WHAT SORT OF MEASUREMENT IS PSYCHOPHYSICAL …rdluce/bio/pre1990/1972/Luce... · Physical Measurement An essential distinction in understanding physi-cal measurement is that between

WHAT SORT OF MEASUREMENT IS PSYCHOPHYSICAL MEASUREMENT?*

R. DUNCAN LUCE 2

The Institute for Advanced Study, Princeton, New Jersey

THE thesis of this article has both a negativeand a positive aspect. The negative oneholds that psychophysical measurement is

not of a character closely analogous to eitherfundamental or derived physical measurement.Whatever loudness may be, I endeavor to establishthat it probably is not a measure much like massor energy or density. In brief, the reason is thatpsychophysical measures do not exhibit any fixedrelation to physical measures and most likely not toone another when examined over individuals. Thisis reflected in the absence of any structure to theunits of psychophysical measures.

The positive thesis is that man—and any otherorganism—is, among other things, a measuring de-vice, in function not unlike a spring balance or avoltmeter, which is capable of transforming manykinds of physical attributes into a common measurein the central nervous system. According to thisview, the task of psychophysics is to unravel thenature of that device. The difficulty in doing sostems from several facts: as compared with man-made, special-purpose devices, higher organismsare both complex and flexible measuring devices;their overall behavior does not clearly suggest thenature of the receding of signals; some responsesdepend on peculiar nonlinear processing of thesensory information; and each individual within aspecies is calibrated somewhat differently.

To flesh out and make this view reasonably clear,we must know something of physical measurementand of psychophysics. Since the details of thenature of physical measurement, especially theabstract theory, are probably not very familiar

1 The preparation of this article was supported by agrant from the Alfred P. Sloan Foundation to The Insti-tute for Advanced Study. It was delivered as a Dis-tinguished Scientific Award Address at the annual meetingof the American Psychological Association, September 1971,Washington, D.C.

2 Requests for reprints should be sent to R. DuncanLuce, The Institute for Advanced Study, Princeton, NewJersey 08S40.

to most psychologists, I devote some space inthis article to them.3 To keep these remarks withinbounds, I am forced to assume a general familiaritywith contemporary psychophysics, to the extentthat mere mention of a method, theory, or result issufficient to bring it to mind. The remainder ofthe article is devoted to showing how poorly thefundamental measurement concepts of physics fitthe structure of psychophysics and how much moresatisfactory is its interpretation as the study ofa measuring device.

Brief Comments on the Nature ofPhysical Measurement

An essential distinction in understanding physi-cal measurement is that between theory and prac-tice—even though practice often rests on a gooddeal of sophisticated physical theory beyond mea-surement theory. Put another way, this is thedistinction between the foundations of measurement,that is, the study of which ordinal attributes can berepresented systematically by which numericalstructures, and the development and calibration ofdevices designed to render such measurement con-venient and, in some cases, practicable. A simpleexample will make the distinction transparent. Thefundamental analysis of weight measurement isbased on having a very large, very refined collec-tion of objects and an idealized, equal-arm panbalance. Placing a finite collection of objects ineach pan and noticing which one drops permits us,in principle, qualitatively to order by weight allfinite collections of objects. Certain empirical gen-eralizations are so familiar as to seem tautological,even though they are not. The most important isthat if A, B, C, and D are mutually disjoint collec-tions such that A is at least as heavy as C, and Bis at least as heavy as D, then the union of A andB will be found to be at least as heavy as the

3 Anyone interested in a fuller discussion of these mattersshould consult Krantz, Luce, Suppes, and Tversky (1971),especially Chapter 10.

96 • FEBRUARY 1972 • AMERICAN PSYCHOLOGIST

union of C and D. With a suitable set of suchgeneralizations as axioms, a classical mathematicaltheorem (Helmholtz, 1887; Holder, 1901; for mod-ern formulations see Chapters 2 and 3 of Krantzet al., 1971) shows how to assign numbers to theobjects in such a way that sums over finite collec-tions of them order them numerically in exactly thesame way as does the pan balance.

This, then, is the basic theory of weight measure-ment, and any proposal for the actual weighing ofobjects must be shown, in principle, to be equiva-lent to it. In practice, pan balances are not terriblyconvenient, although they were common enoughuntil well into this century and can still be found inrural areas all over the world, and so other devicessuch as spring balances are often used. Such abalance is based on the empirical fact that forcessystematically alter the length of a spring. So themeasurement of weight is transformed into anequivalent one of length, which is rather easier tomeasure. The only remaining step is to calibratethe device, that is, to determine from fundamentalmeasurement of weight and of length exactly howlength covaries with weight in the particular springbalance.

In general, then, a measuring device is anyspecially constructed (or naturally occurring) ap-paratus that transforms one attribute into anotherin an (approximately) one-to-one fashion. It is auseful device if we wish to measure one of theattributes and the other one is easy to measure.Often length is the one that is easy to measure.It is, of course, essential to know how to measureeach attribute independently of the device in orderto be able to calibrate it. This last point is some-times overlooked by those who introduce so-calledoperational definitions of attributes. The lack ofa deep measurement analysis of anxiety is not over-come by saying that, in a particular context, anxietyis by definition skin resistance measured in ohmsat a particular location on the body. A clue thatsomething is missing is the failure to discuss thecalibration of the ohm meter in nonarbitrary unitsof anxiety.

Next, let us consider something of the struc-ture of the theory of physical measurement. Oneaspect is that it provides a way to pass from anordinal definition of an attribute to a numericalrepresentation of it. There are now a number ofresults along these lines, the two most important for

physics being extensive measurement, such as de-scribes the meaurement of weight, and additive con-joint measurement (Debreu, 1960; Luce & Tukey,1964), which, for an attribute having two or moreindependent components that affect it, formulatesconditions that are sufficient to construct a numeri-cal representation that is additive over those com-ponents.4 Examples of the latter are the depend-ence of kinetic energy on mass and velocity andthat of mass on volume and substance.

Much emphasis has been placed on the numericalrepresentation of isolated physical attributes, often,I fear, at the expense of overlooking a secondequally important feature of physical measurement.The several—well over 100—attributes of physicsare locked together, forming a rigid algebraic struc-ture which embodies a number of simple, basicphysical laws. It is the existence of this structurewhich makes the method of dimensional analysiswork. Its most obvious manifestation is thefamiliar, if ill understood, composition of units.At present, the structure is six-dimensional in thesense that six attributes can be chosen inde-pendently—one suitable set is length, mass, timeduration, angle, temperature, and charge—and allof the others are monomial functions of these. Thisshows up in the fact that the units of all otherphysical measures can be taken to be of the form

meterx gram" second" radian" degree6 coulomb*,

where the exponents X, /u, etc., are rational numbers.This is truly quite remarkable and bears someanalysis.

The key to constructing this algebra is theexistence of attributes that are both extensively andconjointly measurable. The conjoint theory, whenit applies, constructs interlocked measures on threeattributes. It is a fact of physics that at least twoof the three attributes involved in any conjointstructure are also extensively measurable; that is,empirical operations of addition are possible ontwo of them. This means, then, that at least two-thirds of these attributes have two quite distinctmeasures defined. The one is based on an in-ternal, additive structure of the attribute and theother on its relation to two other attributes. Ofcourse, these duplicate measures are monotonicallyrelated to each other since they preserve the same

*In physics, it is customary to write these conjointmeasures in multiplicative rather than additive form—theonly difference being an exponential transformation.

AMERICAN PSYCHOLOGIST • FEBRUARY 1972 • 97

ordinal structure, but beyond that the measurementtheories alone cannot take us. If, however, certaintypes of qualitative empirical laws hold—they arecalled laws of exchange and of similitude by Krantzet al. (1971)—then these monotonic functions areactually power functions. This means that themeasure of the nonextensive attribute of a conjointtriple can be written as a product of powers of thetwo extensive measures. Assuming that sufficientlymany such laws hold, as they do in physics, one canshow that the resulting algebra has the form thatthe measures of all attributes can be expressed asproducts of powers of the measures of a finite, basicset of extensive attributes.

Those nonextensive attributes that enter into alaw of similitude or exchange can be expressed interms of extensive ones, and so they are often calledderived. An example is density. In this case, massis the conjoint attribute which can be manipulatedby varying volume and substance independently.Both mass and volume are extensive attributes.The law of similitude that holds is a qualitativeanalogue of the usual numerical statement that, fora given substance, the ratio of mass to volume isindependent of the volume. That ratio, which isthe conjoint measure associated to substance, iscalled density. From one point of view it is mis-leading to call such attributes derived because theyare measured fundamentally via conjoint measure-ment procedures; however, they are not extensivelymeasured, and they can be "derived" from otherextensive measures via a law of exchange or ofsimilitude.

A Brief Summary of Psychophysical Measurement

Three quite distinct bodies of theory and datafrom psychophysics have been interpreted as formsof measurement.

1. There are attempts to construct numerical rep-resentations of ordinal data; often this is calledscaling. Perhaps the best known example isShepard's (1966) very successful program forrepresenting ordinal judgments of the similarity ofstimuli as distances in a Euclidean space of mini-mum dimension. A second example is the currentattempt to scale such attributes as loudness bymeans of additive conjoint measurement (Levelt,Riemersma, & Blunt, in press). A third is Pfanzagl's(1959, 1968) proposed theory of bisection which,to my knowledge, has never been applied to data.

Still other examples can be found in Coombs'(1964) book.

2. There are attempts to treat numerical re-sponses to signals as numerical measures of at-tributes. Perhaps the most widely used method ofthis type in psychology is the rating scale, but farmore important for psychophysics is Stevens'(1957) method of magnitude estimation and itsclose relatives, magnitude production and cross-modality matching. The striking regularities of thedata obtained by magnitude methods—not only thefamiliar power functions (Stevens, 1957; Stevens &Galanter, 1957) and the predicted average expo-nents of the matching work (Stevens, 1959, 1966b),but also the representation of intensity-durationexchanges (Stevens, 1966a; Stevens & Hall,1966), the effect of adaptation (Stevens & Stevens,1963), and the close relation to reaction time (Mc-Gill, 1963; Vaughan, Costa, & Gilden, 1966)—haveimpressed many of us and should challenge theoreti-cians to provide theories that encompass both thesemethods and the more traditional ones having to dowith discrimination and detection.

3. There are attempts to interpret estimatedparameters of probabilistic models as measures ofpsychological attributes. This, the classical andmost widespread, approach to psychophysical theo-rizing can be traced back at least to Fechner's(1860) famed rescaling of just noticeable differ-ences, which in recent times has been reappraisedby Luce and Edwards (1958) and Falmagne(1971), generalized in Levine's (1970, in press)studies of uniform and similar systems of functions,and modified by Eisler (1963). A second type oftheory in which parameters can be interpreted asmeasures includes Thurstone's (1947) law of com-parative judgment for paired-comparisons data, itsgeneralizations to other kinds of data (Bock &Jones, 1968; Coombs, 1964), and its close relatives,such as the BTL model (Bradley & Terry, 1952;Luce, 1959). Related models for detection experi-ments include the theory of signal detectability(Green & Swets, 1966; Tanner & Swets, 1954) andthe low threshold model (Luce, 1963). Perhapsthe best known example of such a parameter is d'from detection theory which, because it increasessystematically with signal-to-noise ratio and be-cause it is unaffected by payoffs and various otherchanges in experimental design, is interpreted as ameasure of the detectability of the signal. Still

FEBRUARY 1972 • AMERICAN PSYCHOLOGIST

another type of theory postulates a receding ofsignal intensity into neural pulse trains. The pulserate is an inferred parameter which is interpreted asa psychological measure of intensity. Considerablymore, including references, is said below about suchmodels.

To summarize, psychophysical measurement canbe grouped into the scaling of ordinal data, magni-tude methods, and the estimation of parametersfrom probabilistic models.

Does Psychophysical Measurement ResemblePhysical Measurement?

Current wisdom about psychological measure-ment, most forcefully and carefully articulated bySuppes and Zinnes (1963), sees the following anal-ogy to physical measurement.5

1. The attempts to pass, by means of a formalaxiomatic theory such as extensive or conjoint mea-surement or certain other measurement theories notused in physics, from ordinal data to numericalrepresentations of these data are attempts at funda-mental measurement, comparable to those success-fully carried out in physics. Whether any of theseattempts in psychology is yet successful is de-batable, but according to this view, if some axio-matic measurement theory is shown to fit a psycho-logical attribute, then that attribute has been mea-sured in the same fundamental sense as in physics.The work of Campbell and Masterson (1969) onaversiveness to electric shock in rats and of Leveltet al. (in press) on the loudness of pure tones inhuman beings encourages one to think that somepsychological attributes may fulfill the conditionsof additive conjoint measurement.

2. Magnitude estimation can be seen simply as aspecial case of cross-modality matching in whichthe match is between the number scale and thephysical stimulus. Furthermore, as Krantz (inpress) demonstrated, cross-modality matching maypossibly satisfy ordinal axioms that lead to anumerical representation in which each attribute is

5 Savage (1970), in an extended critique of psycho-physics, argued that psychophysical measurement shouldnot be viewed in the following way. Roughly, he at-tempted to show that loudness and pitch are really physicalmeasures and that the business of psychophysics is thestudy of perceptual abilities. For a brief, critical reviewof his position, see Luce (1971).

some fixed power function of each of the others.If so, magnitude estimation is then no longer a spe-cial category of psychophysical measurement be-cause it has been subsumed as a part of funda-mental ordinal measurement.

3. The probabilistic models from which param-eters are estimated are, in reality, just empiricallaws, comparable to .those of exchange and simili-tude in physics, and so the parameters are simplyderived measures, comparable to density. Ofcourse, the parameters are expressed in terms ofresponse probabilities, but since probability is akind of extensive measurement, the analogy tophysics is said to be quite close—the derived mea-sures are reduced to prior extensive measurement.

All of this is true, in a sense, and yet I claimthat psychophysical measurement is really not verymuch like physical measurement. A key differenceis seen when we ask what is the algebraic structurerelating the psychophysical measures; it should bereflected in the pattern of their units. It is con-ceivable that the set of psychophysical attributeshas a structure itself. If so, it surely willnot be constructed in the same way as the physicalstructure, that is, using extensive and conjoint mea-sures interlocked by qualitative laws, because nopsychophysical attribute has been shown to beextensively measurable. But other structures arepossible, for example, the one resulting from cross-modality procedures. For this to work out so thatall of the units can be expressed as monomialfunctions of a few, it will be necessary for theexponents relating attributes to be independent ofthe subject. Although I know of no data toexclude this possibility, I find it difficult to beoptimistic that it will hold. Clearly, however,Krantz's theory must be checked empirically, and,in particular, it must be decided whether or notthe exponents are invariant over people.

An alternative approach, for which we do havethe data, is to suppose that the psychophysicalmeasures are to be adjoined to the structure ofphysical ones. Were this to work out, loudness, forexample, would be expressed as products of powersof certain physical measures—among them, soundintensity—and its unit would be a product of(rational) powers of the six basic physical unitsand perhaps some psychological units. As a mat-ter of fact, there is considerable evidence frommagnitude estimation that loudness is (approxi-


mately) a power function of intensity, and Leveltet al. (in press) have shown the same result usingconjoint procedures on the two ears. The onlytrouble is that the exponents vary from subject tosubject—over a range from at least .15 to .50—andeven between the two ears of a single subject. Thus,no unit of loudness can be stated, independent ofthe individual, in terms of physical units.

In sum, it seems to me that it is incumbent uponthose who allege a similarity of psychophysicalmeasurement to fundamental physical measurementto show the existence of invariant relations amongthe measures and to develop a coherent system ofunits as has been done in physics. This has notbeen done, and I doubt that it can be.

Let us next turn to the resemblance of inferredparameters of models to derived measures of phys-ics. I claim that three differences are sufficientlymarked to make that resemblance remote. First,the type of law formulated by a theory such assignal detectability theory is considerably morecomplex than the laws of similitude and exchangeused to derive measures in physics, and certainly itis of a wholly different, nonmeasurement character.Second, because of the lack of structure amongpsychophysical measures, there is considerable un-certainty about what units these inferred param-eters should have. In many cases, they are definedin such a way as to be dimensionless, which differsconsiderably from physics. Third, although proba-bility is an extensive measure of sorts, it is a some-what unusual one; moreover, it would be amazingif the different psychophysical measures—for loud-ness, brightness, subjective length, etc.—should allbe derived from the same extensive measure, re-sponse probability, whereas physics requires sixextensive attributes.

All in all, the analogy to the structure of physicaldimensions seems stretched all out of shape, and Idoubt that it will prove useful in guiding futurework.

Is Psychophysics the Study of aMeasuring Device?

Having tentatively rejected the notion that psy-chophysical measurement is comparable to eitherfundamental or derived physical measurement, weare left with the possible analogy to a measuringdevice. The version that I shall urge is that thesensory part of the organism is a mechanism whose

function is to recast all sorts of physical inputs intoa common measure with which the central nervoussystem can deal. In this view, the task of psycho-physics is to discover the nature of that receding, toprovide ways to ascertain the particular calibrationof any given individual, and to understand howthese measurements are processed by the organismso as to lead to responses. The various kinds ofobserved psychophysical data—magnitude esti-mates, comparisons of signals, detection, reactiontime, etc.—should assist us in inferring exactlywhat that receding and processing is.

According to this view, we must discover answersto two quite different questions. The first is thenature of the receding from physical stimuli intosome suitable, internal measure. The model forthis should take into account the stimuli and theorganism, but should be completely independent ofthe tasks we set the organism, the payoffs, and non-sensory variations in the experimental design. Thesecond question concerns the sorts of processingthat are carried out on these measures as thesubject attempts to respond to the questions we askof him. Here the model presupposes the theory ofsensory transduction, and it varies with the experi-ment and the question asked, for surely the sensoryinformation is not processed in the same way whenwe ask for a magnitude estimate as when we askfor a fast detection.

Consider the first problem, the nature of thetransduction. Although most physical measuringdevices transform an attribute to be measured intoa measurement of length, that is obviously not themode of operation in the nervous system. Some ofthe psychophysical theories designed to deal withconfusable stimuli have postulated rather abstractrecodings. Fechner supposed it to be the trans-formation that renders just noticeable differencesequal; Thurstone postulated it to be one thatrenders the randomness of the representation nor-mally distributed; the theory of signal detectabilitysays it is the likelihood ratio of a hypotheticalobservation;6 and the threshold theories assume adiscrete set of ordered, abstract states with somesort of probability distribution over them. Eachof these ideas has one or more drawbacks, amongthem the fact that some are untestable, some are

6 The postulated normality of the distribution of likeli-hood ratio is entirely secondary in the theory of signaldetectability.


limited to a very few experimental designs, and alllack neurophysiological support.

Recently, several psychophysicists (Creelman,1964; Fitzhugh, 1958; Green & Luce, 1967, 1971,in press; Grice, 1968; Kohfeld, 1969; Luce, 1966;Luce & Green, 1970, 1972; McGill, 1963, 1967;McGill & Goldberg, 1968; Siebert, 1965, 1968,1970) have begun to take seriously the neuro-physiological evidence that signal intensity is en-coded as pulse rates in the peripheral neural sys-tem (Kiang, 1965, 1968; Perkel & Bullock, 1968;Rose, Brugge, Anderson, & Hind, 1967). Here,then, is a possible convenient measure in terms ofwhich all information about sensory intensity couldbe encoded—time. As an idealization—althoughneither an essential one nor, to judge by the dataof Rose et al., a bad one—the pulse trains resultingfrom a constant intensity signal are assumed to bePoisson; that is, the times between successive pulsesare independent random variables with an expo-nential distribution. The expected value of theseinterarrival times, which is the reciprocal of thepulse rate, has been shown to decrease systemati-cally with intensity. This means that each inter-arrival time provides the nervous system with anindependent estimate of the signal intensity pro-ducing the pulse train. Of course, the transforma-tion from signal intensity to the Poisson intensityparameter is, in general, nonlinear and may beexpected to vary from person to person and fromear to ear or eye to eye within a person.

Since the processing of receded information lead-ing to responses is almost certainly not carried outat the periphery, these neurophysiological data arenot necessarily relevant to our problem of develop-ing a suitable psychophysical theory. A priori, itis entirely possible that the representation has beenaltered considerably by the time it arrives at thecentral processing location. We assume not. Asa working hypothesis, we suppose that signal in-tensity is represented at the processing center as aPoisson pulse train; however, we do not assumethat the peripheral neurophysiological estimates ofrates necessarily apply to the central process. Ina sense, then, this type of theory is just as abstractas those I mentioned above; however, it has theadvantages of paralleling the peripheral data, ofbeing independent of the nonsensory aspects of theexperiment, and of applying to all experimentaldesigns in which the signal is used.

Some Consequences of Neural Pulse Models

Given our working hypothesis about the sensoryreceding, the problem is to devise plausible de-cision rules that are able to account for all psycho-physical results. Clearly, the working throughof a program of such scope will require the effortsof a number of people over some years. Here Isimply cite a few encouraging results that David M.Green of the University of California, San Diego,and I have obtained during the past two years.

A decision rule operating on a pulse train canfocus either on the number of pulses that occurin a fixed time or on the time required for a fixednumber of pulses to occur. Each ratio provides anestimate of the Poisson parameter. The formerhas the advantage of taking a known amount oftime, but the sample size, and therefore the statisti-cal quality of the estimate, varies; the latter has apredetermined sample size, but the time neededfor the estimate is variable. Models that assumea count of the pulses occurring in a fixed time areknown, naturally enough, as counting models (Mc-Gill, 1967). Those in which a fixed number ofinterarrival times are accumulated are known astiming models (Luce & Green, 1972).

A first question is whether we can dismiss eitherclass of rules. The answer seems to be no—appar-ently both modes of operation can occur. Theevidence for this statement comes from a yes-no,auditory detection experiment using a response-terminated signal and a response deadline (Green& Luce, in press). Responses occurring prior tothe signal onset or after the deadline were fined;those in between were paid off for accuracy. Wevaried both the deadline for a fixed symmetric pay-off matrix and the payoff matrix for a fixed dead-line. It is not difficult to show that the countingmodel implies that the response times should de-pend neither on the signal presented nor on theresponse made. Further, the receiver operatingcharacteristic (ROC curve) in a normal-normalprobability plot should be approximately linear andhave a slope less than one. The timing modelimplies that variation of the deadline should leadto a linear relation between the mean responsetime to noise and the mean response time to signal,and variation of the payoffs should lead to anapproximately linear ROC plot. Moreover, bothfunctions should have the same slope and it shouldbe greater than one—which is virtually unheard of

AMERICAN PSYCHOLOGIST • FEBRUARY 1972 • 10]

1200

1000

800

600

400

200

No

200 400 600 800 1000 1200 200 400 600 800 1000 1200

MRT,sY MRTsN

FIG. 1. Mean reaction time plots using the ordinary deadline procedure. (The first subscript is the stimulus con-dition—noise [n] or signal [s]. The second is the response condition—Yes [Y] or No [N]. The open points were gen-erated by varying the deadline for a fixed symmetric payoff matrix; the solid ones, in the insert, by varying thepayoff matrix for a fixed 600-millisecond deadline.)

for empirical ROC plots constructed from signalsof fixed duration. The mean response times for theconventional deadline procedure are shown in Fig-ure 1, and the corresponding ROC curve isshown in Figure 2. Clearly, of these two models,the counting one is correct. But now con-sider an apparently minor variant of the ex-periment in which the deadline applies only tosignal trials, not to noise ones. The comparableresults are shown in Figures 3 and 4. Clearly, thetiming model is now favored.7

5 10 20 30 40 50 60 70

P(Y|n)FIG. 2. ROC curves, on double probability coordinates,

obtained from a fixed 600-millisecond deadline by varyingthe payoff matrix.

i The close agreement of the mean reaction time andROC slopes, which according to the theory should beidentical, is probably more apparent than real. Withoutgoing into details, the difficulty is that the slope of the ROCcurve is very sensitive to small discrepancies in the proba-bilities near 0 and 1, and the approximation of the gammadistribution by the normal introduces just such discrep-ancies.

We do not yet have any certain idea as to theconditions that elicit counting rather than timingbehavior. My personal guess is that in its ordinaryenvironment an organism simply monitors short,overlapping sequences of interarrival times, attend-ing to marked changes in the average time asevidence for a change in the intensity of that signal.Counting is the preferred mode of operation onlyin the peculiar environment of those experimentshaving fixed, well-marked observation periods.Even then, as we have just seen, the countingbehavior can be replaced by timing behavior. Pre-sumably the reason that the counting mode isavailable, even though it cannot be very useful inthe natural environment, is that both modes of oper-ation require both counting and timing mechanismsin the nervous system. Therefore, models of bothtypes are going to have to be worked out in detail.Since Green and I have emphasized timing ones, Iwill limit myself to some illustrative results fromthese.

In a magnitude estimation experiment, the sub-ject is asked to report numerically the level of in-tensity of each signal. In terms of a pulse-ratemodel, this can be interpreted as asking him toestimate and report the rate. If so, in the timingmode of operation, he collects a fixed number ofinterarrival times (the number being limited bysignal duration or by some form of memory), aver-ages them, and reports the reciprocal.8 Thus, Stev-

8 If the experimenter has set an arbitrary unit, then thereciprocal has to be multiplied by a positive constant.


1400-

-200MRTS (in msec)

FIG. 3. Mean reaction time to noise versus that to signal (combined over reponses)using a variable deadline only on signal trials.

ens' (19S7) data (and numerous later publicationsby him and others) say that the.Poisson parameteris approximately a power function of intensity;moreover, the exponent varies over a fairly widerange with both modality and subject. Stevens(1970) summarized data in which physiological

measures appear to grow with intensity in the sameway as magnitude estimates. The timing theoryalso predicts that the frequency distribution ofthese estimates at any one level of intensity has ahigh tail—of the form t* rather than e~at—whichresult was noted empirically by Luce and Mo

5 10 20 30 40 50

P(Y|n)FIG. 4. ROC curves, on double probability coordinates, obtained from a fixed 600-millisecond deadline on signal trials

by varying the payoff matrix. (The dashed line is the corresponding gamma distribution, which shows how sensitivethe slope is to minor discrepancies in the tails.)


(1965). Our estimate of the number of interar-rival times entering into calculations of magnituderesponses is not very solidly based at the presenttime, but it appears to be in the range from 10 to25.

A timing analysis of the simple reaction-time,random foreperiod experiment is easily described,although not so easily worked out. We assume thateach interarrival time is compared with a criterion,and the first one less than the criterion is inter-preted as evidence for the onset of the signal.Among other things, this leads to a predictionwhich indirectly favors the above power functionconclusion, namely, that for signals of relativelylong duration, the mean reaction time should beof the form MRT = f + B/ME, where f and B arepositive constants and ME is the mean magnitudeestimate of the same signal.9 Vaughan, Costa, andGilden (1966) reported that this holds far bright-ness; however, Mansfield (1970) showed that itbreaks down for durations below 10 milliseconds.By choosing f to be 5 milliseconds less than themean reaction time corresponding to the most in-tense signal, the three sets of loudness data shownin Figure 5 support the same hypothesis.10 In or-der for a counting model to handle these data, wewould have to make a special ad hoc assumptionabout the observation interval changing with signalintensity.

Turning to discrimination experiments, sufficeit to say that the timing models resemble non-constant variance Thurstonian models in the sensethat the total time to collect a fixed number ofinterarrival times is a random variable that is com-pared with another similar time. The major differ-ence is that the distribution of these random vari-ables is gamma rather than normal. A necessaryand sufficient condition for Weber's law to hold inthis model is that the Poisson parameter be apower function of signal intensity. Moreover, thewell-known increase in the Weber fraction at lowintensity has a completely natural explanation in

dB re Threshold0 10 20 30 40 SO 60 7080 90 100 ,

IOOOK I i—I—I—I—r~i—i—i—n~TTTrn 1 1—r-rrrr

9 Of course, the reaction-time theory actually predictsa linear relation between mean reaction time and the re-ciprocal of the Poisson parameter. Combining this withthe theory for the magnitude experiment leads to the pre-diction stated.

10 McGill (1963), using a different estimate for f, con-cluded otherwise; the slope, but not the linearity, of theplot is markedly affected by changes of a few millisecondsin the value selected for f .

100-

10-

EXPERIMENTER _r_ IND. VARIABLE. McGill 132 MEo Snodgrass 114 ME

Chooholle 105 dB

TO 100"

MAGNITUDE ESTIMATION

FIG. 5. Plots of MRT-r versus ME of loudness of1000 Hertz tones from three experiments. (For Chocholle's[1940] data, it is assumed that ME = al"-s. In each case,f was chosen to be 5 milliseconds less than the minimummean reaction time; this is different from McGill's [1963]choice, and it alters radically his conclusion. The data ofJ. G. Snodgrass are unpublished. Reprinted from an arti-cle by Luce and Green from the January 1972 PsychologicalReview. Copyrighted by the American Psychological As-sociation, Inc., 1972.)

the timing model, namely, that the number of inter-arrival times that can be expected to be observedduring a fixed signal interval decreases with signalintensity, and so the precision with which the sig-nal intensity is defined must, of necessity, be re-duced. The constancy of the Weber fractionbeyond a certain intensity suggests either a limitto the number of interarrival times that can be col-lected and stored or an asymptote to the functionrelating the Poisson parameter to intensity. Wedo not yet know which is the better explanation.

Concluding Remarks

My main contention is that existing psycho-physical measures do not exhibit one of the majorinvariances required if they are to have a structureat all comparable to physical measures. I concludethat it is more appropriate to think of sensorysystems simply as devices that recode the intensityof various types of physical signals into a commonneural measure, namely, time between pulses or,equivalently, pulse rate. Assuming that such a


view is correct, what consequence does it have forthe standardization of psychophysical scales suchas is typified by the sone scale of loudness? I in-terpret such standard scales simply as statementsabout the approximate calibration of a typicalperson as a measuring device. When it is said thatloudness is a power function with an exponent of .3,this means that, to some approximation, the mea-suring device effects a power transformation fromsound intensity to a time measure in the nervoussystem and that the average exponent over somepopulation of people is .3. The attempt to attachunits of loudness to such a standard scale is, Ibelieve, completely misguided and is based on themistaken notion that one is discovering fundamentalmeasures comparable to mass, energy, and the like.The value in having standard scales, which is verygreat indeed, depends not on their measurementtheoretic character, but on the fact that they de-scribe how the typical person transduces physicalintensity into the measure used by his central ner-vous system. Knowing this permits us to approachmore rationally social decisions about such mattersas acceptable and unacceptable noise levels.

In closing, let me explicitly cite three things thatwere not said and which I do not intend to imply.First, by saying that psychophysics is concernedwith the workings and calibration of measuring de-vices and not with the discovery of new funda-mental measures, I do not intend to denigrate psy-chophysics; I have emphasized the point only be-cause it seems useful to be quite clear as to whatone is about. Second, I do not imply that myconclusions apply beyond psychophysical measure-ment. I am not at all sure whether other branchesof psychology ultimately will develop fundamentalmeasures distinct from those already known tophysics. A number of apparently ordinal concepts,among them hunger, anxiety, fear, drive, haveso far resisted any systematic measurement analy-sis. Perhaps when we gain more insight intothem, they will assume a status as fundamentalattributes. And third, my conclusions in no wayreflect adversely upon the usefulness of ordinaltheories of measurement in psychology, in particu-lar in psychophysics. Theories such as additiveconjoint measurement, which formulate ordinallycertain forms of independence and show how to con-struct economical representations of it, provideinitial steps toward simple theories of interaction.

All that I urge is that we not impute to such scalesmore structure than we have actually demonstrated.

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