Psychological Review1978, Vol. 85, No. 4, 305-320

Theory of Behavioral Power Functions

J. E. R. StaddonDuke University

Data in operant conditioning and psychophysics are often well fitted by func-tions of the form y — qx'. A simple theory derives these power functions fromthe simultaneous equations dx/x = aif(z)dz and dy/y = a j ( z ) d z , where z is acomparison variable that is equated for the effects of x and y, and Oj and a2 aresensitivity parameters. In operant conditioning, * and y are identified with re-sponse rates; in psychophysics, with measures of stimulus and response. Thetheory can explain converging sets of power functions, solves the dimensionalproblems with the standard power function, and can account for the relationbetween Type I and Type II psychophysical scales.

Many empirical relations take on a linearform when plotted in double-logarithmic co-ordinates. In experimental psychology suchrelations have frequently been reported inoperant studies of choice behavior and rate-dependent drug effects and in experiments onpsychophysical scaling. Power functions of theform y = qx', where q and s are constants, arethe simplest mathematical approximation tothese empirical functions. While there has beenconsiderable theoretical discussion about thepossible processes responsible for the psycho-physical power law, choice of power functionfits in other cases has been largely a matter ofempirical convenience. The theoretical basiseven for the psychophysical power function isstill a matter for dispute.

In this article, I propose a very simple theo-retical basis for power functions. This deriva-tion can be interpreted in terms of behavioralcompetition in operant conditioning situationsand in terms of an internal "sensation" vari-able in psychophysical scaling experiments.The theory predicts that sets of power func-tions will in many cases converge on a commonpoint; it solves problems of dimensional balanceencountered when x and y are measured in the

This research was supported by grants from theNational Science Foundation to Duke University.

I am grateful to R. D. Luce, L. E. Marks, and twoanonymous reviewers for comments on the manuscript.I also thank Marks and L. M. Ward for patient at-tempts to clarify my thoughts on psychophysics.

Requests for reprints should be sent to J. E. R.Staddon, Department of Psychology, Duke University,Durham, North Carolina 27706.

same units; and it is consistent with recentwork on the relation between different typesof psychophysical scales. Data from numerousexperiments on operant behavior and psycho-physical scaling are consistent with the con-vergence prediction.

The derivation is most easily introduced inconnection with a specific example, and this isdone in the first section. The second sectiondescribes its application to rate-dependentdrug effects, and the third section deals withthe psychophysical power law.

Response Competition

The most obvious effect of reward andpunishment is on the level of the rewarded orpunished response. However, for a completeunderstanding of reinforcement, it is necessaryto study not just direct effects on a particularresponse but also indirect effects on the waythat behaviors interact with one another. Themost frequent type of interaction is competi-tion, since the time available for responding ininstrumental (operant) conditioning situationsis generally less than the time needed for alltendencies to action to find full expression(Atkinson & Birch, 1970; McFarland, 1974;Staddon, 1977a; Staddon & Ayres, 1975).Recent theoretical analyses of competitive in-teractions have shown that even quite simplequantitative treatments can sometimes inte-grate a surprisingly wide range of data (Mc-Farland & Sibly, 1975; Staddon, 1977b).However, the topic of response competitionhas received relatively little direct experi-

Copyright 1978 by the American Psychological Association, Inc. 0033-295X/78/8504-030S$00.75

30S

306 J. E. R. STADDON

mental attention, and until recently, few re-sults were available comparable in quantita-tive precision to those from studies of thedirect effect of reinforcement on rates of op-erant responding.

The most direct way to study competition isto look at the level of one activity while thelevel of a second, competing one is varied bysome means known not to affect the level ofthe first activity directly. In practice, however,this method is difficult to apply because acausal factor sufficient to affect one activitymay also have direct effects on the other. How-ever, a recent series of experiments by Nevin(1974a, 1974b) appears to meet this objectionand has yielded data that are susceptible of arelatively simple interpretation. In his mostextensive experiment, Nevin (1974b) studiedthe behavior of hungry pigeons responding forfood reinforcement on two separate responsekeys. Pecks on one key yielded brief access tograin for the first peck after SO sec (fixed-interval [FI] 50-sec schedule). On the otherkey, two colors alternated at 10-sec intervals.Food was available intermittently at differentrates in the presence of each color for peckingon this second key (multiple variable-intervalvariable-interval [VI VI] schedule). The rela-tive frequency of food delivery for pecking oneach of the two multiple-key colors was variedin different conditions of the experiment. Ineach condition, the animals were exposed to theschedules daily until their rates of respondingon each key and to each color had settled downto stable levels.

On FI schedules, responding increases pro-gressively through the time between fooddeliveries, rising to a maximum as the nextfeeding becomes imminent. The pigeons in thisstudy responded in this fashion on the FI key.Responding on the other (multiple) key fol-lowed a complementary course: The pigeonspecked rapidly early in the (FI) interfoodinterval and more slowly toward the end.Food delivery for pecks on the multiple keywas aperiodic. Hence, it is likely that the ob-served temporal variation in multiple-key re-sponding was due primarily to progressivesuppression by Fl-key responding and not toa separate internal clock for responding on themultiple key.

The major result from this study is a set offunctions showing how responding on each ofthe multiple-key colors was suppressed by theprogressively increasing Fl-key respondingthroughout the 50-sec FI interfood interval.The interval was divided into 10-sec blocks,and plots were made of response rate duringcomparable blocks in the presence of each ofthe multiple-key colors, that is, of responserate to one color versus response rate to theother, at comparable times in the 50-sec fixedinterval. These response-response plots were inall cases acceptably fitted by power functionsof the form

y = qx> (1)

where x and y are response rates in the presenceof the two multiple-key colors, and q and 5are parameters that varied with the frequen-cies of reinforcement in the two colors.

Equation 1 can be derived in a number ofways. However, the basis to be presented hereis one of the simplest and may be the mostgeneral. I suppose only that the effect on be-haviors a; or y of changes in the competingbehavior z, the level of FI responding, dependssolely on the initial levels of x or y. Thisassumption can be stated precisely as follows:

and

dx/x = aif(z)dz

dy/y = azf(z)dz.

(2a)

(2b)

In words, these equations simply affirm thata small change, dz, in the level of z producesa change, dx (or dy) in x or y that is propor-tional to the initial level of x, to the changein z, and to some unspecified function of z,f(z). The constants of proportionality «i and#2 represent the sensitivities of x and y, re-spectively, to these changes in z. The fact that/(z) need not be specified simply indicatesthat the relation between x and y (which isabout to be derived) does not depend on theform of the relation between x (or y) and z(except that it be the same for x and y). Ireturn to this point in a moment. Since theinteractions between x or y and z in Nevin's(1974b) experiment are assumed to be com-petitive, /(z) is likely to be a negative functionof some sort.

BEHAVIORAL POWER FUNCTIONS 307

3.2

2.8

2.4In q

2.0

1.6

1.2

Nevin 1974

Null point^

In y

.8 .9

In xFigure 1. The right-hand panel shows lines fitted by Nevin (1974b) to the logarithms of response rates(In y and In x) in two multiple-schedule components at corresponding times in relation to periodic fooddelivered for an alternative response to a second response key. (Each line represents a different com-bination of reward frequencies for responding on the multiple-schedule key.) The left-hand panel showsthe relation between the slopes (s) and intercepts (In q) of the lines in the right-hand panel.

Integrating both sides of Equation 2 yields

In x + Ki = 01 / f(z)dz (3a)

and

In y + Ki. = f(z)tiz. (3b)

Dividing Equations 3a and 3b to eliminateexpressions involving z, rearranging, and ex-ponentiating yields

y = exp(a2#i/ai - K^-x^"\ (4)

where K\ and K?. are constants of integration.Equation 4 is of the same form as Equation

1, but with

q = exp(a2^i/ai - #2) (5)and

s = a<ila\. (6)

Parameter 5, the ratio of sensitivities, repre-sents the elasticity of behavior y with respectto behavior x. It follows from Equations 5 and6 that

In q = sKi - Ki. (7)

Nevin found that both 5 and q varied withthe frequencies of food delivery for respondingto the two multiple-key colors. The simplestassumption is that these changes are solely areflection of changes in a\ and a^, the sensi-tivities of the two responses, so that K\ andKi can be assumed constant across conditions.If this is so, then Equation 7 implies a linearrelation between elasticity parameter 5 inEquation 1 and the logarithm of parameter q.

Nevin (1974b) obtained a number of func-tions that were fit by Equation 1, and heestimated s and q for each. A plot of In qversus s from these data appears as the left-hand panel of Figure 1. The six points are verywell fit by a straight line, with K\ = — 3.96and KI = — 4.69, in accordance with theprediction.

Geometrically, Equation 7 implies that theoriginal power functions from which 5 and qwere obtained should all pass through thepoint In x — — K\, In y — — Kt. The right-hand panel of Figure 1 shows that the data areconsistent with this prediction.1 The point of

1 There is an important caveat to be noted here.Since the zero point of a logarithmic scale correspondsto the unity point of the corresponding linear scale,the location of zero depends on the units of measure-ment. The linear relation of Equation 7 simply repre-sents the obvious fact that the slopes and interceptsof a set of straight lines that go through a point willbe related—positively if the origin is to the right ofthe point of intersection, negatively otherwise. How-ever, since on a log scale the origin can be moved asfar as we please from the region of the empirical datasimply by changing units, a linear relation between sand In q can be forced by measuring rates in sufficientlyextreme units. In this case the "point" through whichthe lines all go is nothing but the region of possiblerates, which is limited at the top by physiologicallimitations and at the bottom by the time during whichbehavior is sampled. Since Nevin's (1974b) data coveralmost the full range of the In x axis on the right ofFigure 1, the convergence shown there is not forced inthis way.

308 J. E. R. STADDON

convergence may be termed the null point,since it represents that pair of response ratesthat is unaffected by changes in a. If therewere no preference for one color over the otherand if the situation were symmetrical in everyother way, one might expect K\ and KI tobe equal. This does not seem to be the case inthis study: K2 was greater than Ki in absolutemagnitude, and the birds tended to respond athigher rates on red (y) compared to green (x),a commonly observed preference in pigeons.

The derivation of Equation 1 just describeddoes not depend on the form of the relationbetween x and z. However, Killeen (1975), inhis discussion of Nevin's result, has assumeda particular relation between x and z (hereconsidered not as a competing response butjust as time, t, in the fixed interval) of the form

x = A\ exp( — t/Ci). (8)

This result can be obtained from Equation 2by taking /(z) = — 1; whereupon, on integra-tion x = a exp(— aiz), which is of the sameform as Killeen's equation. Killeen suggests aparticular identification for the constants A\and C\ that goes beyond the present deriva-tion. Since his model can be considered as aspecial case of the present one, it can makesimilar predictions about the relation betweens and q in Equation 1. However, it also re-quires an exponential relation between x and t,and unfortunately, Nevin's data do not suggesta simple relation. Hence, there is virtue in atheoretical approach that is noncommittalabout the function relating x (or y) and z. Weshall see later that a similar uncertainty existsin the case of the psychophysical law.

Interpretation of Parameters

The parameters a\ and 0,1 in Equation 4have already been interpreted as sensitivities.KI and Kz are scale factors. This can be shownin two ways. First, unless x and y are con-sidered to be dimensionless (e.g., as relativerates), Equation 1 cannot be made to balancedimensionally in its simple form (i.e., with qas an independent parameter) if x and y aremeasured in the same units.2 For example, ifx and y are considered as rates, having thedimensions T~l, then q must have the dimen-sions T*~l for dimensional balance, which im-

plies dependence of q on s. However, if inEquation 4, the dimensions of KI and KI aretaken as In T, the proper dimensional balanceis achieved, as follows:

r-1 = expj>ln (r) - In (T)]- (r~')s;

whence, taking logarithms of both sides,

-In (T) = s In (r) - In (r) - s In (T)

= - In (T).

Hence KI and K^ can be interpreted as thelogarithms of the time bases with respect towhich x and y are measured. The same con-clusion can be arrived at by noting the effectof changing the units of x and y by multipliers,such that x = ax' and y = fty'. Then, a littlealgebra shows that K'\ = K\ + In a andK\ = Kz + In ft.

Rate-Dependent Drug Effects

Many drugs produce effects on responding inoperant conditioning situations that are welldescribed by Equation 1 (Dews, 1958, 1964;Gonzalez & Byrd, 1977). Since there is growingevidence that in such situations there is com-petition between the recorded operant responseand other "schedule-induced" activities, thepresent analysis can be applied.

The usual procedure in these experiments isto compare response rates at equivalent pointsin a periodic schedule, such as fixed interval,between 2 days: a day on which no drug, ora control injection, is administered, and asecond day (typically the next day) when agiven drug dose is administered before theexperimental session. (A few studies havecompared response rates in the presence ofdifferent stimuli that sustain different ratesunder control conditions.) Traditionally, theresults have been displayed as a plot of controldata versus the ratio of drug and control ratesfor each segment of the fixed interval, col-lapsed over all the intervals in each experi-mental session. Such plots are linear in log-logcoordinates, which implies a power relation ofthe same form as Equation 1 between drug

21 am indebted to J. A. Nevin and M. C. Davisonfor pointing out the dimensional problem with Equa-tion 1.

BEHAVIORAL POWER FUNCTIONS 309

i.o

In q

-1.0

-2.0

D P I

OFR

Ft data

In Rn

-iIn R,

Figure 2. The right-hand panel shows lines fitted from Barrett (1974) to the logarithms of response ratesof a single pigeon under drug (Ro; pentobarbital) and no-drug (Re) conditions at comparable pointsin a periodic schedule. (Each line is the result of a different drug dose.) The lines in the left-hand panelshow the relation between the slopes (s) and intercepts (In q) of the fitted lines, both for responding ona key associated with a fixed-interval schedule (FI) and for responding on a key associated with afixed-ratio schedule (FR).

and control rates plotted directly (Gonzalez &Byrd, 1977).

There is considerable evidence that onperiodic schedules of food delivery there iscompetition between the instrumental re-sponse, which occurs with increasing frequencyas the time when food will be available ap-proaches, and other "interim" activities, whichoccur predominantly at the beginning of eachinterfood interval (Staddon, 1977b; Staddon& Ayres, 1975; Staddon & Frank, 1975). Forexample, if the interim activities are preventedin some way, instrumental responding beginsearlier in the interval. These considerationsset the stage for applying the theory to rate-dependent effects. It is necessary only to as-sume that the interim activities exert a sup-pressive effect on the instrumental responsethat is equivalent to the suppressive effect ofFl-key responding on multiple-key respondingin Nevin's (1974b) experiment. If the fre-quency of these interim activities is denotedby z, and the rates of the instrumental re-sponse on control and drug days are denotedby x and y, respectively, then Equation 2 canbe applied directly. Proceeding as before,Equation 7 then describes the relation betweenthe two parameters 5 and q of the power func-tion relating drug and control response rates

on the assumption that the effect of the drugis solely on the sensitivity parameter: a\ isthen the sensitivity parameter under controlconditions, and 02 the sensitivity parameterunder drug conditions. K\ and K% are thereforeassumed to be constant across conditions,that is, drug doses.

The slope and intercept of the functionrelating control and drug rates have beenshown in many experiments with numerousdrugs to vary with drug dose. Figure 2 showsdata from an individual pigeon in an experi-ment by Barrett (1974). The animal wastrained on a schedule in which pecks on onekey (interval key) were reinforced with foodonce every 5 min, providing at least 10 pecksto a second key (ratio key) had occurred(conjunctive fixed-ratio [FR] 10 - FI 5).Various doses of pentobarbital were admin-istered, and the response rates in successive30-sec portions of the fixed interval on controland drug days were compared in the way justdescribed. The left-hand panel of Figure 2shows the relation between 5, the slope, andIn q, the intercept of the power functions ob-tained by Barrett at different drug doses. Twosets of data are shown: one for responding onthe ratio key and the other for responding onthe interval key. In both cases, the points are

310 J. E. R. STADDON

well fitted by a straight line. The right-handpanel of Figure 2 shows the actual powerfunctions fitted to his data by Barrett,3 andas implied by the linear relations in the left-hand panel, they converge on approximatelythe same point. If, as I have proposed, thedrug affects only parameter a, then with nodrug injection, «i = a?, and y should equal x.Hence, from Equation 4, Ki should equal K%.This is approximately true for the ratio datain Figure 2, but lines fitted to the FI dataconverge on a point quite far removed fromequality. This implies that control saline in-jections may have had an effect on K in thisexperiment.

I have examined a total of 14 studies in-volving 21 different drugs, in which rate-dependent relations were obtained (Barrett,1974; Bond, Sanger, & Blackman, 1975;Branch & Gollub, 1974; Byrd, 1975; Dews,1964; Leander, 1975; MacPhail & Gollub,1975; Marr, 1970; McKearney, 1970; Mc-Millan, 1973a, 1973b; Stitzer, 1974; Wenger& Dews, 1976; Wiittke, 1970). These studiescontain a total of 69 sets of power functionssimilar to those shown in Figure 2, obtainedwith six species of animals.4 Fifty-eight ofthese sets comprise three or more functions,permitting a test of the linearity prediction ofEquation 7. No study shows a systematicdeviation from linearity. Twenty-five studies(43%) show linear fits with coefficients ofdetermination (r2) greater than .80; 9 werebetween .60 and .79, 14 between .20 and .59,and 10 less than .20.

Most studies therefore fit the theory quitewell. Those that do not fit (i.e., show r1 valuesless than .6) can be classified into three groups,(a) Some studies fail because the range ofdrug doses used yielded only a small range ofslopes. It is clear from the geometry of thesituation that if many slopes are close to-gether, even small errors in determining slopeswill produce large variations in the points ofintersection of the lines. Hence, a good fit tothe model is not to be expected if the range ofslopes obtained is small, (b) The geometryalso indicates that a poor fit will be obtainedif the units are such that the point of inter-section of the lines (the null point) is in thevicinity of a control rate of unity (zero on thelog scale). In this case, there will be variation

in slope but little variation in intercept, (c)Finally, studies may yield an adequate rangeof slopes, with no common intersection point,that is, clear failure to agree with the model.Careful examination of the 25 or so sets offunctions that yielded low r2 values showedthat only a handful, five or six, are clearlydiscrepant with the model. The clearly dis-crepant results are from studies by Leander(1975, three examples), McKearney (1970,Figure 5, SD), and McMillan (1973a, Table 2,pentobarbital, unpunished; Table 4, imipra-mine, unpunished). As suggested by thestudies cited previously, there are a numberof intermediate cases. All the other cases fallinto Categories a and b.

For most sets of functions, K\ and KI arewithin 20% of each other, as implied by theassumption that different drug doses affectonly parameter a. However, the null point isusually different for different drugs, implyinga dose-independent effect of the drug on K.These characteristics are illustrated by datafrom an extensive study by Leander (1975),which are shown in Figure 3. The figure showslog (control rate) versus log (drug rate) func-tions fitted by Leander to data producedunder different doses of 11 major tranquilizers.Three of the drugs (triflupromazine, trifiuo-perazine, and fluphenazine) fit the modelpoorly; the remaining 8 (and the three injec-tion vehicles) show the predicted linear rela-tions. The null points for the different drugsare clearly different. Other studies show thatthe null point depends on whether respondingis punished or unpunished (McMillan, 1973a,1973b), whether it is associated with a positive(reinforcement associated) or negative (as-sociated with the absence of reinforcement)

3 Barrett's (1974) data were plotted in the usualform: log (drug rate/control rate) versus log (controlrate). I estimated slopes and intercepts from the fittedlines in his Figure 3 and then transformed the functionsto yield the log (drug rate) versus log (control rate)lines shown in Figure 2.

4 In the Wiittke (1970) study, power functions werenot given. I read off the relevant data from his Figure5 and fitted power functions in the usual way. All theother studies presented linear fits of data in log-logplots, usually of control rate versus drug rate/controlrate, occasionally of drug rate versus control rate. Iestimated .s and In q (or log q) from these lines orcomputed them from tables, if available.

BEHAVIORAL POWER FUNCTIONS 311

OrPromazine rChlorpromazine Triflupromazine OpHaloperidol

aDC

O)o

-2log RC

Figure 3. The relation between the logarithm of drug rate (log RU) and control rate (log Re) in a periodicschedule for a number of tranquillizers studied by Leander (1975). [The lines are derived from theslopes and intercepts of lines fitted to plots of 100 log (Ro/Rc) versus log RC by Leander (his Table 1).]

stimulus (McKearney, 1970), and on the typeof reinforcement schedule associated with theresponse (see Figure 2). Generally, the nullpoint is higher on both axes for unpunishedconditions, but this is reversed for some drugs,most notably, morphine. The null point ishigher for responding in the positive stimulus(S°). Individual animals also show differentnull points. Of course, no experiment has at-tempted explicitly to study the effects of theseor any other variables on the null point, sothat these correlations must be accepted withcaution. For example, in most experiments,different doses of the same drug are admin-istered in succession rather than being inter-mixed with doses of other drugs. Hence, effectsof drug type are usually confounded withtime of administration and order: If K valueschange slowly over time, then the sets of dose-response power functions obtained with differ-

ent drugs will tend to show different nullpoints.

The identification of the a and K parameterswith sensitivity and scale factors, respectively,provides an objective basis for the distinctionbetween rate-dependent and rate-independenteffects. If there is a fixed null point, and more-over, K\ and Kt are equal, then the effect ofthe drug is strictly on the a parameter, a purerate-dependent effect. However, if K\ and K^are not equal, then in addition to a dose-related rate-dependent effect, there is a dose-independent and rate-independent effect. Fi-nally, if there is no fixed null point, the differ-ent drug doses have dose-related effects thatare both rate dependent (on the a parameter)and rate independent (on the K parameter).K\ and KZ are approximately equal for mostof the studies discussed here, so that the drugeffects are indeed rate dependent.

312 J. E. R. STADDON

The competition model appears to fit mostof the available data on rate-dependent drugeffects. Most of the apparently discrepantdata can be attributed either to a limited rangeof slopes for the log (control rate) versus log(drug rate) functions or to an infelicitouschoice of units. Most results show a dose-independent effect of drug type on the nullpoint, and data that fail to fit may perhaps beattributable to a dose-dependent effect of thedrug on the null point. The null point appearsto depend on a number of situational factors,but controlled experiments are obviously re-quired to map out these relations in detail.

Psychophysical Law

The sensation produced by a stimulus suchas a sound or a light is not simply proportionalto the stimulus intensity measured in physicalunits. For most stimulus dimensions, the re-sponse measure grows more slowly than phys-ical intensity. On so-called prothetic (intensive)continua, the appropriate relation appears tobe a power function of the same form asEquation 1, where y is the response measure,* the physical stimulus measure, and s anexponent characteristic of the stimulus dimen-sion (Richardson & Ross, 1930; Stevens,1957).6 In a long series of papers, S. S. Stevens(e.g., 1957, 1975) presented evidence and argu-ments for the power relation and establishedthat comparisons between different modalities(stimulus dimensions) generally behave aspredicted by the law. For example, if the rela-tion between perceived intensity and physicalstimulus intensity for continuum A isyA = XAS,and for continuum B, is yB = x/ir, then therelation between A and B when the level of Ais adjusted so that A and B have the sameperceived intensity should be XA = XBrla- Incross-modality matching experiments, the ap-propriate relations between exponents havebeen frequently demonstrated.

The relation of S. S. Stevens' law to twoother types of psychophysical measurement,scales based on just-noticeable differences(JNDS) and so-called partition scales, is still amatter of debate. The present theory hasstraightforward implications for these ques-tions. In some measure, it is a restatement ina convenient form of previous proposals. How-

ever, in this form, it can account for convergentpower functions. It also makes clear the rela-tions between Fechnerian and magnitude scalesimplied by different assumptions about themapping of stimuli onto sensations.

The power law can be immediately derivedfrom the present theory, given the followingidentifications: x is identified with the physicalstimulus and y with the response measure,which may be a number (as in magnitude esti-mation) or some other physical measure (as incross-modal matching). The comparison vari-able 2 must then be identified with sensation,that is, the level of some internal variablethat is affected by both the stimulus x and theresponse y.

The suggestion that number can be con-sidered a sensory dimension was first made byAttneave (1962) and has recently receivedconsiderable support from other sources (e.g.,Rule, Curtis, & Markley, 1970; Teghtsoonian,1974; Wagenaar, 1975). The idea that thecentral sensation variable bears the same rela-tion to stimuli as to responses has also beenfrequently proposed. If that relation conformsto Weber's law (a possibility discussed morefully below), then, according to Teghtsoonian(1974), "if Weber's law holds, then it appliesto both the target continuum plotted on theabscissa, and the matching continuum plottedon the ordinate" (p. 170). The assumption thatstimuli and responses (including number) arerelated to sensation in a similar way is moresymmetrical than the alternative view thatresponses are a direct measure of sensation,but stimuli are not. While symmetry is notproof, the fact that the present theory re-quires a symmetrical assumption should not bea cause for concern.

By the terms of the scaling experiment, thesubject can be assumed to adjust his responsey so that the level of sensation 2 associatedwith it is the same as that associated with thestimulus x. Thus, the expressions involving 2

61 exclude here any consideration of second-order ormolecular effects, such as the biases associated with dif-ferent scaling methods, hysteresis, and sequentialeffects. These require explanation at a different levelthan that attempted here. Refinements of method suchas the widely used threshold correction are also takenfor granted.

BEHAVIORAL POWER FUNCTIONS 313

on the right-hand side of Equations 2a and 2bcan be set equal, and the derivation proceedsas before. The relation between stimulus andresponse dimensions is then given by Equation4. Parameters a\ and #2 are identified with thesensitivities of the individual to the stimulusand response dimensions, respectively. We maysuspect that these parameters are related todiscriminability, and I return to this in amoment. The K parameters are scale factorsas before.

It is obvious that this derivation immedi-ately predicts the results of cross-modalitymatching experiments. Every situation, in-cluding direct magnitude estimation, is re-garded as involving both a stimulus and aresponse dimension, both of which bear asimilar relation to the central sensation vari-able; and s, the exponent of the power func-tion, is already expressed as the ratio of sensi-tivities of the stimulus and response systems.

The major empirical prediction of the pres-ent theory is that in many cases, variables thataffect both the slope and the intercept of thepower function act solely by affecting thesensitivity parameter. Hence, the family ofpower functions so produced should convergeon a null point. J. C. Stevens (1974) has re-cently collected 13 sets of such convergingfunctions. One example, from a study on glareinhibition by S. S. Stevens and Diamond(1965), is shown in Figure 4. In this experi-ment, three subjects adjusted the luminanceof a matching field (seen only by the right eye)so that it matched the apparent brightness ofa target field seen by the left eye. A small,intense glare source was also visible to theleft eye. The experimenters adjusted thebrightness of the target field and thus obtaineda series of matches. The relation betweentarget luminance and matching field luminancewas a power function whose slope and inter-cept (in log-log coordinates) depended on thevisual angle between glare source and target.5. S. Stevens and Diamond obtained a totalof 14 power functions in this way for differentglare angles. I fitted lines (by the method ofleast squares) to the corrected decibel data intheir Table 1 and obtained 14 slope-interceptpairs. Figure 4 shows that these are very wellfitted by a straight line (r2 = .995), with slopeand intercept Ki = - 109 dB, #2 = - 108 dB.

60

50

m

§> 30

20

10Stevens & Diamond

1965

.4 .6 .8 I.O

Figure 4. The relation between the slopes (s) and inter-cepts (10 log q) of power functions fitted by S. S.Stevens and Diamond (1965) to matching data fromcomparisons of the perceived brightness of a matchfield and a target field inhibited by an adjacent glaresource.

KI and KZ are approximately equal, as theyshould be for this symmetrical situation.

Data presented by Marks (1974b) andseveral studies reviewed by S. S. Stevens(1965) can also be treated in this way. How-ever, some studies are available in which thetreatments affected only the intercepts, leavingthe slopes unchanged (e.g., Babkoff, 1976). Interms of the present theory, this correspondsto a change in one or both of the K parameters.In other studies (e.g., Pollack, 1949; replottedin Stevens, 1965), the convergence is lessperfect than that shown in Figure 4, perhapsreflecting some effect on the K parameters.Nevertheless, when there is an effect on slope,the intercept usually changes in the way to beexpected from Equation 7, and the conver-gence prediction is amply confirmed.

J. C. and S. S. Stevens have provided inter-pretations for the null point in various experi-ments. For example, in the S. S. Stevens andDiamond (1965) study, the coordinates of thenull point are approximately equal to theluminance of the glare source, which was 118dB. In other cases, the null point is identifiedwith the stimulus level at which a qualitativechange in sensation occurs, as from loudnessto tickle or pain, or with the system's physio-logical limit, as in maximum heft. In some

314 J. E. R. STADDON

cases, however, the interpretation is less plau-sible. For example, the family of power func-tions obtained by J. C. Stevens and S. S.Stevens (1963) in a study of adaptationconverges on a null point whose luminancecoordinate is between 150 and 160 dB. Thisapproximately matches the luminance of thesolar disc, but J. C. Stevens (1974) acknowl-edges that this identification is far fromcompelling. The present derivation does notpreclude any particular interpretation for thenull point. However, it also allows for the pos-sibility that the null point may have no uni-versal significance and is simply a consequenceof constant scale factors and the generatingprocess described by Equation 2.

Relation to Weber's and Fechner's Laws

Setting /(z) = 1 in Equation 2 yields theequations dx/x — a\dz and dy/y = a^dz, whichare simply the familiar Fechnerian expressionsof Weber's law, with ai and a^ proportional tothe Weber fractions for the stimulus and re-sponse dimensions.6 When integrated, theseequations yield Fechner's logarithmic relationbetween stimulus intensity and sensation (seethe Appendix ). Thus, Equation 2 embracesthe derivation of S. S. Stevens' law fromlogarithmic functions applied to both inputand output (Ekman, 1964; MacKay, 1963;Treisman, 1964; see also Fechner, cited inStevens, 1957). However, this derivation doesnot require that f(z) = 1; hence, it is con-sistent with Phillips' (1964) logical demon-stration that no two-stage (i.e., stimulus-sensation-response) power law model canuniquely define the relation between stimulus(or response) and sensation.

Brentano (e.g., see Stevens, 1975, p. 234)suggested that JNDS yield a sensation that isproportional to the initial sensation levelrather than being constant. This implies thatin Equation 2, f(z) = 1/z. Integrating bothsides of Equation 2a then yields In z = (1/fli)X In x + Ki — (l/fli)^C2, which is a powerfunction but now relating stimulus and sensa-tion (rather than response). It is noteworthythat the same slope-intercept relation holdsfor this one-stage derivation as for the two-stage derivation. Hence, if the response mea-sure is identified directly with a sensation,

as S. S. Stevens (1957) proposed for magni-tude estimation, the convergence property canbe deduced directly. Unfortunately, the simpleone-stage model cannot easily accommodatethe relations between different types of psy-chophysical scales. However, Brentano's as-sumption reappears in the two-stage analysisof these relations, as we shall see.

The Fechner's law form for Equation 2[i.e., f(z) = 1] suggests the hypothesis thatthe sensitivity parameter may be proportionalto the Weber fraction (cf. Teghtsoonian, 1974),although this is not logically forced. It mightbe that JNDS on one continuum, electric shock,say, have a larger effect on sensation thanJNDS on another, say, luminance. This proposi-tion is readily tested, since if at bears a fixedproportion to the corresponding Weber frac-tion, then for any pair of stimulus and re-sponse dimensions, s = a<ija\ = W*/W\; hence,

sWi = Wi, (9)

where 5 is the exponent of the power function,and Wi and W-i are the Weber fractions forthe stimulus and response dimensions, respec-tively. If number is the fixed response dimen-sion (as in magnitude estimation), Equation 9implies that the product of the exponent timesthe Weber fraction for the stimulus dimensionshould be constant. Teghtsoonian (1974) hasprovided evidence that it is. For example, inhis Table 1, he presents values of W and s for

8Auerbach (1971; see also Luce & Edwards, 1958)has shown by adding up JNDS that if Weber's lawapplies to both dimensions in the cross-modalitymatching experiment, then the power law exponentwill be s = In (1 + Wi)/ln (1 + W,), where Wi andW2 are the Weber fractions for the two dimensions.The present derivation yields s = Wi/Wi (= Oi/ds).This apparent contradiction is due simply to the differ-ence between adding up discrete JNDS and adding upJDNS that are assumed to be infinitesimal, as implied byFechnerian integration. The contradiction is imme-diately resolved by noting that the series expansion ofln(\ + W) = W- W*/2 + W/3 - ..., (-K W ̂ 1),since when W is small, terms in W and above can beneglected; whereupon, In (1 + W) = W. Since theactual value for W depends on the percentile criterionused, and this is simply a matter of convention, W canin practice be made as small as we please (cf. Eisler,1963b). If the criterion is chosen to be within thelinear range of the psychometric function, relativevalues of W for different continua will be independentof the criterion.

BEHAVIORAL POWER FUNCTIONS 315

nine perceptual continua. The value of sW\computed from these data ranges from .026 to.034 (compared to a range from .026 to .033for a comparable quantity computed usingAuerbach's method [see Footnote 6]). Note,however, that in terms of the present theory,Teghtsoonian's finding implies only that Wi= kdi, where k is approximately constantacross perceptual dimensions; the finding doesnot require that Fechner's law hold, since /(z)need not be denned for the relation betweenWi and the exponent of the power function tobe fixed.

Thus, neither the power law data norTeghtsoonian's invariance can uniquely definethe form of /(z) in the present theory. How-ever, a choice between the two simple possi-bilities can be made by looking at the differ-ences between what Marks (1974a) has re-cently termed Type I (ratio) and Type II(partition) scales.

Partition Scales

The experiments from which power law re-sults have been produced—magnitude estima-tion and production and cross-modal matching—have been analyzed here as instances ofmatching: The subject is presumed to matchthe sensation produced by a stimulus to thatproduced by another stimulus, either a numberor a stimulus on another dimension. This ap-proach allows for the possibility that otherinstructions may induce the subject to performother operations with his sensations. Thesimplest such operation is "differencing," thatis, the equation of perceptual differences alonga single stimulus dimension.

A more complicated possibility is the com-putation by the subject of sensation ratios.S. S. Stevens, because of his theory of measure-ment, has tended to assume that magnitudeestimation, because it yields a ratio scale,must therefore involve actual judgments ofstimulus or sensation ratios. However, thepresent theory shows that the power law canbe derived by assuming only matching by thesubject. Wagenaar (1975) has also argued forthe matching interpretation on purely em-pirical grounds. Moreover, Schneider, Parker,Farrell, and Kanow (1976), in an experimentexplicitly designed to require judgments of

loudness ratios, showed that their subjectswere actually judging sensation differences.Hence, the assumption that subjects can com-pute sensation ratios is neither required bytheory nor well supported by experiment.

Partition scales are produced when subjectsare required to place stimuli into a fixednumber of categories (category scaling) orto divide a stimulus interval into equal-appearing parts (equisection). The scales thatresult from these operations are nonlinearlyrelated to magnitude scales, as well as beingespecially subject to memory-related "hys-teresis" and stimulus-spacing effects (Ander-son, 1974; Marks, 1974a; Parducci, 1974;Stevens, 1975).

S. S. Stevens and others (e.g., Indow, 1974;Marks, 1974a; Torgerson, 1961) have sug-gested that partition scales are the result of adifferencing operation by the subject. That is,instead of matching sensations (as implied byEquation 2 here) or matching response ratiosto stimulus ratios, perhaps the subject isequating sensation differences. Equation 2makes quite definite predictions about theform of the category scale to be expected,given the differencing assumption and an as-sumption about the form of /(z). In thesimplest case, where /(z) = 1, the predictedrelation is logarithmic (cf. Torgerson, 1961).This can be seen as follows for the case ofbisection :

Let Zi, zs, and zs be the sensation valuesproduced by three stimuli, where Si and st arethe anchor stimuli, and s% is the variablestimulus. Then, if the subject responds to theterms of the bisection experiment by differenc-ing, s% is adjusted so that zi — z-i = 22 — Zj.Integrating both sides of Equation 2a and re-arranging then yields

In x\ — In Xi = In x^ — In x*. (10)

Hence, equal stimulus ratios yield equal sensa-tion differences, a logarithmic scale.

If /(z) = 1/z, a similar set of manipulationsyields

Xi X% ===~ A/2 $" j V -^ *•)

that is, equal sensation differences are definedby equal differences between stimulus valuesraised to a power. Equation 10 is consistentwith Fechner's version of Weber's law, and

316 J. E. R. STADDON

Equation 11 with Brentano's version or witha Fechnerian version of the "near-miss" powerfunction form of the law (see the Appendix).

These results, and others for different as-sumptions about /(z), can be obtained moredirectly by noting that the differencing as-sumption implies that the category scale islinearly related to sensation. Hence, simplyintegrating Equation 2a for a given /(z) atonce yields the form of the category scale. Forexample, if /(z) = 1/z, then,

z = - K0), (12)

where K\ and KQ are constants of integration.The translation from 2 to a particular categoryscale requires a further arbitrary constant Kadded to z to allow for the fact that subjectswill partition stimuli into categories labeled1 — N in the same way as they will intocategories labeled M + 1 — M + N.

These conclusions about the relations be-tween category, JND, and magnitude scales arealso compatible with a general model for intra-individual scale relations proposed by Eisler(1963a; see the Appendix).

Partly because of the sensitivity of partitionscales to context and time effects7 and partlybecause of their equivocal conceptual status (atleast within S. S. Stevens' scheme), there hasbeen disagreement on the proper form forpartition scales. Earlier workers plumped forthe logarithmic form (e.g., Luce & Galanter,1963; Titchener, 1905). However, more recentwork (e.g., Marks, 1968, 1974a; Schneider &Lane, 1963; Ward, 1975) favors a power func-tion of the form

C + K = Ax'', (13)

where C is the category value, sc is the "virtualexponent" (generally smaller than the magni-tude estimation exponent), and A and K areconstants. Given that by the differencingassumption, C is linearly related to sensationlevel z, Equation 13 is then equivalent toEquation 12 and also to Equation A4 (see theAppendix), which is derived from the near-miss power function version of Weber's law.

Thus, the two-stage theory embodied inEquation 2 can be developed in two paralleldirections, each internally consistent and de-pendent on the form of /(z). On the one hand,if Weber's law holds, JNDS are subjectively

equal \_j(z] = 1], and partition scales involvedifferencing, then magnitude estimation andcross-modal matching procedures will yieldpower functions, and partition scaling willyield a logarithmic relation between responseand stimulus variables. On the other hand, ifBrentano's version of Weber's law (sometimescalled Ekman's law) holds [/(z) = 1/z], thenmagnitude estimation and cross-modal match-ing procedures will yield power functions ; andpartition scaling, with an arbitrary constantadded to the category value, will also yieldpower functions.

On the assumption that the power form iscorrect, it is possible to deduce a relation be-tween the exponent of the category scale sc andthe exponent for the magnitude estimation(ratio) scale sm. Comparison of Equations 4 and12 shows that

tfc, (14)

where a^ is the sensitivity parameter for num-ber (the response dimension in magnitude esti-mation). «2 can be estimated by the ratiosm/sc for a variety of continua or from the sc

value for number, given that sc = \/a\. Marks(1968) in his Table 1 gives estimates of sr, andsm for nine pro the tic continua. The ratiosm/sc ( = aj) ranges from 1.42 to 2.54,8 withfour of the values lying between 2.26 and 2.54.The value of !/$„ for the dimension of numberis 1.85. The fact that sc for number is less thanunity is consistent with the fact that categoryscale exponents are reliably lower than theirratio scale counterparts. The values of sc forloudness and brightness were estimated for anumber of conditions differing in the numberof categories used and other procedural details.These estimates vary quite a bit depending onthe condition; for example, sc estimates forbrightness range from .055 to .22. Brightnessand loudness are the two modalities for whichmost information is available, and the agree-ment here is quite good: The median valuesof sc for these two dimensions from Marks'table are .13 and .25 compared with accepted

'This characteristic is consistent with the greatermemory load imposed by the differencing operation.

81 used the median s» estimates for brightness andloudness and the estimate from the linear spacing con-dition for repetition rate.

BEHAVIORAL POWER FUNCTIONS 317

sm values of .33 and .6, yielding a-i values of2.54 in the first case and 2.40 in the second.Hence, the constancy of «2 predicted by thepresent model is confirmed as well as can beexpected.

The predictions of the present theory, onthe assumption that f(z) = 1/z, are in accordwith Marks' (1974a) conclusion that "thepsychophysical functions that relate Type Iand Type II scales to their correspondingphysical scales are in both cases power func-tions, but the exponents that govern Type Ifunctions are typically about twice as large"(p. 358). Thus, Equation 2 provides a naturalrepresentation for the relations among parti-tion, magnitude, and JND scales on the as-sumption that Weber's law holds and thatBrentano's conjecture is correct.

The formulation has some implications forthe near-miss version of Weber's law thatappears to apply to loudness under some con-ditions. If Fechner's assumption is correct,then the theory predicts that the sensationfunction will be a power function; if the rela-tion between sensation and sound intensity isto be positive, then the exponent of the JNDfunction should be negative, as it is (see theAppendix). Comparison of Equations A4 and11 or 12 shows that the exponent of the near-miss power function should be predictablefrom se, since sc = \/a\ = — n. Jesteadt, Wier,Si Green (1977) present data on n for loudnessfor their study and five others. The valuesrange from —.035 to —.125. The predictedvalues range from —.13 to —.50, using the sc

data in Marks' (1968) Table 1; all are toolarge. A possible reason is that Brentano'sassumption about the relation between stimu-lus (x) and sensation (z) may be better thanFechner's; it yields an expression of the form2 = a exp(j3x~") for the near-miss form (seethe Appendix). When »| is small, as is thecase for loudness, and x ranges over two orthree log units, this function is so close to apower function, with exponent somewhatlarger than n \ , that the two may not beseparable empirically.

The present theory shows the power law tobe quite independent of the form of /(z), andonly two simple possibilities have been con-sidered here. Brentano's version appears to be

better than Fechner's, but it may be that stillanother alternative will be better than bothand will resolve the remaining discrepancies.It may even be that /(z) is not fixed, butdepends on contextual factors, a distressingpossibility for classical psychophysicists, butone that is not unlikely, given the strong se-quential and other contextual effects that canbe demonstrated in many scaling situations.

Conclusion

In operant conditioning, power function re-lations between response rates are often foundin situations that involve successive compari-son between each response and a third com-peting class of activities. Examples are certainchoice studies and numerous experiments onrate-dependent drug effects. Equation 2 pro-vides a natural account of this comparisonprocess and explains the convergence propertyof many sets of power functions as well asdealing with the dimensional problems with thesimple power function (Equation 1).

Equation 2 can also provide an account ofthe psychophysical power law, which is inter-preted as the outcome of matching betweenthe sensations produced by the stimulus andresponse. The approach immediately accountsfor converging sets of power functions and,with the aid of a differencing assumption andan assumption about the form of the sensationfunction f(z), can also account for the ap-proximately fixed ratio between the exponentsfor magnitude (Type I) and partition (TypeII) scales. The approach may also predictqualitative properties of the near-miss form toWeber's law, although there are still quantita-tive discrepancies that remain to be resolved.

Luce (1972) has pointed out the lack inpsychophysics of a system of invariant rela-tions among the measures that might justifythe assimilation of the subject to classicalphysics. The simple theory embodied in Equa-tion 2 begins to meet this criticism. However,the auxiliary assumptions (of matching ordifferencing) required to relate the theory todata tend to favor Luce's view of psycho-physics as the study of a measuring instrumentrather than Fechner's more grandiose visionof a field akin to physics.

318 J. E. R. STADDON

References

Anderson, N. Algebraic models in perception. In E. C.Carterette & M. P. Friedman (Eds.), Handbook ofperception (Vol. 2). New York: Academic Press, 1974.

Atkinson, J. W., & Birch, D. The dynamics of action.New York: Wiley, 1970.

Attneave, F. Perception and related areas. In S. Koch(Ed.), Psychology: A study of a science (Vol. 4).New York: McGraw-Hill, 1962.

Auerbach, C. Interdependence of Stevens' exponentsand discriminability measures. Psychological Review,1971, 78, 556.

Babkoff, H. Magnitude estimation of short electro-cutaneous pulses. Psychological Research, 1976, 30,39-49.

Barrett, J. E. Conjunctive schedules of reinforcement.I: Rate-dependent effects of pentobarbital and(^-amphetamine. Jo^^rnal of the Experimental Analysisof Behavior, 1974, 22, 561-573.

Bond, N. W., Sanger, D. J., & Blackman, D. E. Effectsof ^-amphetamine on the behavior of pigeons main-tained by a second-order schedule of reinforcement.Journal of Pharmacology and Experimental Thera-peutics, 1975,194, 327-331.

Branch, M. N., & Gollub, L. R. A detailed analysis ofthe effects of df-amphetamine on behavior underfixed-interval schedules. Journal of the ExperimentalAnalysis of Behavior, 1974, 21, 519-539.

Byrd, L. D. Contrasting effects of morphine on schedule-controlled behavior in the chimpanzee and baboon.Journal of Pharmacology and Experimental Thera-peutics, 1975, 193, 861-869.

Dews, P. B. Studies on behavior. IV. Stimulant actionsof methamphetamine. Journal of Pharmacology, 1958,722,137-147.

Dews, P. B. A behavioral effect of amobarbital.Naunyn-Schmiedeberg's Archiv fitr ExperimentellePathologie und Pharmakologie, 1964, 248, 296-307.

Eisler, H. A general differential equation in psycho-physics : Derivation and empirical test. ScandinavianJournal of Psychology, 1963, 4, 265-272. (a)

Eisler, H. Magnitude scales, .category scales, andFechnerian integration. Psychological Review, 1963,70, 243-253. (b)

Eisler, H., & Montgomery, H. On theoretical andrealizable ideal conditions in psychophysics: Magni-tude and category scales and their relation. Percep-tion &• Psychophysics, 1974, 16, 157-168.

Ekman, G. Is the power law a special case of Fechner'slaw? Perceptual and Motor Skills, 1964, 19, 730.

Falmagne, J. C. The generalized Fechner problem anddiscrimination. Journal of Mathematical Psy-chology, 1971, 8, 22-43.

Gonzalez, F. A., & Byrd, L. D. Mathematics under-lying the rate-dependency hypothesis. Science, 1977,195, 546-550.

Indow, T. Scaling of saturation and hue shift: Sum-mary of results and implications. In M. Moskowitz,B. Scharf, & J. C. Stevens (Eds.), Sensation andmeasurement. Dordrecht, The Netherlands: Reidel,1974.

Jesteadt, W., Wier, C. C., & Green, D. M. Intensitydiscrimination as a function of frequency and sensa-

tion level. Journal of the Acoustical Society of America,1977, 61, 169-178.

Killeen, P. On the temporal control of behavior. Psy-chological Review, 1975, 82, 89-115.

Krantz, D. H. Integration of just-noticeable differences.Journal of Mathematical Psychology, 1971, 8, 591-599.

Leander, J. D. Rate-dependent effects of drugs. II.Effects of some major tranquillizers on multiplefixed-ratio, fixed-interval performance. Journal ofPharmacology and Experimental Therapeutics, 1975,193, 689-700.

Luce, R. D. What sort of measurement is psycho-physical measurement? American Psychologist, 1972,27, 96-106.

Luce, R. D., & Edwards, W. The derivation of subjec-tive scales from just noticeable differences. Psycho-logical Review, 1958, 65, 222-237.

Luce, R. D., & Galanter, E. Psychophysical scaling.In R. D. Luce, R. R. Bush, & E. Galanter (Eds.),Handbook of mathematical psychology (Vol. 1). NewYork: Wiley, 1963.

MacKay, D. M. Psychophysics of perceived intensity:A theoretical basis for Fechner's and Stevens' laws.Science, 1963, 139, 1213-1216.

MacPhail, R. C., & Gollub, L. R. Separating theeffects of response rate and reinforcement frequencyin the rate-dependent effects of amphetamine andscopolamine on the schedule-controlled performanceof rats and pigeons. Journal of Pharmacology andExperimental Therapeutics, 1975, 194, 332-342.

Marks, L. E. Stimulus-range, number of categories,and the form of the category scale. American Journalof Psychology, 1968, 81, 467-479.

Marks, L. E. On scales of sensation. Perception 6"Psychophysics, 1974, 16, 358-376. (a)

Marks, L. E. Spatial summation in the warmth sense.In M. Moskowitz, B. Scharf, & J. C. Stevens (Eds.),Sensation and measurement. Dordrecht, The Nether-lands: Reidel, 1974. (b)

Marr, M. J. Effects of chlorpromazine in the pigeonunder a second-order schedule of food presentation.Journal of the Experimental Analysis of Behavior,1970, 13, 291-299.

McFarland, D. J. Time-sharing as a behavioral phe-nomenon. In D. S. Lehrman, J. S. Rosenblatt, R. A.Hinde, & E. Shaw (Eds.), Advances in the study ofbehavior (Vol. 5). New York: Academic Press, 1974.

McFarland, D. J., & Sibly, R. M. The behavioral finalcommon path. Philosophical Transactions of theRoyal Society (Serial B), 1975, 270, 265-293.

McKearney, J. W. Rate-dependent effects of drugs:Modification by discriminative stimuli of the effectsof amobarbital on schedule-controlled behavior.Journal of the Experimental Analysis of Behavior,1970,14, 167-175.

McMillan, D. E. Drugs and punished responding. I:Rate-dependent effects under multiple schedules.Journal of the Experimental Analysis of Behavior,1973, 19, 133-145. (a)

McMillan, D. E. Drugs and punished responding. Ill:Punishment intensity as a determinant of drug effect.Psychopharmacologia (Berlin), 1973, 30, 61-74. (b)

Nevin, J. A. On the form of the relation between re-sponse rates in a multiple schedule. Journal of the

BEHAVIORAL POWER FUNCTIONS 319

Experimental Analysis of Behavior, 1974, 21, 237-248. (a)

Nevin, J. A. Response strength in multiple schedules.Journal of the Experimental Analysis of Behavior,1974, 21, 389-408. (b)

Parducci, A. Contextual effects: A range-frequencyanalysis. In E. C, Carterette & M. P. Friedman(Eds.), Handbook of perception (Vol. 2). New York:Academic Press, 1974.

Phillips, J. P. N. A note on Treisman's model. QuarterlyJournal of Experimental Psychology, 1964, 16, 386.

Pollack, I. The effect of white noise on the loudnessof speech of assigned average level. Journal of theAcoustical Society of America, 1949, 21, 255-258.

Richardson, L. F., & Ross, J. S. Loudness and tele-phone current. Journal of General Psychology, 1930,3, 288-306.

Rule, S. J., Curtis, D. W., & Markley, R. P. Input andoutput transformations from magnitude estimation.Journal of Experimental Psychology, 1970, 86, 343-349.

Schneider, B., & Lane, H. Ratio scales, category scales,and variability in the production of loudness andsoftness. Journal of the Acoustical Society of America,1963, 35, 1953-1961.

Schneider, B., Parker, S., Farrell, G., & Kanow, G.The perceptual basis of loudness judgments. Percep-tion &• Psychophysics, 1976, 19, 309-320.

Staddon, J. E. R. Behavioral competition in condition-ing situations: Notes toward a theory of generaliza-tion and inhibition. In H. Davis & H. M. B. Hurwitz(Ed$.~),0perant-Paiilovianinteractions. Hillsdale, N.J.:Erlbaum, 1977. (a)

Staddon, J. E. R. Schedule-induced behavior. In W. K.Honig & J. E. R. Staddon (Eds.), Handbook ofoperant behavior. Englewood Cliffs, N.J.: Prentice-Hall, 1977. (b)

Staddon, J. E. R., & Ayres, S. L. Sequential and tem-poral properties of behaviour induced by a schedule ofperiodic food delivery. Beliaviour, 1975, 54, 26-49.

Staddon, J. E. R., & Frank, J. A. Temporal controlon periodic schedules: Fine structure. Bulletin of thePsychonomic Society, 1975, 6, 536-538.

Stevens, J. C. Families of converging power functionsin psychophysics. In M. Moskowitz, B. Scharf, &

J. C. Stevens (Eds.), Sensation and measurement.Dordrecht, The Netherlands: Reidel, 1974.

Stevens, J. C., & Stevens, S. S. Brightness function:Effects of adaptation. Journal of the Optical Societyof America, 1963, 53, 375-385.

Stevens, S. S. On the psychophysical law. PsychologicalReview, 1957, 64, 153-181.

Stevens, S. S. Power-group transformations underglare, masking and recruitment. Journal of theAcoustical Society of America, 1965, 39, 725-735.

Stevens, S. S. Psychophysics. New York: Wiley, 1975.Stevens, S. S., & Diamond, A. L. Effect of glare angle

on the brightness function for a small target. VisionResearch, 1965, 5, 375-385.

Stitzer, M. Comparison of morphine and chlorproma-zine effects on moderately and severely punishedresponding in the pigeon. Journal of Pharmacologyand Experimental Therapeutics, 1974, 191, 172-178.

Teghtsoonian, R. On facts and theories in psycho-physics: Does Ekman's law exist? In H. R. Mosko-witz, B. Scharf, & J. C. Stevens (Eds.), Sensationand measurement. Dordrecht, The Netherlands:Reidel, 1974.

Titchener, E. H. Experimental psychology (Vol. 2; Part2, Instructor's Manual). New York: Macmillan, 1905.

Torgerson, W. S. On distances and ratios in psycho-physical scaling. Ada Psychologica, 1961,19,201-205.

Treisman, M. Sensory scaling and the psychophysicallaw. Quarterly Journal of Experimental Psychology,1964, 16, 11-22.

Wagenaar, W. A. Stevens vs. Fechner: A plea fordismissal of the case. Ada Psychologica, 1975, 39,225-235.

Ward, L. M. Sequential dependencies and responserange in cross-modality matches of duration to loud-ness. Perception &• Psychophysics, 1975, 18, 217-233.

Wenger, G. R., & Dews, P. B. The effects of phency-clidine, ketamine, ^-amphetamine, and pentobar-bital on schedule-controlled behavior in the mouse.Journal of Pharmacology and Experimental Thera-peutics, 1976, 196, 616-624

Wiittke, W. The effects of a!-amphetamine on schedule-controlled water licking in the squirrel monkey.Psychopharmacologia (Berlin), 1970, 17, 70-82.

Appendix

Sensation Functions and Just-Noticeable Difference Functions

The relation between sensation, which ismerely an intervening theoretical variable, andphysical variables can be reduced to simpleterms if our objective is just to explain theoutcomes of various operations (i.e., scalingtasks) applied to the subject and, particularly,to understand the relations among them (cf.Luce & Edwards, 1958). Consider first therelation between the increment in a physicalstimulus that is just noticeable, dx, and thevalue of the base stimulus x. Two simple pos-

sibilities have been reported. First, Weber'slaw states

dx/x = A, (Al)where A is a constant. More recently, thefollowing near-miss form of Weber's law hasbeen reported for the intensity discriminationof pulsed sinusoids (Jesteadt et al., 1977):

dx/x = A(x/xa)n, (A2)

where n is negative and close to zero, and XQis the threshold stimulus value. What do these

320 J. E. R. STADDON

two versions of Weber's law imply about therelation between the hypothetical sensationvariable z and the value of the physical stimu-lus xl

This question can be answered only if someassumption is made about the relation betweena just-noticeable increment in the physicalstimulus and the increment in sensation as-sociated with it. The simplest assumption isFechner's, namely, that all JNDS yield thesame constant increment in sensation level.Given this assumption, it remains simply todefine the physically invariant unit that cor-responds to a JND and integrate the result toobtain the expected relation between a and x.(Numerous papers have been written on thelegitimacy or otherwise of Fechnerian integra-tion [e.g., Eisler, 1963b; Falmagne, 1971;Krantz, 1971; Luce & Edwards, 1958], Thecurrent consensus appears to be that the pro-cedure is legitimate, subject to certain safe-guards.) For Weber's law, Equation Al givesthis at once:

1 JND = dz = ( l / A ) ( d x / x ) ;

hence, integration yields

z+ Ci = (1/04) (In x + C2), (A3)

which is Fechner's law. For the near-missform, rearrangement yields

1 JND = dz = (x0n/A)(dx/xn+1);

hence, integration yields

« + Ci = (*o"A4 )[(*-"/- ») + C»], (A4)

which is a power function of the form a + K\= Kzxm, where K\ and KI are constants, andm is the exponent (which is positive for loud-ness because n is negative).

If Brentano's conjecture is correct, then dzis replaced by dz/z, and Weber's law yields2 = x

l<A . exp(Kt/A — K I ) , a power function.The near-miss form yields an expression of theform z = a exp(J3x~n), where a and /3 areconstants.

Relation to the General PsychophysicalDifferential Equation

Eisler (1963a; Eisler & Montgomery, 1974)has proposed an equation to describe theintraindividual relation between different psy-chological scales (e.g., magnitude and categoryscales). If u and v are the scale values, and<ru(u) and <r,(v) are the corresponding Weberfunctions, estimated by the SDs of intraindi-vidual judgments, the equation is

dudv

(AS)

Eisler's formulation can be reconciled with thepresent one as follows. First, Equation 2 isrewritten in its most general form, which isthe following pair of separable differentialequations:

dx dz ,. , .T7IT = TTTT (A6a)

anddz_

h(y) (A6b)

The functions /'[= I//(a)], g, and h can beregarded as Weber functions in Eisler's sense.For example, if Weber's law holds for all threevariables, then/'(a) = z, g(x) = a\x, and h(y)= aiy. (Note that Equations A6a and A6byield the power law for magnitude estimationonly if Weber's law holds for x and y.) We cantherefore rewrite Equation A6b as follows:

dy dz(A7)

where <ry(y) and <r,(a) are the Weber functionsfor y and z. From arguments in the text, it isclear that y and z are represented (but foradditive constants) by the subject's magnitudeand category judgments, respectively. Hence,granted that the trs can be estimated fromjudgmental SDs, Equation A7 is equivalent toEquation AS, the general psychophysical dif-ferential equation.

Received December 5, 1977 •