JOURNAL OF THE EXPERIMENTAL ANALYSIS OF BEHAVIOR

SUPERIMPOSITION OF RESPONSE-INDEPENDENTREINFORCEMENT

I. S. BURGESS AND J. H. WEARDEN

MANCHESTER POLYTECHNIC AND UNIVERSITY OF MANCHESTER

Studies that have superimposed response-independent reinforcement (or reinforcers scheduled bycontingencies placed on the absence of responding) upon conventional response-dependent schedulesare reviewed. In general, providing alternative sources of reinforcement reduced response rates belowthe levels observed when alternative reinforcement was absent. However, response-rate elevation wassometimes found, particularly when rates of superimposed response-independent reinforcement werelow. Superimposition of schedules providing reinforcers contingent on the absence of respondingusually produced more severe response-rate decrements than superimposition of response-independentreinforcement. A variant of Herrnstein's equation, which assumes that some of the alternative rein-forcers function as if they were delivered by baseline response-dependent source of reinforcement, isin qualitative agreement with the overall body of results obtained, and can predict both increases anddecreases in response rate as resulting from superimposed reinforcers.Key words: response-independent reinforcement, Herrnstein's equation, alternative reinforcement,

fixed time, variable time, differential reinforcement of other behavior

The paradigm that is the focus of interesthere involves superimposition of response-in-dependent reinforcement-scheduled eitherperiodically, according to a fixed-time (FT)schedule, or aperiodically, according to a vari-able-time (VT) schedule-onto a schedule ofresponse-dependent reinforcement that re-mains in force. Such manipulations allow theautomatic measurement of conventional op-erant responses, as contrasted with studies ofFT or VT alone, which usually require ob-servational measures of behavior. Thesemethods also allow response-independent re-inforcement effects to be studied for long pe-riods of time in steady-state conditions, unlikethe studies of transitions from response-de-pendent to response-independent reinforce-ment, during which responding sometimesdeclines almost to zero under the response-independent schedule (e.g., Boakes, 1973). Asshown by Rachlin and Baum (1972), super-imposition studies can also be easily related toresults from response-dependent reinforce-ment schedules via the medium of Herrn-stein's equation (Herrnstein, 1970).The form of Herrnstein's equation that ad-

dresses effects of superimposed response-in-dependent reinforcement is:

Reprints may be obtained from J. H. Wearden, De-partment of Psychology, The University, Manchester M139PL, Great Britain.

R= kr,r, + r2 + rO (1)

Here, R is the rate of the measured operant,r1 is the rate of reinforcement that it produces,and r2 and ro are the rates of reinforcementavailable from alternative sources. The first(r2) is the rate of reinforcement that is ar-ranged by the experimenter independent ofresponding or for activities other than themeasured operant, and r. is the rate of "ex-traneous" reinforcers, including those arisingfrom the environment but not arranged by theexperimenter, and those, such as grooming,arising from the animal's own maintenanceneeds. The constant k reflects the asymptoticrate at which the measured operant can beemitted.The effects of superimposition of response-

independent reinforcement often have beencompared with those of superimposing a con-tingency requiring a period of nonrespondingfor reinforcer delivery (differential reinforce-ment of other behavior, or DRO). There aretwo reasons for this comparison. First, DROschedules might be supposed to reinforce non-responding, or to enhance the emission of re-sponses incompatible with the measured op-erant. To the extent that behavioral effects ofFT and VT depend on this process, FT andVT should affect behavior in the same way asdoes DRO. Second, the study by Rachlin and

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1986, 45, 75-82 NUMBER 1 (JANUARY)

7. S. BURGESS and J. H. WEARDEN

Baum (1972) had a seminal influence in thisarea, and this article drew particular conclu-sions from a comparison of the effects of aDRO and VT schedule superimposed on abaseline of variable-interval (VI) reinforce-ment.

Before proceeding with a review of the re-sults of superimposition studies, a note on no-menclature is required, for in this context therehave been inconsistent uses of conventionalschedule terms. Some writers (e.g., Henton &Iversen, 1978; Rachlin & Baum, 1972) haveused the term concurrent schedule to apply tothose cases in which one schedule is super-imposed upon another. Others (e.g., Lattal &Bryan, 1976; Zeiler, 1976) have used the termconjoint to describe this same arrangement.To circumvent these difficulties in nomencla-ture, we will adhere to a terminology that canconsistently describe all conditions that havebeen used. Consider two schedules, A and B(e.g., VI 3 min and VT 3 min). The operationof superimposing schedule B onto schedule A(which remains in effect) will be written Asup B (e.g., VI 3 min sup VT 3 min). Thebaseline schedule will always be indicated first,the superimposed schedule second. This su-perimposition description refers to scheduledcontingencies and may not represent the ac-tual pattern of reinforcer deliveries.

SUPERIMPOSITION OFRESPONSE-INDEPENDENTAND DRO SCHEDULES UPONRESPONSE-DEPENDENT

SCHEDULESRachlin and Baum (1972) initially trained

pigeons on conjoint VI 3-min VI 3-min sched-ules in which availability of reinforcers fromone of the VI 3-min schedules was signaled;they also exposed the pigeons to either VI 3min sup (tandem VT 3 min DRO 2 s), underwhich a reinforcer was scheduled on averageevery 3 min and delivered when a subsequent2-s period of nonresponding occurred, or VI3 min sup VT 3 min. Rachlin and Baum re-ported that the rate of key pecking declinedordinally as the amount or rate of non-VIreinforcers increased; they concluded that thesource of the alternative reinforcement wasunimportant, because in their experiment thesuperimposed VT and DRO produced verysimilar effects.Two questions that arise from this study

120 -

90 -

60 _

30-

FR180 FR300

120 -

0

0la

c

90

60

30

120 -

90 -

60-30 -

120

90-

60 -\

30

60 180 300

Bird

16

124

158

160

60 180 300

FT Reinforcer* per hour

Fig. 1. Effect of FT superimposition on respondingmaintained by fixed-ratio schedules of reinforcement (datafrom Zeiler, 1979). The baseline schedule was either FR180 (left column) or FR 300 (right column), and rates ofreinforcement delivered by superimposed FT schedulesvaried from 0 to 360 reinforcers per hour.

provide a framework for discussing other re-sults. First, what effects does superimpositionof a DRO or a response-independent schedulehave on response rate or patterning main-tained by a response-dependent baselineschedule? Second, does the scheduling of thesuperimposed reinforcement play some criti-cal role, or is the influential factor merely itsrate of delivery?

As for the first question, the overall patternof results provides considerable support forRachlin and Baum's contention that increas-ing rates of superimposed "alternative" rein-forcement produce progressively greater andgreater decrements in response rate, relativeto situations without arranged sources of al-ternative reinforcement. In a few cases, lowrates of superimposed response-independentreinforcement appear to have rate-enhancingeffects, but in almost all studies some rate ofsuperimposed response-independent or DROreinforcement can be found that maintains alower response rate than the baseline schedulealone.

For example, Edwards, Peek, and Wolfe(1970) studied the effects of fixed-ratio (FR)

76

----o

---o

RESPONSE-INDEPENDENT REINFORCEMENT

sup FT and typically found response-rate dec-rements when the rate of FT reinforcementwas substantially greater than that producedunder the FR schedule alone. Similarly, Lat-tal (1974) varied in an ordinary VI the per-centage of reinforcers that were delivered in-dependently of responding, and found thathigher percentages led to lower response rates.Note, however, that although this procedureholds overall reinforcement rate constant, itconfounds two operations-reducing the rateof response-dependent reinforcement and in-creasing the rate of response-independent re-inforcement-both of which might be expect-ed (after Herrnstein, 1970, and Rachlin &Baum, 1972) to produce decreases in responserate.

Zeiler (1976) studied Fl 3-min sup FT andFl 3-min sup DRO schedules. Parameter val-ues for the superimposed schedules were var-ied over a wide range (10 to 100 s). Both thesuperimposed FT and DRO schedules gen-erally reduced response rates below those ob-taining under Fl alone, with higher rates ofsuperimposed reinforcement usually produc-ing lower response rates. Zeiler (1977) foundsimilar results under variable-ratio (VR) 100and VI 100-s schedules with and without var-ious FT and DRO superimpositions. In a lat-er study, Zeiler (1979) superimposed variousFT schedules upon FR 180 and 300, and var-ious DRO schedules upon FR 60, 180, and300. The results of these manipulations weremore complex than in earlier studies. Figure1, drawn from a table in Zeiler's 1979 study,shows the effect of those FT superimpositions.

Although high FT rates totally suppressedresponding for all subjects, low FT reinforce-ment rates (e.g., FT 80 and 120-s schedules)reliably produced increases in response rateabove that prevailing under FR. The only ex-ception occurred in data from Bird 158 underFR 300. Zeiler did not provide details regard-ing delivery rates of DRO reinforcement, but,in general, DRO superimposition tended toproduce response suppression. Responding onthe FR-300 schedule was eliminated by DROvalues less than 120 s; with FR 180, DROvalues of 80, 40, 20, and 10 s produced totalresponse suppression; with FR 60, only DRO10 s had this effect.

Lattal and Bryan (1976) found similar re-sults when superimposing various FT and VTschedules on FI baselines. Response-rate dec-rements were common, although in some cases

the superimposition resulted in rate increases(e.g., when FI 5 min was changed to Fl 5 minsup VT 2.5 min). Lattal and Boyer (1980)also obtained reliable rate increases by whatwas, in effect, a VT schedule superimposedon FI 150 s. However, instituting variousDRO requirements (with the DRO being ar-ranged according to a VI schedule) usuallyproduced marked decreases in response rates,with rates generally declining ordinally withincreases in the rate of DRO reinforcers de-livered.Thus, the finding of increased response rates

resulting from superimposition of response-independent reinforcement is not unusual, al-though it may be less common than a ratedecrement of the type reported by Rachlin andBaum (1972). The reasons that both effectsmight occur are discussed further below.

Evidence bearing on the second questionposed earlier provides only partial support forRachlin and Baum's (1972) contention thatthe source of superimposed reinforcement haslittle influence over and above its overall rateof delivery. For example, Zeiler summarizedthe results of his 1977 study as follows:

Even at low time requirements where theschedules operate most similarly, the DRO islikely to establish the lower rate. With stilllonger times, DRO consistently generates lessresponding. Then, with the longest times, FTmay establish lower rates than DRO, becausethe DRO requirements are rarely if ever met.(p. 31)Other differences between a DRO and FT

superimposition were found by Zeiler (1979).As mentioned earlier, when the DRO contin-gency was met, and DRO reinforcers weredelivered, a degree of response-rate decrementwith respect to the FR baseline was usuallynoted. Low rates of FT reinforcement, on theother hand, consistently elevated response rateswhen superimposed above the FR schedulesused. In addition, Lattal and Bryan (1976)found differences in response patterning con-tingent on superimposing VT and FT on Fl,and Lattal and Boyer (1980) found differenteffects of a DRO and VT schedule on re-sponding maintained by FI 150 s.

All these results suggest, of course, that thesource of alternative reinforcement can mat-ter. At the same time, it should be emphasizedthat these findings do not constitute a directcontradiction of, or failure to replicate, Rach-lin and Baum's (1972) results, because most

77

I. S. BURGESS and J. H. WEARDEN

of the studies employed schedules very differ-ent from those used by Rachlin and Baum.One particularly striking difference betweenRachlin and Baum's procedures and those ofZeiler (1 976, 1979), Lattal and Bryan (1 976),and Lattal and Boyer (1980) is that Rachlinand Baum's study employed a baseline sched-ule (VI) and superimposed schedules (VT andtandem VT DRO) under which reinforcerdeliveries were aperiodic, whereas the otherauthors employed baseline conditions (Fl andFR), or superimposed schedules (FT or sim-ple DRO) under which reinforcer delivery wasperiodic, or followed some distinct event suchas a substantial period of nonresponding.The elevations of response rate above base-

line levels produced by superimposed re-sponse-independent reinforcement noted herehave usually accompanied changes in char-acteristic response patterns generated by thebaseline schedules, particularly a marked de-crease in postreinforcement pausing. For ex-ample, Lattal and Boyer (1980) found thatthe VT superimposition that produced elevat-ed response rates was generally correlated withdisruption of the typical FI pattern of re-sponding and decreases in proportion of totaltime spent pausing. Zeiler (1979) also notedchanges in response patterning occasioned bysuperimposed FT schedules, and examinationof the cumulative records he presented sug-gests that reduction or elimination of the post-reinforcement pause observed under FR wasa common effect of the superimposed sched-ules.A study by Henton and Iversen (1978) also

illustrates the importance of interactions be-tween the events arranged by the superim-posed schedule and the pattern of behaviormaintained by the baseline schedule. Theseworkers varied the superimposition conditionsimposed on a VI baseline using rats as sub-jects. Their requirements that responses bepresent or that responses be absent in a 1-speriod before superimposed reinforcer deliv-ery had marked effects on response probabil-ity: The requirement of no responding duringthe 1 s before the reinforcer (similar to a DROrequirement) maintained less responding thanthe requirement that a response be present(similar to a variable delay-of-reinforcementrequirement). A simple VT schedule mark-edly decreased lever-press probability, but ob-servation of the animals revealed that this

probability was related to the frequency withwhich previous food deliveries had been con-tiguous with some behavior other than leverpressing. In general, the outcome of a super-imposition condition seemed to depend incomplex ways on the contiguities between le-ver pressing, other behavior, and food deliv-ery.

This type of finding suggests that the out-come of any particular superimposition ex-periment may be predictable only from a com-plex combination of factors. The pattern ofbehavior generated by the baseline scheduleand the requirements of the superimposedcondition may be foremost, but the details ofthe interactions of the two may also be veryinfluential. On the other hand, when a base-line schedule generates responding that is ap-proximately uniform over time (such as a VIschedule), and when superimposed schedulesare aperiodic and impose minimal or zero re-sponse requirements (such as VT or a briefDRO arranged in tandem with VT), then therate of reinforcement delivered by the super-imposed schedule may be a dominant predic-tor of response rate. Such conditions were thoseused by Rachlin and Baum (1972).

HERRNSTEIN'S EQUATIONThe present section seeks to use Herrn-

stein's equation (both in its original form andin a variant developed below) as a frameworkfor discussing the overall pattern of results ob-tained when response-independent and DROschedules are superimposed upon maintainedoperant baselines.

Herrnstein's equation readily encompassesthe finding that a superimposed response-in-dependent schedule, particularly one deliver-ing a high rate of reinforcement, can producea decrease in responding maintained by a re-sponse-dependent schedule that is held con-stant. An alternative source of reinforcementwould contribute to r2 in Equation 1, so for aconstant r, (i.e., a constant rate of reinforce-ment contingent upon the measured operant),increases in the rate of superimposed rein-forcement would produce greater and greaterresponse decrements. This is obviously true ofall variants of Herrnstein's equation (e.g.,McLean & White, 1983; Wearden, 1981) inwhich alternative sources of reinforcementcontribute only to the denominator of theequation.

78

RESPONSE-INDEPENDENT REINFORCEMENT

1.0'-

0

0

0

0

CL

c

0.8

0.6

0.4

0.2

0

pzl

p=O.8

p=0.2

p=O

2 4 6 8 10

r2(r1 units)

Fig. 2. Effects of superimposed response-independentreinforcement predicted by Equation 2. The dotted lineshows response rate in the absence of superimposed re-inforcement. The function with p = 0 is equivalent to theprediction of Herrnstein's equation (Equation 1).

Herrnstein's equation fares less well, how-ever, with several other findings. One of theseis the result that low rates of superimposedVT or FT apparently can elevate responserates. Equation 1 also fails to address the dif-ference usually found between DRO and VT(or FT) conditions when these are superim-posed upon a simple response-dependentschedule. The equation takes into account onlythe overall rate of reinforcement delivered, thuspredicting that DRO and a response-indepen-dent schedule would have identical effects.

It is possible to modify Herrnstein's equa-tion to better accommodate at least some ofthese findings. We begin by considering thenotion of alternative reinforcement. In Equa-tion 1, there are two sources of alternativereinforcement: r2, which is provided by theexperimenter but superimposed upon the re-inforcement schedule for the measured oper-ant, and ro, which is presumed to arise fromvarious other sources. Herrnstein's equationimplicitly assumes that when any particularreinforcer occurs, it unerringly functions as amember of the appropriate class (rl, r2, or ro).ro reinforcers might remain functionally dis-tinct from those produced by measured oper-ant responses, inasmuch as they may be qual-itatively different (e.g., food vs. "leisure," orfood vs. the reinforcing consequences of somemaintenance activity such as grooming), but

food reinforcers delivered according to super-imposed schedules might not always be func-tionally separable from those delivered by thesuperimposed scheduled. For example, in thecase of VI being succeeded by VI sup VT,Equation 1 requires that a subject's behaviorbe controlled independently by r1 and r2. Un-der VI, of course, all food reinforcers operateas r1. Before considering the question of thebasis on which functional separation of rein-forcers may occur, we shall consider some for-mal consequences of failure of this separation.Assume that a certain proportion (p) of the

r2 reinforcers function as r1. Assume furtherthat, because of a previous training history onsome conventional response-dependent sched-ule, all scheduled r1 reinforcers function as rl.Then the effective rate of reinforcement for r1becomes ri + pr2, and the effective value of r2becomes (1 - p)r2. Assuming that ro remainsconstarnt, Herrnstein's equation is thus mod-ified to produce:

k(r1 + pr2)r1 + pr2 + (1 -p)r2 + rO

(2)

In Equation 2, all terms remain as inEquation 1 except for the addition of p, avariable expressing the proportion of r2 rein-forcers functioning as rl, and thus taking val-ues from 0 to 1. p might thus be interpretedas the degree of "generalization" from r2 torl, or the degree of "confusion" between r2and r, (although it should be noted that sub-jects are assumed never to be "confused" aboutthe source of r1 reinforcers). When p = 0,Equation 2 obviously reduces to Equation 1;when p = 1, all food reinforcers function asr1, regardless of source.

Figure 2 illustrates several properties ofEquation 2. For the purposes of calculation,response rates are expressed as a fraction ofthe asymptote expressed as k in Herrnstein'sequation. The parameter ro is maintained ata constant value of 0.5r,, and r1 is held con-stant at 1 to simulate the conditions in whichthe rate of reinforcement produced by themeasured operant remains constant. The pa-rameter r2 is varied across a range from 0.1r,to 1Or, and p is also varied.

Figure 2 shows that Equation 2 behaves inseveral distinctly different ways depending onp value. When p is large, increasing r2 con-sistently increases response rates above the

79

I. S. BURGESS and J. H. WEARDEN

1.0- 1.0

~ 0.8 - - 0.82C

0.6- 0.6

p

0.4- 0.4

0ffi 0.2- - 0.2

0 0

2 4 6 8 10

r2( r1 units)

Fig. 3. Effects of superimposed response-independentreinforcement predicted by Equation 2 when p (filledsquares) is varied as rate of superimposed reinforcementvaries. The dotted line shows response rate in the absenceof superimposed reinforcement.

level obtaining when r2 is absent. Low valuesof p, on the other hand, produce consistentlydeclining values of response rate as r2 in-creases, and this obviously also applies whenp = 0, and Equation 2 reduces to Equation 1.The Appendix shows that variation in r2

produces consistently increasing or consistent-ly decreasing response rates except when p =

r1/(r1 + ro), in which case response rate re-

mains constant as r2 changes. Thus, the pro-cesses embodied in Equation 2 permit in-creases, decreases, or no change in responserate with r2 variation, depending on p. Theydo not, however, permit certain r2 values toincrease response rate, and others to decreaseit, if p remains constant.The question of what factors might influ-

ence p is considered in greater detail below,but to further develop the present analysis wepropose that one possible influence is rate ofalternative reinforcement itself. If this is lowrelative to the baseline r, rate, then superim-position of response-independent reinforce-ment produces only a slight change in theoverall rate of reinforcement. It also may pro-duce very little disturbance to other aspects ofthe schedule, such as the pattern of contigui-ties between measured operant responses, oth-er types of behavior, and food delivery. On theother hand, with higher rates of superimposedreinforcers, overall rates of reinforcement maychange markedly from the response-depen-dent phase, and substantial numbers of rein-forcers may be contiguous with behavior other

than the measured operant. Such differencesbetween high and low rates of superimposedreinforcement may tend to make reinforcer"misfunction" (i.e., r2 functioning as r1) muchmore likely when superimposed reinforcers aredelivered at lower rates. Thus, we might sup-pose that p and r2 are negatively correlatedwith low r2 rates accompanying high valuesof p, and high r2 rates accompanying low val-ues of p. The consequences of such a devel-opment are illustrated in Figure 3.

In the calculations for Figure 3, p progres-sively decreases as r2 increases. This resultsin low r2 values reliably increasing responserates above the level obtained when r2 is zero,whereas high r2 rates reliably produce re-sponse decrements. This pattern of results israther similar to that obtained by Zeiler (1979)and by Lattal and Boyer (1980).

Consideration of the factors determiningwhether food reinforcers function as r1 or r2also enables this modified version of Herrn-stein's equation to deal, at least qualitatively,with the effects of DRO and VT superim-position. Reinforcers delivered according tosimple response-independent schedules likeVT are delivered without regard for re-sponses, and can thus follow measured oper-ant responses in the way that reinforcersscheduled by the prevailing response-depen-dent schedule can. Reinforcers provided byDRO schedules, on the other hand, can bedelivered only when at least a minimum pe-riod of nonresponding has occurred (a condi-tion that never arises under a simple response-dependent schedule), and this requirementmay make them less liable to misfunction thanVT reinforcers. In general, then, in terms ofEquation 2, DRO reinforcers may have alower p value than VT reinforcers, and as aconsequence may produce greater responsesuppression for a given r2 level. This is theresult obtained by Zeiler (1979) and by Lattaland Boyer (1980). On the other hand, whena small DRO requirement is scheduled aperi-odically, and the comparison response-inde-pendent schedule is also arranged aperiodi-cally, p values for DRO and VT reinforcersmay be very similar, with consequently simi-lar effects on prevailing response rate (e.g.,Rachlin & Baum, 1972).

DISCUSSIONIn a consideration of this modification of

Herrnstein's equation, a few general points

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RESPONSE-INDEPENDENT REINFORCEMENT 81

should be made. First, the modification em-ployed here is not the only one possible. Forexample, it might seem more logically consis-tent to assume that reinforcers misfunction inboth directions (i.e., r, sometimes functions asr2, as well as r2 functioning as rl, as in Equa-tion 2). However, our calculations show thatsuch cross-misfunction with constant p in bothdirections produces predictions very much atvariance with data. It seems necessary to as-sume that a higher proportion of r2 reinforcersfunction as r, than the other way around. Thisassumption is to some extent arbitrary, but itmay be justified by prior history of training.If subjects have had long experience with asingle response-dependent schedule such as VI,then any delivered food reinforcers may havea tendency to function as rl, regardless ofwhether or not they are response-dependent.

Second, we have not attempted to fit ourEquation 2 to any actual results obtained, be-cause no study has provided data in an ap-propriate form. As shown here, however,Equation 2 is qualitatively consistent withmany trends in obtained data, and deals betterthan Equation 1 with data such as those shownin Figure 1.

Third, the use in our model of a parameterthat represents the extent to which reinforcersmisfunction in terms of their source is not aradical step, and is consistent with several re-cent arguments about reinforcement-scheduleperformance. There appears to be growing in-terest in the notion that such performancemight be explained by imperfect adherence tosome simple process such as the matching law(Wearden, 1983). Our Equation 2 similarlydescribes the effect of imperfect adherence toHerrnstein's original equation (Equation 1),but portrays the "imperfection" as systematicdeparture.

Fourth, apart from some discussion of theway differences in the events arranged byDRO and response-independent schedulesmight be manifested in terms of our Equation2, we have not examined here the generalquestion of how patterns of contiguities be-tween responses and reinforcers (or other lo-cal events such as changes in reinforcer per-iodicities) should be interpreted in terms ofHerrnstein's equation. The relation betweenmolar accounts of behavior, which emphasizeoverall response and reinforcement rates oversubstantial periods of time, and local events isa difficult and controversial subject which

cannot be fully treated here. (See Baum, 1981,and Thomas, 1983, for two contrasting dis-cussions.) However, it should be noted that aformulation such as Equation 2 may not beable to account for detailed local patterns ofperformance when response-independentreinforcers are superimposed (e.g., Henton &Iversen, 1978), although it may deal ade-quately with overall rates of measured oper-ants under such conditions.What factors might determine the value of

the reinforcement misfunction parameter (pin Equation 2) in some particular study? Onepossibility is that a reinforcer misfunctions interms of source to the extent that it is notdistinguishable from reinforcers delivered byother sources. Operations that might tend tomake the food reinforcers procedurally ar-ranged as r1 and r2 discriminably different(e.g., arranging that the baseline and super-imposed reinforcers differ in some way, bybeing of different type or magnitude, beingdelivered in different locations, or being dif-ferentially signaled) might thus tend to alterp in predictable ways.

Even in cases in which the baseline andsuperimposed reinforcers are not procedurallydistinct, it is possible that certain features ofthe operant contingencies may determine thefunctional nature of reinforcers. For example,reinforcement misfunction may be less com-mon during VI sup VT-under which thereinforcers provided by VI must immediatelyfollow responses whereas the VT reinforcersneed not-than when VT is superimposedupon response-dependent schedules thatthemselves permit variable relations betweenresponses and food delivery (e.g., see Burgess& Wearden, 1981).

Techniques such as that of Killeen (1978)may prove useful for distinguishing and char-acterizing functions of reinforcement sources.In his procedure, the basis for reinforcementof one of two responses was whether a priorevent was response-dependent or response-in-dependent. The present Equation 2 addressessimilar discriminations, in which the responsedependency to be discriminated relates to re-inforcement itself.

REFERENCES

Baum, W. M. (1981). Discrimination of correlation. InM. L. Commons & J. A. Nevin (Eds.), Quantitativeanalyses of behavior: Vol. 1. Discriminative properties of

82 I. S. BURGESS and J. H. WEARDEN

reinforcement schedules (pp. 247-256). Cambridge, MA:Ballinger.

Boakes, R. A. (1973). Response decrements producedby extinction and by response-independent reinforce-ment. Journal of the Experimental Analysis of Behavior,19, 293-302.

Burgess, I. S., & Wearden, J. H. (1981). Resistance tothe response-decrementing effects of response-inde-pendent reinforcement produced by delay and non-delay schedules of reinforcement. Quarterly Journal ofExperimental Psychology, 33B, 195-207.

Edwards, D. D., Peek, V., & Wolfe, F. (1970). Inde-pendently delivered food decelerates fixed-ratio rates.Journal of the Experimental Analysis of Behavior, 14,301-307.

Henton, W. W., & Iversen, I. H. (1978). Classical con-ditioning and operant conditioning: A response patternanalysis. New York: Springer-Verlag.

Herrnstein, R. J. (1970). On the law of effect. Journalof the Experimental Analysis of Behavior, 13, 243-266.

Killeen, P. (1978). Superstition: A matter of bias, notdetectability. Science, 199, 88-90.

Lattal, K. A. (1974). Combinations of response-rein-forcer dependence and independence. Journal of theExperimental Analysis of Behavior, 22, 357-362.

Lattal, K. A., & Boyer, S. S. (1980). Alternative rein-forcement effects on fixed-interval performance. Jour-nal of the Experimental Analysis of Behavior, 34, 285-296.

Lattal, K. A., & Bryan, A. J. (1976). Effects of con-current response-independent reinforcement on fixed-

interval schedule performance. Journal of the Experi-mental Analysis of Behavior, 26, 495-504.

McLean, A. P., & White, K. G. (1983). Temporal con-straint on choice: Sensitivity and bias in multipleschedules. Journal of the Experimental Analysis of Be-havior, 39, 405-426.

Rachlin, H., & Baum, W. M. (1972). Effects of alter-native reinforcement: Does the source matter? Journalof the Experimental Analysis of Behavior, 18, 231-241.

Thomas, G. V. (1983). Contiguity and contingency ininstrumental conditioning. Learning and Motivation, 14,513-526.

Wearden, J. H. (1981). Bias and undermatching; im-plications for Herrnstein's equation. Behaviour Anal-ysis Letters, 1, 177-185.

Wearden, J. H. (1983). Undermatching and over-matching as deviations from the matching law. Journalof the Experimental Analysis of Behavior, 40, 333-340.

Zeiler, M. D. (1976). Positive reinforcement and theelimination of reinforced responses. Journal of the Ex-perimental Analysis of Behavior, 26, 37-44.

Zeiler, M. D. (1977). Elimination of reinforced behav-ior: Intermittent schedules of not-responding. Journalof the Experimental Analysis of Behavior, 27, 23-32.

Zeiler, M. D. (1979). Reinforcing the absence of fixed-ratio performance. Journal of the Experimental Analysisof Behavior, 31, 321-332.

Received January 25, 1985Final acceptance September 26, 1985

APPENDIXFrom Equation 2,

R - k(r, + Pr2)ri + pr2 + (1 - p)r2 + rO

or

R = k(r1 + pr2)r, + r2 + rO

The slope of the function produced by plottingresponse rate (R) against r2 (dR/dr2) is thus,by the quotient rule,

dR _ k[(r1 + r2 + r.)p - (r, + pr2)]dr2 (r, + r2 + ro)2

The denominator must always have a positivesign, because it is a square, so the sign for theexpression dR/dr2 is determined by the nu-merator alone. dR/dr2 is therefore positivewhen

pr, + prO- r, > O,

or when

p > rir, + rO

Similar reasoning shows that dR/dr2 is neg-ative when

p < r,ri + rO

and is zero when

p= r1r1 + rO

The expression for the numerator of dR/dr2 does not contain any powers greater thanone and can therefore only be monotonicallyincreasing, monotonically decreasing, or zeroas r2 is varied with the other terms constant.