Empirical Tests of Scale Type Individual Ratings

265

Empirical Tests of Scale Typefor Individual RatingsRainer Westermann

University of Göttingen

This article describes eight studies that tested em-pirically the hypothesis that rating procedures lead tointerval-scale measurements for each single subject. Inorder to enhance the probability of obtaining intervalscales, subjects made numerical ratings and were de-liberately instructed to choose their responses so thatthe algebraic differences between numbers representedthe subjective differences between the correspondingobjects with respect to the attribute under study. Thisapproach is based on axiomatic measurement theory.It is exemplified by a study from clinical psychologi-cal research pertaining to the subjective fear arousedby each of 160 objects or situations. Any subject’sratings are regarded as interval-scale measurements ofhis or her individual degree of fear if the testable ax-ioms of a finite, equally-spaced difference structureare satisfied empirically. These axioms pertain to ordi-nal judgments on differences, and they are tested em-pirically by deriving statistical hypotheses and using arefined significance-test method as an error theory.For the eight studies criteria were chosen primarily toavoid accepting false interval-scale hypotheses at theexpense of relative high risks for false rejections.Nevertheless, empirical data allow acceptance of thehypothesis for 54 of the 114 subjects. As a conse-quence, for at least half of the subjects, this ratingprocedure seems to result in interval scales.

Both the use of rating scales and their (implicit)interpretation as interval-scale measurements aretypical for a large part of psychological research

in different areas. Dawes (1972, pp. 95-96) andSurber (1984) gave some illuminating examplesfrom social and developmental psychology, re-

spectively.There are many different types of rating scales

ranging from small sets of verbally labeled re-sponses (e.g., agree, undecided, disagree) to nu-merical or graphical ratings. It is a common featureof all rating procedures, however, that they do notinclude checks for response consistency. For thatreason, ratings are not representational measure-ments, but only index measurements in the sensegiven by Dawes (1972).

It is important to note that the same result holdsfor most magnitude or ’ ’ ratio ’ ’ scalings in the senseof Stevens (1960). These methods are frequentlyused not only in psychophysics, but also in socialsciences (Lodge & Tursky, 1982). Usually, sub-jects are instructed to assign a number to each ob-ject so that subjective ratios are represented. Gen-erally, no consistency checks can be made in thesecases (cf. Dawes, 1972).

According to Dawes (1972), index measure-ments can be evaluated only in terms of their use-fulness, for example, in terms of predictive valid-ity ; and he stressed that there is no sense in askingwhat scale type is attained by a particular indexmeasurement technique. It seems obvious that thisposition may be detrimental for both measurementtheory and substantial research. When rating andmagnitude scales are excluded from any measure-

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266

ment theoretical analysis, the common (implicit)interpretation of rating scale values as interval scalescan be neither justified nor criticized from a meth-odological point of view. In other words, therewould be no criteria for differentiating between asound and a nonsensical statistical and numerical

interpretation of ratings or magnitude estimations.In addition, this exclusion certainly lessens boththe impact and importance of measurement theoryfor psychology as a whole, and it supports the

tendency to consider measurement theory a some-what esoteric discipline.

However, Dawes’ position is not the only wayto attack the problem. On the contrary, it is indeedpossible to analyze ratings and other index mea-surements from a measurement theoretical point ofview. To be concrete, a method is presented hereto test the hypothesis that any subject’s ratings ordirect estimations are of interval-scale level. This

method has been applied in eight studies, the resultsof which are presented below. In comparison withother approaches (cf. Anderson, 1982; Birnbaum,1982; Orth & ~7e~e~er, 1983), this testing pro-cedure tries to minimize the number of necessary

assumptions and relies exclusively on qualitativeand relatively simple judgments.

Method

Each of the eight studies consisted of two parts.First, subjects rated a set of items according to aprespecified attribute. On the basis of these re-sponses, special sets of items were selected for eachsubject to test the interval-scale hypothesis. In thesecond part, each subject made judgments withrespect to these sets of items.

In Table i , the eight studies are characterized(1) by the number of subjects for which the interval-scale hypothesis was tested, (2) by the number ofitems rated by the subjects, and (3) by the attributeto be judged. In Study l , nine subjects gave directratings of their personal fear with respect to 160objects or situations. In Study 2, the likeablenessof personality trait adjectives was rated (Wester-m~~r~9 1984). Studies 3 and 4 were follow-up stud-ies of Study 1. In Study 3 the same kind of fear-

arousing were assessed as in Study 1 ~ whereasin Study 4 fear-arousing situations were describedin more detail in one or two sentences. In Study 5job characteristics of psychologists were judgedwith respect to their attractiveness for psychologystudents. In Study 6 different combinations ofcommunicators and sources were scaled with re-

spect to their trustworthiness. Studies 7 and 8 dealtwith belief strength concerning political opinionsand the respondent’s personal situation at the uni-versity, respectively.

For a more detailed description of the methodof testing hypotheses concerning the scale type ofratings and other index measurement, the first studyis taken as an example.

Behavior therapists use fear inventories both asclinical and research instruments (~rria~dell9 1980;Wolpe, 1973). The best known fear inventories arethe various forms of the Fear Survey Schedule (FSS)presented, for example, by ~l®lp~ and Lang (1964), 9Suinn (1969), Lawlis ( 1971 ) and Wolpe ( 1973) (fora review, see ~~ck ~ Schr6der, 1977, and Sei-denstucker & Weinberger, 1978). The FSS-M fromWolpe and Lang (1964), for example, consists of72 items such as &dquo;noise of vacuum cleaner,&dquo; 9 6 6 n~its99

9

and &dquo;1®c~l~~~~ foolish.&dquo; The items &dquo;refer to thingsand experiences that may cause fear or other un-pleasant feelings&dquo; (p. 28). Subjects are asked toplace each item in one of five categories labeledverbally from &dquo;not at all&dquo; to &dquo;very much.&dquo; Thus,this method results only in a rather coarse place-ment of fear-arousing stimuli in five ordered cat-egories. To obtain a finer ordering and quantifi-cation, some authors recommend using ratingmethods with a larger number of possible responses(Baade, Burck, Koebe, & Zummvenne, 1980; Os-wald, 1980). Then, the assumption is usually madethat each subject’s numerical responses can be in-terpreted as interval-scale measurements.From various Fear Survey Schedules, 160 items

were compiled. Each subject was asked to judgehis or her personal magnitude of fear with respectto each object or situation. Judgments were givenin terms of a rating scale ranging from 0 to 100.The two end points were defined verbally: 0 meansto be completely free of fear, and 100 means the



267

maximum fear the subject can imagine. In addition,subjects were asked to give their responses so thatequal subjective fear differences were representedby equal numerical differences. For the sake ofclarity, a ruler-like scale from 0 to 100 was pre-sented graphically.The resulting individual numerical assignments

(scale values), 9 S, were used to construct a set ofitems by means of which the interval-scale hy-pothesis could be tested for each subject (cf. We-stermann, 1982, 1983, 1984).To be concrete, the two critical necessary con-

ditions for interval-scale measurement were tested,which are formulated as axioms of finite, equally-spaced difference structures: Equal-Spacing andMonotonicity. A complete axiomatic definition ofa finite, equally-spaced difference structure was

given by Krantz, Luce, 9 ~uppes & Tversky (1971).The most important point for the present studies isthe fact that axioms defining difference structurespertain to an order relation -- between pairs ofstimuli. Thus, (a, b) = (c, J) may mean, for ex-ample, that (according to a person’ judgment) thedifference between objects a and b with respect toa certain attribute is not less than the respectivedifference between the two other objects c and d.

the Equal-Spacing ~~~®

For numerical assignments resulting from a rat-ing procedure, there is usually a large number ofquadruples of stimuli with the property that theabsolute difference between the scale values of the

first two stimuli equals the corresponding differ-

Table 1Characteristics of the 8 Empirical Studies

Testing the Interval-Scale Hypothesisfor Individual Ratings

=========================================================



268

ence between the other two stimuli:

with Sa, Sb, S~9 and S, denoting the numbers as-signed to stimuli a, b, c, and d9 respectively. If asubject’s numerical responses are indeed interval-scale level, the subjective difference between thefirst two stimuli (a, b) must be equal to the sub-jective difference between the other two stimuli(c, ~9 ~ which case the set of stimuli can be thoughtof as being part of a so-called standard sequence.A standard sequence is an ordered set of stimuli

satisfying the Equal-Spacing Axiom of finite,equally-spaced difference structures, that is, hav-ing equal (subjective) intervals between adjacentstimuli (Krantz et al., 1971.).As a consequence, testing whether the subjective

difference between a and b, on the one hand, and

c and d, on the other, are equal is a test of theEqual-Spacing Axiom. In Studies 1 and 2, 20 dif-ferent quadruples satisfying Equation I were ran-domly chosen for each subject in order to test theEqual-Spacing Axiom;’ the numbers and scale val-ues of the stimuli selected for Subject 1 are listedin Table 2. Note that the sum of the scale values

for the first pair is always not less than the corre-sponding sum for the second pair. This allows atest of the conflicting hypothesis that a numericaldifference of, say, 10 points means a smaller sub-

1Random samples of items and quadruples are appropriate foroverall tests of the axioms and the interval-scale hypothesis.For more specific tests, researchers may restrict the correspond-ing populations, or they may deliberately choose the sets andsubsets in which they are primarily interested.

Table 2Test of the Equal Spacing Axiom

for Subject 1 in Study 1. -.---------...---.-



269

jective difference in upper regions of the scale thanin lower regions. Actual presentation, however,was determined randomly for each subject.To avoid any problems with indifference or

equality judgments, subjects in Studies I and 2

were forced to judge which of the two differenceswas larger. These responses were coded as &dquo;I-~9’

or &dquo;L,&dquo; indicating whether the difference betweenthe pair with the higher or the lower scale valueswas judged larger. If the Equal-Spacing Axiomholds, both possible responses have the same prob-ability, that its, 9

This hypothesis can be tested by a binomial orsign test. However, with n = 20 replications, con-ventional significance levels, such as .05, lead tovery high Type 2 error probabilities. It is importantto note that any Type 2 error leads to falsely ac-cepting the axiom as valid and, eventually, to falselyinterpreting the numerical assignments as interval-scale measurements. Any Type 1 errors, on the otherhand, can only result in interpreting interval-scalevalues as ordinal scales. Thus, small risks of Type 2errors seem to be more important than small sig-nificance levels for the problem at hand (cf. We-stermann & Hager, 1983a, 1983b). As a conse-quence, it was decided to reject the null hypothesisof equal probabilities when the frequencies of re-sponses of either type were not greater than 7 ornot less than 13, which corresponded to proportionsof .35 and .65, respectively. These cutoff pointsled to probabilities of .26 and .10 for Type 1 andType 2 errors, respectively (for a &dquo;large&dquo; effectsize in the sense of Cohen, 1977).

In Studies 3 through 8 subjects were allowed tojudge the differences to be equal. To test the Equal-Spacing Axiom, half of these responses were as-signed to both the response categories H and L.The number of quadruples used to test this axiomwas reduced from 20 to 15. The Equal-SpacingAxiom was considered satisfied if the proportionof Type H responses was between .33 and .67.

These critical values led to error probabilities ofor = .12 and J3 = . 16 for ‘6very large&dquo; effect sizesof g = .30.

the Monotonicity Axiom

Monotonicity, as formulated both in finite andinfinite difference structures, is a necessary con-dition for interval-scale level measurement per-taining to sets of six stimuli, which are symbolizedas e, f, ~, 1, M, and n. In the case of only twopossible responses, the Monotonicity Axiom re-duces to the following condition:

If the difference between e and f is judged largerthan the difference between I and ~, and if thesame is true for / and in comparison with m and~a, then the difference between e and g ~aa~st bejudged larger than the difference between I and n.

As a consequence, three quadruples have to bejudged for a single test of the Monotonicity Axiomwith respect to a set of six stimuli [e, f, ~, , a~c9 n]. oFor each subject in Studies 1 and 2, three such setsof six stimuli were selected randomly under theconstraint that both

holds. In Studies 3 through 8, five such sets wereselected to test monotonicity. Table 3 shows, asan illustration, the numbers and scale values of thethree sets of stimuli selected for Subject in Study 1.The first pair is always the pair with the largerdifference in scale values, that is, (e9~, (f, g), or(e, ~). When the subject judged the correspondingdifference to be larger than the difference betweenthe other two stimuli, the response was coded as&dquo;T,&dquo; or otherwise, as &dquo;U.&dquo;

If the Monotonicity Axiom holds, the same typeof responses should be given in the case of all threequadruples derived from one set of six stimuli, thatis, a T-T-T or a U-U-U sequence of responses isa positive result with respect to the validity of theMonotonicity Axiom. When there are Type T re-sponses for both the first and the second quadruplein Expression 3, but a Type U response for thethird one, or vice versa, this is a negative result.When there is one Type T response for one of thefirst two quadruples and a Type U response for the



270

Table 3Test of the Monotonicity Axiom

for Subject 1 in Study 1

*Significantly larger than p = .5 ( a = .10).

other, the Monotonicity Axiom is not testable. Inthese cases the premise of the axiom is not given,and it does not make sense to ask whether the

conclusion is satisfied empirically.In the present studies, there were, at most, five

opportunities to test the Monotonicity Axiom.Therefore, it was not possible to derive decisionrules in terms of a significance test. Instead, in

Studies 1 and 2, this axiom was considered satisfiedif there was a positive result for at least one of thethree sets, and if there were no negative results forthe remaining sets. In Studies 3 through 8, five setsof stimuli were used to test monotonicity, and theaxiom was considered satisfied if there were more

positive than negative results.

the Consistency Condition

Each quadruple presented to test monotonicityled to another kind of consistency check betweenthe direct ratings in the first part of the study andthe ordinal judgments on differences in the secondpart. If differences between numerical ratings doindeed represent (the order of) subjective differ-ences, then the differences between the stimuli e,

f, and g were expected to be judged larger than thedifferences between 1, M, and n. Taking randomfluctuations into account, Type T responses wereexpected to be more probable than Type U re-sponses, that is 9

This prediction was complementary to the predic-tion derived from the Equal-Spacing Axiom forstimulus pairs with equal differences in scale val-ues. Although the test of this ordering conditionwas dependent on the test of the Monotonicity Ax-iom, it can be considered as a third way of testingthe interval-scale hypothesis.

It is clear that Expression 5 can again be testedby a binomial test. In this case, however, any Type 1

error may lead to an unwarranted interpretation ofthe ratings as interval-scale measurements. For thatreason the significance level should be chosen rel-atively low at the expense of power. For the currentexample there were only nine replications, how-ever. Under these circumstances it was decided to

set a equal to .09, which resulted in a power of1 ~3 = .60 for &dquo;large&dquo; effect sizes ®f ~ - .25(cf. Cohen, 1977). In Studies 3 through 8 the Con-sistency Condition could be tested with 15 repli-



271

cations. The condition was accepted if the propor-tion of Type T responses was at least .72, whichresulted in error probabilities of out = .06 and f3= .31 (for an effect size of g = .25).

In order to avoid erroneous interpretations ofratings as interval-scale measurements, the inter-val-scale hypothesis for a subject was accepted onlyif all three conditions (Equal-Spacing, Monoton-icity, and Consistency) were considered as satis-

fied. This is a very strict criterion, but it seems

difficult to give justifications for more lenient cri-teria on the basis of significance tests or probabilitytheory. The total set of quadruples selected for anysubject to test equal-spacing, monotonicity, andconsistency was presented in written form and inrandom order. In addition, the order of the twopairs of stimuli forming a quadruple and the orderof the items within each pair were determined ran-domly.

Results

As an illustration, the results for Subject 1 ofStudy 1 are reviewed. Table 2 contains the re-

sponses of this subject with respect to those 20

quadruples selected to test the Equal-Spacing Ax-iom. The proportion of responses of Type H is .55.So, the Equal-Spacing Axiom is regarded as sat-isfied for this individual.The responses pertaining to monotonicity and

order consistency are shown in Table 3. There isa positive result for the first set of stimuli with

respect to the Monotonicity Axiom. Subject 1 hasgiven Type T responses to all three quadruples.Set 2, however, does not allow a test of monoton-icity because there is a T response to the first, buta U response to the second quadruple. The sameis true for the third set of stimuli. According to thecriteria specified above, the Monotonicity Axiomis considered as satisfied empirically. The Con-sistency Condition is satisfied, also. As shown inTable 3, Subject 1 gave Type T responses in sevenout of nine cases. Taken together, all three tests

speak in favor of the interval-scale hypothesis forthis subject.The results of Study 1 are summarized in Ta-

ble 4. According to the criterion described above,the Equal-Spacing Axiom was not satisfied for Sub-jects 2, 4, 7, and 8. In addition, orderings of dif-ferences did not appear to be consistent for Subjects

Table 4Tests of the Interval-Scale Hypothesis for

All Subjects in Study 1

Note. The Axioms or conditions are considered as~

satisfied (+), not satisfied (-), or not testable (o)according to the criteria specified in the text.



272

4 and 7, and there were deviations from monoton-

icity for Subjects 2 and 4. Adopting the strict cri-terion specified above, the interval-scale hypoth-esis was rejected for these four subjects, with atleast one rejection in the tests of the three condi-tions. For the other five of the nine subjects, how-ever, there was no negative result. As a conse-quence, the ratings for these five subjects can betentatively considered as interval-scale values.The results of all eight studies are summarized

in Table 5. A total number of 114 subjects wererun. The Equal-Spacing Axiom was satisfied by80% of the subjects, the Consistency Condition bymore than 70%, and the Monotonicity Axiom by67%. All three conditions can be regarded as sat-isfied for 54 of the 114 subjects.

Discussion

According to the criteria specified above, about50% of the subjects seemed to have given ratingsat the interval-scale level. This proportion may seemlow; two facts, however, have to be taken intoaccount. First, all these studies were conducted

with students participating in an undergraduate coursefor experimental psychology, so that the subjects’motivation to give careful responses might havebeen only moderate. Second, decision criteria werechosen in order to hold the risk of falsely acceptinginterval-scale level for any subject’ ratings at rea-

sonable low levels, so that the risk of falsely re-jecting the interval-scale hypothesis must be con-sidered relatively high. In other words, these testswere rather severe (Meehl, 1967; Popper, 1975;Westermann & Hager, 1983a, 1983b).

Therefore, these results seem to corroborate thehypothesis that a considerable proportion of sub-jects were able to give ratings at the interval-scalelevel. Because of the limited scope of these studies,further severe tests of this hypothesis would beworthwhile.

In applied psychological research, however, thereis not generally the opportunity to check scale prop-erties for each of the subjects by the methods pre-sented here and to discard those subjects that failedthe test. Nevertheless, this result may have some

consequences for applied psychological research-ers who plan to use rating scales.

Beginning from the result that at least about 50%of the subjects in these studies seemed to be ableto give ratings at the interval-scale level, research-ers can try to maximize the corresponding propor-tions in their own studies. First, subjects should beprovided with a proper response format, such asthe ruler-like scale from 0 to 100. Second, subjectsmust be deliberately instructed how to use numbersand numerical differences to represent subjectivemagnitudes and differences. Third, researchers

should try to motivate their subjects to judge care-fully in accordance with the instructions. Under

Table 5Results of the Studies Testing the Interval-Scale

Hypotheses for Individual Ratings



273

these conditions, researchers can reduce their riskof erroneous conclusions when interpreting all re-sulting ratings as interval-scale measurements, andthey will have a better justification for using par-ametric statistical tests, for averaging individualratings to derive mean scale values, and so forth.As has been shown in this article, contrary to

the position held by Dawes (1972), it is possibleto answer questions concerning the scale type ofratings in a reasonable manner. Thus, usefulnessis not the only criterion to evaluate ratings, andDawes’ classification of ratings as index measure-ments may be somewhat misleading.

There remains a basic difference, however, be-tween such interpretations of ratings as, say, in-terval-scale measurements and genuine fundamen-tal and representational measurements at the interval-scale level in the sense given by Suppes and Zinnes(1963) and by Dawes (1972). Interval-scale inter-pretations of ratings usually are based on the resultsof other studies testing the interval-scale hypothesisfor other individuals’ rating. In addition, accordingto the present procedure, the axioms and necessaryconditions for the desired interval-scale represen-tation are tested, not in the course of the scalingprocedure, but post hoc, after the ratings have beengiven.

For these reasons it would be preferable to speakof a &dquo;quasi-representational&dquo; interval-scale mea-surement when ratings are interpreted as intervalscales. This new category of &dquo;quasi-representa-tional measurements&dquo; is introduced to encouragea clear distinction between ratings and their inter-pretation, on the one hand, and the well-knownclasses of index, derived, representational, andfundamental measurements, on the other hand.

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Acknowledgments

Thanks are due to Heinz Liebeckfor a list of fecc~°-ca~°or~s-ing stimuli; S. Gunkel, C. Kordowski, Y. Missios, S.Anton, B. Giermann, N. Th6lking, K. Lampe, A. Bettges,S. Drangmeister, C. Sch~afer, W. Storch, S. Floer, D.Kucllcacek, E. Andres, S. Schemat, M. 5’c~M~zwc~~, K.Engel, G. Czechatz, S. Pollmann, H. -J. Froinm, and B.Rumpffor conducting the empirical studies; and to twoanonymous reviewersfor helpful comments on can earlierdraft of this ca~°ticle. Special appreciation is cxpressed toKornelia Junge and Eckehard Kuhlmeifor computerpro-gramming and assistance in collecting and analyzing thedata.

A~~~®~°~~ Address

Send requests for reprints or further information to Rai-ner Westermann, Institut für Psychologie, Universität

Göttingen, Goßlerstr. 14, D-3400 Göttingen, West Ger-many.



Empirical Tests of Scale Type Individual Ratings

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