Perception & Psychophysics1979, Vol. 26 (4), 255-264

Nested models of relative judgment:Applications to a similarity averaging model

ROBERT F. FAGOTUniversity of'Oregon, Eugene, Oregon 97403

A theoretical analysis of the Eisler and Ekman (1959) model of similarity judgments forunidimensional continua is presented, based on a general model of relative judgment. Thisgeneral model assumes that judgments are mediated by perceived relations of pairs of stimuli,that there exists a transformation of the judgmental response that is a function of the sensoryratio of the two stimuli, and that response bias-operates in a multiplicative manner. Threestructural conditions are presented, each imposing constraints on the structure of observedjudgments. The structural conditions define three nested models of relative judgment, withthe second a weakened version of the first, and the third a weakened version of the second.The special virtue of the general model is that it is applicable to a variety of judgmentaltasks (e.g., ratio estimation, similarity, pair comparison), the key being derivation of theresponse trans/ormation conforming to the structural conditions. The structural conditions thusconstitute necessary conditions for several different judgmental models. The theory was firstapplied with success to ratio estimation judgments (Fagot, 1978), and this paper applies thegeneral model to the Eisler and Ekman similarity "averaging" model. Empirical tests werecarried out on published data for pitch, darkness, visual area, and heaviness judgments.Although the strong form of the model presented by Eisler and Ekman was rejected, weakenedversions were generally supported by the data. These results were similar to those obtainedfor ratio estimation (Fagot, 1978), and are interpreted to be very promising for the generalmodel of relative judgment.

Eisler and Ekman (1959) have proposed an appeal­ingly simple model of similarity judgments for uni­dimensional continua, namely,

where Sab is the judged similarity between the twopercepts and tp is a scaling function. Similarity isdefined such that 0 ~ Sab ~ 1, with Sab = 1 for"perfect" similarity or identity, and Sab = 0 forperfect dissimilarity. Equation 1 thus assumes thatthe judged similarity of two stimuli is the ratio of thesubjective magnitude of the smaller stimulus to thearithmetic mean of the magnitudes of the two stimuli.

It should be noted that although the Sab are inter­preted as similarity judgments, Equation 1 is not ageneral similarity model in the sense used by mostinvestigators who try to account for similarity judg­ments of stimuli varying on several dimensions.Equation 1 is applicable only to those situations inwhich stimuli vary with respect to a single attribute,such as pitch, brightness, heaviness, etc.

Various studies have made different interpretationsof tp. To distinguish these, we refer to Equation 1

Work on this paper was carried out, in part, during theauthor'S tenure as a Fellow at the Netherlands Institute forAdvanced Study, Wassenaar, Holland.

as a similarity scaling model only when tp is inter­preted as a similarity scale such that scale valuesmay be inferred from the observed similarity judg­ments Sab.Equation 1, as a scaling model, will bereferred to as a similarity averaging model (SAM).On the other hand, if tp represents a scale derivedfrom some other scaling method, e.g., ratio estima­tion, Equation 1 will be referred to as a similarityfunction.

A choice between these two interpretations cannotbe made easilyon the basis of previous empirical studiesEkman, Engen, Kunnapas, and Lindman (1964) andEkman and Sjoberg (1965) accepted the relationgiven by Equation 1 as established, based on empiri­cal studies by Eisler (1960), Eisler and Ekman (1959),and Ekman, Goude, and Waern (1961). However,the latter two studies treated Equation 1 strictly as asimilarity function, and although Eisler and Ekmanderived a similarity scale from Equation I, andreported a good fit of the similarity scale to a scalederived from fractionation, there was no attempt totest Equation 1 as a scaling model. Tests of Equa­tion 1 treated as a similarity function may have nodirect bearing on the adequacy of this model equa­tion as a scaling model. This will be clearer after thetheoretical analysis presented in the next section, butthe point follows from the fact that the two inter­pretations entail possibly different scaling functions.In the interpretation of Equation 1 as a scaling

Copyright 1979 PsychonomicSociety, Inc. 255 0031-5117/79/100255-10$01.25/0

256 FAGOT

and

(8)

(2)

(3)

(4)

(b = 2,3, ... , n-l). (9)

Rab = {Jtpa/tpb'

Rab = {Jbtpa/tpb'

ratios tpa/tpb and possibly bias parameters, i.e.,Rab = f(tpa/tpb, (Jb). Three specific model equationswere considered:

Ratio constancy (C3):

It can be shown that Equation 2-the C model­entails Cl, C2, and C3 (but C2 and C3 are entailedby Cl); Equation 3-the CB model-entails C2 andC3; and Equation 4-the RB model-entails C3. Afurther condition, monotonicity (see Fagot, 1978),is entailed by each of the three models.

The bias parameters in Equations 3 and 4 areexpressed entirely in terms of observables:

Ratio consistency (Cl): Rae = RabRbe (5)

Product constancy (C2): RabRbd = RaeRed (6)

where {J and {Jb may be interpreted as bias parametersand tp is the scaling function.' Equations 2, 3, and 4are referred to as the classical (C), constant bias(CB), and relative bias (RB) models, respectively.The bias parameter could account for variations dueto experimental arrangements, such as position ortime-order effect.

Three structural conditions, each imposing con­straints on the structure of the observed ratio esti­mates, were presented:

Not all three structural conditions need to be satis­fied to justify scale construction. Given that mono­tonicity is satisfied: if Cl is satisfied, then Equation 2can be used to construct a ratio scale. If Cl is vio­lated but C2 and C3 are satisfied, then Equation 3can be used to construct a ratio scale with {J estimatedfrom Equation 8. Finally, if Cl and C2 are violatedbut C3 is satisfied, then Equation 4 can be used toconstruct a ratio scale with {Jb estimated fromEquation 9.

Thus, the structural conditions define nested modelsof relative judgment. Starting with the weakestmodel (RB) (i.e., weakest in the sense that it placesweakest constraints on the structure of the data),successive strengthening is obtained by adding on

model, the scale tp is derived from similarity judg­ments Sab' whereas in its interpretation as a similarityfunction, the scale tp is derived from some otherscaling method, for example, ratio estimation. Thus,if Equation 1 is rejected (or accepted) with tp a ratioestimation scale, this result bears on the issue of therelationship betwen similarity judgments and ratioestimation judgments, but has no necessary bearingon whether a similarity scale tp exists, satisfyingEquation 1 (the scaling model interpretation).Necessary conditions for the existence of a similarityscale tp satisfying Equation 1 are presented in thenext section, and are expressed entirely in terms ofobserved similarity judgments Sab.

Sjoberg (1966) carried out the first test ofEquation 1 interpreted as a scaling model, based ondata from judgments of pitch, darkness, visual area,and heaviness, presented in Eisler (1960), Eisler andEkman (1959), and Ekman, Goude, and Waern(1961). Sjoberg concluded that the data did not satisfyEquation 1. The theoretical basis of Sjoberg's testsmust await some theory to be presented later. How­ever, it should be pointed out at this stage that, inspite of Sjoberg's conclusions, the data cited abovewill be interpreted in the present paper as generallysupportive of Equation I as a fundamental similarityscaling model provided response biases are admitted(Fagot, 1978).

As suggested above, the difficulty in evaluatingEquation 1 based on prior research has been thefailure in most cases to distinguish between a simi­larity function and a similarity scaling model, and thelack of a theoretical analysis of Equation 1 as ascaling model. The key question to pose in inter­preting this equation as a scaling model is: Whatrelations among observed similarity judgments areentailed by Equation 1 if tp is a similarity scalingfunction? The answer would provide a set ofnecessary conditions on observed Sab if a similarityratio scale tp exists satisfying Equation 1.

Fagot (1978) posed a similar question for alterna­tive ratio estimation models: i.e., what relations musthold among observed ratio estimations if a givenmodel is assumed. He presented three structural con­ditions, each placing constraints on the structure ofratio estimation data. The main theoretical contribu- ,tion of the present paper is to show how the Fagot(1978) theory of relative judgment can be extendedto the SAM. The theory will then be tested againstthe published data cited above.

THEORY

The ratio estimation theory (Fagot, 1978) appliesto reported ratio judgments of stimulus a to stimulus b(relative to a defined attribute) denoted by Rab, andassumes that the response Rab is a function of sensory

NESTED MODELS OF RELATIVE JUDGMENT 257

and

and that bias operates in a multiplicative manner(Equations 2, 3, and 4). Hence, applying theseassumptions to judgments of similarity and the SAMgiven by Equatiion 11, we get

where a and ab permit different magnitude biases forratio estimation and similarity judgments. Equa­tions 12 and 13 are equivalent to Equation 1 com­plicated by the multiplicative operation of bias onlpa; i.e., the similarity judgment of stimulus a relativeto stimulus b converts lpa in Equation 1 to alpaentailing Equation 12, and to %lpa entailing Equa­tion 13.

Equation 12 is of the same mathematical form asEquation 3, and hence Equation 12 entails that C2and C3 (but not Cl) hold for similarity transformsS~b. Equation 12 defines the constant bias similarityaveraging model (SAM-CB).

Equation 13 is of the same mathematical form asEquation 4, and hence Equation 13 entails that C3(but not Cl and C2) holds for similarity transformsS~b. Equation 13 defines the relative bias similarityaveraging model (SAM-RB).

Thus it is seen that Cl, C2, and C3 provide directlytestable consequences of the SAMs expressed entirelyin terms of observed similarity transforms S~b.

The similaritybias parameters a and % are expressedentirely in terms of observed similarity transformsS~b as given by Equations 8 and 9, respectively, withS~b substituted for Rab.

The theory of relative judgment described abovecan be stated in more general terms, as follows,for some judgment Jab:

(13)

(12)

conditions one at a time, such that a weaker modelis nested within a stronger model (RB is nested withinCB, and CB within C).

A common procedure for testing scaling modelshas been to estimate scale values, usually by a leastsquares procedure, and test goodness-of-fit ofobserved data (e.g., responses Rab) to values repro­duced using the estimated scale values. The disad­vantage of this approach is the very large numberof parameters used (scale values), and the obscuringof systematic trends in the data. By contrast, thestructural conditions are parameter-free conditionsthat are necessary for the existence of the scalingfunction lp satisfying the model equation (Equation 2,3, or 4). There is the added benefit that violationof a structural condition uncovers systematic errors.

The value of the CB and RB models might bequestioned on the grounds that the addition of para­meters simply results in more mathematical freedomto account for data. However, comparative tests ofthe models (including the C model) would be carriedout on the structural conditions, which, as notedabove, are parameter-free. Hence, the bias parametersin the CB and RB models do not gain mathematicalfreedom to account for data in the usual sense, sincethese parameters are not used in goodness-of-fit tests.Fagot (1978) showed that Cl was generally violatedfor ratio estimation data, a finding justifying aban­donment of the classical theory of ratio estimation.However, the CB and RB models salvage the classi­cal theory that the response is a function of thesensory ratio by the simple revision of furtherassuming that response bias operates in a multiplica­tive fashion producing two further weakened versionsof the classical model.

The theory of relative judgment summarized abovewill now be applied to the SAM (Equation 1) throughintroduction of bias parameters.

The SAM given by Equation 1 may be written inthe form

and defining S~b = Sab/(2 - Sab)' we have

Equation 11 is of the same mathematical formas Equation 2; and hence, since Equation 2 entailsthat Cl, C2, and C3 hold for Rab' Equation 11entails that Cl, C2, and C3 hold for similaritytransforms S~b (i.e., substitute S~b for Rab in Equa­tions 5, 6, and 7). Equation 11 will be referred to asthe classical similarity averaging model (SAM-C).

The theory of relative judgment assumes thatstimulus a is judged relative to stimulus b, that theresponse is a function of the sensory ratio lpa/%,

(14)

(15)

where~ is the bias parameter. The function g deter­mines the model: g(~) = 1 entails the C model,g(~) = e entails the CB model, and g(~) =~ entails the RB model. The function f determinesthe response transformation conforming to the struc­tural conditions. To apply the theory to a givenscaling model, it is only necessary to derive theresponse transformation f satisfying Equation 14.For example, for the SAM, f(Sab) = Sab/2 - Sab= S~b, leading to Equations 11, 12, and 13 as specialcases of Equation 14applicable to the SAM.

As another example, consider the probabilitymodel

(10)

(11)

258 FAGOT

and his version of the SAM is

and hence Vb/Sb = {3b, the bias parameter for theratio estimation RB model; and Equation 17 entailsthat

response to be a direct reaction to the stimulus,i.e., that subjects' judgments are mediated by sensa­tions evoked by single stimuli. The view expressedin the relative judgment theory is that subjects judgepairs of stimuli, not single stimuli, a view consistentwith relation theory as formulated by Krantz (1972),based on a proposal by R.N. Shepard that subjects'judgments are mediated by perceived relations ofpairs of stimuli. Similarity is an obvious propertyof pairs of stimuli, and for ratio estimation, theperceived relation is interpreted to be sensation ratios.Unlike relation theory, the theory of relative judg­ment formulated in the present paper accounts forbiased judgments due to the interactive effect ofstimuli. A flaw in Krantz' elegant theory is his in­corporation of Cl as an axiom. For ratio estimation,Cl has been shown to fail for a variety of percep­tual attributes (Fagot, 1978).

Judgment-Judgment (J-J) RelationsCliff (1973) distinguishes three basically different

types of scales: magnitude scales, category scales,and discrimination scales. Magnitude scales arebased on "ratio methods," such as magnitude esti­mation, fractionation, ratio estimation, and magni­tude production. Much research has been carriedout on scale relations, including a very extensive studyby Stevens and Galanter (1957) on the relation be­tween ratio and category scales. Of interest in thepresent paper is the class of methods generating"magnitude scales" and in particular the relationbetween ratio estimation magnitude scales and simi­larity scales based on the SAMs, including the cor­responding J-J relations between Rab and Sab orS~b' The SAM scales have in common with magnitudescales a reliance on direct numerical estimation andthe production of ratio scales provided consistencytests are satisfied. Do SAM scales belong to the classof magnitude scales? If the answer is yes, then aSAM scale should be related to a ratio estimationscale by a similarity transformation, i.e., should beidentical except for possibly different units. Denotingthe similarity (SAM) scale by lIJs and the ratioestimation scale by 'tIr, the scale relation hypothesisis that lIJs = KlIJn where K is a positive constantcorresponding to change of unit.

The scale relation hypothesis will be expressed interms of a judgment-judgment (J-J) relation, i.e.,a functional relationship between ratio estimates Raband similarity transforms S~b' A J-J relation cor­responds to a psychosensory relation as defined byMarks (1974).

J-J relations are derived assuming that one of thebias models (C, CB, or RB) holds for both Raband S~b' Assuming lIJs = KlIJn the followingresults hold:

(17)

(16)

(18)

(19)

Related Scaling TheorySjoberg (1966) points out that scaling theories

generally assume strict perceptual invariance, i.e.,that only one perceptual magnitude corresponds toeach stimulus. This assumption is embodied in Equa­tions 2, 3, and 4 for ratio estimation and Equations 1,11, 12, and 13 for similarity judgments. Sjoberg(1966) proposes as an alternative a variable/standard(VS) model that assigns a different scale value to astimulus depending on its status as a comparison(variable) or standard stimulus. The Sjoberg modelfor ratio estimation is simply

where Va and V~ refer to the scale values of stimulusa as a variable stimulus, and Sb and S6 refer tothe scale values for stimulus b as a standard stimulus.

It can be shown that the Sjoberg VS model andthe RB model are data equivalent, i.e., they are con­sistent with exactly the same finite sets of basic"observation" statements (Adams, Fagot, & Robinson,1970). Specifically, the VS model entails C3 (but notCI or C2). Equation 16entails that

(Bradley & Terry, 1952; Luce, 1959), where Pab isthe probability that stimulus a is chosen to stimulus band V is a scaling function. The response transfor­mation is P~b = Pab/1- Pab, and g(9t» = 1, i.e.,P~b = Va/Vb' Hence, a necessary condition for theexistence of a scaling function V satisfying Equation 15is that the response transformation P~b conformsto C1, which is a well-known testable consequenceof Equation 15. Furthermore, f(9t» = a,~ providesextensions of this theory that require satisfaction ofC2, C3 for P~b, and sets of data may exist for whichCl is violated and C3 or C2 and C3 are satisfied.

and hence V6/S6 = 111>, the bias parameter for theSAM-RB.

Furthermore, if the C model is correct, thenV6/S6 = 1 for all b; and if the CB model is cor­rect, then V6/S6 = a for all b.

Classical psychophysical scaling models (e.g., ratioestimation and magnitude estimation) consider a

NESTED MODELS OF RELATIVE JUDGMENT 259

(1) If the C model holds for both Rab and S~b,

thenstitutions ab = a, [3b = [3; and for the C model,by Gl'b = [3b = 1.

(2) if the CB model holds for both Rab and S~b,

then

Tests of Structural Conditions for the SAMsMonotonicity and the three structural conditions

were tested on similarity transforms S~b by usingdata from published studies for four attributes­pitch, darkness, visual area, and heaviness. Experi­mental procedures for these studies are describedbriefly in Kunnapas and Kunnapas (1973), and refer­ences to the original studies are given in Table 1.

Monotonicity was perfectly satisfied for each ofthe four attributes. Ratio estimation judgments werealso made by subjects for three of the four attributes­darkness, visual area, and heaviness-with noviolations of monotonicity observed (Fagot, 1978).

Each structural condition was tested in the formY = X, where Y and X are the two sides of theequation for each condition (e.g., for Cl, Y =S~c and X = S~bSbc)' Each condition was analyzedin two ways: (1) agreement, in the sense of reliability,between Y and X as measured by the intraclasscorrelation coefficient (ICC) (Bartko, 1976); and(2) tests for the presence of systematic errors in theplot of Y as a function of X. Simple t tests forpaired differences, Yi - Xi, were carried out to testfor the presence of systematic errors. For furtherdetails on the analysis, see Fagot (1978),2

The ICCs and results of the tests for the presenceof systematic errors are presented in Table 1.

Figures 1-4 give plots of each structural conditionon similarity transforms for each of the four studies.In each plot, the dotted line is the identity line Y :;= X.

Table 1 shows that for all four attributes thereliability is appreciably lower for Cl, and thatreliabilities for C2 and C3 tend to be quite high.

In general, the tests for systematic errors in Table 1and Figures 1-4 are encouraging for C2 and C3 butfail to support Cl and hence the SAM-C. Inparticular: (1) For darkness, the TSE was significantfor all three structural conditions, but Figure 2 showsvery marked systematic errors for Cl , yet only slightsystematic error for C2 and C3 with a reasonablyclose fit to the identity lines. (2) The TSE rejectedCl in three cases, but not for visual area. But notethe marked systematic deviations for Cl in Figure 3,not detected by the TSE with only 5 degrees of

(21)

(20)

(22)

(24)

(23)

(3) if the RB model holds for both Rab and S~b,

then

Note that even if both types of judgment arebiased, the simple J-J relation given by Equation 20holds if the bias magnitudes are equal (a = [3 inEquation 21 or ab = [3b for all b in Equation 22).

Equations 20, 21, and 22 place strong constraintson the relation between similarity and ratio estimates.Note that in testing the fit of Equations 21 and22 there are no free parameters, since a, [3 in Equa­tion 21 and Gl'b, [3b in Equation 22 are independentlyestimated from their respective judgment experi­ments and operate as known constants.

Kunnapas and Kunnapas (1973, 1974) assumedthat similarity judgments were a power functionof empirical ratio estimations:

It is clear that Equation 23 postulates a J-J relation,not a scaling model, and hence is not a logicalcomparator to the SAMs.

Equation 23 does, however, suggest that the simi­larity and ratio estimation scales may not be equiva­lent, except for change of unit, as assumed inderiving Equations 20, 21, and 22. If we assume thatthe similarity scale tlJs is a power function of the ratioestimation scale tlJr, i.e., tlJs = +tIJ~, then the RBmodel entails that

where y is the only parameter to be estimated. Forthe CB model, Equation 24 is modified by the sub-

Table IIntraclass Correlation Coefficient (ICC) and Tests for Systematic Error (TSE)

ICC TSE

Attribute Reference Cl C2 C3 Cl C2 C3

Pitch Eisler & Ekman (1959) .854 .898 .924 t ,.. ,..,..Darkness Ekman, Goude, & Waem (1961) .856 .987 .988 tt t ,..,..Visual Area Ekman, Goude, & Waem (1961) .888 .979 .946 ,..

*,..

Heaviness Eisler (1960) .932 .986 .988 tt *,..

"Nonsignificant. **p <.05. t» <.01. tt» < .001.

260 FAGOT

PITCH

'.

CI

DARKNESS/

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L_----l._~L,--.L---"----"-----L----'---~

9

2

.3

6

5

.7

.4

8

u.0

(Cf)

In general, the tests of the structural conditionsare interpreted as providing a firm basis for rejectingthe SAM-C in the form proposed by Eisler andEkman (1959), but as giving moderately good sup­port for the bias forms of the SAM as given byEquations 12 and 13.

••

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CI

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6

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A

o 3'0(<fl

u, 0

(<fl

.",U

(<fl

"'S' «,oc Sbd

Figure 1. Comparative fit of structural conditions for pitchsimilarity transforms (Eisler &Ekman, 1959).

98.7.6.5.43.2

C2

C3

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Figure 2. Comparative fit of structural conditions for darknesssimilarity transforms (Ekman, Goude, & Waern, 1961).

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freedom. Figure 3 shows a good fit for C2 and C3.(3) The TSE for heaviness was highly significant forCl but not significant for C2 and C3. Figure 4 showsvery marked systematic errors for C 1 but good fitsand absence of systematic errors for C2 and C3.(4) Figures 1-4 show the same pattern: large system­atic errors for C1 with substantially improved fitsand reductions of systematic errors for C2 and C3.Except for pitch, systematic errors for Cl are in thesame direction (S~bSbc > S~c) and in the samedirection observed for ratio estimation for judgmentsof nine attributes, including darkness, visual area,and heaviness (Fagot, 1978). The reversal in the caseof pitch is perplexing.

NESTED MODELS OF RELATIVE JUDGMENT 261

HEAVINESS

.7.6

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.5

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.4

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S~c S'bdFigure 4. Comparative fit of structural conditions for heaviness

similarity transforms (Eisler, 1960).

Turning to the results for the regression modelY = bX, we see that the estimate of the slope (6)of the best-fitting line through the origin is estimatedby cr/{3 extremely well-the error is .002 and .003for darkness and visual area, respectively .

Plots of S~b as a function of Rab for darknessand visual area are given in Figures 5 and 6,respectively. The dotted line is the function S~b =

.5

.4

"0 .3,U«(J)

u,0

.6«(J)

.2

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§~c S'bdFigure 3. Comparative fit of structural conditions for visual

area similarity transforms (Ekman, Goude, &Waern, 1%1).

VISUAL AREA

Tests of J-J RelationsTests of the J-J relations were carried out for

darkness, visual area, and heaviness, but not forpitch, since for the latter the method of fractionationwas used, providing estimates of theoretical lIJa/%'not empirical ratio estimates Rab.

Table 2 presents results of the regression analysisof S~b on Rab' Note that for the regression modelY = b.X + bo the hypothesis Hs.b, = 1 was rejectedin all three cases; hence, the simple J-J relationentailed by the C model (Equation 20) must berejected.

The hypothesis Ho:bo = 0 was not rejected fordarkness and visual area; hence, for these two attri­butes, regression through the origin is tenable.

262 FAGOT

Table 2Regression of S~b (Y) on Rab (X)

Regression Model

Y=b, X+b o

Y=bX

Attribute ho h,Darkness .010 .954Visual Area -.018 .954Heaviness -.043 .897

"Nonsignificant. tp < .01. tt» < .001.

b, = I

tttttt

**t

.969

.921

.830

a/[3

.971

.924

.918

(a/(3) Rab entailed by the CB model (Equation 21).Note that the fit is, in each case, very good withoutany indication of systematic error.

These results support the J-J relation entailed bythe CB model for darkness and visual area. The fitsare particularly impressive considering the fact thatthe slope (a/(3) was determined in independent scalingexperiments and not estimated via the regressionmodel.

For a test of the J-J relation entailed by the RBmodel (Equation 22), plots (not presented) of S~b/

ab as a function of Rab/(3b were constructed. Thefits for darkness and visual area were somewhatbetter than given in Figures 5 and 6 for the CBmodel, but further analysis of the RB model is notnecessary since the stronger constraints placed on theJ-J relation by the CB model are supported.

Note that the results in Table 2 are not supportivefor heaviness: as already indicated, Hs.b, = 1 isrejected; Ho:bo = 0 is also rejected, and a/{3 deviatesappreciably from 6. The plot of S~b as a functionof Rab is presented in Figure 7. Note that all the

points lie below the dotted line Sf = (a/{3)R, ex­hibiting large systematic errors. The results forheaviness may be due to the untenability of theassumption that the similarity and ratio estimationscales are equivalent except for change of unit.Therefore, the power function hypothesis embodiedin Equation 24 was tested for the CB model in logform:

log(S~b/a) = y log(Rab/(3). (25)

Figure 8 shows a plot of log (S~b/a) as a functionof log (Rab/(3), where the dotted line is the identityline. The fit is very appreciably better than that ofFigure 7, giving some support to the hypothesis thatfor heaviness, unlike the cases of darkness and visualarea, the scales are related by a power function.

DISCUSSION

The authors of the studies listed in Table 1accepted Equation 1 as established, based on their

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o 4 5 6 7 8 9 10

ROb

Figure 5. Graph of the J-J relation for darkness (Ekman, Goude, Figure 6. Graph of the J-J relation -for visual area (Ekman,& Waern, 1961) entailed by the CD model (Equation 21). Goude, & Waern, 1961)entailed by the CD model (Equation 21).

NESTED MODELS OF RELATIVE JUDGMENT 263

10

Figure 7. Graph of the J-J relation for heaviness (Eisler, 1960)entailed by the CD model (Equation 21).

based in part on a failure to distinguish betweenEquation 1 interpreted as a similarity function and asa scaling model. In the following discussion, webegin with the evaluation of the SAMs as scalingmodels.

Among the early studies cited, only Sjoberg (1966)carried out an empirical test of Equation 1 interpretedas a scaling model. Sjoberg based his tests on the VSmodel, and deduced that if Equation 1 was correct,then V/S = 1; i.e., each stimulus has only onescale value (perceptual invariance). Using this test,Sjoberg rejected Equation 1 for all four sets of datalisted in Table 1. However, as shown in the sectionon Theory, V/S = 1 is a constraint only for SAM-C,and hence the Sjoberg analysis does not providea basis for rejecting the similarity bias models (SAM­CB and SAM-RB). Sjoberg's rejection of Equation 1is paralleled in the present paper by the rejectionof Cl, and both analyses agree that the SAM-Cis an inadequate similarity model. But the reasonableconformity of the four sets of data to C2 and C3illustrated in Figures 1-4 support the tenability of thesimilarity bias models.

Similar results were obtained for ratio estimation,particularly the clear rejection of Cl (Fagot, 1978).The results for ratio estimates and similarityjudgmentssupport the general judgment model given by Equa­tion 14, but indicate clearly that g(6t,) "* 1.

Four other studies tested Equation 1 as a simi­larity function for ratio estimation and similarityjudgments. These included Eisler (1960), for heaviness;Ekman, Goude, and Waern (1961), for darknessand visual area; Sjoberg (1971), for circular area andheaviness; and Franzen, Nordmark, & Sjoberg(Note 1) for pitch. Data from the latter two studiesare not available to the author, but the analyses forboth studies led to the rejection of Equation 1 asa similarity function with tp interpreted as a ratioestimation scale. Furthermore, the darkness, visualarea, and heaviness data (Table 1) are clearly incon­sistent with the simple J-J relation given by Equa­tion 20. Of course, Equation 1 cannot be unequivo­cally rejected as a similarity function since it is pos­sible that the relation may hold if tp is a scale derivedfrom some method other than ratio estimation.

Going beyond the analyses made in the studiescited above, we see that Figures 5 and 6 showreasonably good support for the CB-model form ofthe J-J relation (Equation 21) for darkness and visualarea. Hence, for these two attributes, it appears thata similarity scale generated by the CB model is amagnitude scale. However, the clear rejection of Cland Equation 20 shows that neither the ratio estima­tion scale nor the similarity scale is acceptable ifresponse bias is not taken into account.

Figure 8 gives good support for the CB form of theJ-J relation for heaviness if it is assumed that thesimilarity and ratio estimation scales are related by

1098.75 6

Rob

43.2.1

7

a

o

01o

-I

-4

e":, -2-0(f)

-3

HEAVINESS

data for pitch, darkness, visual area, and heaviness.Yet Sjoberg (1966) rejected Equation 1 based on hisanalysis of these data; Sjoberg (1971) rejectedEquation 1 for new data for judgments of circulararea and heaviness; and Franzen, Nordmark, andSjoberg (Note 1) also rejected Equation 1 based onpitch judgments. These inconsistent conclusions are

9

8

Figure 8. Graph of the log-log form of the J-J relation forheaviness (Eisler, 1960) entailed by the CD model under theassumption that the similarity and ratio estimation scales arerelated by a power function (Equation 25).

264 FAGOT

a power function. However, Figure 7 indicatesconclusively that the ratio estimation and similarityscales do not belong to the same family of magnitudescales (If/s /- Ku»). This negative result for heavinessis perplexing in view of the positive results for dark­ness and visual area.

The fits in Figures 5, 6, and 8 are more impres­sive taking into account that the bias parametersare not estimated via the regression of S~b on Rab'but estimated independently in different scalingexperiments, and operate as known constants inEquations 21 and 25. The confirmation of the J-Jrelations entailed by the CB model gives further sup­port to the general judgment model (Equation 14).

The classical psychophysical scaling view considersa response to be a direct reaction to the stimulus.Data from studies reviewed in this paper call for arejection of this classical view, at least as appliedto ratio estimation and the SAM. The classical posi­tion is reflected in Equation 2 for ratio estimationand in Equations 1 and 11 for the SAM, and bothequations are generally inconsistent with data. Theview expressed in this paper is that judgments aremediated by perceived relations of pairs of stimuli(or triples in the case of interval judgments), but thatjudgments may be biased due to the interactive effectof stimuli, including order effects of various kinds.Equations 2 and 11 are consistent with the relativejudgment view only in the absence of response bias.Results with ratio estimation (Fagot, 1978) and uni­dimensional similarity judgments as reported in thispaper are promising for the use of the CB andRB models in the construction of ratio scales ofsensation in the presence of response bias. The generalmodel given by Equation 14 shows that the theorycan be extended to other judgmental tasks.

REFERENCE NOTE

I. Franzen, 0., Nordmark, J., & Sjoberg, L. A study of pitch.Goteborg Psychological Reports, University of Goteborg, Sweden,Vol. 2, No. 12, 1972.

REFERENCES

ADAMS, E. W., FAGOT, R. F., & ROBINSON, R. E. On theempirical status of axioms in theories of fundamental measure­ment. Journal ofMathematical Psychology, 1970,7,379-409.

BARTKO, J. J. On various intraclass correlation reliability coef­ficients. Psychological Bulletin, 1976, 83, 762-765.

BRADLEY, R. A., & TERRY, M. E. Rank analysis of incompleteblock designs. I. The method of paired comparisons. Biometrika,1952,39,324-345.

CLIFF, N. Scaling. Annual Review of Psychology, 1973, 24,473-506.

EISLER, H. Similarity in the continuum of heaviness with somemethodological and theoretical considerations. ScandinavianJournal ofPsychology; 1960, 1,69-81.

EISLER, H., & EKMAN, G. A mechanism of subjective similarity.Acta Psychologica, 1959, 16, 1-10.

EKMAN, G., ENGEN, T., KUNNAPAS, T., LINDMAN, R. A quan­titative principle of qualitative similarity. Journal of Experi­mental Psychology, 1964,68,530-536.

EKMAN, G., GOUDE, G., & WAERN, Y. Subjective similarity intwo perceptual continua. Journal of Experimental Psychology,1961,61,222-227.

EKMAN, G., & SJOBERG, L. Scaling. Annual Review of Psychol­ogy, 1965, 16,451-474.

FAGOT, R. F. A theory of relative judgment. Perception &Psychophysics, 1978,24,243-252.

KRANTZ, D. H. A theory of magnitude estimation and cross­modality matching. Journal of Mathematical Psychology,1972, 9, 168-199.

KUNNAPAS, T., & KUNNAPAS, U. On the relation between simi­larity and ratio estimates. Psychologische Forschung, 1973,36, 257-265.

KUNNAPAS, T., & KUNNAPAS, U. On the mechanism of sub­jective similarity for unidimensional continua. American Jour­nal ofPsychology, 1974,87,215-222.

Lues. R. D./ndividual choice behavior. New York: Wiley, 1959.MARKS, 1. E. Sensory processes, the new psychophysics.

New York: Academic Press, 1974.SJOIJER(i, L. Unidimensional similarity revisited. Scandinavian

Journal ofPsychology; 1966, 7,115-120.SJOBERG, L. Three models for the analysis of subjective ratios.

Scandinavian Journal ofPsychology, 1971, 12,217-240.STEVENS, S. S., & GALANTER, E. H. Ratio scales and category

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NOTES

I. For the sake of simplicity of notation, 1ji is used here todenote the scaling function for both the SAMs and the ratioestimation models. However, this is not intended to imply thatthe scales are necessarily identical, and, in consideration ofjudgment-judgment relations below, where scale relations are anissue, the scale functions are distinguished.

2. To insure independence of data points, a method was usedthat deleted some stimulus triples in the test of Cl, and somestimulus tetrads in the test of C2. See Fagot (1978, footnote 2)for further explanation.

(Received for publication May 14, 1979;revision accepted August 28,1979.)