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SENSITIVITY AND BIAS UNDER CONDITIONS OF
EQUAL AND UNEQUAL ACADEMIC TASK DIFFICULTY
D EREK D. R EED AND B RIAN K. MARTENS
SYRACUSE UNIVERSITY
We conducted an experimental analysis of children’s relative problem-completion rates across two workstations under conditions of equal (Experiment 1) and unequal (Experiment 2) problem difficulty. Results were described using the generalized matching equation and were evaluated for degree of schedule versus stimulus control. Experiment 1 involved a symmetrical choice arrangement in which the children could earn points exchangeable for rewards contingent on correct math problem completion. Points were delivered according to signaled variable- interval schedules at each workstation. For 2 children, relative rates of problem completion appeared to have been controlled by the schedule requirements in effect and matched relative rates of reinforcement, with sensitivity values near 1 and bias values near 0. Experiment 2 involved increasing the difficulty of math problems at one of the workstations. Sensitivity values for all 3 participants were near 1, but a substantial increase in bias toward the easier math problems was observed. This bias was possibly associated with responding at the more difficult workstation coming under stimulus control rather than schedule control. DESCRIPTORS: choice, effort, generalized matching equation, math problem completion, schedule control
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Matching theory predicts that the relative rate of responding across two alternatives should match the relative rate of reinforcement associ- ated with those alternatives (Herrnstein, 1961, 1970). Shifts in the allocation of behavior across response alternatives away from perfect match- ing are regarded as biased responding (Baum, 1974). Research into biased responding suggests several reasons for these shifts, including (a) discrepancies between the number of obtained and programmed reinforcers, (b) qualitatively different schedules of reinforcement (i.e., fixed interval vs. fixed ratio; LaBounty & Reynolds, 1973), and (c) preference for a particular dimension of reinforcement associated with one response alternative (e.g., Neef & Lutz, 2001).
Not surprisingly, the role of response effort has been discussed as an explanation for biased responding across response alternatives. Previ- ous research has suggested that children’s choice in response allocation, particularly academic engagement, is often a function of the effort required to complete a task as well as competing sources of reinforcement for each alternative (e.g., Lannie & Martens, 2004; Neef & Lutz, 2001; Neef, Shade, & Miller, 1994; Weeks & Gaylord-Ross, 1981). For example, Cuvo, Lerch, Leurquin, Gaffaney, and Poppen (1998) found that when participants were asked to engage in a number of choices involving work requirements with equal schedules of reinforcement, they consistently chose the less effortful alternative. Similarly, Bradshaw, Rud- dle, and Szabadi (1981) demonstrated that when participants were asked to pull levers in exchange for points, they demonstrated an increase in response rate for the less effortful alternative despite equal rates of reinforcement programmed across both levers. One way to evaluate biased responding between concurrent alternatives is to fit the generalized matching equation to log ratios of
This study was conducted by the first author in partial fulfillment of the requirements for the MS in School Psychology at Syracuse University. We thank Florence D. DiGennaro, Tanya L. Eckert, William J. Hoyer, and Laura Lee McIntyre for their assistance throughout the duration of this project. Address correspondence to Derek D. Reed, 320 Rindge Ave. #101, Cambridge, Massachusetts 02140 (e-mail: [email protected]). doi: 10.1901/jaba.2008.41-
JOURNAL OF APPLIED BEHAVIOR ANALYSIS 2008, 41, 39–52 NUMBER 1 ( SPRING 2008)
behavior (B) and contingent reinforcement (R) across two response alternatives (Baum, 1974, 1979):
log
B 1
B 2
~ a log
R 1
R 2
z log b:
The parameter a represents sensitivity to reinforcement, or the extent to which unit changes in reinforcer ratios are accompanied by unit changes in response shifting. The param- eter b represents bias toward a particular response not accounted for by the reinforce- ment schedules in place. Both experimental and descriptive research has shown the generalized matching equation to provide a good description of children’s choice behavior (e.g., Borrero & Vollmer, 2002; Mace, Neef, Shade, & Mauro, 1994; Martens, Halperin, Rummel, & Kilpatrick, 1990; Mar- tens & Houk, 1989; Shriver & Kramer, 1997). For example, Mace et al. arranged concurrent variable-interval (VI) schedules of reinforce- ment contingent on math problem completion for 3 adolescents with learning and behavior problems. The percentage of time allocated to problems on the richer schedule was related to the percentage of reinforcers obtained on that schedule for each participant using the gener- alized matching equation. A linear function provided a good fit to the data, accounting for 56% to 98% of the variance in relative time allocation. Despite the use of modeling, changeover delays, signaled reinforcement, and equivalent sets of math problems, significant undermatching (mean slope 5 .19) and biased responding (mean intercept 5 .65) was ob- served for each participant. Although apparently robust as a description of behavior–environment relations, no applied studies have systematically investigated the effects of task difficulty on responding in applied situations through use of the generalized matching equation’s bias parameter. Given that children typically are expected to complete assigned work of varying difficulty during different instructional activities, it is important to understand how task difficulty interacts with
competing sources of reinforcement to affect response allocation across concurrent alterna- tives. Thus, the purpose of the present study was twofold. First, we evaluated the sensitivity and bias parameters of the generalized matching equation with typically developing, elementary- aged students using procedures similar to those of Mace et al. (1994, Experiment 1). This analysis was conducted to establish a baseline description of the distribution of behavior across a two-alternative choice arrangement of math problems in which response effort was constant across alternatives. In Experiment 2, we manipulated response effort for one alter- native using the same participants and methods as Experiment 1. The goal of this manipulation was to evaluate the effects of response effort on the relative rate of math problem completion through the bias parameter of the generalized matching equation.
METHOD
Participants and Setting Children from three third-grade regular public education classrooms were recruited to participate in Experiments 1 and 2. Criteria for participation in the study included (a) the absence of any formally diagnosed disability, no prescribed medications, and no teacher-report- ed behavior problems or identified problems with off-task behavior; and (b) demonstration that problem-completion rates were sensitive to a simple reinforcement contingency as mea- sured by a brief experimental analysis (described below). Three children met criteria for the study and participated in both experiments: Gina, a girl aged 8 years 9 months, and Andy and Nate, boys aged 9 years 2 months. In addition, all children were behind grade level for fluency in math, based on curriculum-based assessment (Shapiro, 1996) screening probes, and principal report. Each 4-min session was conducted in an empty conference room located in the partici- pants’ school building. Two workstations,
40 DEREK D. REED and BRIAN K. MARTENS
item picture) was identified as the preferred item. Preferred items were ranked in terms of most to least chosen. The top three items were identified as highly preferred, with the median three of the remaining five items identified as moderately preferred. These moderately and highly preferred items were pooled to constitute the project store (i.e., the back-up reinforcers in which obtained tokens may be exchanged for later in the study). Pens, pencils, and Skittles were identified as the most preferred items, with M&Ms, stickers, and erasers identified as the moderately preferred items. During experimental sessions, the highly preferred items had a token exchange rate of eight tokens per item, and the items identified as moderately preferred had an exchange rate of four tokens. Tokens were delivered by placing a bead on an upright wooden dowel according to concurrent VI VI schedules. Both the token and dowel were color-coordinated with the worksta- tion (blue or orange). Concurrent reinforcement schedules. Partici- pants were allowed to choose which stack of math problems to complete and to switch between workstations at any time. Three paired VI schedules were prepared as follows: VI 100 s VI 20 s, VI 60 s VI 60 s, and VI 20 s VI 100 s, with the first value in each pair representing the schedule value for the blue workstation and the second value representing the schedule value for the orange workstation. At the end of each session, participants were given the opportunity to redeem any tokens earned for their choice of items from the project store. Each workstation was equipped with a compact disc (CD) player programmed to signal the availability of reinforcement for completing problems at that workstation. The sound of a cash register opening was recorded on the orange workstation CD according to a random number generator based on each of the three VI schedules (4 min for each schedule). The sound of clapping was recorded on the blue workstation CD. The sounds at the two workstations were offset by a minimum of 5 s
to allow participants time to switch worksta- tions and complete a problem. Auditory signals were used to help students discriminate rein- forcer availability at each workstation and to bring responding under schedule control. Instructions and pretraining. Prior to each session, the experimenter read the directions explaining that each participant would have an opportunity to earn tokens by completing math cards at each workstation, as well as the maximum number of tokens possible at each alternative. Immediately following the direc- tions and prior to the first session, the experimenter demonstrated the procedures for 1 min. For Gina, an additional coaching session was conducted between Sessions 9 and 10 due to increasing variability in responding across the initial nine sessions. During the coaching session, Gina was told by the experimenter that she could switch between the workstations whenever a tone sounded (these data were not included in any analyses and are not reported in the Results).
Procedural Fidelity and Interscorer Agreement An additional observer collected data on procedural fidelity (occurrence and nonoccur- rence of each step of the experimental protocol) and interscorer agreement during 94% of sessions. Agreement was assessed by having two observers independently score each prob- lem and count the total number of reinforcers delivered upon the completion of each exper- imental session. In addition, agreement as to whether the stability criteria were met in the time-series figure (see below) was assessed based on independent judgments (both occurrence and nonoccurrence) by the researchers. This was done by having two researchers review the time- series plots after the primary researcher graphed each session’s data and then independently check a box on a data sheet if stability criteria had or had not been met. Procedural fidelity and interscorer agreement were determined by calculating the number of instances of agree- ments, dividing by the number of instances of
42 DEREK D. REED and BRIAN K. MARTENS
agreements plus instances of disagreements, and multiplying by 100%. For all participants, agreement was 100% and procedural fidelity was 99% (range, 96% to 100%).
Data Analysis
Stability criteria. Data collection began under the VI 100-s VI 20-s schedule, and participants remained exposed to this schedule until respond- ing met stability criteria of three consecutive data points that favored responding toward the richer schedule (e.g., above 50% of problems completed at the blue workstation when the blue worksta- tion had the richer reinforcement; Mace et al., 1994), with all three data points having a range of 10% or less. When the schedules were equal, the required range was still 10% or less for variability, but all three data points had to fall above the range of the three stable data points for the leaner (blue) alternative from the previous condition. These stability criteria were applied to all phases of the study for both experiments. Fitting the generalized matching equation. To fit the generalized matching equation to the data, the number of reinforcers obtained from the blue workstation was divided by the number of reinforcers obtained from the orange work- station. In addition, a blue response ratio was calculated by taking the number of digits correct at the blue workstation divided by the number of digits correct at the orange worksta- tion. We chose digits correct rather than digits attempted as the primary dependent variable for two reasons. First, digits correct involves a combination of accuracy and fluency and has been shown in previous research to be a meaningful (i.e., sensitive and valid) measure of academic performance (e.g., Shapiro, 1996). Second, through the preliminary assessments, we were able to identify instructional level math problems that students were able to complete with high accuracy but low fluency. Similar to other concurrent-operants research with hu- mans, only terminal data (final three data points in each phase that met the stability criteria) were used in fitting the generalized matching equa-
tion in an effort to ensure analysis of steady- state responding (e.g., Baron & Perone, 1998; Critchfield, Paletz, MacAleese, & Newland, 2003; Johnston & Pennypacker, 1993). Discrimination indexes. To assess if partici- pants’ responding was under stimulus control of the reinforcement signals at each workstation, discrimination indexes (Pierce & Epling, 1999) were computed. Because auditory tones signaled when the next problem completed would be reinforced for either schedule, respective dis- crimination indexes were calculated by dividing the number of reinforced problem completions at a given workstation by the total number of problems completed at that workstation during each session.
RESULTS
Experiment 1 Figure 1 depicts the percentage of behavior allocated toward the blue workstation during each experimental session of Experiment 1 for each participant. The percentage of behavior allocated toward the blue alternative was calculated by dividing the number of correctly completed math problems at the blue worksta- tion by the total number of correct problems completed during the session (orange plus blue). Recall that for each schedule require- ment, the first VI schedule value refers to the schedule in place for the blue alternative and the second VI schedule value refers to the schedule in place for the orange alternative. Gina and Nate completed a higher percentage of prob- lems correctly at the blue workstation during each 4-min session as the relative rate of reinforcement for completing these problems increased. The mean percentage of behavioral allocation towards the blue alternative for Gina was 34% (range, 0% to 85%) under the VI 100-s VI 20-s schedule, 46% (range, 28% to 64%) under the VI 60-s VI 60-s schedule, and 80% (range, 79% to 81%) under the VI 20-s VI 100-s schedule. Nate’s mean percentages of behavioral allocation towards the blue alterna-
SENSITIVITY AND BIAS 43
ratio of correct problem completion (blue or orange, y axis) was plotted against the log ratio of tokens earned (blue or orange, x axis), with each terminal data point depicted as an open circle in a scatter plot. The parameters of the regression line are presented along with the variance accounted for by the generalized matching equation in the lower right corner of each panel. Both Gina’s and Nate’s performances approx- imated strict matching, with sensitivity-to-rein- forcement parameters of 0.91 and 1.12, respec- tively. Moreover, Gina and Nate demonstrated little or no bias to either alternative, with bias parameters of 2 .05 and 2 .12, respectively. These data suggest that there were no systematic preferences toward either alternative not account- ed for by the programmed schedules of rein- forcement. Finally, the generalized matching equation accounted for a high proportion of the variance in responding for Gina and Nate (99% and 90%, respectively). Andy’s responding yielded a low sensitivity-to-reinforcement param- eter of 0.25, which suggested indifference to the schedules. These data suggested that Andy’s relative problem completion rates were not under schedule control, consistent with his undifferen- tiated pattern of responding (Figure 1). Figure 3 shows discrimination indexes of responding to each workstation by each partic- ipant across all sessions in Experiment 1. If the participants’ responding was under stimulus control (via the auditory signals), increasing
discrimination index values that approached 1. across sessions in each phase would be likely, suggesting that the participants essentially sit and wait for the reinforcement signal before working a problem. Gina and Nate revealed relatively low discrimination index values under all conditions, with the exception of Sessions 7 and 9 for Gina. Thus, relative problem- completion rates by these 2 students appeared to be under schedule control rather than stimulus control. By contrast, Andy’s discrim- ination index values reflected indifference at the concurrent VI 60-s VI 60-s values; however, when completion of either problem set was reinforced on the VI 20-s schedule, Andy’s responding shifted accordingly. Given Andy’s strategy of switching after each problem and his relative-problem completion rates approximat- ing 50% under all conditions (see Figure 1), it is possible that his high discrimination index values under the rich VI 20-s schedules resulted from a higher number of contiguous behavior– reinforcer pairings as a function of his switching strategy. That is, the pattern of switching workstations after each problem completion in combination with a rich reinforcement schedule likely produced a higher proportion of rein- forced problems by chance.
Experiment 2 Experiment 2 consisted of a response-effort manipulation in which easy math problems
Figure 2. Results of fitting the generalized matching equation to participant data from Experiment 1. Each data point represents one session after meeting the stability criteria. Gray diagonal lines indicate strict matching.
SENSITIVITY AND BIAS 45
were presented at the blue workstation and difficult math problems were presented at the orange workstation. Figure 4 depicts the per- centage of behavior allocated toward the blue alternative during each experimental session in Experiment 2 for Gina, Andy, and Nate. All
participants demonstrated fairly immediate and clear increases in responding toward the easier blue alternative in accordance with the concur- rent-operants design. Specifically, Gina had mean percentages of allocation towards the blue problems of 35% (range, 20% to 58%), 71% (range, 62% to 76%), and 94% (range, 93% to 96%) across the VI 100-s VI 20-s, VI 60-s VI 60-s, and VI 20-s VI 100-s schedules, respec- tively. Andy and Nate responded with similar patterns of allocation towards the blue work- station, with mean percentages of blue problems correct of 45% (range, 33% to 62%), 53% (range, 42% to 64%), and 91% (range, 82% to 96%) for Andy; and 46% (range, 26% to 64%), 83% (range, 83% to 85%), and 95.5% (range, 95.2% to 95.7%) for Nate, respectively, across the three schedule pairings. The matching plots of Gina, Andy, and Nate are presented in Figure 5. Least squares linear regression lines are presented in each panel as black lines, indicating the degree to which each participant matched behavior to scheduled reinforcement rates as predicted by the gener- alized matching equation. In addition to the regression lines, a dotted gray line representing the regression line of Experiment 1 is included in each plot to allow within-subject compari- sons of matching parameters across Experiments 1 and 2. For Gina, a slight increase in sensitivity to reinforcement was observed from Experi- ment 1 (0.91) to Experiment 2 (1.06). Andy’s sensitivity to reinforcement increased greatly from Experiment 1 (0.25) to Experiment 2 (1.08), shifting to near-perfect matching. A slight decrease in sensitivity to reinforcement was observed for Nate, decreasing from 1.12 in Experiment 1 to 0.80 in Experiment 2. However, Nate’s sensitivity-to-reinforcement parameter of 0.80 was still in the typical range for human participants (Kollins, Newland, & Critchfield, 1997). More important, all 3 participants’ matching analyses suggested a positive shift in bias towards responding at the easier blue workstation as a
Figure 3. Discrimination indexes for each participant in Experiment 1. Indexes near 1.0 suggest high degrees of stimulus control, and indexes near .5 or below suggest indifference. The blue workstation was associated with the leaner schedule (VI 100 s; filled circles) in the first phase and the denser schedule (VI 20 s) in the last phase.
46 DEREK D. REED and BRIAN K. MARTENS
relative reinforcement rates across workstations. The manipulation to problem difficulty, how- ever, produced a positive bias for completing problems at the easier blue workstation that was not accounted for by the reinforcement sched- ules. This bias was associated with responding that may have been influenced by the schedule- correlated discriminative stimuli at the more difficult orange workstation, in that participants completed problems at this workstation only when reinforcement was signaled.
DISCUSSION
Experiments 1 and 2 investigated the degree to which children’s academic behavior con- formed to the generalized matching equation. In addition, Experiment 2 investigated the role of response effort in influencing the bias parameter of the generalized matching equa- tion. In Experiment 1, relative rates of correct problem completion closely approximated the relative rates of reinforcement across worksta- tions for 2 participants (Gina and Nate), whereas the 3rd participant (Andy) showed little sensitivity to relative reinforcement rates. In addition, in Experiment 1 all 3 participants displayed high variance-accounted-for estimates and little to no evidence of stimulus control, suggesting that the generalized matching equa- tion provided an accurate description of the
data. These data support a small body of research on the applicability of matching theory to academic performance (Mace, McCurdy, & Quigley, 1990; Mace et al., 1994; Neef et al., 1994). In Experiment 2, response effort was manip- ulated by increasing the difficulty of math problems at the orange workstation. This manipulation increased the bias parameter of the generalized matching equation (in the direction of the easy blue workstation) for all 3 participants relative to Experiment 1, al- though the obtained discrimination indexes suggested that responding at the more difficult orange workstation was under stimulus control of the reinforcement signal as opposed to the programmed schedule and effort manipula- tions. These results lend further evidence that response effort contributes to deviations from matching in applied settings (Baum, 1979; Kollins et al., 1997; Neef et al., 1994). In addition, these results suggest that increases in response effort may lead to discriminated responding at the more effortful alternative. Interestingly, the participants appeared to complete difficult problems only when rein- forcement for that response was signaled. A comparison of responses allocated to the blue workstation between Experiments 1 and 2 (Table 1) revealed a definitive shift during the effort manipulation for each of the 3 participants.
Figure 5. Results of fitting the generalized matching equation to participant data from Experiment 2. The regression lines from Experiment 1 are included as gray dotted lines to allow within-subject comparisons between Experiments 1 and 2.
48 DEREK D. REED and BRIAN K. MARTENS
Specifically, across participants the mean increase in responding to the blue workstation from the schedule conditions in Experiment 1 to Experi- ment 2 was 17.5% (SD 5 13.4). This increase constitutes a significant difference in allocation between the two experiments, t(8) 5 3.92, p 5 .004 (one tailed). In light of the increase in the bias parameter of the generalized matching equation and the discrimination indexes, these data provide further evidence that the effort manipulation successfully biased responding toward the less effortful alternative in accordance with previous behavioral accounts of effort manipulations (Bradshaw et al., 1981; Cuvo et al., 1998; Lannie & Martens, 2004; Neef & Lutz, 2001; Neef et al., 1994; Weeks & Gaylord-Ross, 1981). Matching theory assumes that alternative sources of reinforcement are highly discrimina- ble (Davison & McCarthy, 1988). Although auditory cues were used to enhance discrimina- tion in the present study, reinforcer delivery alone also can acquire discriminative properties (Ferster & Skinner, 1957). The role of discriminated responding in the bias parameter of the generalized matching equation might be clarified in future research by examining within- session distributions of responding under con- ditions of unequal effort. Approaching academic performance and related behaviors with an understanding of the influence of task effort on responding may prove to be beneficial for instructional design as well as for behavioral programming. For example, when students are assigned work to complete, they are placed in a choice situation that is partly under the control of task difficulty as well as the reinforcement contingencies associated with each alternative (e.g., Lannie & Martens, 2004). Previous research has shown that children choose not to complete difficult tasks when given less effortful alternatives (e.g., Berkowitz & Martens, 2001; Neef & Lutz, 2001). Conversely, assigning work that is appropriate to students’ ability level in terms of difficulty can promote high levels of task
Figure 6. Discrimination indexes for each participant in Experiment 2. Indexes near 1.0 suggest high degrees of stimulus control, and indexes near .5 or below suggest indifference. The blue workstation was associated with the leaner schedule (VI 100 s; filled circles) in the first phase and the denser schedule (VI 20 s) in the last phase.
SENSITIVITY AND BIAS 49
behavior (Bradshaw, Szabadi, & Bevan, 1979; Critchfield et al., 2003), which limits one’s ability to analyze response distribution in the context of aversive conditions (e.g., high effort, punishment). In studies such as this, in which response effort is manipulated within a choice arrangement, a model of matching that incor- porates punishment may better account for the variance in the observed distribution of behavior. Third, although we used a concurrent- operants design to demonstrate experimental control over relative response rates, there was no within-subject replication of effects due to the effort manipulation. That is, the effects of increased problem difficulty on the bias param- eter of the generalized matching equation were compared across experiments. Similarly, the schedule comparisons progressed in an identical manner for all participants (i.e., VI 100 s VI 20 s, VI 60 s VI 60 s, VI 20 s VI 100 s). The effects of this history as a threat to the internal validity of the obtained results cannot be ruled out in the current investigation. Future research should evaluate different schedule progressions either within or across participants. Fourth, the present study employed the relative rate of correct math problem comple- tion as the dependent variable in all analyses. Although the problems were designed to be at students’ instructional level (i.e., completed with high accuracy but low fluency), the more difficult math problems in Experiment 2 required more time and effort to complete. As a result, using the number of correct math problems may have underestimated each par- ticipant’s ability to match responding to reinforcement. For example, a participant may have allocated responding toward one alterna- tive for a relative amount of time that matched the relative rate of reinforcement but did not complete any math problems correctly, render- ing the assumption that no behavior towards that alternative was emitted. Thus, using time spent at each alternative as a response output may have provided a more accurate description
of behavior compared to either digits correct or digits attempted. Direct analyses of these different response types as the target behavior in matching analyses may prove to be beneficial in future investigations. Future research into the use of the general- ized matching equation to evaluate academic performance should perhaps investigate time allocation in addition to discrete task completion to allow a more precise description of choice (see Baum & Rachlin, 1969). In addition, basic research using the generalized matching equation with human participants should consider the generality of its methods and interpretations to applied settings. Specifically, more research into the influence of punishment and aversive control should be investigated in basic laboratory settings to better equip applied researchers with equations to describe the effects of punishment on choice in natural settings. Finally, basic research on choice should address the roles in which instructions (i.e., rules), feedback, and discriminative stimuli contribute to deviations in matching, in both within-subject and between-subjects designs. Such information would provide researchers and practitioners alike with valuable information about the factors that contribute to behavior change and sensitivity to reward.
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Received November 7, 2006 Final acceptance August 8, 2007 Action Editor, Henry Roane
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