a situation in which a practitioner is interested in on-task behavior and attention associated with on-task behavior, while noting that many different topographies of off-task behavior may occur (e.g., talking to a neighbor, staring out the window). In this situation, all of these possible off-task responses are also associated with some kind of reinforcement, although it may not be specifcally captured by the measurement system the practitioner has chosen to employ in her observations. An estimate of the sum of all on- and off-task behaviors constitute k, with R

e serving as an estimate for the sum of all reinforcers

associated with the off-task responses (i.e., those not specifcally captured in the measurement system). In short, deriving these ftted parameters (k and R

e ) accounts for the assumption that

only one response and source of reinforcement are identifable with the measurement system in place, with the recognition that other responses and sources of reinforcement are present but may not be captured by this measurement system:

Using simple algebra and multiplying both sides by k to iso- late B

1 , we are left with Herrnstein’s (1970) single-alternative

matching equation:

where B 1

represents the rate of the target response (in the above example, on-task behavior), R

1 represents the rate of reinforce-

ment associated with B 1 (e.g., attention for on-task behavior),

Figure 5. A hypothetical single-alternative matching law plot depicting hyperbolas derived from low (dashed) and high (solid) R

e values.

and k and R e represent free parameters that are derived by ftting

a hyperbola. A hyperbola is an open curve that, in the case of matching, curves upward away from the origin and continues infnitely, decelerating until it appears to be a fat horizontal line nearly parallel to the x-axis, which is the asymptote to the data points. That is, k and R

e are not known to the researcher until

a nonlinear best-ft line is produced. When matching data are plotted and Equation 5 is used to analyze the data, the best-ft line is a negatively accelerated hyperbolic curve (see Figure 5). In a negatively accelerated curve, changes in x-values initially result in large changes in y-values, but as x-values continue to increase, the proportional change in the y-values decrease. In other words, the hyperbolic curve starts accelerating steeply (i.e., more closely resembling a vertical line) but then deceler- ates less until there is little change in the y-values relative to the increasing x-values (i.e., more closely resembling a horizontal line).

In Herrnstein’s single-alternative equation (i.e., the quanti- tative law of effect; 1970), reinforcement rates are the x-values, with rates of the target behavior comprising the y-values. The parameter k represents a constant property of behavior (e.g., the effort associated with the response, the speed at which the response can be emitted) that governs the maximum amount of behavior that can be emitted during an observation period. Thus, in

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