Perception & Psychophysics/977. Vol. 2/ (3).2/7-226

Magnitude scales, category scales, and the generalpsychophysical differential equation

HENRY MONTGOMERYUniversity ofGoteborg, Goteborg, Sweden

The general psychophysical differential equation (GPDE) which relates scale values and Weberfunctions for two subjective variables was tested statistically with data from a number ofcategory rating and magnitude estimation experiments. In all tests, it was investigated whetherthe GPDE is compatible with the assumption that the Weber function of the category scale isconstant. This assumption implies that the category scale may be regarded as a discriminationscale. The tests were carried out by estimating the Weber function of the category scale by means ofthe GPDE and testing these estimates for constancy. For group data, there were fairly smallsystematic deviations from constancy across different experiments in the Weber functions thatwere estimated by the GPDE, and the hypothesis of a constant estimated Weber function couldnot be rejected statistically (p > .05) in most experiments. For individual data the estimatedWeber functions deviated from constancy in idiosyncratic ways, and these deviations werestatistically significant (p < .05) for most subjects.

This work was supported by the Swedish Council for SocialScience Research. I am indebted to Dr. Hanes Eisler for hisvery helpful comments on the manuscript. Requests for reprintsshould be sent to Henry Montgomery, Department of Psychology,University of Goteborg, Fack , S 400 20 Goteborg 14, Sweden.

In a series of studies, Eisler and his co-workershave demonstrated that the category scale is close tothe Fechner integral of the Weber function of themagnitude scale (Eisler, 1962, 1963b c, 1965; Eisler& Montgomery, 1974; Eisler & Ottander, 1963;Montgomery & Eisler, 1974). This is to say that thecategory scale is predicted by integrating the equation

where K and tp denote, respectively, category andmagnitude scale values for the same set of stimuli,k is a constant assumed to correspond to a constantWeber function for the category scale, and ollJ{tp)corresponds to the Weber function of the magnitudescale (usually computed as intraindividual SDs of themagnitude estimates). The form of the Weber func­tion of the magnitude scale has been described aslinear (Eisler, 1962, 1963b) or parabolic(Montgomery, 1975; Montgomery & Eisler, 1974).

Equation 1 corresponds to a special case of thegeneral psychophysical differential equation (theGPDE; Eisler, 1963a, 1965; Eisler, Holm, &Montgomery, Note 1) which expresses the followingrelation

where x and y denote scales values of subjectivevariables for the same set of stimuli, and Ox(x) andOy(Y) their Weber functions. As can be seen, theFechnerian model corresponds to the case with aconstant Oy(Y) in Equation 2. This case of the GPDEis of particular interest since a constant Oy(Y) impliesthat the y variable can be considered a discrimina­tion scale provided certain other assumptions of theunderlying scaling model are valid (cf. Eisler &Montgomery, 1974; Torgerson, 1958, Chap. 10).

As mentioned above, the Fechnerian model allowsgood prediction of category scale values. From thisit might be inferred that the GPDE is valid for therelationship between category and magnitude scales,and, furthermore, that the category scale is a dis..crimination scale. However, the assumption of aconstant Weber function for the category scale seemsto be contradicted by empirical results. The SDs forcategory ratings are typically greatest in the middleand decrease toward both ends. Eisler andMontgomery (1974) suggested that this discrepancybetween theory and empirical data could be explainedin terms of bias or distortion in the Weber functionof the category scale. The primary cause of this biaswas assumed to be that the more extreme stimuliare recognized more often than stimuli which arecloser to the middle of the stimulus range (cf.Garner, 1952). In the Eisler and Montgomery (1974)study, an attempt was made to minimize this biasby (a) constructing a set of stimuli that were equallydiscriminable, and (b) constructing a set of stimuliwith equal response ambiguity (Attneave, 1959) forall stimuli. For group data, the Weber function ofthe category scale proved approximately constantfor both spacings except that the SDs correspondingto the two end categories were too low. For individual

(1)

(2)dy Oy(Y)

dx = ox(x)'

217

218 MONTGOMERY

data, however, there were in some cases rather largedeviations from constancy in the Weber functionsof the category scale. The approximately constantWeber functions obtained for the category scales(particularly for group data) together with ratheraccurate predictions yielded by the GPDE andFechnerian integration suggested a distinctionbetween the genotypic, i.e., unbiased, Weber func­tion (assumed to be constant for the category scale)and the phenotypic, i.e., empirically obtained,Weber function.

Thus, the Fechnerian model as well as the assump­tion of a constant genotypic Weber function of thecategory scale seem to be well supported by empiricalresults, particularly for group data. However, as willbe shown below, objections can be raised againstthe procedures that have been used for testing thevalidity of the Fechnerian model (Equation 1) andthe GPDE in general (Equation 2). The typical proce­dure in those previous tests has been to integrateEquations 1 or 2, to solve for one of the subjectivevariables, and to fit the equation obtained toempirical data. The computation of the integralshas been based on data that were collapsed overseveral subjects with an exception for a series ofindividual experiments in the Eisler and Montgomery(1974) study. The goodness of fit of these integralshas been evaluated only by eye or by purely descrip­tive measures. The following objections can be raisedagainst this procedure:

(1) The procedure is insensitive. In the Eisler andMontgomery (1974) study, the fit of integrals ofEquation 1 (the Weber function for the categoryscale assumed to be constant) and of Equation 2(empirical Weber function for both subjective vari­ables) were compared for a number of group andindividual experiments on category rating and magni­tude estimation. The difference in the fit of the twointegrals proved small, despite the fact that the devia­tions from constancy in the Weber function for thecategory scale were occasionally rather large. Sinceintegrating has a certain data smoothing consequence,thereby leading to a decrease in sensitivity, studyingEquation 2 directly rather than after integrationshould give a clearer picture of the correctness ofthe GPDE (cf. Mashour, 1964).

(2) The use of data that were collapsed over severalsubjects is questionable, especially for the Weberfunction of the magnitude scale, since the form ofthis Weber function can vary markedly betweenindividuals (Eisler & Montgomery, 1974).

(3) It is difficult to draw more definitive con­clusions about the goodness of fit of predictionsallowed by the GPDE without using statistical tests.

The aim of the present study was to test theFechnerian case of the GPDE in a way that takesthe above objections into consideration. This wasdone for a number of group and individual experi-

ments (cf. Table 1) by estimating the value of k(Equation 1) for each pair of adjacent stimuli foreach subject separately. In the individual experi­ments, the sampling variance of each k value wasestimated and k was tested for constancy by meansof an F test. In the group experiments, a nonpara­metric test (Friedman's two-way analysis of variance)was used for testing the hypothesis of constant kvalues for all pairs of adjacent stimuli.

The data analysis outlined above require that thesame subjects be employed for category rating andmagnitude estimation and that individual data beavailable. The analysis was carried out with datafrom all experiments reported by Eisler and his co­workers that fulfilled these requirements. The experi­ments are described in some detail in Table 1. Asmentioned above, data from group and individualexperiments were used. The group experimentsrepresent four different continua, viz, loudness ofwhite noise, softness of white noise, intensity of smellof amyl acetate, and length of lines. In the individualexperiments, all subjects judged the loudness ofwhite noise.

DATA ANALYSIS

Estimation of k ValuesSolving for k in Equation 1 yields

(ta)

In both the group and individual experiments, thevalue of k in this equation was estimated for eachsubject separately, and for each pair of adjacentstimuli nand n + I according to

k' = Kn+l - Kn . (Sn+l + sn.\ (3)~n+l - ~n 2 J

where K and ~ denote arithmetic means of categoryratings and magnitude estimates, .and s the samplestandard deviation of the magnitude estimates.

The quantity (Kn+ 1 - Kn)l(~n+ 1 - ~n) inEquation 3 is an estimate of the derivative of thecategory scale with respect to the magnitude scale.It should be noted that there could be two kinds oferrors in this estimate of the derivative, viz,systematic errors and stochastic errors.

An attempt to assess the systematic error was madewith values from explicit functions obtained byintegration of Equation 1 for two cases of combina­tions of Weber functions, viz, linear-constant andparabolic-constant. The numerical values of thesefunctions were taken from two experiments on loud­ness of white noise, viz, Eisler (1962, linear-constantcombination of Weber functions) and Montgomery

MAGNITUDE AND CATEGORY SCALES 219

Denota­tion inText Reference

Table IExperiments Employed in the Test of the Fechnerian Model (Equation 1)

Group Experiments

Continuum Remarks

1: Log spacing, ambient noise not controlled

2: So ne spacing, ambient noise not controlled

Stimulus exposure .12 secStimulus exposure as long as the subject wishedEqual response ambiguity (ERA) spacingEqual discriminability (ED) spacingED spacing

Loud ILoud 2Loud 3Soft 1Soft 2Soft 3Smell ISmell 2Smell 3Line ILine 2ERAED IED 2

Eisler. 1962Eisler, 1962Eisler, 1962Eisler, 1962Eisler, 1962Eisler. 1962Eisler. 1963bEisler, 1963bEisler, 1963bEisler, 1963cEisler, 1963cEisler & Montgomery, 1974Eisler & Montgomery, 1974Eisler & Montgomery, 1974

Loudness of white noiseLoudness of white noiseLoudness of white noiseSoftness of white noiseSoftness of white noiseSoftness of white noiseIntensity of smell of amyl acetateIntensity of smell of amyl acetateIntensity of smell of amyl acetateLength of linesLength of linesLoudness of white noiseLoudness of white noiseLoudness of white noise

777777 3: Log spacing, soundproofed chamber5 1-3: Different standards in magnitude estimation559977

15

Initialsof Ss Reference

Individual Experiments

Continuum Remarks._---~...---_.._---

K.J., D.M.M.W.,B..l.D.F., M.L.H.S., E.E.

T.J., A.N.B.E.,I.S.

Eisler & Montgomery, 1974 Loudness of white noise

Eisler & Montgomery, 1974 Loudness of white noiseEisler & Montgomery, 1974 Loudness of white noise

Eisler & Montgomery, 1974 Loudness of white noise

7 ERA spacing

7 ED spacing7 ED spacing, stimuli presented in ascending and

descending series15 ED spacing

Note-itv: denotes the number of ratinX categories ill each category rating experiment.

where s{, denotes the variance. V is a function of

computed from Equation 3. However, the stochasticerror for the latter estimates of the derivative seemedto be more homogeneous than for the derivativesthat were computed by parabolic interpolation.

(4)evevOX' ox'1 J

Statistical AnalysisGroup experiments. For each group experiment,

the hypothesis of constant k ' values for all pairsof adjacent stimuli was tested with Friedman's two­way analysis of variance (Subjects by Pairs of Ad­jacent Stimuli). No parametric test was used, sincethe distribution of k ' values were markedly skewedand heterogeneous.

Individual experiments. For each of the 12 subjectsin the individual experiments (Eisler & Montgomery,1974), k ' was tested for constancy by means of anF test. The sampling variance in each k ' value wasestimated by using error calculus. The computationswere based on the following equation (cf. Beers,1953, p. 29).

and Eisler (1974, parabolic-constant combination ofWeber functions). The difference between theestimated derivatives and derivatives computed bydifferentiating the explicit function was about IltJofor practically all scale values in both combinationsof Weber functions.

The stochastic error in the estimated derivativeswas estimated by means of error calculus (cf. belowunder Statistical Analysis) for each subject and pairof adjacent stimuli in the individual experiments inthe Eisler and Montgomery (1974) study. The medianof the relative standard error

where Sd denotes the standard error of the estimatedderivatives, was 0.173 and the quartile deviation was0.057. Thus, the stochastic error in the estimatedderivatives was rather large. A more efficientestimate of the derivative could not be found,however. Spline derivatives (Greville, 1969) had stillgreater estimated stochastic errors for the presentdata, whereas derivatives computed by parabolicinterpolation (Hildebrand, 1956) had estimatedstochastic errors that varied on approximately thesame level as the errors in those derivatives that were

220 MONTGOMERY

the variables x., X 2 ... xn, and r denotes the productmoment correlation. For the present data, s{r cor­responds to the sampling variance of k ' and dV/ dXicorresponds to the partial derivates of k' withrespect to the variables on the right-hand side ofEquation 3. These partial derivatives can be ob­tained by logarithmic differentiation of Equation 3(cf. Hald, 1952, p. 250). Differentiating

In k' = In(Kn+ 1 - Kn) - In(tlln+l - tIIn)

dk' dk'dSn+ 1 . dSn Sn + I . Sn

(k' )2Sk2, - (sK2 + SK2- K - K n+1 nn+ 1 n

corresponding to rsn+ ISn was neglected since thenumerical values of

proved to be very small compared to the other errorterms.

The above assumptions imply that only two of thecovariance terms need to be computed, i.e., the termscorresponding to rKn+ IKn and rtpn+ Itpn. Inserting theseterms as well as the variance terms correspondingto Kn, Kn+ 1, tIIn, tIIn+ 1, Sn, and Sn+ 1 into Equation 4yields, after rearranging,

(7)

(6)

(5)

1

1

Kn + l - Kn '

+ 1n(sn+ 1 + sn) - In2

_1.~k' dKn

dk'k' dKn + 1

yields

dk'

k' dtlln+ 1 tIIn+ 1 - tIIn (8)+ ( k' )2(S~n+1 + s~n

tIIn + 1 - tIIn

The following assumptions were made about thecorrelations between the variables in Equation 3. Thecorrelation between the K and til variables andbetween K and s were assumed to be zero since Kand til derive from different experiments. The cor­relation between the arithmetic means and thestandard deviations of the same samples of observa­tions will be zero if the observations are normally dis­tributed (cf. Cramer, 1945, p. 382). Since the magni­tude estimates were approximately normally distributedin the individual experiments, it was assumed thatrtpn+ ISn+ 1 and rtpnsn were approximately zero.Because of this assumption, it seemed reasonablealso to assume that rtpn+ISn and rtpnsn+1 wereapproximately zero. Finally, the covariance term

-._-------

1 dk'k' dtpn tpn+ 1 - tIIn

1 dk' 1---=k' dsn+ 1 sn+ 1 + sn

As can be seen from Equation 12, the above assump­tions imply that the error variance of k' can bepartitioned into three additive components whichare related to the variances of Kn+ I - Kn,tIIn + 1 - tIIn, and (sn+ 1 + sn)l2, respectively.

The error variance of k ' was estimated accordingto Equations 12 for each individual and for eachpair of adjacent stimuli. In the following discussion,this estimate of sk' is denoted as s' ~ r , As notedabove, Kn, Kn+}, tIIn, and tIIn+ 1 correspond toarithmetic means of single observations. The vari­ance of these arithmetic means was estimated accord­ing to s2/(N - 1), where N = number of trials,whereas the variance of Sn and Sn + I, respectively,was estimated according to s2/[2(N - 1)] (cf.McNemar, 1950, p. 60). The estimation of rKn+IKnand rtpn+llfIn was dependent on the order in whichthe stimuli were presented. For 10 of the 12 subjectsthe 10 different stimuli were presented in 50 successiveblocks, where each block consisted of all 10 stimulipresented in random order. The correlations forthese 10 subjects were computed between the singleobservations from the two stimuli within each pairof adjacent stimuli over all blocks of stimuli. [Notethat correlations between single observations can be

(12)(k' )2+ (s~n+ I + s~n)·

Sn+1 + Sn(9)

(11)

(10)

1

sn+l + Sn

1 dk'

k ' dSn

and

expected to be equivalent to correlations betweenarithmetic means (Beers, 1953, p. 31)]. For the tworemaining subjects (subjects H.S. and E.E.), the 10different stimuli were presented in 84 successiveblocks of ascending or descending stimuli, whereeach block consisted of 5, 6, or 7 stimuli (cf. Eisler& Montgomery, 1974; Eisler & Ottander, 1963). Forthese two subjects, rKn+IKn and rljJn+lljJn werecomputed between arithmetic means of samples ofobservations from stimulus nand n + 1, respectively,where each sample consisted of seven blocks whichwere drawn randomly without replacement untilall blocks had been drawn. The correlations werethus computed over 12 samples of observationsbetween the means of the observations from the twostimuli within each pair of adjacent stimuli.

As mentioned above, the hypothesis of constantk' values was tested by means of an F test for eachof the 12 subjects in the individual experiments.This F test was based on the ratio between theestimated population variance of k' between thenine different pairs of adjacent stimuli and thearithmetic mean of s' ~, (Equation 12) within thenine different pairs of adjacent stimuli. The degreesof freedom for the within and between variances,respectively, were assumed to be 441 and 8. (Thetotal number of observations was considered to be450, since there were 9 pairs of adjacent stimuli and50 observations from each stimulus in the categoryrating and magnitude estimation task, respectively.)

It can be noted that the above procedure for testingthe Fechnerian case of the GPOE (Equation 1) in­volves two steps: (a) estimating values of k in Equa­tion 1 and (b) conducting an F test of the hypothesisthat these estimated k values are constant. In orderto check the adequacy of this procedure for testingEquation 1, a computer simulation was carried out.The simulation was based on data from three sub­jects (B.J., M.L., and T.J.) which were used forconstructing cases in which Equation 1 yieldedconstant population values of k for all nine pairsof adjacent stimuli. The population values of tp (meanmagnitude estimates) and of sK (50s of categoryratings) were the same as the correspondingempirically obtained values for the three subjects.The population values of stp, i.e., otp in Equation 1,were obtained by letting otp(tp) be defined by aparabola which was fitted to the empirical Weberfunction of the magnitude scale for each of the threesubjects. The population values of K (mean categoryratings) were derived by inserting the fitted parabolasinto Equation 1 and solving the equation for K. The"category ratings" and "magnitude estimations" inthese constructed cases were normally distributed,and the population values of the correlations rIjJIl +1tpnand rx., IKn were the same as the correspondingempirical correlations for the three subjects. From

MAGNITUDE AND CATEGORY SCALES 221

these distributions, a distribution of 1,000 F ratioswas generated for each case by drawing randomsamples of 50 "category ratings" and "magnitudeestimations," respectively, and computing theF ratio for each sample over all the nine pairs ofadjacent stimuli. The distributions of F ratiosthereby generated was then compared to theF distribution that obtains for the degrees of freedomthat were used in the test described above. It wasfound that the 95th and 99th percentiles in these F dis­tributions corresponded to the 93.4, 96.7, and 94.3percentiles and the 97.6, 99.0, and 98.2 percentiles,respectively, in the simulated distributions. Conse­quently, the results of the simulation suggest thatthe present test of the Fechnerian case of the GPOEis valid within acceptable limits.

RESULTS

Group ExperimentsFigure 1 shows for each group experiment medians

of the k' values as a function of the category scale(cf. the circles in Figure 1). The medians were takenover all subjects for each pair of adjacent stimuliand are in the following discussion denoted as Mdj . .The horizontal straight lines in Figure 1 representmedians of Mdj . over all pairs of adjacent stimuli.In most experiments, there is a considerable vari­ability in Mdk' around this horizontal line. Foralmost all experiments, the Mdk' values for the ex­treme pairs of stimuli lie below the horizontal line.No other common trend can be seen in the deviationsof Mdk' from the horizontal line.

Friedman's two-way analysis of variance indicateda significant effect of pairs of adjacent stimuli onk I (p < .05) in 5 of the 14 experiments (cf. Table 2),viz, in Experiments Loud 2, Soft 2, Line 1, ED I,and ED 2. The strongest significant effect was ob­tained in Experiment ED 2, in which the subjectsrated the stimuli on a IS-point category scale. TheMdk' values in this experiment tend to increasefrom the lower end of the category scale toward apoint corresponding to the 10th rating category andthen to decrease toward the upper end of thecategory scale. This result corroborates the tentativeconclusion drawn in the Eisler and Montgomery(1974) study that the IS-point category scale isnot a discrimination scale in the sense of having aconstant genotypic Weber function.

The significant effects in Experiments Loud 2,Soft 2, and ED I are probably at least partially dueto the fact that some stimuli were closely bunchedin these experiments, particularly the stimuli in thelouder region of the stimulus range. This is becausethese stimuli were so close together that many sub­jects gave the same category ratings for adjacentstimuli, which implies that k' became zero in these

222 MONTGOMERY

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Category scale

Figure 1. Theoretical and empirical measures of the uncertainty in category ratings as a function of the corresponding categoryscale (arithmetic means of the two category scale values from each pair of adjacent stimuli) for the group experiments listedin Table I (circles: medians of k I (Equation 3) over all subjects for each pair of adjacent stimuli (MDk , ) ; squares: intraindividual5Ds). The asterisks correspond to those pairs of adjacent stimuli for which k I was equal to zero for more than 50010 of the subjects.The straight horizontal line in each panel represents the median of Md k , over all pairs of adjacent stimuli.

cases, provided that lIln + 1 - lIln was different fromzero, which was practically always the case. (Notethat the subjects were only allowed to use integers inthe category rating task.) The points in Figure 1 thatare marked with an asterisk correspond to those casesin which k I was equal to zero for more than 50010of the subjects. If the points marked with an asteriskare excluded, there is a barely significant effect ofpairs of adjacent stimuli on k I in Experiment Soft 2

and no significant effect in the other two experiments(cf. Table 2).

The significant effect in Experiment Line 1 is moredifficult to account for. There were relatively fewcases of k I = 0 in this experiment, and the functionwhich relates Mdk' to the category scale is mostirregular. This function bears some resemblanceto the corresponding function in Experiment Line 2,in which an almost significant effect was obtained.

MAGNITUDE AND CATEGORY SCALES 223

Category scale

Figure 2. Theoretical and empirical measures of the uncertaintyin category ratings as a function of arithmetic means of categoryscale values over all experiments on loudness or softness exceptExperiment EO 2 (circles: averages of Md k · ; squares: averagesof intraindividual 50s).

0.3

0.1

0.5

in Figure 2 were obtained by taking medians acrossdifferent experiments of Mdk· and intraindividualSOs of category ratings, respectively, and smoothing­these medians over three successive values (movingaverages). In panel A of Figure 2, the medians wereobtained by rank ordering the category scale valuesin each experiment and then taking the median overall pairs of adjacent stimuli with the same ranks.(The number of stimuli was 10 in all experiments.)The medians corresponding to panel B were obtainedby rank-ordering the arithmetic means of the twocategory scale values from each pair of adjacentstimuli across all eight experiments and then takingthe median over the eight values corresponding toranks 1-8, 9-16, etc .. to ranks 65-72. (The categoryscale values in Experiments ERA and ED 1 werefirst reduced by 1 in order to make the category scalesin these experiments cover the same range of numbersas the category scales in the other experiments.)

Both panels of Figure 2 show the same trends:The averages of Mdk ' are very close to constancybut with somewhat toO low values at the extremesof the category scale. The averages of the intra­individual SOs are greater than the averages of Mdk'and decrease from the middle of the stimulus rangetoward both ends.

Individual ExperimentsFigure 3 shows, for each subject in the individual

experiments, k ' as a function of the category scale(cf. the circles in Figure 3). The horizontal straightline in each panel represents the median of the k 'values over all pairs of adjacent stimuli. As in thegroup experiments, there is a considerable variabilityin the estimates of k around the horizontal line.However, in contrast to the group experiments, thereis no common trend in the deviations from the hori­zontal line. It can also be seen from Figure 3 thatthe k ' values vary within approximately the sameregion as the SOs of the category ratings (cf. thesquares in Figure 3).

The estimates of the sampling error in k ' (sO aswell as the empirical SDs of the category scale (SK)

are introduced in Figure 3 as vertical lines around

Experiment df x2

Loud I 8 10.31Soft I 8 9.54Loud 2 8 22.90**Loud 2t 7 13.88Soft 2 8 23.12**Soft 2t 7 14.50*Loud 3 8 7.80Soft 3 8 15.50Smell I 5 4.16Smell 2 5 5.75Smell 3 5 3.90Line I 13 25.72*Line 2 13 22.51ERA 8 8.82ED I 8 18.32*ED It 7 14.06ED 2 8 26.81 **

Table 2Friedman's Two-Way Analysis of Variance of k ' (Equation 3)

as a Function of Pair of Adjacent Stimuli in the GroupExperiments Listed in Table I

----

[Pairs of stimuli for which k' was equal to zero for more than50% of the subjects excluded from the test. *p < .05 **p <01

It can be noted that the Weber function of themagnitude scales in the two experiments on lengthof line was linear only for the lower part of thestimulus range, and that the tIJ intercept was closeto zero. Similar Weber functions were also obtainedin Experiments ED I and ED 2, but the relationbetween the category and magnitude scale was muchmore curvilinear in these two experiments than in theexperiments on length of line.

Estimates of k were also computed from groupdata values of the variables in Equation 3 (geometricmeans of the arithmetic means of the magnitudeestimates from each subject, arithmetic means of thecategory ratings over all subjects, and intra­individual SOs computed as described in Eisler,1962). Apart from being somewhat greater thanMdk" there were no clearly systematic deviationsbetween these estimates of k (not shown here) andthe medians of the individual k ' values.

The squares in Figure 1 correspond to intra­individual SOs of the category ratings as a functionof the category scale. These SOs were computed in away that was analogous to the computation of k 'and Mdj », For each pair of adjacent stimuli nandn + 1, the quantity (SK + SK

n~ )/2, where K denotes

category ratings and s standard deviation, wascomputed for each subject, and the median ofthis quantity was taken over all subjects for eachpair of adjacent stimuli. In all experiments, the intra­individual SOs tend to be greater than Mdk" Thistendency is most conspicuous in the smellexperiments.

In Figure 2, an attempt was made to summarizethe results from all experiments on loudness or soft­ness, except Experiment ED 2 (the only experimentwith a l S-point category scale). The ordinate values

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3.T

heoreticaland

empirical

measures

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uncertaintyin

categoryratings

asa

functionof

thecorresponding

categoryscale

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individualexperim

entslisted

inT

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errorof

thesevalues.

each value of k I and SD, respectively [cf. also thewithin variances (s~) in Table 3]. It can be seen thatsk' varies considerably from subject to subject. Insome cases, there is also a considerable intraindividualvariation in sk' (e.g., subjects D.M., M.W., andD.F.).'

Table 3 shows that there was a significant effect(p < .05) of pairs of adjacent stimuli on k I for 8 ofthe 12 subjects. It should be noted, however, that3 of these 8 subjects used a 15-point category scale,which is probably not a pure discrimination scale(cf. Eisler & Montgomery, 1974). No general trendscan be seen in the deviations from constancy in k ' .For individual subjects, the following trends can benoted: decreasing toward the upper end of the cate­gory scale (subject T.J.), increasing toward the upperend of the category scale (subject A.N.), Uvshape(subject D.F. and possibly subject H.S.), and invertedV-shape (subjects M. W. and B.E.).

Figure 3 demonstrates that these individual trendsin the k I values for some subjects seem to beparalleled by corresponding trends in the empiricalSDs of the category scale (cf. subjects D.M., M.W.,E.E., T.J., and B.E.). Table 3 gives the productmoment correlations between k ' and (SKn + I + sKn)l2.As can be seen, these correlations are positive for10 out of the 12 subjects and significant (p < .05)for 4 subjects. It merits attention, however, that forsome subjects there seem to be clearly systematicdeviations between k I and the empirical SDs of thecategory scale (cf. particularly subjects H.S. andD.F.).

DISCUSSION

In most group experiments, the Fechnerian caseof the GPDE could not be rejected statistically.However, there may have been fairly great samplingerrors in the theoretical uncertainties (Mdj .) thatwere estimated by means of the GPDE, since, whenthe Mdk' values did not deviate significantly fromconstancy, there was generally a considerable vari­ability in these values across different pairs of ad­jacent stimuli. On the other hand, there were fairlysmall systematic deviations from constancy in theMdk, values across different experiments (cf. Fig­ure 2). From this, it might be tentatively concludedthat the Fechnerian case of the GPDE is at leastapproximately valid for group data from categoryratings and magnitude estimation experiments, and,moreover, that a category scale may be close to adiscrimination scale when the scale values are col­lapsed over several subjects.

For individual subjects, there were significantidiosyncratic deviations from constancy in theestimates of k. That is, the Fechnerian model wasdisconfirmed for most subjects in the individualexperiments. Thus, the present test of the GPDE

MAGNITUDE AND CATEGORY SCALES 225

Table 3Statistical Analysis of k' (Equation 3) as a Function of Pair ofAdjacent Stimuli in the Individual Experiments Listed in Table 1

Subject s' Fw

KJ. .075 .65 .710*D.M. .040 2.12* .718*M.W. 10.104 .01 .506BJ. .039 1.10 -.156H.S. .006 5.96*** -.007E.E. .002 2.28* .766*M.L. .008 1.97* .3060.1'". .025 2.25* .177TJ. .063 4.25 *** .687*A.N. .050 2.79** .513B.E. .049 4.37*** .577LS. .071 .89 .274

Note-df for all F tests = 8,441,- df for all r values = 8.*p < .05 **p < .01 ***p < .001

does not generally support the assumption thatcategory scales for individual subjects are pure dis­crimination scales in the sense of having a constantgenotypic, i.e., unbiased, Weber function whichcan be predicted by means of the GPDE.

It might be speculated that the individual trendsin the k I values reflects corresponding trends in thegenotypic Weber functions of the category scale.This would be the case if the GPDE is valid forgenotypic Weber functions and if the empiricalWeber function of the magnitude scale coincideswith the genotypic Weber function, i.e., is free frombias. In this connection, it is interesting to note thatthe k I values in the individual experiments tended tobe positively correlated with the empirical SDs ofthe category ratings. Thus, the GPDE in its generalform (Equation 2) accounted for part of the variancein the empirical SDs of the category ratings in theindividual experiments. If the k I values reflect theform of the genotypic Weber function, then thisresult suggests that the genotypic Weber function ofcategory scales for individual subjects may covarywith the corresponding phonotypic, i.e., empiricalWeber function. It should be noted, however, thatfor group data as well as for some individual datathere seemed to be clearly systematic deviationsbetween phenotypic Weber functions and Weberfunctions that were predicted by the GPDE (cf.Figure 2 and Figure 3, subjects H.S. and D.F.).

It can be concluded that neither the Fechnerianmodel (Equation 1) nor the GPDE in general (Equa­tion 2) may be strictly valid for individual categoryrating and magnitude estimation data. It should beborne in mind, however, that predictions of categoryscale values for individual subjects from theFechnerian model or from the GPDE in generalhave been found to be close to empirical categoryscale values (Eisler & Montgomery, 1974). Thus,although the GPDE may not be strictly valid forindividual category and magnitude scales, it may still

226 MONTGOMERY

be a powerful instrument for predicting the relationbetween the two types of scales for individualsubjects.

REFERENCE NOTE

1. Eisler, H., Holm, S., & Montgomery, H. Is the generalpsychophysical differential equation an approximation? Reportsfrom the Psychological Laboratories, University of Stockholm,1973, No. 386.

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HALD, A. Statistical theory with engineering applications. NewYork: Wiley. 1952.

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NOTES

I. The stimulation of test of the GPDE indicated that thistest is relatively insensitive to heterogeneous variances. The rangeof variation for the population values of s~, (Equation 12)in the three simulated cases was 0.007-0.142, 0.007-0.074, and0.017-0.312, respectively.

(Received for publication July 13, 1976;revision accepted November 14,1976.)