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JOURNAL OF THE EXPERIMENTAL ANALYSIS OF BEHAVIOR 1999, 71, 91–113 NUMBER 1 (JANUARY)

PSYCHOPHYSICS OF REMEMBERING

K. GEOFFREY WHITE AND JOHN T. WIXTED

UNIVERSITY OF OTAGO, NEW ZEALAND ANDUNIVERSITY OF CALIFORNIA, SAN DIEGO

We present a new model of remembering in the context of conditional discrimination. For proce-dures such as delayed matching to sample, the effect of the sample stimuli at the time of remem-bering is represented by a pair of Thurstonian (normal) distributions of effective stimulus values.The critical assumption of the model is that, based on prior experience, each effective stimulus valueis associated with a ratio of reinforcers obtained for previous correct choices of the comparisonstimuli. That ratio determines the choice that is made on the basis of the matching law. The standarddeviations of the distributions are assumed to increase with increasing retention-interval duration,and the distance between their means is assumed to be a function of other factors that influenceoverall difficulty of the discrimination. It is a behavioral model in that choice is determined by itsreinforcement history. The model predicts that the biasing effects of the reinforcer differentialincrease with decreasing discriminability and with increasing retention-interval duration. Data fromseveral conditions using a delayed matching-to-sample procedure with pigeons support the predic-tions.

Key words: remembering, conditional discrimination, forgetting functions, discriminability, rein-forcer probability, delayed matching to sample, pigeon

The foundation for a psychophysical anal-ysis of remembering was laid over a centuryago, with Fechner’s quantitative analysis ofsensation and Ebbinghaus’ experimentalanalysis of memory. Fechner (1860) had pro-posed that sensory experience is a logarith-mic function of stimulus intensity. Althoughthe function relating experience and environ-ment has since been interpreted as followinga power law (Stevens, 1961) Fechner’s fun-damental contribution was to render sensoryexperience amenable to scientific analysis.Fechner had also described a theory for dis-crimination that predated the more recentdevelopment of signal-detection theory(Link, 1994). The field of psychophysics now

Earlier versions of the model described in this paperwere presented to the meetings of the Society for Quan-titative Analyses of Behavior, Washington, D.C., June,1995, and the Behavior Symposium, Christchurch, NewZealand, August, 1996. Preparation of the manuscriptwas facilitated by the generous hospitality provided byPeter Killeen during the first author’s leave at ArizonaState University. Killeen’s constructive comments consid-erably improved the manuscript. We are grateful to An-gela Ruske for her outstanding assistance during the con-duct of the experiments reported here, to members ofour laboratory groups, especially Emily Cooney and Deir-dra Dougherty, for their helpful discussions, and to BarryDingwall, for his essential technical expertise.

Requests for reprints should be addressed to the firstauthor at the Department of Psychology, University ofOtago, Dunedin, New Zealand (E-mail: [email protected] or [email protected]).

offers well-established methods for quantify-ing the changes in behavior that result frommeasurable changes in the physical environ-ment.

Ebbinghaus (1885/1964) showed thatchanges in remembering were amenable toquantification. His measure of retention, thepercentage savings in time to relearn a list ofnonsense words, decreased logarithmically asretention interval increased over a period ofseveral weeks. It has since been shown thatEbbinghaus’ data are better described by apower function (Anderson & Schooler, 1991;Wixted & Ebbesen, 1991). Ebbinghaus’ earlyattempt to quantify memory nevertheless es-tablished the possibility that rememberingmay be amenable to analysis in the sameterms as sensing and perceiving.

The aim of the present paper is to offer ananalysis of remembering that follows the gen-eral approach of signal-detection theory andto apply it to remembering in nonhuman an-imals. The analysis assumes that the effect ofthe stimulus can be represented in terms ofdiscriminal processes of the kind suggestedby Thurstone (1927). Unlike decision-theo-retic approaches, however, a decision criteri-on is not assumed; instead, the animal’schoice is assumed to be influenced directlyby the payoff ratio. This approach makes theinteresting prediction that the effect of pay-offs in biasing an animal’s choice depends on

92 K. GEOFFREY WHITE and JOHN T. WIXTED

the discriminability of the stimuli to be dis-criminated.

Detection Models of Recognition

The treatment of remembering in thesame terms as sensory discrimination was notdeveloped until 80 years after Ebbinghaus,when Murdock (1965) suggested that remem-bering was a matter of discriminating familiarfrom novel events (see also Banks, 1970;Lockhart & Murdock, 1970; Parks, 1966). Animportant assumption was that familar andnovel items were assumed to vary in ‘‘mem-ory strength’’ or ‘‘familiarity,’’ with the meanof the distribution of novel events at zero andthe mean of the distribution of familiarevents at a value of memory strength greaterthan zero. The familiarity of an event increas-es with training or practice. The increase infamiliarity (or item strength) tends to followa power function of the number of trainingtrials (Anderson, 1995). According to the the-ory of signal detection (Green & Swets,1966), the task of deciding whether a partic-ular event is familiar is accomplished by usinga decision rule: Events of memory strengthgreater than a criterion value are categorizedas familiar and hence are responded to as re-membered, whereas those of weaker strengththan the criterion are categorized as novel.

Discriminability and Bias

The signal-detection approach to recogni-tion generated an extensive empirical litera-ture that has benefited from perhaps themost influential theoretical assumption ofpsychophysics: the independence of discrim-inability and response bias. The approachyields a measure of discriminability or mem-orability, measured by the distance d9 be-tween the means of the distributions of noveland familiar events. Memorability is indepen-dent of the location of the criterion value fordeciding between familiar and novel. The lo-cation of the criterion value can be used as ameasure of response bias, that is, the tenden-cy to report familiar versus novel indepen-dently of the discriminability of familiar fromnovel. A major determinant of the location ofthe criterion is relative payoff, although spe-cific rules relating criterion value to relativepayoff have yet to be developed (Macmillan& Creelman, 1991). If the payoff favors re-porting an event as familar, the criterion is

adjusted so that more events are reported asfamiliar. In principle, and in the absence ofcriterion variance, the analysis prescribed bythe signal-detection approach leaves the mea-sure of discriminability untainted by errantchanges in response tendencies, and revealspure memorability.

Criterion Location

Egan (1975) and Macmillan and Creelman(1991) have summarized possibilities forchoice of a decision rule for models that relyon the assumption of a decision criterion. (a)A probability matching rule was proposed byParks (1966) and elaborated by Thomas andLegge (1970) and Thomas (1975). The prob-ability matching rule is assumed to apply incases in which observers have incomplete in-formation about underlying distributions incontrast to the ‘‘ideal observer’’ who has com-plete information and is thus able to utilize alikelihood ratio criterion. For symmetricalpayoff matrices, the probability matching ruleassumes that the observer reports occurrenceof the target items with a probability thatmatches their a priori occurrence. Thus, forexample, if Stimulus A occurs on 80% of thetrials and Stimulus B occurs on 20% of thetrials, the criterion will be placed on the de-cision axis in such a way that the probabilityof reporting Stimulus A is 80%. Creelmanand Donaldson (1968) showed that for judg-ments of line length, changes in the priorprobability of the stimuli did not affect dis-criminability (i.e., the distance between thedistributions) but did affect the placement ofthe decision criterion such that the highlytrained subjects matched response propor-tions to relative stimulus probability. Becausecorrect responses produced monetary re-wards, it is also possible that response pro-portions were sensitive to the relative mone-tary payoff. In other studies, responseproportions undermatched prior stimulusprobability (Dusoir, 1975). (b) An alternativedecision rule is one that maximizes expectedvalue. If expected value is to be maximized,the rule takes account of both the prior prob-abilities of occurrence of the stimuli and thepayoff matrix. It assumes that the decisionrule is based on the likelihood ratio, that is,the ratio of probabilities of occurrence of onesample versus the other (Egan, 1975). Healyand Kubovy (1981) orthogonally varied pay-

93PSYCHOPHYSICS OF REMEMBERING

off and prior probability in a numerical cat-egorization task. They reported that priorprobabilities had larger effects on the likeli-hood ratio criteria than did payoff, but thata probability matching rule that included aconstant based on the payoff matrix could ac-count for their data.

Conditional Discrimination

The separation of discriminability frombias in psychophysics has a parallel in thestudy of conditional discrimination learning,where the effects of the discriminative stimulimay be separated from the effects of the dif-ferential reinforcer probabilities that main-tain the discrimination (Nevin, 1981; White,1986; Williams, 1988). In the context ofchoice procedures, differential reinforce-ment biases the choice towards one alterna-tive versus another. Variation in the reinforc-er probabilities for two choices results in apower function relation between the ratios ofthe choices and the reinforcer ratios (Baum,1974). Variation in the disparity of the dis-criminative stimuli of a conditional discrimi-nation allows an assessment of the extent towhich stimulus disparity limits the discrimi-nation (Nevin, 1969). In conditional discrim-inations, however, the effects of the reinforc-er differential are modulated by stimulusdisparity. When choice relies on both stimu-lus and reinforcer differences, large stimulusdifferences attenuate the effect of the rein-forcer differential and small stimulus differ-ences amplify the reinforcer effect (White,1986). Signal-detection procedures, whichare also classed as conditional discriminations(McCarthy & White, 1987), are associatedwith a similar problem. The extent of discrim-inability may modulate the effects of the re-inforcers that otherwise bias choice towardsone or the other alternative. In situations inwhich discriminability is expected to vary, theeffect of the reinforcer differential may inter-act with the effects of the discriminative stim-uli.

Discriminability Decrement in Remembering

The interaction of reinforcer effects withstimulus effects is especially evident in re-membering by virtue of its primary charac-teristic, the systematic decrement in discrim-inability with increasing retention-intervalduration. Remembering (or forgetting) in-

volves continual changes in discriminabilityor stimulus control. At very short retentionintervals, remembering is easy and the bias-ing effects of differential payoffs are attenu-ated, whereas at very long retention intervals,remembering is difficult and the effects ofdifferential payoffs are magnified (Wixted,1989). The model we describe here addressesthis issue and at the same time deals with an-other more fundamental difficulty associatedwith the earlier signal-detection approach toremembering, namely the problem of speci-fying how the subject’s decision rule is deter-mined by the location of the criterion on thememory strength continuum.

The model we describe does not rely onthe assumption of a criterion. It combines thetime-honored assumption that stimulus ef-fects are represented by random values drawnfrom normal density functions, as in Thur-stone’s (1927) discriminal processes, with themore recent generalization that the prefer-ence or choice between alternatives is a func-tion of the payoffs they produce (Baum,1974; Herrnstein, 1970).

Interaction of Reinforcer andStimulus Control

Quantification of the effects of payoffs onremembering is easily achieved in terms ofthe matching law. Jones and White (1992) ex-amined performance in a standard delayedmatching-to-sample procedure in which fiveresponses to a red or green sample stimulusinitiated a delay interval of variable duration,followed by a choice between red and greencomparison stimuli. The probability of rein-forcers for correct (matching) choices follow-ing red and green sample stimuli was variedover several conditions. When the probabilityof a reinforcer for a correct choice of red washigher than for a correct choice of green, thetendency to choose red was greater than thetendency to choose green. More generally,the ratio of red to green choices followingred and green samples was a power functionof the ratio of reinforcers obtained by correctred versus green choices.

The result reported by Jones and White(1992) is summarized in Figure 1, for 1 oftheir birds, X1. Panel A shows the standarddecrement in matching accuracy with increas-ing delay-interval duration. Matching accu-racy was assessed in terms of a measure of


Fig. 1. Data for Bird X1 replotted from results reported in Appendix A of Jones and White (1992). In (A), theforgetting function is defined by the decrease in discriminability as a function of delay-interval duration. In (B), theslope of the function relating the log ratio of red to green choices to the log ratio of red to green reinforcers steepenswith increasing delay, as summarized in (C).

discriminability (see below). Panel B showsthat the tendency to choose red versus greenis a function of the ratio of reinforcers ob-tained by red versus green choices, as ex-pected from the power function version ofthe matching law (Baum, 1974). The slopesof the functions in Panel B are a measure ofthe exponent of the power function relatingthe ratio of choices to the ratio of reinforcers.The exponent is usually interpreted as an in-dex of sensitivity to reinforcement (Davison& McCarthy, 1988). Panel C summarizesthese slopes and, more importantly, indicatesthat sensitivity to reinforcement is a functionof delay-interval duration. The slopes of thefunctions in Panel B gradually increase withincreasing retention-interval duration.

The result that the biasing effect of the re-inforcer ratio is weak at short delays when dis-criminability is high and is strong at long de-lays when discriminability is weak has aparallel in an earlier result for conditionaldiscriminations (White, 1986; White, Pipe, &

McLean, 1985). Here, reinforcer control ofthe choice was weak for an easy line-tilt dis-crimination and strong for a difficult discrim-ination. Nevin, Cate, and Alsop (1993) havealso reported a stronger effect of varying re-inforcer probability for a smaller disparity be-tween discriminative stimuli in a discrete-tri-als conditional discrimination. That is, thereis a parallel between reinforcer sensitivity ef-fects for perceptual and memorial proce-dures. If a discrimination is easy because oflarge stimulus disparity or a short retentioninterval, there is little effect of varying thereinforcer ratio. If a discrimination is diffi-cult, as with small stimulus disparity or longretention intervals, behavior is very sensitiveto variation in the reinforcer ratio. This resultis consistent with the model of delayedmatching-to-sample performance describedby Wixted (1989). According to this model,the influence of sample stimuli and the ratioof reinforcers obtained by correct choiceshave separate effects that depend on the de-


lay interval. At short delays the conditionaldiscrimination is influential and the reinforc-er effect is weak. At long delays when the con-ditional discrimination is weaker, the rein-forcer effect is strong.

Response MeasuresThe measures of discriminability and bias

that we adopt are, respectively, the geometricmean of the ratio of correct to error respons-es following each sample and the geometricmean of the ratio of red to green choices fol-lowing each sample. These measures can bederived from different theoretical assump-tions, but can otherwise be treated as theory-free measures of performance (as is our pref-erence here). The measures were originallyproposed by Luce (1963) in the context ofchoice theory, and as anticipated by Nevin(1969), Davison and Tustin (1978), and Nev-in (1981) in the context of behavioral detec-tion theory. From a theory-free perspective,the measures reflect the likelihood that thesubject makes correct responses versus errors,or reports one alternative versus the other. Bytaking ratios, the measurement scale is notconstrained in the way that, for example, aproportion scale is bounded by 1.0. The dis-criminability measure (log d, or log a) is lin-early related to the discriminability measured9 derived from signal detection theory, andsatisfies the requirement that both hits andfalse alarms contribute to the measure of dis-criminability (Macmillan & Creelman, 1991,pp. 11–13). The discriminability measure iscalculated by taking the logarithm (base 10)of the ratio of correct (c) to error (e) respons-es following each sample (subscripts r and gfor red and green samples) and averagingthem according to

discriminability,

log d 5 .5·log[(c /e )·(c /e )]. (1)r r g gThe tendency or bias to choose red versus

green is calculated by averaging the loga-rithms of the ratios of choices of red to choic-es of green following each sample. In Panel Bof Figure 1, it is plotted as a function of thered to green reinforcer ratio, according to

log red/green choices

5 0.5·log[(c /c )·(e /e )]r g r g

5 a log(R /R ) 1 c. (2)r g

Consistent with the generalized matching law,the log ratio of red to green choices is a lin-ear function of the log ratio of reinforcersobtained by correct red versus green choices(Rr, Rg). The slope of the function, a, mea-sures sensitivity of choice to variation in thereinforcer ratio, and c is a constant describingoverall (unexplained) preference for one orthe other choice alternative.

The result that the biasing effects of thereinforcer ratio depend on the delay suggeststhat discriminability and bias may not be in-dependent, as otherwise assumed by the stan-dard signal-detection approach or early ver-sions of the behavioral detection approach(e.g., Davison & Tustin, 1978; but see Alsop& Davison, 1991). Both approaches assumeseparate and independent influences of thediscriminative stimuli and factors that gener-ate bias. In the signal-detection model, thelocation of a decision criterion along the ev-idence variable (the determinant of bias) isindependent of the distance between signaland noise distributions (the determinant ofdiscriminability). In the behavioral detectionmodel, the biasing effects of stimulus dispar-ity on choice are independent of the biasingeffects of the reinforcers obtained by thechoices. In a more recent version of the be-havioral detection model (Alsop & Davison,1991; see also Nevin et al., 1993, especiallyEquation 8), although the two free parame-ters describing stimulus and reinforcer effectsare said to be independent, sensitivity to re-inforcement, as measured by Equation 2, re-flects the joint effects of stimulus differenceand differential reinforcement under someconditions. An interesting issue for the be-havioral detection model is that it assumesthat the choice between two alternatives fol-lowing one sample is determined by the ratioof reinforcers obtained by correct choices fol-lowing both samples. That is, it treats the mul-tiple concurrent schedule as if it were a con-current schedule. McLean and White (1983)have claimed that in multiple concurrentschedules (including detection procedures),the reinforcers obtained by correct choicesfollowing one sample do not influence choicefollowing the other sample (with the excep-tion of reallocation of extraneous reinforcersfrom one time to another). An alternative as-sumption was made by Nevin (1981) and isconsistent with Luce’s (1963) choice theory:


Fig. 2. (A) assumed distributions of stimulus effectfor red and green sample stimuli (discriminal processes).(B) decrement in discriminability with increasing delayinterval that characterizes the forgetting function. (C)greater variance (and hence greater overlap) of the dis-criminal processes is illustrated for a long delay intervalcompared to a short delay.

The choice following one sample is influ-enced by the reinforcers obtained by correctchoices following that sample relative to thegeneralized effects of reinforcers obtained bycorrect choices following the other sample.

A Criterion-Free Model of Remembering

We model the interaction between discrim-inability and the biasing effects of reinforcersby assuming that the individual chooses be-tween available response options on the basisof which is more likely to be reinforced in agiven instance. The model may be character-ized as a behavioral theory of remembering,in that choice is directly determined by thereinforcer ratio and no decision criterion isassumed.

Distributions of stimulus effect. The effect ofthe sample stimulus presumably varies fromtrial to trial. The stimulus effect at the timethe choice is required is assumed to be a ran-dom value drawn from a normal density func-tion distributed along a dimension of stimu-lus effect. This dimension may be interpretedas Thurstone’s (1927) psychological continu-um (Luce, 1994), and the distributions cor-respond to Thurstone’s discriminal processes.Consecutive occurrences of a stimulus alwaysinvolve some variation in their effect on be-havior (Green & Swets, 1966), and we assumethat this is particularly the case when the stim-ulus is temporally separated from the behav-ior. At present we are not committed to theview that the variation in stimulus effect is theresult of a psychological process; more simply,the stimulus effect is a reflection of the en-vironment (Fetterman, 1996; White, 1991).The distribution of stimulus effect is specifiedat the time of remembering. The passage oftime weakens the discrimination and contrib-utes to the trial-by-trial variability in stimuluseffect. Thus the discriminal processes repre-sent the potential stimulus effect at a givendelay interval. This issue is discussed furtherin the General Discussion section below.

This assumption is illustrated in Figure 2(Panel A). The distance D between the meansof the discriminal processes is directly relatedto stimulus disparity. Overall higher levels ofdiscriminability are reflected in larger valuesof D. The variances of the distributions areusually assumed to be equal, but unequal var-iances may result from specific manipulationsof the attributes of the to-be-remembered

events. In studies of human recognitionmemory, for example, receiver-operatingcharacteristics that relate hit rates to falsealarm rates are typically asymmetrical. That is,there are different variances for the normaldistributions for new and old items on thefamiliarity or memory strength dimension(Ratcliff, McKoon, & Tindall, 1994; Yoneli-nas, 1994).

The assumption depicted in Figure 2 (Pan-el A) follows the standard detection approach(cf. White & Cooney, 1996; Wixted, 1993), inthat the discriminal processes have equal var-iance and are separated by a distance that isrelated to stimulus disparity. A specific as-sumption we make, however, is that the vari-ance of the distributions increases monoton-ically with increasing delay-interval duration.This assumption is illustrated in Figure 2(Panel C). Two pairs of discriminal processesare shown, both with the same distance D be-tween their means. The pair with less overlapillustrates the case at a short retention inter-val. The pair with the greater overlap, owing


Fig. 3. Probability distributions of stimulus effect(Panel A) multiplied by reinforcer probabilities of .7(green) or .3 (red) (Panel B).

to larger standard deviations, illustrates thecase at a longer retention interval. Thus thereduction in discriminability with increasingretention-interval duration (Figure 2, PanelB) is the result of a continuous increase inthe standard deviations of the discriminalprocesses as the retention interval lengthens.At this stage it is not necessary to specify thefunction relating variance to delay interval,because the main aim of the model is to pre-dict an inverse relation between discrimina-bility and sensitivity to reinforcement. That is,the model predicts that reinforcer sensitivityincreases as discriminability decreases. Thereduction in discriminability can be achievedby increasing the variances of the distribu-tions or by decreasing the distance betweentheir means (or both).

Distributions of reinforcer probability. In ex-perimental procedures, reinforcing conse-quences follow accurate remembering with adefined probability. Each distribution on thecontinuum of stimulus effect is associatedwith a given reinforcer probability. For ex-ample, assume that the left distribution inFigure 3 (Panel A) describes the variation instimulus effect at a given delay interval forthe green sample, and assume that the rightdistribution describes the variation in stimu-lus effect at the same delay for the red sam-ple. (Note that the abscissa is scaled in z-scoreunits, although the variance of the distribu-tions may differ from 1.0.) Also assume that

the reinforcer probability for correctly re-porting green is .7 and is .3 for correctly re-porting red. The distributions of stimulus ef-fect in Panel A may be translated intodistributions of reinforcer probability shownin Figure 3 (Panel B) simply by multiplyingthe values for the green distribution by .7 andthe values of the red distribution by .3. Theresulting distributions are distributions of re-inforcer probabilities along the dimension ofstimulus effect. (Because the probabilities arenot required to sum to 1.0, the distributionsare not probability density functions.) Eachof the possible values of stimulus effect x istherefore associated with a pair of reinforcerprobabilities, one for the green distribution,p(Gx), and one for the red distribution, p(Rx).

Choice determinants. We assume that at a giv-en instant or on a particular trial, the effectof a prior event is represented by a stimulusvalue x randomly determined by either of thediscriminal processes in Panel A of Figure 3.The individual’s task is to choose green orred on each trial. Assuming that the individ-ual has prior experience in the procedure,for a particular value of x choices of red havebeen reinforced with probability p(Rx), andchoices of green have been reinforced withprobability p(Gx). We suppose that on eachtrial (i.e., for each x) the relative tendency tochoose red versus green is directly deter-mined by the ratio of probabilities of rein-forcers, p(Rx)/p(Gx). This assumption followsthe matching law, according to which choiceproportions match reinforcer proportions(Herrnstein, 1970). Our application of thematching assumption differs from the usualuse of the matching law, however, in that wepredict the tendency to choose one of twoalternatives on a single trial on the basis ofrelative reinforcers accumulated over a largenumber of prior trials.

It should be noted that unlike various ver-sions of detection theory, our model does notincorporate the notion of a decision criterion,and, indeed, could be said to be free of de-cision rules in that the response tendency isdetermined directly by relative reinforcer fre-quency. In signal-detection models, the x axiscan be interpreted as a ratio of likelihoodsthat either signal may have been presented(Macmillan & Creelman, 1991). A decisioncriterion is established at a specific likelihoodratio, which remains constant over all trials


(within the bounds of criterion variance). Forvalues of x greater than the decision criteri-on, one response alternative is chosen, andfor values of x smaller than the criterion, theother alternative is chosen. In the presentmodel, choice is not related to the likelihoodratio because it depends on the current ratioof reinforcer probabilities that change with x.The probability of choosing one versus theother alternative varies over the x axis and isnot all or none, as in the detection models.Whereas the adoption of a constant decisioncriterion in the signal-detection models re-sults in the derivation of independent mea-sures of discriminability and bias, the assump-tion that choice is determined by thereinforcer ratio for a given value of x in thepresent model allows the prediction that biasand discriminability interact.

Separate measures of discriminability andbias can be derived from the model, whichsimulates events in a delayed matching-to-sample procedure on a trial-by-trial basis. Themodel generates frequencies of correct anderror responses separately for trials with redand green prior events as a function of therelative reinforcer frequencies that vary overtrials as a result of varying the values of stim-ulus effect.

Predictions from the Model

Predictions from the model were generat-ed by running many simulations, each for5,000 trials of a delayed matching-to-sampleprocedure. The discriminal processes are de-fined by normal (or, in practice, logistic ap-proximations to normal) distributions withvariances Sr2 and Sg2 and means of 20.5D(green) and 10.5D (red). The parameter Dis disparity on the stimulus effect dimension.To model an experimental procedure, initial-ly a red or green sample stimulus is chosenwith a particular probability (usually .5). Forthe chosen sample, say red, a value x is ran-domly selected from the associated distribu-tion. In the simulation, x is specified in z-score units, so that x 5 Sr * zi 1 D/2, or if thegreen distribution is selected, x 5 Sg * zi 2D/2, where zi is a randomly selected z score,that is, a value randomly selected from a nor-mal distribution with a mean of 0 and a stan-dard deviation of 1.

To summarize thus far, on each trial a redor green sample is chosen with a signal pre-

sentation probability of .5, and a value x israndomly sampled from the stimulus effectdimension according to the probability den-sity function defined for each sample stimu-lus. The reinforcer distributions are definedby multiplying the discriminal processes bythe reinforcer probabilities associated withthe distributions. For a given value of x, theproportion of the heights of the resulting dis-tributions (at x) directly predicts the proba-bility of choosing red versus green on thattrial. The heights (or probabilities) are givenby p(Rx) 5 pr * N(D/2, Sr) and p(Gx) 5 pg *N(D/2, Sg). These distributions represent theprobabilities that a reinforcer is arranged onthe red and green choice alternatives for agiven value of x. For example, assume that pr(the probability of reinforcement for a cor-rect red response) and pg (the probability ofreinforcement for a correct green response)are both 1.0 (cf. Figure 3, Panel A). Furtherassume that x on a given red trial happenedto equal 0 (i.e., x on this trial happened tofall at the intersection of the two distribu-tions). In the past for that value of x, a rein-forcer was as likely to be set up on red as ongreen, so an unbiased bird would be equallylikely to choose either alternative. On thenext red trial, imagine that x equaled D/2(i.e., x fell at the mean x value for red trials).Note that, at that point, the height of the reddistribution exceeds that of the green distri-bution by about 6 to 1. That is, in the past,this value of x has occurred on red trialsabout six times as often as green trials. Be-cause pr and pg are both 1.0, this means that,for this particular value of x, a reinforcer issix times as likely to be arranged on the redchoice alternative. Thus, according to thematching law, an unbiased bird would beabout six times as likely to choose red overgreen given this value of x.

The process works the same way when prand pg are unequal. For example, assume thatpr 5 .3 and pg 5 .7. Multiplying both distri-butions by these values yields the distribu-tions shown in Figure 3 (Panel B). Assumeonce again that on a given red trial the valueof x happened to equal 0. Although that valueis as likely to occur on a red trial as on agreen trial (as before), it is no longer the casethat a reinforcer is as likely to be arrangedon the red choice alternative as on the greenchoice alternative. Instead, as illustrated in


Fig. 4. Values of discriminability predicted by themodel as a function of increasing standard deviation withdisparity D set at 1 (top panel) and as a function of in-creasing disparity with standard deviations set at 1 (bot-tom panel).

Figure 3, a reinforcer is now about twice aslikely to be arranged on the green alternativeas on the red alternative. This can be mosteasily appreciated by noting that the heightof the green distribution at 0 is about twicethe height of the red distribution (Figure 3,Panel B). According to the matching law, anunbiased bird will, under these conditions, betwice as likely to choose green as red. Notethat the indifference point (i.e., the value ofx that yields indifference between red andgreen) has shifted to the right. A bird thatwas previously indifferent between red andgreen at an x of 0 (when pr and pg bothequaled 1.0) and that is now indifferent be-tween red and green when x is about 10.5(the indifference point when pr and pg equal.3 and .7, respectively) has exhibited somesensitivity to reinforcement. As described inmore detail later, the degree to which thatindifference point shifts with changes in thescheduled probabilities of reinforcement isdirectly related to D and the standard devia-tions of the distributions. In general, thegreater the overlap between the distributionsby virtue of smaller D or larger standard de-viations, the more a given change in pr andpg changes the indifference point. This is an-other way of saying that the model predictsthat sensitivity to reinforcement will be in-versely related to discriminability. The exper-iments described later provide a test of thatprediction.

For the first 100 trials of the simulation, thereinforcer frequencies associated with thatvalue of x are given by the programmed prob-abilities. However, for later trials, the ob-tained reinforcer proportions (in the simu-lation) were used to compute reinforcerprobabilities. That is, for a particular value ofx, the choice of red or green on that trial waspredicted by the ratio of obtained reinforcersfor that value of x on previous trials. If, forexample, x equals 1, and the number of ob-tained reinforcers for choosing red andgreen in the past for an x of 1 equaled 20and 40, respectively, then on the current trialthe probability of choosing red would be 20/(20 1 40), or .33. That is, the simulated birdswere assumed to match.

Once the choice response, red or green, isselected, reinforcer occurrence is deter-mined by whether the sample stimulus onthat trial was red or green and whether a re-

inforcer had been set up on a probabilisticbasis. Accordingly, the result of a particulartrial would include the sample selected, thechoice response made, and whether a rein-forcer was obtained, just as in a standard ex-perimental procedure. The values of discrim-inability, log d, obtained from the simulationsfor different values of the standard deviationswith D 5 1 (top panel) and for different val-ues of the disparity parameter, D, with Sr 5 Sg5 1 (bottom panel) are shown in Figure 4.Whereas the change in predicted discrimi-nability with increasing standard deviationtends to mimic a forgetting function, the in-crease in discriminability with increasing D isvirtually linear.

Figure 5 shows the results of two sets ofsimulations, conducted as described above,and just as if an actual delayed matching-to-sample procedure was being conducted. Forone set, the disparity parameter was set at D5 4, as might be the case for an overall easierdiscrimination. For the other set, D was set at1, as might be the case for a harder discrim-ination. The standard deviation values wereSr 5 Sg 5 1. For both sets, 12 pairs of rein-forcer probabilities for correct red and greenchoices were varied over values ranging from.9, .1 to .1, .9. From the 5,000 trials of eachsimulation, frequencies of reinforcers for cor-rect red and green choices were used to cal-


Fig. 5. Predictions of the model for low (D 5 1) andhigh (D 5 4) levels of disparity and constant standarddeviations, for log ratios of red to green choices as a func-tion of log ratios of red to green reinforcers obtained bythe correct choices.

Fig. 6. The relation between sensitivity to reinforce-ment predicted by the model and discriminability pre-dicted by the model, for instances when changes in bothwere produced by varying the standard deviation with D5 1 (top panel) and varying the disparity parameter D,with SD 5 1 (bottom panel).

culate log ratios of red to green reinforcers.Corresponding frequencies of red and greenchoices on red-sample and green-sample tri-als were used to calculate the mean of the logratio of red to green choices following thered sample and the log ratio of red to greenchoices following the green sample. Thismeasure is the same as the bias measure pro-posed by Luce (1963) and Davison and Tus-tin (1978), and is described by Equation 2above. The log ratio of choices from the sim-ulations was satisfactorily described by a lin-ear function of the log ratio of reinforcers,consistent with the generalized matching law(Baum, 1974). That is, the present modelpredicts a power function relation betweenchoice and reinforcer ratios.

Three aspects of the predictions in Figure5 are noteworthy. First, the present modelpredicts undermatching in the conditionaldiscrimination, on the basis of the assump-tion that the simulated bird matches choiceproportions to reinforcer proportions at eachvalue of x on the stimulus effect continuum.Second, with decreasing distance between thediscriminal processes (decreasing D), the re-inforcer ratios are predicted to cover a widerrange (despite entering the same values ofprogrammed reinforcer probabilities foreach D). Third, the function for the easy dis-crimination is flatter than the function forthe hard discrimination. That is, sensitivity ofthe choice ratio to changes in the reinforcer

ratio, as given by the slope of the straight line,is smaller for the easy discrimination.

Predicting the Interaction BetweenDiscriminability and Bias

Figure 6 shows predictions from the modelfor sensitivity to reinforcement and discrimi-nability, one plotted as a function of the oth-er. These results were obtained in the sameway as for Figure 5 but with two pairs of re-inforcer probabilities (pr and pg) for each in-stance of standard deviation and D (pr/pg 5.2/.8 or .8/.2). The top panel shows the in-verse relation between reinforcer sensitivityand discriminability when standard deviationswere varied over 14 values with D 5 1. Thebottom panel shows the inverse relation ob-tained by varying D over 17 values with thestandard deviations of the discriminal pro-cesses set at 1. Each 5,000-trial simulationgenerated a matrix of obtained response andreinforcer frequencies, which were used tocalculate log ratios of red to green reinforcersand log ratios of red to green choice respons-es. Reinforcer sensitivity was estimated fromthe slope of the function relating ratios of


Fig. 7. The relation between sensitivity to reinforcement and discriminability at different delay intervals for in-dividual birds (shown by different symbols) in the study by Jones and White (1992) and the regression line for thepredictions from the present model (from Figure 6, top panel).

choices to ratios of reinforcers (cf. Figure 5).Discriminability was calculated from the logratios of correct to error responses (Davison& Tustin, 1978; Luce, 1963; McMillan &Creelman, 1991). Figure 6 shows the inverseand virtually linear relation between reinforc-er sensitivity and discriminability predicted bythe model. The results in the top panel ob-tained when the the standard deviations wereincreased, illustrate the pattern expectedwhen retention-interval duration is increased.This is the interaction evident in the data re-ported by Jones and White (1992). The re-sults in the bottom panel are what might beexpected when other factors that influenceoverall task difficulty are manipulated.

Figure 7 shows the data for individual birdsfrom Jones and White (1992) along with theregression line that best fits the predictionsfrom our simulation for standard devationvaried with the disparity parameter arbitrarilyset at 1 (shown in the top panel of Figure 6).The predictions are generally consistent withthe data.

Sources of Discriminability

An important assumption of the model isthat the distance D between the means of the

discriminal processes is influenced by severalsources of discriminability. These include thewavelength disparity (for example) of thesample stimuli. Retention-interval durationalso influences predicted discriminability, butby increasing the variances of the distribu-tions. Other factors that influence discrimin-abilty are the fixed-ratio requirement for sam-ple-key responding and sample-presentationduration (White, 1985). We now report a setof experimental conditions for pigeons work-ing in a delayed matching-to-sample proce-dure in order to examine further the inverserelation between discriminability and the bi-asing effects of the reinforcer differential. Inall cases, retention-interval duration was var-ied in order to generate further evidence forthe inverse relation predicted when standarddeviations of the discriminal processes are in-creased. In other conditions, factors influenc-ing discrimination difficulty were included inorder to assess the relation predicted whenthe distance between the means of the dis-criminal processes changes. Reinforcementrate was determined by the independentprobability that a correct choice was followedby a reinforcer. Correct choices that were notfollowed by reinforcers had no other conse-


quence, consistent with the typical methodfor arranging delayed matching-to-sampleprocedures and successive discriminations. Inthe latter, for example, when responses inone component are reinforced at variable in-tervals, nonreinforced responses have no oth-er consequence that distinguishes them fromthe nonreinforced responses in the othercomponent in which extinction may be ar-ranged. In some very few studies in which allcorrect choices in delayed matching to sam-ple are followed by hopper illumination andonly some also produce grain (McCarthy &Davison, 1991), hopper illumination couldserve as a conditional reinforcer, and its prob-ability could be confounded with the proba-bility of the food reinforcer.

EXPERIMENT 1:ABSOLUTE RATE OF

REINFORCEMENTIn the signaled magnitude effect described

by Nevin and Grosch (1990), accuracy is over-all higher on trials in which larger reinforcersfollow correct choices than on trials in whichsmall reinforcers follow correct choices. Thetwo types of trials are differentially signaled.The result was confirmed by McCarthy andVoss (1995) and Jones, White, and Alsop(1995). Our generalization of the result isthat accuracy may be overall higher whenhigher absolute rates of reinforcement are ar-ranged in delayed matching to sample.

We compared the effects of overall rich ver-sus overall lean reinforcement rate in a de-layed matching-to-sample procedure. Weasked whether we could replicate the effectreported by Jones and White (1992) and pre-dicted by our model, in which sensitivity toreinforcement increased with increasing de-lay-interval duration, and whether the inverserelation between reinforcer sensitivity anddiscriminability was generalizable to the dif-ferent performance levels generated by themanipulation of absolute reinforcer rate.

METHODSubjects

Three adult homing pigeons with prior ex-perience in delayed matching-to-sample pro-cedures were maintained within 12 g of 80%of their free-feeding body weights by supple-mentary feeding with mixed grain following

experimental sessions. Water and grit werefreely available in the home cages. The colo-ny room was naturally illuminated with a pho-toperiod of approximately 14:10 hr.

Apparatus

A light- and sound-attenuating experimen-tal chamber, 33 cm wide, 33 cm deep, and 34cm high, was painted matte black. A ventila-tion fan provided masking noise. Three trans-lucent response keys, each 2.9 cm in diame-ter, were mounted on one wall 9.2 cm centerto center and 25.6 cm above the grid floor.Each key could be lit red or green and couldbe operated by a minimum force of 0.15 N.A central hopper opening below the centerkey and 5 cm from the grid floor allowed 3-saccess to wheat. The only illumination in thechamber was provided by red or green oncenter or side keys or by a white hopper lightduring 3-s wheat presentations.

Procedure

Sessions were conducted 7 days per week.Each session consisted of 80 trials, separatedby 15-s intertrial intervals (ITIs) during whichthe chamber was dark and responses were in-effective. On each trial, a fixed ratio (FR) offive responses to a red or green sample stim-ulus presented on the center key terminatedthe sample and initiated a dark delay period.The delay lasted for 0.2 s, 1 s, 4 s, or 12 s,with delay durations mixed within sessions.Presentation of red and green comparisonstimuli on side keys followed the delay. A sin-gle choice response to one of the side keysdarkened both keys. Correct (matching)choices produced 3-s access to grain with agiven probability. Unreinforced matching re-sponses and incorrect choices produced 3-sblackout periods, followed by the dark ITI.The order of red and green sample stimuli,whether the red and green comparison stim-uli appeared on the left or the right, and theorder of delay intervals were random withineach session, with the constraint that eachcombination of sample, comparison-stimuluslocation, and delay occurred equally oftenand with no more than three consecutive tri-als with the same sample stimulus.

Reinforcer probabilities were fixed at cer-tain values for each of 20 sessions per con-dition. In five conditions, the probabilitieswere overall rich, and in five conditions they


Table 1

Reinforcer probabilities for correct choices of red andgreen comparison stimuli. 20 sessions were conductedfor each condition.

Condition Order Red Green

Rich FR 5 15-s ITI12345

1239

10

.3

.9

.6

.96

.24

.9

.3

.6

.24

.96

Lean FR 5 15-s ITI678910

45678

.1

.3

.2

.075

.3

.3

.1

.2

.3

.075

FR 1 15-s ITI1112131415

1112131514

.3

.075

.3

.1

.2

.075

.3

.1

.3

.2

FR 1 1-s ITI16171819

16171819

.3

.075

.3

.1

.075

.3

.1

.3 Fig. 8. Discriminability as a function of delay-intervalduration for rich and lean reinforcement-rate conditionsin Experiment 1.

were overall lean. The probabilities areshown in Table 1, which indicates that insome conditions, correct red choices were re-inforced with higher probabilites than werecorrect green choices, and vice versa for oth-er conditions. Table 1 shows the reinforcerprobabilities for all experiments reported inthe present paper; Conditions 1 to 10 con-tributed to Experiment 1.

RESULTS

Data analyses were based on frequencies ofresponses and reinforcers obtained at eachdelay summed over the last eight sessions percondition. Eight sessions allowed a maximumresponse frequency of 80 correct responses inthe cells of the 2 3 2 matrix for red andgreen responses following red and green sam-ple stimuli at each delay, for each reinforcer-ratio condition. Figure 8 shows that when re-sponses were pooled over the five differentreinforcer conditions, discriminability (log d,calculated according to Equation 1) de-creased with increasing delay, and was overall

higher when the overall reinforcer rate wasrich.

The functions in Figure 8 are standard for-getting functions, with a clear reduction indiscriminability with increasing delay dura-tion (White, 1985). For the pooled data foreach bird at each delay, the frequencies ofcorrect red and correct green responses wereeach higher in the rich reinforcement con-ditions than in lean, and the frequencies ofred and green error responses were each low-er in rich than in lean conditions (except forBird S3 at the two shortest delays). Accord-ingly, for each bird, the forgetting functionfor the lean reinforcement conditions wasconsistently lower than that for the rich con-ditions, although the difference was small. Asimilarly small improvement in overall accu-racy as a result of increasing overall reinforce-ment probability has been reported byBlough (1998) for a matching-to-sample pro-cedure with no delays and categories of huesas samples.

The effect of variation in the reinforcer


Fig. 9. Log ratio of red to green choices as a function of log ratio of red to green reinforcers for the richreinforcer-rate conditions of Experiment 1, as a function of delay interval. Values of the parameter estimates for theslope (reinforcer sensitivity) and intercept are shown.

Fig. 10. Log ratio of red to green choices as a function of log ratio of red to green reinforcers for the leanreinforcer-rate conditions of Experiment 1, as a function of delay interval. Values of the parameter estimates for theslope (reinforcer sensitivity) and intercept are shown.

probability ratio is shown in Figure 9 (for therich reinforcement conditions) and Figure 10(for the lean reinforcement conditions).Here, the log ratio of red to green choicesprovides a measure of the tendency to choosered versus green, and was calculated accord-ing to Equation 2. For both rich and lean re-

inforcer probabilities, the same effect as wasreported by Jones and White (1992) is clearin Figures 9 and 10. At the short delays, theslopes of the matching lines are relatively flat,and at the long delays they are steep.

The slopes, which provide an index of sen-sitivity to reinforcement, are compared in Fig-


Fig. 11. Sensitivity to reinforcement (values of theslopes in Figures 9 and 10) as a function of delay-intervalduration for rich and lean reinforcer-rate conditions ofExperiment 1.

ure 11 for rich and lean reinforcement con-ditions. With the exception of the leancondition for Bird S1, as the delay lengthens,reinforcer sensitivity increases. This result isconsistent with an inverse relation betweenreinforcer sensitivity and discriminability.There was no consistent difference in rein-forcer sensitivity between rich and lean con-ditions, as indicated by the overlapping stan-dard error bars in Figure 11. In the contextof the inverse relation between reinforcersensitivity and discriminability that is beingexplored here, this result is perhaps not sur-prising in view of the small difference in dis-criminability between the rich and lean con-ditions. In addition, although the generaloutcome of the model is an inverse relationbetween reinforcer discriminability and sen-sitivity, the model predicts that variation inthe absolute reinforcer rate should have noeffect on sensitivity because the same propor-tions of obtained reinforcers are maintainedfor rich and lean conditions.

EXPERIMENT 2:SAMPLE RATIOREQUIREMENT

In Experiment 2 we examined the inter-action between reinforcer sensitivity and thechange in discriminability caused by decreas-ing the sample-key ratio requirement. A stan-dard result is that discriminability is overallhigher at all delay intervals when more re-sponses are required to the sample stimulus(Roberts, 1972; White, 1985). The same 3birds and the same procedure as for the leanconditions in Experiment 1 were used. An ad-ditional five reinforcer probability conditionswere conducted, but with a ratio requirementof FR 1 for sample-key responding. Com-pared to the function for the FR 5 samplerequirement in Experiment 1, an overall low-er level of discriminability was expected forthe FR 1 conditions, together with an overallhigher level of reinforcer sensitivity.

METHOD

The 3 birds of Experiment 1 continued un-der the same procedural conditions as in thelean reinforcer probability conditions of Ex-periment 1, but with just a single response tothe red or green sample stimulus being re-quired to initiate the delay interval. Table 1

shows the order in which Conditions 11 to 15were conducted. As for the earlier conditions,each was conducted for 20 sessions. Analyseswere based on response and reinforcer fre-quencies summed for the last eight sessionsof each condition.

RESULTS

Figure 12 shows the forgetting functionsfor the FR 1 and FR 5 conditions, for datapooled over the five reinforcer conditions ineach set for each bird. The result shows thetypical overall reduction in discriminabilitythat results from decreasing the sample-keyratio requirement (Roberts, 1972; White,1985). Figure 13 shows that the matching lawfunctions for the FR 1 conditions behaved invery much the same way as for the FR 5 leanconditions in Experiment 1 (Figure 10). Theslopes of the functions tended to increasewith increasing delay-interval duration.

Figure 14 summarizes the change in rein-forcer sensitivity over delay-interval durationfor the FR 1 and FR 5 conditions. With theexception of the 0.1-s and 4-s delays for BirdS1 and the 12-s delay for Bird S2, reinforcer


Fig. 12. Discriminability as a function of delay-intervalduration for conditions with sample ratio requirementsof FR 5 (from Experiment 1) and FR 1 (Experiment 2).

Fig. 13. Log ratio of red to green choices as a function of log ratio of red to green reinforcers for the FR 1 ratio-requirement condition of Experiment 1, as a function of delay interval. Values of the parameter estimates for theslope (reinforcer sensitivity) and intercept are shown.

sensitivity at the different delays is lower forthe FR 5 conditions than for the lower dis-criminability FR 1 conditions. The tendencyfor reinforcer sensitivity to increase with in-creasing delay duration, which was reportedby Jones and White (1992) and was observedin the two sets of conditions in Experiment1, is just as apparent for the FR 1 conditionsof Experiment 2 for each of the 3 birds

EXPERIMENT 3:INTERTRIAL INTERVAL

DURATIONAn inconsistency between our result and

model predictions and some data reported byMcCarthy and Davison (1991) and McCarthyand Voss (1995) has left us with an interestingpuzzle. They reported that reinforcer sensitiv-ity decreased with increasing delay-interval du-ration. McCarthy and Davison varied rein-forcer probability over three values in adelayed matching procedure in which thesamples were different brightnesses. For de-lays of 0 s, 1 s, 3 s, and 25 s, respectively, re-inforcer sensitivities averaged .75, .52, .56,and .37 (with standard errors of about .09).This result is not consistent with those re-ported by Jones and White (1992) and in thepresent Experiments 1 and 2, and is incon-sistent with the expectation that when control


Fig. 14. Sensitivity to reinforcement (values of theslopes in Figures 10 and 13) as a function of delay-inter-val duration for FR 1 and FR 5 sample ratio-requirementconditions in Experiments 1 and 2.

by the conditional discrimination is minimal,choice should be governed by the reinforcerratio (cf. Wixted, 1989).

In the study by McCarthy and Davison(1991), each condition was conducted withjust one delay interval in each session (where-as delays were mixed within sessions in thepresent study). In addition, they employed areinforcement procedure in which a reinforc-er for a correct choice on one alternativecould not be obtained until the reinforcerthat had been set up for a correct choice onthe other alternative had been obtained in aprevious trial. This procedure resulted in areduction in total obtained reinforcers withincreasing delay, where total reinforcers ob-tained at the 25-s delay were about half thoseat the three shorter delays. Futhermore,Jones and White (1992) have shown that thereinforcement scheduling procedure, per-haps along with the use of just one delay persession, results in a large response bias forchoice between left and right keys.

The procedure used by McCarthy and Voss(1995) was more similar to that used by Jones

and White (1992), except for (a) use of theprocedure to control the ratio of obtained re-inforcers, (b) presentation of the hopperlight following every correct response, and(c) the use of a short ITI, among other dif-ferences. Their general result was consistentwith that reported by McCarthy and Davison(1991), namely a decrease in reinforcer sen-sitivity with increasing delay. In some instanc-es for long delays when there was zero dis-criminability, there was also zero reinforcersensitivity. That is, variation in the reinforcerratio for correct choices had no effect on thechoice, a puzzling result. In the experimentsby McCarthy and her colleagues, there weregenerally low levels of discriminability, com-pared to those reported by Jones and White(1992). In the experiments by McCarthy andher colleagues, short intertrial intervals wereused, which are known to lower discrimina-bility (Edhouse & White, 1988; Roberts, 1972;White, 1985). Therefore, in Experiment 3 wecompared the effect of a 1-s ITI to the effectof a 15-s ITI using the same procedure as forExperiment 2.

METHOD

The subjects, apparatus, and procedurewere the same as those in Experiment 2 withthe FR 1 sample ratio requirement exceptthat the intertrial interval was reduced to 1 s.Four ratios of reinforcer probabilities wereused. The reinforcer probabilities and the or-der of conditions are given in Table 1. Twentysessions were conducted for each condition.Analyses were based on response and rein-forcer frequencies summed over the last eightsessions for each condition.

RESULTS

Figure 15 shows that the usual ITI effecton the forgetting functions was obtained. Thefunctions with the 1-s ITI exhibit overall lowdiscriminability, compared to the functionsfor the 15-s ITI from Conditions 11 through15 of Experiment 2. The matching law func-tions in Figure 16 are systematic, but com-pared to those in Experiments 1 and 2, ex-hibit steep slopes at all delays.

Figure 17 compares the change in rein-forcer sensitivity with increasing delay-inter-val duration for the 1-s ITI conditions to thefunction for the corresponding 15-s ITI con-ditions. Whereas the function for the 15-s ITI


Fig. 15. Discriminability as a function of delay-intervalduration for ITIs of 15 s (Experiment 2 data) and 1 s(Experiment 3 data).

Fig. 16. Log ratio of red to green choices as a function of the log ratio of red to green reinforcers obtained forcorrect choices as a function of delay, for the 1-s ITI condition of Experiment 3.

increases, the function for the 1-s ITI decreas-es. We therefore suppose that short ITIs maybe responsible for the high reinforcer sensi-tivity at short delays seen here and in the dataof McCarthy and Voss (1995) and McCarthyand Davison (1991). Note, however, that atlonger delays reinforcer sensitivity remainshigh, whereas in the data reported by Mc-Carthy and Voss and McCarthy and Davisonit drops to near-zero levels. That is, even witha short ITI we were unable to produce theresult seen in the studies by McCarthy andVoss and McCarthy and Davison.

The overall high levels of reinforcer sensi-tivity in Experiment 3 are predicted by thepresent model in that discriminability wasoverall very low. But in order to predict thehigh sensitivity at the shortest delays, an ad-ditional assumption is needed. One possibil-ity is that with short delays, short ITIs, andvery low discriminability levels, there may bea stronger proactive effect of the reinforcedresponse from the preceding trial. This situ-ation was modeled by running simulations asabove in order to generate a set of reinforcersensitivities that corresponded to the similarlysmall discriminability values obtained for the1-s ITI condition. The additional assumptionwas incorporated by amplifying the tendencyto make a choice on trials preceded by a re-inforcer, in the direction of the previously re-


Fig. 17. Sensitivity to reinforcement as a function ofdelay interval for conditions with 15-s ITIs (Experiment2) and 1-s ITIs (Experiment 3). Triangles in the panelfor the mean data are the predictions from the model.

Fig. 18. The relation between reinforcer sensitivityand discriminability in Experiments 1 and 2. For eachbird, individual data points are shown for the four delayintervals for each of four sets of conditions combiningFR 1 or FR 5 with rich or lean reinforcer rates and 1-s or15-s ITIs. The straight line is the prediction of the modeltaken from Figure 6 (top panel) and also drawn in Fig-ure 7.

inforced response, by an arbitrarily chosenfactor of five. (By comparison, the level ofleft-right bias at long delays in the study byMcCarthy & Davison, 1991, ranged from 1.26to 8.91, and in the study by McCarthy & Voss,1995, they ranged from 1.15 to 5.99.) Theresulting predictions are shown as trianglesalong with the mean data in Figure 17. Thepredicted values are satisfactorily close to theobtained values for reinforcer sensitivity forthe 1-s ITI condition. Our model thereforecopes well with the present data.

SUMMARY OF RESULTS

The main approach that we took to evalu-ating the present model was to vary the levelof discriminability in delayed matching tosample by varying delay-interval duration andother factors, and to ask whether there wasan inverse relation between reinforcer sensi-tivity and discriminability as predicted by themodel. In order to summarize the relationbetween reinforcer sensitivity and discrimi-nability for Experiments 1 and 2, in whichthere was clear variation in discriminability,

Figure 18 shows values of reinforcer sensitiv-ity for each delay and each of the three setsof conditions, plotted against the correspond-ing values of discriminability (12 points perbird). The predicted straight-line function in-cluded in each panel is exactly the same func-tion as in Figure 7, in which the data fromthe study by Jones and White (1992) wereplotted. That is, the same predicted functionis shown with the data for the different birdsand the three different procedures in thepresent study. Using the straight line in Fig-ure 18 to predict values for reinforcer sensi-tivity, the slopes of the regression (throughthe origin) relating obtained to predicted val-


ues were 1.03, 0.90, and 1.07 for Birds S1, S2,and S3, respectively. The results of Experi-ment 3 were not included in Figure 18 be-cause the lack of variation in discriminabilitydid not contribute to assessment of the in-verse relation predicted by the model. Nev-ertheless, the overall high levels of reinforcersensitivity in Experiment 3 (Figure 17) areconsistent with the present model in that theyare expected at low discriminability levels.When the data from Experiment 3 are plot-ted in Figure 18, they cluster at the lowestlevels of discriminability, and around the linepredicted by the model. Because the samepredicted function as in Figure 7, generatedby varying standard deviations with D arbi-trarily fixed at 1, is drawn for the differentbirds in Figure 18, the correspondence of thedata to the predicted function provides con-vincing evidence that our model successfullypredicts the interaction between reinforcersensitivity and discriminability.

GENERAL DISCUSSION

To summarize the main point of the modelwe propose here, the individual’s trial-by-trialtendency to choose one or the other com-parison stimulus in a delayed matching pro-cedure is determined by the relative proba-bility of previously obtained reinforcersassociated with the value x of the stimulus ef-fect on the current trial. The value x is thestimulus effect at the time of remembering.The variation in stimulus effect is defined interms of a pair of hypothetical discriminalprocesses or distributions. With increasing re-tention-interval duration, the variances of thedistributions (discriminal dispersions) in-crease. Discriminability is predicted to de-crease as a result of the increased overlap be-tween the distributions. Discriminability mayalso decrease as a result of other factors thatlead to a reduction in the separation of themeans of the distributions.

The model can be characterized in termsof a conditional discrimination in whichthere is a continuum of values of the discrim-inative stimuli. That is, although only twostimuli are actually used (red and green),their effects vary from trial to trial, therebycreating a distribution of effective stimulusvalues for each stimulus. Each effective stim-ulus value has associated with it a unique re-

inforcement history. Thus, instead of twophysical stimuli, the model takes account ofa whole range of effective stimuli that happento be normally distributed. But the model isotherwise just the familiar matching law. Themodel predicts that the effect of the reinforc-er ratio in biasing choices is amplified by de-creasing the level of discriminability that oc-curs with longer retention intervals. Forexample, if the distributions overlap exten-sively, variations in the arranged reinforcerratio result in large changes in obtained re-inforcer ratios and hence choice ratios acrossthe range of stimulus effect values. But withvirtually nonoverlapping distributions, ob-tained reinforcer ratios and hence choice ra-tios uniformly favor red for x values underthe red distribution and uniformly favorgreen under the green distribution, relativelyindependently of variation in the arrangedreinforcer ratios. In the experiments report-ed here, there was clear evidence for this pre-diction, and also for the more general pre-diction that factors that decrease overalllevels of discriminability increase sensitivity toreinforcement.

Our behavioral theory of rememberingbears obvious similarities to signal-detectiontheory, but it differs in one critical respect.Detection theory assumes that the individualarrives at a decision by setting a criterionsomewhere along the stimulus effect contin-uum. If x on a given trial exceeds that crite-rion, the pigeon chooses red; otherwise,green is selected. According to this account,every trial involves a decision on the part ofthe bird (namely, does x exceed the criterionor not?). The decision criterion plays no rolein the present nonmediational theory of re-membering. Instead, on each trial, the bird’sbehavior is governed by the history of rein-forcement for choosing red or green underthe prevailing conditions. If, given a value ofx, a response to green has been reinforcedmore often than a response to red, the birdwill be more likely to choose green than redaccording to the matching law.

The version of the model described abovehas two parameters, one for the distance be-tween the discriminal processes and anotherfor their variance. Further possibilities in-clude different variances for the two distri-butions, as may occur in delayed matchingfor asymmetrical sample stimuli (Wixted &


Dougherty, 1996). Bias and undermatchingin the effect of the reinforcer ratio on thetrial-by-trial choice probability may also be in-cluded. These factors have not been incor-porated here because they involve addingfree parameters to an otherwise simple mod-el. Quantitative fits of the model to datawould be enhanced, however, by inclusion ofthe additional parameters.

The present model bears some apparentsimilarity to detection models in that both as-sume distributions of stimulus effect along apsychological continuum, following Thur-stone’s notion of discriminal processes (Luce,1994). The question of how to interpret thiscontinuum is of importance to the issue ofhow the present model differs from othermodels of recognition memory.

Signal-detection theory was applied to rec-ognition memory by Parks (1966) as suggest-ed by Murdock (1965) and later by Banks(1970) and Lockhart and Murdock (1970).Snodgrass and Corwin (1988) give a clearsummary of these models and discuss theproblem of measuring response bias in rec-ognition memory. In the model described byParks, old and new items vary in their degreeof ‘‘familiarity’’ or, in other models, ‘‘memorystrength.’’ Familiarity or memory strengthvaries from item to item according to a nor-mal distribution. The mean of the probabilitydistribution for old items is greater in famil-iarity or memory strength than that for newitems, if old is discriminable from new. Thedimension along which item strength variesis a psychological continuum (Luce, 1994), inthat it reflects variation in how stimuli may berepresented in the nervous system or storedin long-term memory. As applied to thematching-to-sample paradigm or similar two-alternative choice procedures, however, thereis little sense in defining a zero point formemory strength or familiarity, unless thestimulus effect continuum is interpreted asextending from ‘‘greenness’’ to ‘‘redness.’’

In detection models, the dimension is por-trayed as a decision axis (Green & Swets,1966). A decision based on a criterion valueon the dimension provides the main basis forthe recognition response. But in the presentmodel, there is some difficulty in identifyingthe continuum as a decision axis, because nocriteria are assumed. One possibility is to de-fine the stimulus effect dimension in terms

of a composite of the factors that influencethe overall level of performance. That is, thedisparity parameter D may be related to wave-length disparity (for a hue discrimination) inaddition to factors such as the FR sample re-quirement or ITI duration. A related issue iswhether the stimulus effect dimension shouldbe construed in sensory or memorial terms,an issue related to our notion that stimuluseffect is defined at the time of remembering.Although a memorial interpretation is an ob-vious possibility, the notion that the effect ofthe temporally distant sample stimulus maybe direct is consistent with a sensory inter-pretation (White, 1991). That is, if all of thevariation in stimulus effect can be attributedto factors in the environment, a ‘‘direct re-membering’’ approach can be sustained. Itmay be more plausible, however, to adopt theview that organismic processes contribute tostimulus variation. In either case, the stimuluseffect dimension seems to be related to hue(e.g., for red and green sample stimuli), inwhich case the means of the stimulus effectdistributions should not drift towards eachother with increasing delay, as was assumedby White and Cooney (1996). For this reason,it was assumed in the present model that dis-tribution variances increased with increasingdelay, with the location of the means remain-ing constant. A mediational view consistentwith signal-detection interpretations of thestimulus effect dimension is that a value x ona given trial provides ‘‘evidence’’ on the basisof which the choice response is made. Forexample, for red versus triangle samples (forstimuli on different dimensions), the value ofx provides evidence for the redness versus tri-angularity of the previously presented sam-ple. As it happens, the present model is silenton these alternatives, but does emphasizethat the stimulus effect value on which thechoice is conditional in the model is definedat the time of remembering, consistent withthe treatment of remembering as delayedstimulus control.

As a final note, we offer a comment on thescope of the model. The aim was to presenta model that incorporated the effects of thereinforcer ratio in delayed matching-to-sam-ple procedures, or other remembering pro-cedures in which the choice alternatives wereexplicit. The general prediction that the re-inforcer effect becomes more influential as


the delay interval lengthens has been notedbefore (Wixted, 1989) and was given a quan-titative basis in the present model. Althoughthe present paper has focused on remember-ing, the model may be extended in future ap-plications to account for stimulus–reinforcerinteractions in signal-detection proceduresand may be compared to other possible mod-els in relation to the prediction of such in-teractions (Nevin et al., 1993). It is of interestto note that the model described by Alsopand Davison (1991, Equation 3) predicts thatreinforcer sensitivity decreases with increas-ing stimulus discriminability when discrimi-nation between the choice alternatives isimperfect. Future research involving manipu-lation of comparison–stimulus disparity in de-layed matching and signal-detection proce-dures (cf. Nevin et al., 1993; White, 1986)might assist in determining whether the pres-ent model and that described by Alsop andDavison are equally successful in accountingfor the inverse relation between reinforcersensitivity and discriminability described inthe present paper.

Given programmed reinforcer probabili-ties, the present model predicts obtained re-inforcer frequencies as well as response fre-quencies, and on the basis of an assumptionof strict matching of the ratio of choice prob-abilities on a given trial to reinforcer proba-bilities (given a value of x), the model pre-dicts the power function relation betweenresponse and reinforcer ratios. Also to be ex-plored in future analyses, the model predictschange in performance from the early trialsin which the reinforcer effects are variableowing to few obtained reinforcers, to the‘‘steady-state’’ performance shown here. Themain prediction offered by the model, as em-phasized in the present paper, is the inverserelation between discriminability and rein-forcer sensitivity. Although it is possible to in-corporate in the model factors such as con-stant response bias and proactive effects ofreinforcers on prior trials, the simplest ver-sion of the model has two free parameters,one for the distance between the distributionmeans and one for the standard deviation ofthe distributions. Although there is no freeparameter that describes the reinforcer ef-fect, the model predicts systematic variationin reinforcer effect on the basis of factors af-fecting discriminability.

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Received April 29, 1998Final acceptance August 13, 1998

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