Perception & Psychophysics 1981,29 (4),371-382 Towards a behavioral theory of bias in signal 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 stimulus 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 sense-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 Psychology, 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).