& 1981,29 (4),371-382 Towards a behavioral theory of in detection

DIANNE McCARTHY and MICHAEL DAVISON University ofAuckland, Auckland, New Zealand

A behavioral model for performance on signal-detection tasks is presented, It is based on a re­ lation between response and reinforcement ratios which has been derived from both animal and human research on the distribution of behavior between concurrently available schedules of re­ inforcement, This model establishes the ratio of obtained reinforcements for the choice responses, and not the probability of presentation, as the effective biaser in signal-detection re­ search, Furthermore, experimental procedures which do not control the obtained reinforce­ ment ratio are shown to give rise to unstable bias contours. Isobias contours, on the other hand, arise only from controlled reinforcement-ratio procedures,

The theory of signal detection (Peterson, Birdsall, bias (e.g., stimulus-presentation probability and pay­ & Fox, 1954; Tanner & Swets, 1954; van Meter & off). Dusoir, reviewing the then-current theories of Middleton, 1954)holds the promise of extracting two bias (e.g., Broadbent, 1971; Hardy & Legge, 1968; independent measures to describe behavior in a de­ Healy & Jones, 1973; Luce, 1963; Parks, 1966;Thomas tection task. The two measures are stimulus discrim­ & Legge, 1970; Treisman, 1964), found no measure inability, a measure of the subject's ability to tell two of bias satisfying the above requirements. stimulus conditions apart, and bias (or criterion), a Here, we review a behavioral approach to bias measure of how performance can be changed by non­ which: (1) unlike signal-detection theory, does not sensory motivational or payoff variables. Most re­ depend upon any a priori distribution assumptions; search in contemporary psychophysics has placed (2) relates the change in behavior to a change in a primary emphasis upon the sensory performance of measurable independent variable; and (3) is based on human subjects, and attempts to relate stimulus pa­ a well-documented empirical relation in concurrent rameters to the physical properties of the stimuli, in­ schedule choice behavior-the generalized matching dependently of bias, are well documented. As a re­ law (Baum, 1974). Our interpretation results in a bias sult, rather less effort has been expended in the search expression similar to that proposed by Luce (1963), for a bias parameter which remains invariant with who obtained a bias parameter, b, and a discrimina­ changes in discriminability (Dusoir, 1975; Luce, bility parameter, t'J, by applying choice theory (Luce, 1963). 1959) to the standard signal-detection paradigm. The term "bias" (or "criterion") has frequently Like Luce, we also obtain a discriminability-free in­ been used in both an explanatory and a descriptive dex of bias without relying upon any assumed under­ -often with serious confusion (Treisman, lying theoretical distributions. Furthermore, our 1976). In addition, as Dusoir (1975) pointed out, model separates this bias measure into a constant and there is no generally accepted way of measuring bias a variable component. and, hence, there is little agreement on the true shape The paper opens by tracing the development of the of empirical isobias contours. Dusoir suggested the generalized matching law (Baum, 1974)in the experi­ need for a measure of bias which was unaffected by mental analysis of choice behavior, and follows this changes in variables which, on a priori grounds, with the presentation of a model for signal-detection might be expected to change only discriminability performance based on the application of this law to (e.g., stimulus values). The measure must, however, the standard detection-theory payoff matrix (Davison be affected by operations which should manipulate & Tustin, 1978). The remainder of the paper is de­ voted to a discussion of the implications of this model for the measurement of bias and the generation of The research reported here was supported entirely by the New isobias contours in animal psychophysics. In addition, Zealand University Grants Committee, to which organization we we will show how the bias problem, as specified by continue to be most grateful. We thank the Associate Editor, Dusoir (1975), can be seen as a problem in defining Dr. A. Kristofferson, and an anonymous reviewer for helpful comments, and also Michael Corballis for constructive discussion. biasing variables in relation to experimental pro­ Requests for reprints may be sent to Dianne McCarthy, Depart­ cedures. We close with a brief discussion concerning ment of , University of Auckland, Private Bag, the extent to which both animal and human psycho­ Auckland; New Zealand. physics can be described by a behavioral theory.

Copyright 1981 Psychonomic Society, Inc. 371 0031-5117/81/040371-12$01.45/0 372 McCARTHY AND DAVISON

Generalized Matching Law inforcement allocation (Baum, 1974; Lander & Irwin, The generalized matching law (Baum, 1974) is a 1968). Stable performance generally conforms to the quantitative description of how stable performance generalized matching law equation (Baum, 1974): in concurrent (CONe) and multiple (MULT) sched­ PA= C (RA)a ules is affected by changing the reinforcements, or P R ' (1) payoffs, for the two (or more) performances. In con­ B B current schedules, two responses are available simul­ taneously and each is reinforced on a defined sched­ where PA and PB are the number of responses emit­ ule. For example, in a concurrent variable-interval ted in the two components, and RA and RB are the variable-interval (CONC VI VI) schedule, each of the number of reinforcements obtained from the compo­ two responses to two manipulanda is reinforced aperi­ nent schedules. The exponent a reflects the sensitivity odically. A typical manipulation would be to vary of the response ratio to changes in the ratio of ob­ the mean intervals of the two VI schedules, so that tained reinforcements (Lander & Irwin, 1968). The different numbers of reinforcements would be ob­ value of c describes inherent bias (Baum, 1974; tained for the two responses. Other common manip­ McCarthy & Davison, 1979), a constant preference ulations would be to vary the magnitudes, delays, over all experimental conditions unaffected by changes and types of reinforcements (see de Villiers, 1977). in the obtained reinforcement distribution between A multiple schedule is similar, except that the sched­ the two alternatives. ules are successively presented to the subject for a The values of a and c are obtained from the slope period of time with a distinctive stimulus associated and intercept, respectively, of a least squares line fit­ with the operation of each schedule. ted to the logarithmic data, that is: Studies of control by schedules of reinforcement have attempted to specify quantitatively how stable­ log ( ~:) =a log (~:) + log c. (2) state responding in the components of concurrent and multiple schedules may be controlled by the rein­ forcements obtained in each component schedule. If the ratios match (a =c = 1), as they would accord­ The method of assessing the sensitivity of behavior ing to the strict matching law (Herrnstein, 1970), to changes in reinforcement is to plot the ratio of the then a line of unit slope would pass through the point number of responses emitted on each alternative as a (0,0); see Line A in Figure, 1. That is, when reinforce­ function of the ratio of the number of reinforcements ments are distributed equally between the two alter­ obtained on each alternative (Baum, 1974; Baum & natives (unit ratio), an equal number of responses Rachlin, 1969;Staddon, 1968). will be emitted on each alternative. If least squares linear regression lines are fitted be­ Baum (1974) noted that two kinds of deviation tween the logarithm of the response ratio and the from strict matching, in terms of a or c, are observed: logarithm of the obtained reinforcement ratio (Fig­ the subject may over- or underestimate reinforce­ ure 1), the slope of the fitted line measures the sensi­ ment differences between the alternatives for various tivity with which response allocation changes with re- reasons, yielding a value of a different from unity. This is called overmatching or undermatching, de­ pending on its direction. An undermatching relation, 1·0 for example, is shown as Line C in Figure 1. Or the subject may be inclined to over- or underrespond to o one alternative or the other, independent of rein­ forcement. This is inherent bias, and is represented ~ by a nonunit value of c. Such a constant bias, given 0:: c by the antilog of the intercept, shows up as a constant W If) displacement from the matching diagonal, as indi­ Z o cated by Line B in Figure 1. Frequently, combina­ o a.. tions of both undermatching and inherent bias are re­ If) ported (Baum, 1974). w 0:: Undermatching. After training to a criterion of stability (typically 15 to 30 h), the values of the ex­ <.:) o ponent a in Equation 1 are between .80 and 1.0 for ---l-1·0L-~ J..- ---J CONC VI VI schedule performance (Lobb & Davison, -1·0 o 1·0 1975; Myers & Myers, 1977), and about .33 for mul­ LOG REI NF RATIO tiple schedule performance (Lander & Irwin, 1968). Other schedule combinations yielding values for a of Figure 1. The logarithm of response ratios as a function of the logarithm of reinforcement ratios. Line A represents strict match­ less than unity include concurrent fixed-interval ing. Line 8 represents biased matching performance. Line C rep­ variable-interval (CONC FI VI) (Lobb & Davison, resents undermatching. 1975; Nevin, 1971; Trevett, Davison, & Williams, BIAS AND SIGNAL DETECTION 373

1972), MULT FI FI and MULT VI VI (Barron & Matching Model of Signal Detection Davison, 1972; Lander & Irwin, 1968), and CHAIN By applying the generalized matching law (Equa­ FI FI (Davison, 1974) schedules. Undermatching has tion 1) to the standard detection-theory payoff ma­ also been found with concurrent differential­ trix, Davison and Tustin (1978) derived a measure of reinforcement-of-low-rate schedules (Staddon, 1968). , independent of discriminability and Response ratio has been found to undermatch mag­ analogous to that of signal-detection theory. The fol­ nitude of reinforcement (Schneider, 1973; Todorov, lowing section traces the development of this ap­ 1973). Also, for CONC VI VI schedules when quali­ proach. tatively different reinforcers are arranged for each In the standard yes-no detection task, the subject component, response ratios have been found to un­ is trained to emit one response (P ,) when one stimu­ dermatch reinforcement ratios (Hollard & Davison, lus (S,) is presented and another response (P 2) when a 1971; Mathews & Temple, 1979). different stimulus (S2) is presented. With two stimuli Inherent bias. In the generalized matching law, an and two responses, four possible outcomes are de­ inherent bias (log c in Equation 2) refers to a con­ fined. Figure 2 is a general stimulus-response matrix stant preference for one response over another inde­ showing the events in a typical yes-no detection pro­ pendent of the reinforcement ratio (Baum, 1974). cedure. In this figure, W, X, Y, and Z refer to the Such an inherent bias can arise out of the equipment cells of the matrix. For example, with P denoting re­ (e.g., one response manipulandum more difficult to sponses and R denoting reinforcers, Pw is the number execute than another) or out of the subject (perhaps of responses in cell Wand R, is the number of rein­ a preference for responding to one color rather than forcements obtained in cell Z. S, and S2 are two dis­ to another). criminative stimuli which may be related on the same In general, inherent bias is therefore a nonunit ra­ physical dimension, or which may be related by one tio of a preference-controlling variable across the two having an additive property to the other (e.g., noise, responses, and this ratio remains constant when an­ signal-plus-noise), or which may be unrelated. other variable is manipulated. A further example P, and P2are the two choice responses which may might be slightly different reinforcement durations be emitted (e.g., "yes" and "no," or a left-key re­ (4 sec vs. 3 sec of grain for pigeons, say) for the two sponse and a right-key response). Correct responses responses. If this occurred in an experiment in which are P, in the presence of SI (Pw, hits) and P2 in the the numbers of reinforcements for the two responses presence of S2 (Pz' correct rejections). Incorrect re­ were varied over experimental conditions, an inher­ sponses are P2in the presence of SI (Px, misses) and ent bias (excluding other sources of bias) of .5 x log PI in the presence of S2 (Py , false alarms). In general, (4/3), or .06, would be expected (Schneider, 1973; correct responses are reinforced with, perhaps, food Todorov, 1973). The multiplier, .5, reflects the gen­ or brain stimulation for animal subjects and money, eral sensitivity of choice behavior to reinforcement points, or feedback for humans. Sometimes payoffs duration changes. for correct responses are given each time a correct re­ In the generalized matching law, the term "bias" sponse is emitted (e.g., Hume, 1974a, 1974b; Hume has generally been used only in its constant sense. When, however, a behavior-controlling variable is not kept constant (to provide a bias), but is system­ atically varied, it changes behavior with a certain sen­ sitivity (a in Equation 2). The sensitivity parameter a measures the relation between changes in a biaser and changes in behavior. When one biaser is varied and another is kept constant, precise estimates of both the value of the constant biaser and the sensi­ w X tivity with which behavior changes with the varied biaser can be obtained from a number of data points. HIT MISS While these could be estimated from only two data points, it is clearly preferable to use a least squares fit. Exactly the same situation applies in signal-detection theory. Point estimates of discriminability and "bias" Y Z (or criterion) may be obtained from each condition comprising two choice proportions (hits and false FALSE CORRECT alarms), but it is preferable to vary systematically one ALARM REJECTION biaser to obtain more precise estimates of the other, constant, biaser. For example, the payoff matrix may Figure 2. The matrix of stimulus and response events in a typi­ be varied (i.e., criterion changed) to obtain an esti­ cal yes-no detection procedure. The cells of the matrix are denoted mate of discriminability. by W, X, Y, and Z. 374 McCARTHY AND DAVISON

& Irwin, 1974), and sometimes it is given intermit­ bility. Adding Equation 4 to Equation 3 to remove tently on VI schedules (McCarthy & Davison, 1979)' the effects of discriminability gives: or on probabilistic variable-ratio schedules (e.g., Elsmore, 1972; McCarthy & Davison, 1979, 1980a, 1980b; Stubbs, 1976). Usually, incorrect responses have no consequence or are punished in some way (e.g., time-out with animals; Hume, 1974b). The This equation relates behavior in the presence of the detection-theory payoff matrix, on the other hand, two stimuli to the combined effects of inherent bias is a description of the arranged contingent events (re­ (log c), and the biasing effect of changing the rein­ inforcers and punishers) for the various responses in the situation. forcement ratio in the two stimuli. Thus: Davison and Tustin (1978) viewed the yes-no de­ tection task as two concurrent reinforcement-extinction arIOg(~:)+ loge schedules, each operating under a distinctive stimu­ lus. The generalized matching law suggests that if the two stimuli, S, and Sl' are indistinguishable, the dis­ ~Og (::~ = .5 log (::)J (6) tribution of total left and right responses would be determined by the distribution of reinforcers for left and right responses (cells Wand Z in Figure 2). We call the measure on the right-hand side of Equa­ tion 6 response bias (McCarthy & Davison, 1979), Davison and Tustin suggested that, as the stimuli be­ and it provides the same behavioral estimate of "bias" come more discriminable, performance would move (or criterion) as that used in signal-detection theory. toward P, in S, and toward P in Sl (Figure 2), and 1 The left-hand side of Equation 6 specifies the en­ that this movement could be described by a further vironmental conditions which produce the response additive quantity in a generalized matching law equa­ bias. Thus, a discriminability-free estimate of re­ tion (Equation 2). Thus, Davison and Tustin pro­ posed two generalized matching law equations to de­ sponse bias is given by: scribe behavior in the presence of each of the two Vi Pw P y stimuli. Response bias = _._) (7) When only correct responses are reinforced (i.e., (Px Pz Rx = Ry = 0), Davison and Tustin wrote: This measure is equivalent (in terms of the matrix in Given S,: Figure 2) to the reciprocal of that given by Luce (1963), namely: log (::) =~tlog (~) +logd+logc, (3) b =[pr(N/S) . Pr(N/n)lVi Pr(Y/s) Pr(Y/n)J '

where Pr(N/s) is the probability of a miss, Pr(N/n) is the probability of a correct rejection, Pr(Y Is) is log (::) =~zlog (~:) -logd+logc, (4) the probability of a hit, and Pr(YIn) is the probability of a false alarm. where P and R denote number of responses emitted Unlike Luce's (1963) measure, however, the pres­ and number of reinforcements obtained, respec­ ent behavioral approach distinguishes clearly be­ tively, and the subscripts refer to the cells of the ma­ tween two sources of response bias: (1) arising trix in Figure 2. from different numbers of reinforcements for the The parameters ar, and arz are the sensitivities of two choice responses (or different magnitudes, etc.; behavior to changes in reinforcements, and log c is McCarthy & Davison, 1979), and (2) a constant bias inherent bias. Log d measures the discriminability of (log c), which may arise from either the requirements the two stimuli (Davison & Tustin, 1978; McCarthy of the experiment (e.g., response production vs. re­ & Davison, 1979). As stimulus separation increases, sponse omission; different forces required to operate behavior moves toward P, in the presence of S" and response manipulanda) or from the subject itself. away from P, in the presence of Sl. Since the numer­ Such a constant bias is termed inherent bias. Both ators in both Equations 3 and 4 are the P, response sources of response bias-reinforcement bias and in­ category, log d is positive in Equation 3 and negative herent bias-are subsumed under the rubric of "cri­ in Equation 4. terion" in signal-detection research. Our approach If we assume art = arz= a-, Equations 3 and 4 can partials response bias (or criterion) into a variable also be used to specify how a measure of response (i.e., reinforcement) component and a constant (i.e., bias may be obtained independently of discrimina- inherent) component. BIAS ANDSIGNALDETECTION 375

Equations 3 and 4 can also be used to obtain a both animal and human subjects has shown that, measure of stimulus discriminability uncontaminated while a subject's ability to discriminate between two

, by response bias. Again assuming Clr, =Clr 2 Equation 4 stimuli remains constant, he may be induced to change can be subtracted from Equation 3 to eliminate the his response probabilities in the presence of each of effects of reinforcement bias and inherent bias on de­ the two stimuli. In these studies, response bias (or tection performance. Thus: criterion) has commonly been manipulated by vary­ ing the probability of presenting one of the two stim­ uli (e.g., Clopton, 1972; Elsmore, 1972; Galanter & log (;:) - log (;:) = 2 log d. (8) Holman, 1967; Hume, 1974a, 1974b; Hume & Irwin, 1974; Markowitz & Swets, 1967; Schulman & An estimate of discriminability, independent of re­ Greenberg, 1970; Swets, Tanner, & Birdsall, 1961; sponse bias, is, therefore, given by: Terman & Terman, 1972). We have shown, however, that response bias is not a function of stimulus­ presentation probability but, rather, is a function of W d= (P .~)Y2 (9) the relative reinforcement frequency obtained for the P P x y choice responses. In signal-detection research with animals, rein­ Davison and Tustin (1978) noted that discrimina­ forcements for correct responses are often arranged bility, as measured by Equation 9, was identical to in one of two ways: reinforcement for every correct discriminability indices used by some signal-detection response or reinforcement delivered probabilistically theorists (e.g., Luce, 1963) and equivalent to that for correct responses. In both of these procedures, used by others (e.g., Green & Swets, 1966). Luce the number of reinforcements obtained for a given (1963), for example, measures discriminability as: choice response covaries with stimulus-presentation probability (SPP). In other words, changing the _[pr(N/S) . Pr(YIn) Y2 probability of presenting the two stimuli changes the 11 - Pr(YIs) Pr(N/n) distribution of reinforcers between the response cate­ gories PI and P2. For example, when SPP is .9, 9/10ths which, in terms of the matrix in Figure 2, is the re­ of the reinforcers will be obtained for correct re­ ciprocal of the expression given by Equation 9 of the sponses emitted in the presence of one of the two Davison and Tustin model. stimuli. Thus, variations in SPP are confounded with In essence, then, Davison and Tustin (1978) showed variations in the obtained reinforcement ratio, and how independent measures of response bias and dis­ the reported biasing effect of SPP could be simply criminability can be derived from an analysis of de­ the result of the changing distribution of reinforce­ tection performance in terms of the matching of re­ ments. sponse ratios to reinforcement ratios. Perhaps the We investigated this possibility by examining the most salient features of their model, in relation to the performance of pigeons trained to detect differences present paper, include a partialing out of the sources between two light intensities under three experi­ of response bias, in particular, the influence of in­ mental procedures (McCarthy & Davison, 1979). All herent bias and a clarification of the role of rein­ three procedures were standard signal-detection yes­ forcement bias in detection theory. 1 no analogues in which the center key of a three-key

operant chamber was lit by either 8 1 (33 cd/m-) or Implications of the Matching Model of Detection S2 (7 cd/m-) according to set probabilities. These Performance for the Measurement of Response probabilities ranged from .1 to .9 in steps of .2. Fol­ Bias in Animal Psychophysics lowing presentation of either SI or S2 on the center In a series of experimental studies using pigeons as key, a peck on the center key turned on the two side subjects, we have investigatedthe of this match­ keys which were illuminated red (left) and green (right). ing approach to signal-detection performance. Of On S. trials, when the more intense stimulus was pre­ particular concern for the present paper are our at­ sented on the center key, a peck on the left key was tempts to manipulate responsebias by varying stimulus­ defined as "correct." On S2 trials, when the less in­ presentation probability and obtained reinforcement tense stimulus was presented on the center key, a ratios, and to uncover the role played by experimen­ peck on the right key was "correct." tal procedures in the generation of isobias contours. Correct responses produced either a 3-sec maga­ In addition, we will show how the relation commonly zine light or, in addition, 3-sec access to wheat. In­ found between obtained and optimal response biases correct responses (that is, pecks on the left key after in detection theory can be explained in terms of the S2 presentations and pecks on the right key after 8 1 Davison and Tustin (1978)model. presentations) produced 3-sec blackout during which Role of stimulus-presentation probability as a all chamber lights were extinguished, and responses biaser. Contemporary psychophysical research using were ineffective. The center-key light remained on 376 McCARTHY AND DAVISON until food, magazine light, or blackout had been pro­ ified. Typically, in the normative version of detec­ duced, after which a new trial began. A noncorrec­ tion theory, the critical value of the likelihood ratio tion procedure was used throughout the experiment designated P, is chosen to maximize expected payoff the probability of occurrence of either SI or S1 on the (Green & Swets, 1966). The expected value of a de­ center key being independent of accuracy on the pre­ cision outcome represents the value or cost associ­ vious trial. ated with that outcome, weighted by the probability The difference between the three procedures was of that outcome's occurring. Thus, if a subject chooses the way in which food reinforcement was arranged a d:cision rule which maximizes expected payoff, his for correct responses. The first procedure was a stan­ optimal likelihood-ratio criterion, Popt> is represented dard signal-detection design in which the probability by the expression: of occurrenceof SI on the center key (SPP) was varied, and the number of reinforcements obtained (Rw' Rz) for the two correct responses (Pw, Pz) was allowed to covary with changes in the ratio of the frequencies of stimulus presentation. This procedure is an uncon­ w~~r: trolled reinforcement-ratio procedure, and it is typ­ Pr(S1).and Pr(SI) represent the a priori prob­ abilities of stimulus presentation, Vz and Vware the ical of most signal-detection research. In the second values associated with correct responses (P and P procedure, SPP was again varied, but equal numbers z w), of reinforcements were obtained for the two correct and Vy and Vx are the costs associated with incorrect responses (P and P ). The subscripts refer to the responses. The third procedure held SPP constant at y x cells of the matrix in Figure 2. .7, and the number of food reinforcements obtained Such values and costs are, however, poorly defined for correct responses was varied. Procedures 2 and 3 by signal-detection theorists (e.g., Egan, 1975). Hume are thus controlled reinforcement-ratio procedures, (1974a, 1974b)specified value and cost as the number because the reinforcement ratio is set and cannot of brain stimulations for each correct response and covary with either response ratios (preference) or the dur~tion of blackout for each incorrect response, SPP. respectively. The obtained reinforcement ratio, on In this experiment, as in all animal psychophysical the other hand, is a variable which has not been con­ studies, the subjects were trained on each experi­ sidered by detection theorists. The detection-theory mental condition until a strict criterion of stable per­ payoff matrix, denoting arranged payoff, specifies formance was reached. Data were collected over five ~he values a~d costs of the payoffs. It does not spec­ sessions of this stable performance. Training then ify the relative frequency at which the payoffs occur commenced on the next experimental condition with (i.e., obtained payoff). Obtained, rather than ar­ changed reinforcement (or stimulus) parameters. ranged, reinforcement ratio is the critical variable We found that behavior changed reliably in Pro­ both in research on matching and in the present be­ cedure 1, in which both the reinforcement ratio and havioral model of signal detection. As we have sug­ SPP were varied, and in Procedure 3, in which the gested, it is the ratio of obtained reinforcers as well relative reinforcement ratio alone was varied. These as ~heir momentary value (e.g., quality, ma~nitude, behav~oral changes were shown as positive slopes for or Immediacy), which determines reinforcement bias. Equations 3 and 4, and significant trends in the point In terms of the results reported above (McCarthy estimates of response bias (Equation 7) as a function & Davison, 1979), Equation 10 implies that, as the of the logarithm of the obtained reinforcement ratio. value of the payoff remained constant in all three H~wever, no significant trend occurred in the point ~rocedures, response bias should simply be a func­ estimates of response bias in Procedure 2 when SPP non of Pr(SI)/Pr(S1)' This implies that response bias alone was varied. should have changed in Procedures 1 and 2 but not As SPP manipulation produced no systematic in Procedure 3. Clearly, then, Equation 10 is not change in the relative frequencies of the choice re­ c~:msistent wi~h our results, which showed response­ sponses, we concluded that SPP was not a biaser bias changes In Procedures 1 and 3, but not in Pro­ per se, and that its apparent biasing effect arose from cedure 2. changes in the obtained reinforcement ratio. In other It remains, therefore, to consider response bias as words, the variation of the obtained reinforcement in the Davison and Tustin (1978)model, that is: ratio alone controlled or biased performance. Varia­ tions in SPP did, however, allow measures of stim­ ulus discriminability (Equation 9) to be extracted Response bias = Clr log (~:) + log c. (11) from the data. As expected, discriminability re­ mained constant across the three procedures. . Optimal bias. In signal-detection theory, optimal Here, ~ is a free parameter which may take any value, biases which maximize some aspects of payoff (such but WhICh would be expected to fall between .5 and as overall reinforcement probabilities) are often spec- 1.0 (Baum, 1979) on the basis of extensive research BIAS AND SIGNAL DETECTION 377

on schedule control. If a standard yes-no detection 10.---- -t-r-r ----,~ procedure with reinforcement for each correct re­ 8 .­ sponse is used, the reinforcement ratio (Rw/Rz) will 5 -- approximate the ratio Pr(Sl)/Pr(Sz), and a strategy .0 such as maximizing expected value will give an a, 3 value of 1.0. A value of a, of less than 1.0 would rep­ 2 resent a failure to maximize expected utility and, per­ haps, the use of a different strategy. While, in a CI::l.. sense, the a parameter is weak, it does allow changes to r 0 in response bias in signal-detection experiments to be w ·8 z A-IV related directly to another extensive area of research­ « ·5 ....- schedule control. It is worthwhile noting that, in this t- .- CD A-VI latter area, no particular strategy, such as maximiz­ 0 3 ... A-VII ing expected value, has ever been demonstrated (e.g., 2 Herrnstein & Heyman, 1979). A-IX In human psychophysical experiments, subjects generally fail to behave exactly in the way specified ·1 lL-.._L-.l...-.L1...l...-.L1.._-..l.---.J-..l...... l.....J'---'--' ·1 2 3 58 10 2 3 5 8 10 by Equation 10. Typically, when f30pt is relatively OPTIMAL large, the bias exhibited by the subject (f3obt) is not as ! high as the optimal bias, and when f30pt is relatively Figure 3. Data reported by Hume and Irwin (1974). The rela­ small, f30bt is not as low as the optimal bias. Such tion between obtained response criteria ({Jobl) and optimal response conservative behavior has been attributed to the sub­ criteria ({JOP1)' The coordinates are logarithmically spaced, and the major diagonal represents perfect correspondence between ob­ ject's awareness in psychophysical settings that the tained and optimal criteria. (Copyright 1971 by the Society for experimenter's principal interest is in a sensory pro­ the Experimental Analysis of Behavior, Inc.) cess so that he believes that all "yes" or all "no" re­ sponses are either not possible or not desired (Green there is an undermatching relation (slope less than & Swets, 1966). unity) between f30pt and f30bh and there are large in­ Rather than defining behavior as suboptimal, how­ herent biases (e.g., log c of .27 for Rat AVI). Never­ ever. it is more profitable to define the expected de­ theless, these results can be described by Equation 11 gree of bias by Equation 11 above. This expression of the Davison-Tustin model. does not specify strict matching between expected Independence of response bias from discrimina­ and obtained bias, but, rather, allows the possibility bility. As noted by Dusoir (1975), current measures of undermatching (a, < 1.0) for a more general state­ of "bias" (or criterion) in signal-detection theory ment of the relation between response and reinforce­ cannot be unequivocally shown to be independent of ment ratios. Certainly, undermatching is the norm discriminability. The present model assumes that the with various biasers (e.g., reinforcement magnitude; behavioral effects of response bias and discrimina­ Schneider, 1973, and Todorov, 1973), and it may bility are additive in logarithmic terms (Equations 3 also be the norm with various schedule combinations and 4), and hence, there is no interaction between (e.g., CONC FI VI; Lobb & Davison, 1975, and these two independent variables. There can be, how­ Trevett, Davison, & Williams, 1972). ever, many alternative models which do not treat re­ Any general description of detection performance sponse bias and discriminability as simply additive in must account for different biasers produced by dif­ their effects on behavior. Rather, they include inter­ ferent response and schedule combinations, and must actions between these two variables. allow for the possibility of undermatching. Figure 3 In signal-detection theory, for instance, it has long shows a typical plot of obtained bias as a function of been accepted that uncontrolled cognitive factors optimal bias (Hume & Irwin, 1974). Optimal bias is (e.g., criterion variance) can contaminate estimates here defined as the ratio of the probability of one stimulus's occurring to the probability of the other stimulus's occurring on any trial, which, in terms of Table 1 the present model, assuming an uncontrolled­ Slopes and Intercepts for the Relation Between Obtained reinforcement ratio procedure, is correlated with the Response Criteria and Optimal Response Criteria as Plotted in Figure 3 obtained reinforcement ratio, Rw/Rz. Figure 3 shows clearly that the obtained bias (fJobt) Subject Slope Intercept undermatches optimal bias (f3opt) , as would be ex­ AVI .84 .27 pected from schedule-control research. Extracting the AIV .72 .14 data from Figure 3, the slopes and intercepts of the AVII .67 -.01 relation are shown in Table 1. As Table I shows, AIX .96 -.09 378 McCARTHY AND DAVISON of discriminability. The detection-theory concept of 1 0 criterion, or response bias, is formulated in such a A ,­ way as to make measures of discriminability invari­ .-/ ant with changes in criterion. The location of the / 06 .--. .- criterion depends upon the value and cost of making Y=0·50X+009 ./~ correct and incorrect responses and on the prob­ ability of stimulus presentation (Equation 10). While the criterion is assumed to be stable in signal-detection o 2 SE 0 05 /'~__~ tasks, if it is not, it becomes an additional source of variance in the statistical model (Tanner & Swets, 1954; Treisman, 1977; Wickelgren, 1968). In animal If)-0·2 • y :: 0·47X+0 07 psychophysical studies, for example, there are nu­ « SE . 0·06 merous reports of uncontrolled bias shifts (e.g., Irwin CO ...... & Terman, 1970;Terman & Terman, 1972). w -0 6L.S'--_L..-_...... JL.....-_---...I__---J We investigated empirically the independence of If) -1·6 -08 0 0·8 16 response bias and discriminability in the present LOG OBTAINED REINF RATIO model by examining the biasing effect of the rein­ 6(L forcement ratio on detection performance at two dif­ If) ferent levels of discriminability(McCarthy & Davison, ~ 1·0 1980b). Here, the reinforcement ratio was varied for l:) B both an easy and a difficult discrimination to see o whether similar reinforcement sensitivities were ob­ -J 0.6 .-. tained at both discriminability levels. Y:: O·BOX + 014 /,J:f Again, using a standard signal-detection yes-no SE' 007 ", analogue, six pigeons were trained to detect differ­ 0,' 2 "" ences between two white stimuli, 51 and S2, differing o in duration and arranged probabilistically on the cen­ "i'" ter key of a three-key operant chamber. For the easy 0----<> discrimination, SI was 5 sec and S2 was 30 sec. For -0 2 Y::0·49X+0·07 the difficult discrimination, 51 was 20 sec and S2was SE 0·06 30 sec. The probability of occurrence of SI on the center key (SPP) was varied from .1 to .9 in steps of -0·6L&._---L.__-I-__.J--_---' .2, for both pairs of stimuli. The obtained reinforce­ -1·0 -0·5 0 05 10 ment ratio for correct responses emitted on the two LOG STIMULUS PRES. RATIO side keys (left after S.. right after S2) was uncontrolled, allowing it to covary with changes in stimulus­ Figure 4. Point estimates of response bias (Equation 7) for tbe presentation probability. S-sec vs. 3O-sec conditions (unf"illed circles) and the 20-sec vs, JO-sec conditions (filled circles). A shows these point estimates as Discriminability, as measured by Equation 9, was a function of the logarithm of the obtained reinforcement ratio, indeed different for the two discriminations. For the and B as a function of the logarithm of the stimUlus-presentation 5-sec vs. 30-sec conditions, the mean value of dis­ ratio. The best-fitting straight line by the method of least squares criminability (log d), averaged across the six birds, and the standard error (SE) of the estimate are shown for each was 1.48, a significantly higher value than that ob­ stimulus condition. tained for the 2O-sec vs. 30-sec conditions (mean = .39). In addition, there was no systematic change in ment ratio from the arranged reinforcement ratio for discriminability as a function of the obtained rein­ the difficult discrimination. This deviation can be forcement ratio for either the easy or the difficult dis­ seen by plotting the point estimates of response bias crimination. as a function of the logarithm of the stimulus­ Point estimates of response bias (Equation 7), on presentation ratio, as in Figure 4b. The slope for the the other hand, significantly increased as a function difficult discrimination here is greater than that for of the obtained reinforcement ratio for both discrim­ the easy discrimination (slopes = .80 and .49, respec­ inability levels (Figure 4a). While bias estimates were tively). However, when both sets of data were plotted similar for the two discriminability levels, the range as a function of the obtained reinforcement ratio over which response bias varied was larger for the (Figure 4a), no differences in the slopes were found difficult discrimination than for the easy discrimina­ (slope =.47 for the easy discrimination; slope = .50 tion (Figure 4a). This result is attributable to the use for the difficult discrimination). This result, then, of an uncontrolled reinforcement-ratio procedure, again underlines the importance of recognizing the with resulting deviations of the obtained reinforce- obtained reinforcement ratio, and not the probability BIAS ANDSIGNALDETECTION 379 of stimulus presentation, as the effective biaser. In addition, it provides support (as have most of our experiments) for the assumption of equal reinforce­ ment sensitivities (i.e., ar, =ar,) in Equations 3 and 4. However, as we later point out, equality may not al­ ways be the case and, in certain situations, behavior following one stimulus may be more (or less) sensi­ tive to reinforcement variation than that emitted fol­ l- lowing a different stimulus. I Isobias and alloiobias. Parametric variation of re­ « inforcement schedules has received little considera­ U- tion in signal-detection research. For an adequate 0 a isosensitivity contour, an independent variable which >- controls or biases behavior is clearly needed, and, I- 1-0 as noted above, these variables have been well specified -.J CD by concurrent-schedule research. The list includes re­ « 0·8 inforcement rate, reinforcement magnitude, imme­ CD diacy of reinforcement, quality of reinforcement a::0 0·6 (de Villiers, 1977), and response requirements Q.. (Beautrais & Davison, 1977; Davison & Ferguson, 0·4 B 1978). Recent research in this laboratory on response force suggests that, because of its extremely fast ef­ 0-2 fect on behavior, this biaser may be particularly ef­ a~-=-,,="---::::-'-:--::-'-=--::-'-=---' ficient. a 02 0·4 0-6 0-8 1-0 The present model views response bias as a func­ PROBABILI TY OF A tion of the obtained reinforcement ratio rather than of the arranged payoff matrices. Thus, it can specify FALSE ALARM true iso- (equal-) bias contours, and these will be gen­ Figure 5. Predicted isobias contours under various assumptions. erated only in the case in which the ratio of reinforce­ In Graph A, the obtained reinforcement ratio is controlled (see text). The value of (log c +log (Rw/Rz)] is 0 (line 1), .3 (line 2), ments obtained for correct responses is kept constant. .6 (line 3), and -.3 (line 4). In Graph B, the obtained reinforce­ Isobias contours predicted by this model for such a ment ratio is not controlled, and is a function of changing prefer­ controlled reinforcement-ratio procedure are shown ence. Line 1 shows a predicted isobias contour when inherent bias in Figure 5a for various degrees of bias. Response is .25, and lines 2, 3, and 4 show the further development of the bias is given by Equation 11, and comprises inherent contour as changing preference affects the obtained reinforce­ ments for the two choices. bias (log c) and bias due to the reinforcement ratio. Thus, a response bias of .3 may represent any com­ bination of inherent and reinforcement bias. The procedure with the above parameters. From here on, shape of the predicted isobias contour is the same as the reinforcement ratio is uncontrolled, and it is the Dusoir (1975) drew for Luce's (1963) model. How­ inherent bias of .25, and consequent inequality of ever, Luce's model was written for the case in which obtained reinforcements for the two choices, that the obtained reinforcement ratio covaried with the moves the contour toward the upper-right corner of response ratio, that is, the standard signal-detection the ROC space. Line 2 is predicted by the first appli­ procedure. cation of Equations 3 and 4, Line 3 by the second, Our model makes very different predictions for and Line 4 by the third. Thus, as training progresses such an uncontrolled reinforcement-ratio procedure. with a particular pair of stimuli, the data point will When reinforcement ratios are not controlled, and move progressively toward the upper-right corner of stimulus-presentation probability is held constant, the ROC space. The complete alloiobias contour, ob­ changes in behavior will change reinforcement ratios tained from a number of stimulus pairs, will take the obtained, which will again change behavior, and so shape shown in Figure 5b, that is, the shape usually on. Some idea of the expected alloiobias (varying, reported in the literature, and the shape predicted by different bias) contour can be obtained by applying a signal-detection theory maximizing expected value Equations 3 and 4 a number of times. The results are approach (Dusoir, 1975). (Likewise, if inherent bias shown in Figure 5b. Here, we have assumed that was - .25, successive applications of Equations 3 and stimulus-presentation probability is .5, the initial rein­ 4 would move the contour in Figure 5b progressively forcement ratio is 1.0, and ar is 1.0, and that the toward the lower-left corner of the ROC space.) subject has an inherent bias of .25 to response PI Preference is clearly moving faster with the poor (Figure 2). Line 1 is a controlled equal-reinforcement discriminability stimuli (near the major diagonal) 380 McCARTHY AND DAVISON

than with good discriminability. In theory, the pro­ sponse bias. As we have shown, response bias in ani­ cess should stop with exclusive choice for response mal research is unaffected by SPP variation alone, p to but if the sulJjectis insensitiveto small reinforcement­ and extreme SPP values do not produce response ratio changes, the curve will stabilize with a strongly biases when equal payoff is given for the two correct negative decelerated shape. It is, of course, possible responses. Is it more parsimonious to assume, then, that adventitious changes in reinforcement ratios ob­ that humans and animals are different in the effect tained may take the alloiobias line across the minor that presentation frequency has on response bias? diagonal. The point we wish to make, however, is Because the relevant experiment has not been carried that the alloiobias function is predictable from a con­ out with humans, we assume that the same principles sideration of the obtained reinforcement ratios, apply, and that, in the human case, insufficient at­ whereas it is not predictable from stimulus-presentation tention has been given to specifying the reinforcers probabilities. In addition, the problems with uncon­ involved. trolled bias shifts in detection-theory research (e.g., In human experimentation, the reinforcers (money, Hobson, 1975; Terman & Terman, 1972) can now be grades, etc.) are often delivered non-contingently, as seen as problems in defining biasing variables in re­ a means simply of providing and maintaining behav­ lation to experimental procedures. ior. The differential reinforcers reside in the instruc­ tions, and these do have a significant bearing on the Implications for Human Psychophysics strategies adopted. Such instructions as "Report the The present approach to bias in signal detection presence of the tone only when you are absolutely has concentrated almost entirely on animal studies. sure it is present" obliquely specify that the fre­ To what extent do the results discussed apply to hu­ quency, or probability, of payoff for the response man psychophysics? The answers to this question are "absent" is greater than that for the response "pres­ problematical, and seem to depend rather more on ent." It is thus a biasing operation, as is explicit re­ one's theoretical stance than on data. The traditions inforcement variation with animals. Variations of of behavioral and psychophysical research are differ­ SPP in conjunction with an instruction such as "Say ent, and they have never been adequately combined. yes when the tone is present and no when it is absent" Most research in animal psychophysics has simply are an oblique variation of reinforcement frequency. been carried out to extend or generalize principles But what of the instruction, "You will occasionally from human research. We, of course, are taking an receive 10 cents each time you report the tone when opposing view: that animal research can contribute it is present, and when you report it absent and it is generally to psychophysics and can highlight theo­ indeed absent. You have 100 trials in which to make retical difficulties which arise purely from using hu­ as much money as you can"? If the frequencies of man subjects. reinforcement for "yes" and "no" were kept con­ From our radical behaviorist viewpoint, it is neces­ stant while SPP was varied, would response bias sary (except when using respondents) to maintain the change? Likewise, if SPP was held constant and behavior of the animal explicitly through reinforce­ reinforcement frequency for the two reports was ment. Motivational or, more strictly, reinforcement varied, would response bias change? variables must thus be clearly specified and be ar­ We thus argue that animal detection-theory re­ ranged adequately to maintain detection perfor­ search is, in some sense, more informative than hu­ mance. In the absence of reinforcement, detection man research because it forces us better to specify performance ceases. More than this, differential rein­ the variables involved. Only human research can tell forcement is required as shown by Davison and us if this is the correct approach. McCarthy (1980). From our own research, and by At the data level, the commonalities between the implication from published research on concurrent results of animal and human experimentation are schedules, we know many things about reinforce­ enormous, as we have shown. The problems that ment effects: that the degree of deprivation should signal-detection theory has had with "bias" (or cri­ not affect either response bias or discriminability terion), and the form of the isobias contour (Dusoir, (except when different reinforcers are used for the 1975), to take one instance, has been due to a very two correct responses); that overall reinforcement large extent to the inadequate specification of the rate does not affect either response bias or discrim­ variables that lead to bias, as Dusoir pointed out. inability; that the presence or absence of punishment Another particular example can be cited. The rather for errors, in the context of reinforcement for correct oft-occurring nonunit slope of the isosensitivity func­ responses, does not change response bias or discrim­ tion on double z coordinates has been ascribed to cri­ inability; that reinforcement for errors decreases dis­ terion variance (e.g., Wickelgren, 1968). From our criminability; that response bias can be predictably theoretical standpoint, because we see response bias changed by a wide range of variables. as determined largely by environmental (reinforce­ On the other hand, in our research we have yet to ment) contingencies, we must expect the variance in see any unambiguous effects of expectancy on re- response bias as also being so determined. It is an em- BIAS AND SIGNAL DETECTION 381 pirical question, then, whether trial-to-trial or session DAVISON, M. C., & FERGUSON, A. The effects of different com­ to-session variance in obtained reinforcement ratios ponent response requirements in multiple and concurrent sched­ can decrease the slope of the isosensitivity function. ules. Journal of the Experimental Analysis of Behavior, 1978, 29,283-295. From our viewpoint, then, criterion variance does DAVISON, M., & McCARTHY, D. Reinforcement for errors in a not simply exist-it is manipulable and investigat­ signal-detection procedure. Journal ofthe Experimental Analysis able. Equally, from our theoretical standpoint, a ofBehavior, 1980,34,35-47. nonunit slope implies that behavior following one DAVISON, M. C., & TUSTIN, R. D. The relation between the stimulus is more sensitive to reinforcement variation generalized matching law and signal-detection theory. Journal of the Experimental Analysis of Behavior, 1978, 29, 331-336. than is that emitted following another (i.e., aT, 1= aT, DE VILLIERS, P. Choice in concurrent schedules and a quantitative in Equations 3 and 4). Indeed, we have found this in formulation of the law of effect. In W. K. Honig & J. E. R. some experiments as yet unpublished, and we are Staddon (Eds.), Handbook of operant behavior. Englewood presently investigating the conditions under which Cliffs, N.J: Prentice-Hall, 1977. Dusora, A. E. Treatments of bias in detection and recognition this inequality occurs. models: A review. Perception & Psychophysics, 1975, 17, In general, then, our approach to bias and its op­ 167-178. eration could have some wide applications in psy­ EGAN, J. P. Signal detection theory and ROC analysis. New chophysics. It does not attempt to supplant conven­ York: Academic Press, 1975. tional detection theory; rather, it attempts to provide ELSMORE, T. F. Duration discrimination: Effects of probability of stimulus presentation. Journal ofthe Experimental Analysis it with an adequate theory of response bias derived ofBehavior, 1972,18,465-469. from an area in which such effects have been very ex­ GALANTER, E., & HOLMAN, G. L. Some invariances of the isosen­ tensively researched. As we have stated elsewhere, sitivity functions and their implications for the utility function the study of stimulus and reinforcement effects should of money. Journal of , 1967, 73, 333-339. not proceed separately when each approach requires GREEN, D. M., & SWETS, J. A. Signal detection theory and psy­ a consideration of the other. Thus, in addition to chophysics. New York: Wiley, 1966. clarifying the role of reinforcement in psychophysics, HARDY, G. R., & LEGGE, D. Cross-modal induction of changes our work is also addressed to the adequate measure­ in sensory thresholds. Quarterly Journal ofExperimental Psy­ ment of stimulus effects in the situations conven­ chology, 1968,20,2~29. HEALY, A. F., & JONES,C. Criterion shifts in recall. Psychological tionally studied under the heading of choice and Bulletin, 1973,79,335-340. schedule control. Such cross-fertilization between HERRNSTEIN, R. J. On the law of effect. Journal of the experi­ two highly developed areas of psychology will surely mental Analysis ofBehavior, 1970, 13,243-266. lead to benefits for both. HERRNSTEIN, R. J., & HEYMAN, G. M. Is matching compatible with reinforcement maximization on concurrent variable in­ terval, variable ratio? Journal of the Experimental Analysis of REFERENCE NOTE Behavior, 1979,31,209-223. HOBSON, S. L. Discriminability of fixed-ratio schedules for pigeons: 1. Nevin, J. A., Jenkins, P., Whittaker, S., & Yarensky, P. Effects of absolute ratio size. Journal of the Experimental Signal detection, differential reinforcement and matching. Paper Analysis ofBehavior, 1975,23,25-35. presented at the meeting of the Psychonomic Society, Washington, HOLLARD, V. D., & DAVISON, M. C. Preference for qualitatively D.C., November 1977. different reinforcers. Journal of the Experimental Analysis of Behavior, 1971,16,375-380. REFERENCES HUME, A. L. Auditory detection and optimal response biases. Perception &Psychophysics, 1974, 15, 425-433. (a) BARRON, B., & DAVISON, M. Performance in multiple fixed­ HUME,A. L. Optimal response biases and the slope of ROC curves interval schedules. Journal of the Experimental Analysis of as a function of signal intensity, signal probability, and relative Behavior, 1972, 17, 375-379. payoff. Perception & Psychophysics, 1974, 16, 377-384. (b) BAUM, W. M. On two types of deviation from the matching law: HUME, A. L., & IRWIN, R. J. Bias functions and operating char­ Bias and undermatching. Journal ofthe Experimental Analysis acteristics of rats discriminating auditory stimuli. Journal ofthe ofBehavior, 1974,21,231-242. ExperimentalAnalysis ofBehavior, 1974,21,285-295. BAUM, W. M. Matching, undermatching, and overmatching in IRWIN, R. J., & TERMAN, M. Detection of brief tones in noise by studies of choice. Journal of the Experimental Analysis of rats. Journal of the Experimental Analysis of Behavior, 1970, Behavior, 1979,32,269-281. 13, 135-143. BAUM, W. M., & RACHLIN, H. C. Choice as time allocation. LANDER, D. G., & IRWIN, R. J. Multiple schedules: Effects of the Journal of the Experimental Analysis of Behavior, 1969, 12, distribution of reinforcements between components on the dis­ 861-874. tribution of responses between components. Journal of the BEAUTRAIS, P. G., & DAVISON, M. C. Response and time al­ ExperimentalA nalysis ofBehavior, 1968, 11, 517-524. location in concurrent second-order schedules. Journal of the LoBB, B., & DAVISON, M. C. Preference in concurrent interval Experimental Analysis ofBehavior, 1977, 1S, 61-69. schedules: A systematic replication. Journal ofthe Experimental BROADBENT, D. E. Decision and stress. London: Academic Press, Analysis ofBehavior, 1975,24,191-197. 1971. LUCE, R. D./ndividual choice behavior. New York: Wiley, 1959. CLOPTON, B. M. Detection of increments in noise intensity by LUCE, R. D. Detection and recognition. In R. D. Luce, R. R. monkeys. Journal of the Experimental Analysis of Behavior, Bush, & E. Galanter (Eds.), Handbook of mathematical psy­ 1972,17,473-481. chology (Vol. I). New York: Wiley, 1963. DAVISON, M. C. A functional analysis of chained fixed-interval MARKOWITZ, J., & SWETS, J. A. Factors affecting the slope of schedule performance. Journal ofthe Experimental Analysis of empirical ROC curves: Comparison of binary and rating re­ Behavior, 1974,21,323-330. sponses. Perception & Psychophysics, 1967, 2, 91-100. 382 McCARTHY AND DAVISON

MATHEWS. L. R.•& TEMPLE, W. Concurrent schedule assessment TANNER, W. P .• JR., & SWETS, J. A. A decision-making theory of of food preference in cows. Journal ofthe Experimental Analy­ visual detection. Psychological Review, 1954.61,401-409. sis ofBehavior. 1979,32,245-254. TERMAN, M., & TERMAN, J. S. Concurrent variation of response MCCARTHY, D., & -DAVISON, M. Signal probability, reinforce­ bias and sensitivity in an operant-psychophysical test. Perception ment, and signal detection. Journal ofthe Experimental Analy­ & Psychophysics, 1972,11,429-432. sis ofBehavior. 1979, 32, 373-386. THOMAS, E. A. C.• & LEGGE, D. Probability matching as a basis McCARTHY, D., & DAVISON, M. On the discriminability of stim­ for detection and recognition decisions. Psychological Review, ulus duration. Journal ofthe Experimental Analysis ofBehavior, 1970,77,65-72. 1980,33, 187-211. (a) TODOROV, J. C. Interaction of frequency and magnitude of rein­ McCARTHY, D., & DAVISON, M. Independence of sensitivity to forcement on concurrent performances. Journal of the Experi­ relative reinforcement rate and discriminability in signal detec­ mental Analysis ofBehavior, 1973,19,451-458. tion. Journal of the Experimental Analysis of Behavior, 1980, TREISMAN, M. The effect of one stimulus on the threshold for 34.273-284. (b) another: An application of signal detectability theory. British MYERS, D. L., & MYERS, L. E. Undermatching: A reappraisal of Journal ofStatistical Psychology, 1964,17, 15-35. performance on concurrent variable-interval schedules of rein­ TREISMAN. M. On the use and misuse of psychophysical terms. forcement. Journal of the Experimental Analysis of Behavior, PsychologicalReview, 1976,83,246-256. 1977,27,203-214. TREISMAN, M. On the stability of ds• PsychologicalBulletin, 1977, NEVIN, J. A. Rates and patterns of responding with concurrent 84.235-243. fixed-interval and variable-interval reinforcement. Journal of TREVETT, A.J., DAVISON, M. C., & WILLIAMS. R. J. Performance the Experimental Analysis of Behavior. 1971, 16, 241-247. in concurrent interval schedules. Journal of the Experimental PARKS, T. E. Signal detectability theory of recognition Analysis ofBehavior, 1972,17.369-374. performance. PsychologicalReview, 1966,73,44-58. VAN METER, D., & MIDDLETON, D. Modern statistical approaches PETERSON, W. W., BIRDSALL, T. G., & Fox, W. C. The theory of to reception in communication theory. Transactions ofthe IRE signal detectability. Transactions ofthe IREProfessional Group Professional Group on , PGIT-4, 1954. on Information Theory, PGIT-4. 1954. WICKELGREN, W. A. Unidimensional strength theory and com­ SCHNEIDER, J. W. Reinforcer effectiveness as a function of rein­ ponent analysis of noise in absolute and comparative judgments. forcer rate and magnitude: A comparison of concurrent per­ Journal ofMathematical Psychology, 1968,5, 102-122. formances. Journal of the Experimental Analysis of Behavior, 1973.20.461-471. SCHULMAN, A. I., & GREENBERG. G. Z. Operating characteristics NOTE and a priori probability of the signal. Perception & Psycho­ physics, 1970,8,317-320. I. Nevin, Jenkins. Whittaker. and Yarensky (Note I) proposed a STADDON, J. E. R. Spaced responding and choice: A preliminary model similar to that reported here. Their model, which assumes analysis. Journal of the Experimental Analysis of Behavior, 3r= 1 and c = I, is thus a special case of our model. The relative 1968, 11,669-682. merits of these two models. and the necessity for the generalized STUBBS, D. A. Response bias and the discrimination of stimulus model. are discussed in Davison and McCarthy (1980). duration. Journal of the Experimental Analysis of Behavior, 1976,25,243·250. SWETS, J. A., TANNER, W. P., JR., & BIRDSALL. T. G. Decision (Received for publication December 7. 1979; processes in perception. PsychologicalReview, 1961,68,301-340. revision accepted February 9, 1981.)