journal of experimental psychology...

JOURNAL OF EXPERIMENTAL PSYCHOLOGYMONOGRAPH

Vol 102, No 3, 543-561 March 1974

THE NONADDITIVITY OF PERSONALITY IMPRESSIONS

MICHAEL H BIRNBAUM2

University of California, Los Angeles

Ratings of the hkableness of persons described by 2 adjectives showed con-sistent violations of additive and constant-weight averaging models Theeffect of either adjective varied directly with the Hkableness of the otheradjective Monotonic rescaling could remove the interactions, raising thetheoretical question of whether interactions were due to nonlineanty in therating scale or to nonadditive integration of the information Four experi-ments illustrate new methods for distinguishing these interpretations Thefit of the subtractive model for ratings of differences in hkableness between2 adjectives supported the validity of the response scale, in addition,ratings of homogeneous combinations were linearly related to subtractivemodel scale values Judgments of differences in hkableness between pairsof hypothetical persons, each person described by 2 adjectives, were ordmallyinconsistent with additive models, confirming the interpretation that theinteractions are "real" and should not be scaled away Theoretical andmethodological implications are discussed

This research is concerned with howimpressions of personality are formed.This topic, introduced by Asch (1946), has

1 This article is based on a dissertation submittedto the University of California, Los Angeles, inpartial fulfillment of the requirements for the PhDdegree in psychology The author was supportedby a National Defense Education Act Title IVfellowship, Computing funds were received fromthe Campus Computing Network, University ofCalifornia, Los Angeles Additional support wasprovided by the Center for Human InformationProcessing, through National Institute of MentalHealth Grant MH-1S828 Special thanks are dueto Allen Parducci for his advice on this researchand many helpful comments on earlier versionsof this paper Thanks are also due to Claince TVeit for her assistance with these experiments andto the following people for their suggestionsNorman H Anderson, Bonnie G Birnbaum,Dwight Riskey, Barbara J Rose, and Chris ThomasPortions of this research were presented at theMathematical Psychology meetings, Miami, Sep-tember 1970

2 Requests for reprints should be sent to MichaelH Birnbaum, who is now at the Department ofPsychology, Kansas State University, Manhattan,Kansas 66506

recently received new analytic attention(Anderson, 1968b, 1971, 1972, 1974, inpress) The theoretical problem is to ex-plain how the information provided by aset of adjectives is combined to form anoverall impression In the majority of theresearch to be reported here, S's task wasto read 2 adjectives, imagine a person whowould be described by both adjectives,and rate how much he would like sucha person.

Figure 1 provides a schema for theanalysis of impression formation The 2stimulus adjectives are referenced by theindices, i and j The psychological rep-resentations of the adjectives, st and s3,are combined by the integration function,I, to form the psychological impression,^,j, which is transformed by the responsefunction, J, to the overt response, RtJ.

There are 3 problems to be solved.(a) finding the appropriate stimulus rep-resentation (s), (b) determining the inte-gration function (J), and (c) finding the

543

544 MICHAEL H BIRNBAUM

_.. OVERTRESPONSE

FIGURE 1 Schema for discussing research oninformation integration

relationship (J) between the overt re-sponses and the integrated impressionsAny theory of impression formation mustdeal, at least implicitly, with all 3 of theseproblems

ADDITIVE AND CONSTANT-WEIGHTAVERAGING MODELS

A simple linear model provides anexample of such a theory The adjectivesare assumed to be represented as singlevalues on a likableness continuum The/ function, defined on all adjective com-binations, would be a linear combinationof the values of the adjectives

where wi and Wa are the weights reflectingposition in the set, s, and s3 are the scalevalues of the adjectives, and w0so is thepostulated initial impression, reflectingwhat the impression would be in the ab-sence of information

Anderson (1962) pointed out that if setsof adjectives were constructed accordingto factorial designs, then the analysis ofvariance test for interactions provides apowerful test of Equation 1 If the ad-jectives were to change their meaning non-Imearly in combination, if the I functionwas nonadditive, or if the judgment func-tion (/) was nonlinear, then significantinteractions would be expected If theywere nonsignificant, there would seem tobe no reason to postulate such complica-tions The data for the majority ofAnderson's (1962b) 12 5s appeared to bein rough agreement with Equation 1

The constant-weight averaging model (An-derson, 1968b) is a special case of Equa-tion 1 that requires the weights to sumto 1, the additive model places no restric-tion on the sum of the weights Criticaltests of averaging vs adding models (inwhich the number of items in the set arevaried) favor the averaging formulationover the additive (Anderson, 1971, 1974,in press) However, when the number ofitems in the set is held constant, as inthe present experiments, the models areequivalent Therefore, the present articleuses the term additive to include bothadditive and constant-weight averagingmodels

Anderson's (1968b) review of the evi-dence for the constant-weight averagingmodel concluded that

The model always fits the data quite well, butthere are almost always small, significant discrep-ancies Inspection of the data has failed to revealthe origin of the discrepancies, they may reflectsome fundamental error in the model, or theymay result from remaining shortcomings in theexperimental technique [p 736]

Recent studies have shown large dis-crepancies from the constant-weight aver-aging model for several information inte-gration tasks Interactive models havebeen proposed to account for these non-additivities (Anderson, 1972; Birnbaum,1972a, 1972b, 1973, Birnbaum, Parducci,& Gifford, 1971, Lampel & Anderson,1968) However, there are 2 possibleinterpretations of the discrepancies thathave been observed in these studies (a)the integration (7) of information is notan additive or simple averaging process,or (b) the judgment function (/) is non-linear The next section shows howprevious conceptualizations cannot dif-ferentiate these alternatives

MODEL TESTING AND MEASUREMENT

Assuming Validity of Responses

When the J function is assumed to belinear, models of impression formation canbe tested by comparison of theoreticalpredictions with the raw data. Functionalmeasurement (Anderson, 1970) finds the

NONADDITIVITY OF PERSONALITY IMPRESSIONS 545

stimulus representation in accord with themodel to be tested These stimulus pa-rameters are then used as the basis forstatistical tests of fit If either the modelor the response scale was incorrect, itwould show up as a significant discrepancyWhen the data fit the model, the fit isusually interpreted as joint support forboth the model and the response scale

Ordinal Tests

When the / function is only assumedto be strictly monotonic, it becomes moredifficult to discriminate among differentmodels Conjoint measurement analysis(Krantz & Tversky, 1971) describes con-ditions that ideal data would have tosatisfy to be ordinally consistent with thetheory For example, crossover interac-tions would be ordinally inconsistent withadditive models, for no monotonic trans-formation could make the data fit themodel. In this case, everyone agrees thatthe model should be rejected It shouldbe noted that ordinal violations of ad-ditivity will show up as significant inter-actions in the analysis of variance Theproblem arises when significant interactionsoccur in the absence of ordinal violations.

Assuming Validity of the Model

When the additive model is assumed tobe valid, then the analysis of variancetests the linearity of the / functionA nonlinear / function will produce sig-nificant interactions even though the un-derlying integration is additive In theabsence of ordinal violations of the model,it is possible to find a monotonic trans-formation, J~l, which rescales the data toadditivity

When Both Model and Response Scale arein Doubt

Krantz, Luce, Suppes, and Tversky(1971, p 445) have taken the extremeview that when discrepancies from ad-ditivity can be removed by monotonictransformation, they should be attributedto nonhneanty in the response scale, ratherthan to nonadditivity of the integration

function. However, when a monotonictransformation would bring otherwise con-tradictory data into line with the model,the status of the model remains uncertain.This procedure assumes the validity of themodel; hence, the existence of such atransformation does not mean that themodel is validated. To resolve this diffi-culty, it is necessary to place transforma-tions within the scope of psychologicaltheory and to provide additional con-straints which determine their appropriateapplication

ADVANCES IN MODEL TESTING

These 4 experiments provide a progres-sive sequence that systematically elimi-nates the additive or constant-weightaveraging models The experiments illus-trate novel approaches to model testingthat remove the difficulty of decidingwhether or not to rescale the data Sincethese techniques will be of interest topsychologists in many areas, they areoutlined briefly below

The first 2 experiments apply criteriaof stimulus and response scale consistency.Experiment I assumed the validity of apriori values for the adjectives obtainedin previous work (Anderson, 1968a) andrequired that the integration model yieldscale values that are linearly related.Experiment II investigated the effects ofdifferent response procedures, thought toproduce different / functions. By a prin-ciple of convergent operationism (Garner,Hake, & Eriksen, 1956), if similar inter-actions are obtained with different responseprocedures (operational definitions of S^),then the interpretation that the inter-actions are "perceptual" is enhanced

Experiment III illustrates how the si-multaneous evaluation of 2 or more inte-gration processes can provide the leverageto define the concept of an appropriatetransformation (Birnbaum, 1972b, Birn-baum & Veit, 1974) Stimulus scaleinvanance requires that the scale values(s in Figure 1) be independent of taskResponse scale invanance requires that theJ function be independent of task.

In Experiment III , 5s performed 2 in-


tegration tasks, rating the difference mhkableness between the 2 adjectives aswell as the hkableness of a person de-scribed by the combination Since thesame stimuli and response scale wereemployed for both tasks, response scalemvanance requires that the same /-in-verse transformation be applied to bothtasks The stimulus scale convergencecriterion defines an appropriate rescalmgof the combination ratings as one whichboth makes an hypothesized model fit thedata and leads to the derivation of scalevalues that agree with those derived fromthe difference task

Although rescalmg of the data mightmake the model fit a single set of data,the transformation that reduces the inter-actions implies which psychological dif-ferences are greater than or equal toothers Experiment IV obtained directratings of these differences, and it testedwhether these ratings are qualitatively con-sistent with the transformation that makesthe data additive

Experiment IV required only the ordinalinformation in the data to reject the ad-ditive and constant-weight averaging mod-els This leverage was provided by acompound integration task in which 5*8rated the difference in hkableness betweenpairs of hypothetical persons, each de-scribed by a pair of adjectives The ratingscan be rescaled to fit the subtractivemodel, this rescalmg may or may notmake the additive model of impressionformation fit Thus, difference judgmentscan be used as a basis for response re-scaling to provide a scale-free test of theconstant-weight averaging model of im-pression formation

EXPERIMENT I TEST OF ADDITIVEMODELS

The first experiment was designed touncover the nature of the discrepanciespreviously observed Although the basicprocedures were similar to those of theresearch described by Anderson (1968b),these were modified to permit a clearerassessment of the expected interactionsThus, only 2 factors were employed, but

each was represented by a greater numbeiof levels covering a wider range Thispermitted a greater variation of within-set range

Method

The 5s were presented with pairs of personality-trait adjectives and were instructed "your task isto imagine a person who would be described byboth of the traits and judge how much you wouldlike such a pe'rson " The 5s recorded their judg-ments in numerical form, using 1 of 9 ratings foreach pair 1 = dislike very very much, 2 = dislikevery much, 3 = dislike, 4 = dislike slightly, 5 =neutral (neither like nor dislike), 6 = like slightly,7 = like, 8 = like very much, and 9 = hke veryvery much

Subjects The 5s were 300 University of Cali-fornia, Los Angeles, undergraduates fulfilling a re-quirement in introductory psychology One hun-dred different 5s served in each replicate of theexperiment

Stimuli The adiectives were taken from Ander-son's (1968a) list of SSS common personality-traitadjectives Each stimulus replicate consisted ofa set of 25 adjective pairs produced from a 5 X 5(Adjective A X Adjective B) factorial design The5 levels of hkableness of each adjective factorwere separated by steps of approximately 1 28 onAnderson's 0-6 normative scale The adjectivesare printed (with normative values in parentheses)in the margins of Figure 2

Procedure The 25 adjective pairs were printedin 1 of 6 random orders on the same page withthe instructions and the labeled response scaleThe adjectives in each pair appeared side by side,with the adjective from Factor A on the left forhalf of the forms The 5s were instructed to readthrough the entire list before beginning to recordtheir ratings

Results and Discussion

Additive and constant-weight averagingmodels The 3 panels of Figure 2 showthe mean judgment of each adjective pair,averaged over all 5s within the replicateThe mean judgments are plotted as a func-tion of the normative value for the ad-jective from Factor A, with a separatecurve for each adjective from Factor BThe slope of the curves represents theeffects of the Factor A adjective Thevertical differences between the curvesrepresent the effects of the Factor Badjective According to additive or con-stant-weight averaging models, the curvesin each panel should be parallel; that is,the effect of one adjective should not


|S si SJ?= as; Is

Trustworthy(5.39)Lightlieorted(424)

Solemn(2.89)

Listless(1.69)

Mm(.371

MJKKK A

Dependctlo(536)Refined(4.22)Unpredictable(2.90)

en

Cruel(40)

ISiS

Understanding(5.49)Self-tonWenl(4.21)Sly(2.91)

Noisy(173)

Obnoxious(48)

0 1 2 3 4 5 6 0

ADJECTIVE A NORMATIVE VALUE

2 3 4 5 6

FIGURE 2 Mean ratings of hkableness of pairs of adjectives Experiment I(Anderson's, 1968a, normative values for individual adjectives are listedabove and to the right of each panel)

depend upon the particular adjective withwhich it is paired. Instead, the curves ineach panel diverge to the right, the effectof either adjective appears to be propor-tional to the Hkableness of the other ad-jective describing the same person

This divergence was characteristic of thedata for most of the individual 5s 88%88%, and 89% of the 5s in the respectivereplicates showed this form of interactionIn Experiments I-IV between 85-95% ofthe single 5s resembled the group data.The A X B interactions are highly sig-nificant for each of the replicates; F valuesfor these interactions are indicated in thepanels of Figure 2 Of course, the F ismuch larger when the data from all 3replicates are combined, F (16, 4752)= 72 80 If the response scale of thisexperiment is assumed to be valid, theinteractions shown in Figure 2 can beinterpreted as evidence against the ad-ditivity of impressions.

Although the A X B interaction vanesbetween stimulus replicates, F (32, 4752)= 8.28, this higher order interaction issmall in comparison with the similarityof the nonadditivities. There are, how-ever, a few peculiarities that appear todepend upon the meanings of the particularadjectives For example, SELF-CONFIDENT& MALICIOUS is less likable than SHY &

MALICIOUS although SELF-CONFIDENT ismore likable than SHY in combinationwith other traits A SELF-CONFIDENT,malicious person may be perceived as morelikely to carry out malicious actions thana SHY one But the divergent interactionoccurs in all 3 replicates and representsthe mam source of difficulty for the ad-ditive models

Multiphcatwe model The multiplicativemodel, *„ = stsj, predicts that the curvesin each panel should be a diverging fanof linear functions. The interaction shouldbe located entirely in the bilinear com-ponent (Anderson, 1970) Although themajor portion of the variance of the inter-action (81 6%, 56 2%, and 86 9% for the3 replicates) was located in the bilinearcomponent, the residuals were also statis-tically significant, Fs (15, 1485) = 6 36,20 68, and 3 26 The nature of the dis-crepancy can be seen most clearly in thesecond panel of Figure 2 According tothe multiplicative model, the curves shouldbe linear; instead, the upper curves arenegatively accelerated relative to the lowercurves. The other 2 replicates show thesame discrepancy, but to a lesser degree.

Range model. The range model was fitto the data (averaged over replications)following the general procedures outlinedin Birnbaum et al (1971), and using


Anderson's (1968a) normative values asestimates of the scale values The rangemodel, using just one additional parame-ter, accounts for some 80% of the varianceleft unaccounted for by the constant-weight averaging model

Averaging model with differential weightsThe mean absolute discrepancy was only05, indicating a very good fit to 15 points

using 10 parameters The best-fit estimatesof scale values for the 5 levels were 1 49,302, 524, 648, and 783; the best-fitweights were 1 92, 1 18, 67, 73, and 90Fewer parameters are required with theassumption of the validity of Anderson's(1968a) values and by estimating theweights as a polynomial function of scalevalue. Although this procedure uses moreparameters than the range model, the fitwas not as good

Rescalmg. Interactions in the analysisof variance might be due to a nonlinearrelationship between the impressions ofhkableness and the overt responses ofthe 5s Thus, the actual impressions maybe additive, but the overt responses maybe related only ordmally to the theoreti-cally correct response scale The pos-sibility of response scaling was investigatedusing 2 techniques of data transformation

The first technique assumes that Ander-son's (1968a) normative values for thesingle items are appropriate estimates ofthe scale values and that judgments ofpairs of items of equal value should belinearly related to the judgments of thesingle items (Birnbaum, 1972a) Thetransformation is then estimated as a poly-nomial by a least squares criterion Theadvantage of this technique is that al-though it is capable of producing a radicalrescaling, it will not eliminate "real" inter-actions that depend upon differences inwithin-set range, if the a priori scales areappropriate As can be seen in Figure 2,the judgments of pairs of minimal within-set range (connected by dashed lines) arealready nearly linearly related to norma-tive ratings of single values If anything,this function (treated in this procedure asthe / function of Figure 1) is slightlynegatively accelerated, so that the rescalingthat makes it linear would actually in-

crease the magnitude of the divergentinteraction

If the divergent interaction is real, thenthe distribution of psychological impres-sions is somewhat positively skewed (asgiven by the projection of the data pointson the ordmate of each panel of Figure 2)Positive skewing would lead to a nega-tively accelerated judgment functionaccording to range-frequency theory(Parducci & Perrett, 1971) This sug-gests that contextual effects operating onthe / function would tend to counteractthe effects of true interactions Birnbaumet al (1971, Experiment V), demonstratedthat manipulation of the frequency dis-tribution of physical means of sets ofpsychophysical stimuli influences the formof the interaction between the componentsin a manner predictable from range-fre-quency principles of judgment

The second transformation procedureassumes that the integration is additive(or constant-weight averaging) and at-tempts to transform the data to fit theadditive model (Kruskal, 1965) Themonotone analysis of variance (MONA-NOVA) computer program (Kruskal &Carmone, 1969), applied to the meanjudgments, greatly reduced the percentageof total variance in the interaction (from5 0% to 4%) Thus, the data appearroughly consistent with the constant-weightaveraging or additive model at an ordinallevel The fact that the data can betransformed to fit raises a difficult theo-retical problem Is the nonparallelism inFigure 2 due to nonadditive integrationof information or to a nonlinear responsefunction?

If the additive model is assumed to becorrect, the MONANOVA analysis indi-cates that the judgments are a positivelyaccelerated function of the impressionsAdditionally, it would mean that the scalevalues for adjectives presented in pairsare a negatively accelerated function ofAnderson's (1968a) values for the sameadjectives presented singly. The positivelyaccelerated function for J derived fromthe MONANOVA analysis can be inter-preted by range-frequency theory (Par-ducci & Perrett, 1971) to indicate that

NONADDITIVITY OF PERSONALITY IMPRESSIONS 549.

the psychological distribution of stimuli isnegatively skewed Kanouse and Hanson(1972) have also hypothesized that thedistribution of evaluative stimuli is nega-tively skewed, but based their argumentson other considerations.

In short, Experiment I demonstratesthat ratings of hkableness are inconsistentwith the additive models However, 2 in-terpretations are consistent with the data.The first assumes that the ratings arevalid measures of psychological impressionsand concludes that the integration of in-formation is nonadditive The second as-sumes the validity of the constant-weightaveraging (or additive) model and con-cludes that the responses are a positivelyaccelerated function of the impressionsBoth interpretations can account equallywell for the data of Experiment I Con-sequently, the following experiments weredesigned to discriminate them

EXPERIMENT II VARIATION OFRESPONSE SCALES

The earlier experiments yielding datamore consistent with the additive modelshave used other procedures for obtainingS's response (see Anderson, 1974, in press)The interactions observed in Experiment Imay be due to nonlineanty in the J func-tion that depends upon the particularprocedure S uses to indicate his impres-sions. Therefore, Experiment II testedthe additive models using several differentprocedures for responding to assess whetherthe interaction obtained in Experiment Iis specific to the 9-category rating scaleused in that experiment Each of the 4conditions of the present experiment testeda different interpretation of how the in-teractions might depend upon the methodfor responding: reversing the scale, end-point anchoring with 20 categories, line-mark responses, and matched pairs

Although the matched pairs procedurehas not been used before, it has the ap-parent advantage that it does not requirea metric response from the 5 Thus, thisprocedure avoids the argument that metricresponding may induce & to integrateseparate implicit responses to the indi-

vidual adjectives rather than forming anoverall impression before responding

MethodAs in Experiment I, Sa read pairs of personahty-

trait adjectives and judged how much they wouldlike hypothetical persons who would be describedby both adjectives The conditions differed withrespect to the instructions for the response Eachset of instructions was used in a separate smallexperiment, with different 5s

For the reversed scale, 1 = like very very muchand 9 = dislike very very much The 5s wereconsequently instructed to rate how much theywould dishke the hypothetical persons The 20 5s>rated the adjective pairs of both Replicates 1 and 2of Experiment I

The 20-category scale was anchored by instruc-tions that 1 represented the hkableness of a personwho would be described by MEAN, PHONY, MALI-CIOUS, OBNOXIOUS, and LIAR, 20 represented thehkableness of SINCERE, LOYAL, INTELLIGENT, UNDER-STANDING, and DEPENDABLE. The 5s were in-structed to judge each adjective pair relative tothese end anchors and to assign a numeral be-tween 1 and 20 to represent the appropriate positionrelative to the end values The 34 5s rated the25 adjective pairs of Replicate 2

For the linemark response, 5s were instructedto indicate each judgment of hkableness by makinga short vertical mark on a line so that the lengthbetween the margin and the mark would be pro-portional to the hkableness The 46 5s in thiscondition judged the adjective pair of Replicate 2

The list for the matched pairs procedure wasconstructed from Anderson's (1968a) normativedata Each pair contained 2 adjectives of equalscale value The 22 pairs covered the baselinerange 35-5 60, in 25-category steps The 50 5swere instructed to judge each of the adjectivepairs of Replicate 2 by selecting the pair of ad-jectives from the list of 22 pairs, "most nearlyequal in likeableness to the pair you are judging "The value of 5's response was taken to be thenormative value of the adjective pair selected by 5The 22 pairs are listed in Birnbaum (1972b)

Supplementary scahng These 22 adjective pairsand the 30 adjectives used in Experiment I wereprinted in random order Each of 100 5s judged3 aspects of each adjective or pair (a) the hkable-ness of a person who would be described by theadjective or pair, (6) the acttwty of such a person,and (c) the range of hkableness of persons whowould be described by the adjective (s) The 5sjudged all of the adjectives on one aspect beforeproceeding to the next task Nine-point scaleswere used for all 3 scaling tasks, with 9 = likevery very much, very very active, or very verywide range of possibilities

Results and DiscussionFigure 3 shows the mean responses,

averaged across 5s, in each condition.


I

2

3

h-

=5 5

|s

7

8

UJ5—j3 36

I 31o£26e 21<ujS 16

9 CATEGORY REVERSED SCALEF06.304) 1572

0.38

LINEMARK RESPONSEF(I6,720) 2206

040

3 2

20 CATEGORY ANCHORED SCALEr (16,528) -18 22

MATCHED PAIRSF(I6,784) 857 536

0 1 2 3 4 5 6 7 0 1 2 3 4 5 6 7

ADJECTIVE A NORMATIVE VALUE

FIGURE 3 Mean judgments of hkableness of pairs of adjectives Experi-ment II (Each panel shows results for a different response procedure )

Additive and constant-weight averagingmodels again predict parallel curves foreach panel Instead, the interactions areall of the same general form (divergence)as m Figure 2, and the F values listed ineach panel show that they are all highlysignificant

The ordmate of the upper left-handpanel of Figure 3 has been relabeled fordirect comparison with Figure 2. Thedata show the same interactions, andratings of "dishkableness" appear to be alinear function (with negative slope) ofratings of "hkableness " Since reversingthe scale did not reverse the interaction,the deviation from additivity cannot beattributed to anything like preference forsmaller numbers The interaction for theanchored scale (upper right-hand panel)

shows the same divergent form as thoseshown in the other panels and is highlysignificant The percentage of varianceassociated with the interaction is actuallylarger for the hnemarks condition than forany of the other conditions The posi-tively accelerated relationship betweenhnemarks and category ratings may ac-count for the larger interaction.

The matching procedure assumes thatthe normative values obtained by Anderson(1968a) are valid estimates of scale valuesOne check on this technique is to seewhether the pairs having minimal within-set range are linearly related to Ander-son's scale values As can be seen byconnecting these points (the diagonal) inthe lower right-hand panel of Figure 3,this assumption is supported. Never-


theless, the interaction has the samedivergent form as the others and is highlysignificant. The results for this non-numerical response technique do not sup-port the objection that perhaps 5's inte-gration strategy is affected by the numericalresponse procedures typically used in thistype of study The error terms for thismethod were greater than for any of theother methods, perhaps because 5s mustjudge the pairs of items defining the scalein addition to the usual stimulus pairs

In summary, the mam conditions ofExperiment II show that variation of ex-perimental procedures for responding didnot eliminate the divergent interaction.Each procedure can be considered an al-ternative operational definition of the im-pression. By the principle of convergingoperations (Garner et al , 1956), the di-vergent interactions in Figure 3 supportthe interpretation that impressions of per-sonality are nonadditive In order toretain the additive model, it would benecessary to assume that the responses forall of these procedures are nonhnearlyrelated to the impressions

Supplementary scaling One purpose ofthe supplementary scaling was to checka previous prediction (Birnbaum, 1972a)that items of lower value have narrowerdispersions Successive interval scale val-ues and dispersions (Torgerson, 1958) wereobtained for each adjective and adjectivepair. Mean ratings were linearly relatedto Anderson's normative values and toThurstone scale values derived from thesame data In agreement with the dis-tributional interpretation, Thurstone scalevalues were positively correlated with dis-persion values (r = .65) Since each ofthese dispersions was produced by thedifferences between the responses of dif-ferent 5s to the item, they are probablynot the best estimates of the distributionof meaning for the adjectives. Only if itis assumed that different Ss select a ran-domly sampled "person" from the dis-tribution of "persons" who possess the traitwould the Thurstone dispersion reflectthis dispersion directly. Nevertheless, thecorrelation between value and dispersionseems consistent with this prediction.

Subjective estimates of the range ofhkableness seem a more direct measure ofdispersion. These were highly correlatedwith mean ratings of hkableness (r = 91)The plot of this relationship showed thatalthough range is a nearly linear functionof hkableness for the dishkable traits,neutral and positive traits are all seen asimplying similar wide ranges of likable-nesses This finding is consistent with thefact that the interactions in Figure 2 ap-pear to be located in combinations thatinclude dishkable traits. In a post hocanalysis of the 2 X 2 subdesign containingonly positive adjectives for Experiment I,the interaction was nonsignificant (F < 1).The 2 X 2 subdesign containing the nega-tive traits had a significant interaction,F (1, 297) = 5 93.

In summary, the supplementary scalingsuggests that adjectives should be repre-sented by distributions, with the lowervalued items having smaller variance

Supplementary test of activity. One in-terpretation of the crossover interactionobserved in Experiment I, Replicate 3(Figure 2) is that the activity componentof the meaning of one adjective can mul-tiply the evaluative component of theadjective with which it is paired. Hence,a SELF-CONFIDENT MALICIOUS person ismore actively malicious than a SHY one.Similarly, an IMPULSIVE CRUEL person maybe more likely to act cruel than an UNTIDYone (Experiment I, Replicate 2)

This hypothesis—that one adjective canbehave like an adverb—was further in-vestigated by having 130 additional 5s ratethe hkableness of 36 hypothetical persons,each described by 2 adjectives producedfrom a 6 X 6 factorial design. The ad-jectives for the first factor varied inhkableness (RUDE, IMMATURE, TROUBLED,DIRECT, INTELLECTUAL, and HONEST) Theadjectives for the second factor were rela-tively neutral in hkableness, but 3 were"active" (IMPULSIVE, AGGRESSIVE, andCHANGEABLE) and 3 were "passive" (QUIET,HESITANT, and SHY).

Consistent with the prediction, the hka-bleness effect of an adjective was greaterwhen paired with an active adjective.The interaction was large and significant,


F (25, 3225) =, 14 49, and even showedreliable crossovers, for example, HESITANT& HONEST is less likable than AGGRESSIVE& HONEST although HESITANT & RUDE ISmore likable than AGGRESSIVE & RUDE

Ratings of the activity of the 30 ad-jectives used in Experiment I were un-correlated with ratings of hkableness ofthe same adjectives but were correlatedwith the Thurstone dispersions (r = 43)The activity hypothesis is not sufficient toaccount for the overall divergent inter-actions in Figures 2 and 3, but it appearsto explain some of the second-order effects(which would otherwise be called "pecu-liarities") for particular adjective combi-nations in Experiment I.

EXPERIMENT III SUBTRACTIVEPREFERENCE VERSUS ADDITIVE

COMBINATION

Experiment III had the following ob-jectives- (a) to evaluate a subtractivemodel for preference judgments reflectingthe difference in Hkableness between 2 hy-pothetical persons, each described by oneadjective; (b) assuming that the sub-tractive model were to fit, to comparescale values for the preference task withthose obtained for the usual combinationtask, in which 5s judge the hkableness ofa person possessing both traits; (c) to usethe scale values obtained for the preferencetask as the basis for rescahng the datafrom the combination task, (d) to use thescale values for the preference task toevaluate Anderson's normative values forsingle items; and (e) to evaluate the effectsof a change in instructions for the com-bination task, in which the adjectives aregiven equal importance and accuracy.

The subtractive model asserts that pre-ference ratings can be represented as thealgebraic differences between the values ofthe items-

*.,D = 5, - s,, [2]

where ^4JD is the psychological difference

in hkableness (preference for Stimulus iover j), and ss and s, are the scale valuesof these 2 stimuli If the subtractivemodel can be fit to the data, then the

adjectives can be located as points on aunidimensional hkableness continuum

Two principles of scale convergence canbe applied to this experiment (a) Re-sponse scale invanance With the sameset of stimuli and the same response pro-cedure, the judgment function is assumedto be independent of task (b) Stimulusscale invanance • With the same set ofstimuli, the scale values of the stimuli areassumed to be independent of task, Prin-ciple a implies that whatever transforma-tion is applied to fit the data to the ad-ditive model for the combination taskshould also be applied to the data forthe difference task. Principle b impliesthat irrespective of whatever transforma-tions of the responses fit the data for the2 tasks to their respective models, thescale values derived from the subtractivemodel should be linearly related to thescale values for the same adjectives incombinations Stimulus scale invanancecan be considered a necessary (but notsufficient) condition for establishing mean-ingful scales and psychological laws Afailure of scale convergence can providethe leverage for rejecting either the sub-tractive or additive model, even thoughthe 7 functions are unknown

Method

St^muh The stimuli were the adjective pairsof Experiment I

Subjects The 5s were 90 University of California,Los Angeles, undergraduates, 30 serving in eachreplicate

Procedure Each S performed both tasks, withhalf of the 5s in each replicate performing thedifference task first There were no effects oftask order

In the difference task, 5s were instructed toimagine 2 different people, each described by oneof the adjectives of each pair, and then to judgethe difference in hkableness between the 2 personsThe 9-pomt scale had labels varying from 1 = likethe person (described by the adjective) on theleft very very much more than the person (de-scribed by the adjective) on the right, throughS = like both persons equally, to 9 = like theperson on the right very very much more thanthe person on the left.

The procedure for the combination task was inall respects identical to that of Experiment I,except that the instructions were modified to em-phasize that the adjectives should be considered


fe 5

I "

i 2;*s 9

REPLICATE IF(I6,464) - 95

REPUCATE 2F(I6,464) 265

'REPLICATE 3F06.464) 109

REPLICATE IF(I6,464)=I063

/ 2 3 5 6 0 / 2 3 4 5 6 0 1

ADJECTIVE A NORMATIVE VALUE2 3 4 5

FIGURE 4 Mean ratings of difference in likableness between 2 hypo-thetical persons, each described by a single adjective (upper panels), andmean ratings of likableness of hypothetical persons, each described by thesame 2 adjectives (lower panels). Experiment I I I

of equal importance and accuracy, each wordhaving been contributed by a different acquaintance"who knows the person well "


Difference task The upper panels ofFigure 4 show the mean judgments of thedifference in likableness Because ratingsreflect the difference between Factors Aand B, the slopes are positive, but theorder of the curves is negative.

Since the subtractive model is a specialcase of an additive model, the parallelismof the curves supports its validity. Onlythe interaction for Replicate 2 is signifi-cant, and examination of the figure in-dicates that the interaction is minor andmainly due to one point. Since Anderson's(1968a) normative values are plotted onthe abscissa, the near linearity of thesefunctions supports the validity of thenormative values as estimates of scalevalues. Similarly, the vertical separationsbetween the curves are nearly proportionalto differences in Anderson's scale values.

The good fit of the subtractive modelsupports the validity of the category ratingscale and indicates that the adjectives canbe located on a umdimensional likablenesscontinuum.

Combination task The lower panels ofFigure 4 show the mean judgments of thelikableness of persons described by bothadjectives The data appear to replicateExperiment I closely, including the pecu-liarities for ratings of particular adjectivecombinations The interactions for thisexperiment diverge in the same fashionas Experiment I and are highly significantfor each replicate, as indicated by the Fvalues printed in the panels of the figure.The F values are smaller than those forExperiment I because there were fewer >Ssin this experiment The change in in-structions, emphasizing equal importanceand accuracy, appears to have had nodiscernible effect on the interactions

Scale convergence Both the stimulusand response scale invariance assumptionswould require that either the subtractive


model for differences or the constant-weight averaging model for combinationsmust be rejected Response scale invan-ance implies that any transformation ofthe ordmate of the lower panels must beapplied to the upper panels as well Be-cause the subtractive model fit the datadirectly, any nonlinear transformationwould induce an interaction. Therefore,if the validity of the subtractive modeland the principle of response scale mvan-ance are assumed, the constant-weightaveraging model must be rejected

The fit of the subtractive model impliesthat the marginal means for these dataconstitute an interval scale for the ad-jectives (Anderson, 1970) However, whenthe combination data were rescaled toadditivity using Kruskal's (1965) MONA-NOVA, scale values derived from thisprocedure were nonhnearly related to thosefor the same adjectives derived from thesubtractive model Therefore, the prin-ciple of stimulus scale invanance impliesthat this transformation would not beappropriate

Further support for the validity of therating scale is provided by the findingthat ratings of homogeneous combinations(adjectives of similar normative value) arenearly linearly related (slightly sigmoidal)to subtractive model scale values. Hence,the assumption of stimulus scale invananceimplies that the J function is roughlyindependent of task Thus, 3 proceduresagree—Anderson's (1968a) normative val-ues, subtractive model scale values, andratings of homogeneous combinations areall nearly linearly related

This analysis also indicates that a simplemultiplicative model, *„ = s,$,, would beinappropriate for the combination task, itwould incorrectly predict that judgmentsof adjectives of equal value are a positivelyaccelerated quadratic function of the scalevalues for the difference model However,a geometric averaging model (square rootof the product of the scale values) wouldstill be consistent with the subtractivemodel scale values

The data of Experiment III indicatethat with the same stimuli, 5s, and general

experimental procedure, ratings of prefer-ence are consistent with a subtractivemodel, but ratings of combinations areagain inconsistent with the additive models.This suggests that the interactions are notdue to trivial details of experimental pro-cedure If scale values for the adjectivesare assumed to be independent of task,the fact that the ratings of homogeneouscombinations are a nearly linear functionof the scale values for the difference ratingsimplies that it would be inappropriate torescale the ratings of combinations Inorder to retain the additive model, itwould be necessary either to reject thesubtractive model (in spite of its good fitto the data in Figure 4), or assume thatboth the scale values and the responsefunction depend upon the task.

EXPERIMENT IV QUALITATIVENONADDITIVITY

Experiment IV was designed to providea test of additivity that would requireonly the ordinal information in the data.The evidence against the additive andconstant-weight averaging models obtainedin the first 3 experiments is illustrated inFigure 5A, where the difference in ratingbetween a2&2 and a%bi is greater than thedifference between dibs and a\b\ An ad-ditive model requires that these differencesbe subjectively equal If the J functionis assumed to be linear, then differencesin judgment are proportional to differencesin the impressions, therefore, the diver-gence (previously shown in Figures 2, 3,and 4) would be contrary to additivemodels However, if / were positivelyaccelerated, impressions might be additiveThe same ratings have been transformedto parallelism in Figure SB Although thedifference in rating (Figure 5A) due tothe change from 63 to bi depends onwhether a\ or az was in the same set,the difference in psychological value (Fig-ure SB) is assumed by additive models tobe constant.

Experiment IV tested this possibility byasking 5s to judge directly the differencesin hkableness between integrated impres-sions. The success of the subtractive

NONADDITIVITY OF PERSONALITY IMPRESSIONS

A. a555

NONADDITIVE ADDITIVE

98

gj 7

1 6^ 51 4

b, -

«£<_>CO

ce

ce

c.a I bj

0. 'Q| b|

a b

a,b2 \

a b

a,b2

ajb, \,\

FIGURE 5 A Schematic diagiam of nonadditive data B Schematicdiagram of additive impressions (following transformation) showing predic-tion for difference judgments (note that differences in transformed ratingsare assumed to represent psychological differences) C Matrix of hypo-thetical data representing ratings of the difference in hkableness betweenthe person described by the pair of adjectives for the column and the persondescribed by the pair for the row, assuming nonadditive impressions butsubtractive preferences D Matrix of hypothetical data for column minusrow differences, assuming additive impressions and subtractive preferences[Matrix entries for Panels C and D are equal to J(d) + S, where d is thedifference between the ordmate values for Panels A and B ]

model in Experiment III suggests that itcan be used to infer the impressions of thecombinations It is only necessary to as-sume that ratings of differences in likable-ness are strictly monotomc to differencesin psychological impressions If the inter-action is real, then the preference for a^b^over aj)i should exceed the preference fora-iia over ajti, as illustrated by the lengthof the dashed arrows in Figure 5A If theinteraction reflects only a nonlineanty inthe J function, however, then the 2 prefer-

ences should be equal, as shown by thedashed arrows of Figure SB.

The matrices m the lower panels, ofFigure 5 illustrate 2 patterns of data thatmight be obtained for ratings of the dif-ference in hkableness between 2 persons,each described by a pair of adjectivesFigure 5C shows the pattern that wouldbe obtained if the integration of informa-tion were nonadditive Each pair ofcircled entries is an example of a predictedrank-order comparison The dashed ar-


TABLE 1MEAN RAIINGS OF HIE DIFFERENCE IN LIKABLENESS BEiWEEN PAIRS OF HYPOTHETICAL PERSONS

Person on the left

Replicate 1PHONY & MEANPHONY & TRUSTWORTHYSINCERE & MEANSINCERE &. TRUSTWORTHY

Replicate 2DECEITFUL & CRUELDECEITFUL & DEPENDABLEHONEST & CRUELHONF.ST & DEPENDABLE

Replicate 3MALICIOUS & OBNOXIOUSMALICIOUS & UNDERSTANDINGLOYAL & OBNOXIOUSLOYAI & UNDERSTANDING

Person on the right

PHONY &MEAN

4913833 541 66

DECEITFUL &CRUEL

503

365389209

MAI ICIOUS &OBNOXIOUS

503

377358198

PHONY &TRUSTWORTHY

634

507489260

DECEITFUL &DEPENDABLE

6 15492526246

MALICIOUS &UNDERSTANDING

635500485255

SINCERE &MEAN

631494494251

HONEST &CRUEL

6264885062 12

LOYAL &OBNOXIOUS

6375 25506240

SINCERE &TRUSTWORTHY

792742746501

HONEST &DEPENDABLE

820

7 89795483

LOYAL &UNDERSTANDING

8207587625 12

rows represent the same comparison as thedashed arrows m Figure 5A Since theratings represent judgments of differencesbetween impressions labeled by the columnand the row entries, the direction of theorder is reversed below the diagonalFigure 5D represents the type of patternthat would be obtained if the adjectivescombined additively. Each pair of circledentries would be equal Both matrices inFigure 5 were generated by assuming thesubtractive model for preference (Equa-tion 2) However, the rank order of thematrix entries would remain the same ifsubjected to a strictly monotonic trans-formation Thus, the data can be rescaledto fit the subtractive model without pre-cluding the ordinal test for the additiveintegration models for impression forma-tion (Birnbaum, 1972b)

For example, if the interaction shownin Figure 2 is real, then the judged dif-ference in likableness between someonewho IS UNDERSTANDING & LOYAL andsomeone who is UNDERSTANDING & MALI-CIOUS should exceed the judged differencebetween someone who is OBNOXIOUS &LOYAL and someone who is OBNOXIOUS &

MALICIOUS But if adjectives were inte-grated according to an additive or con-stant-weight averaging model, then the 2differences would be equal

MethodThe 5s were instructed to judge the difference

in likableness between pairs of hypothetical persons,each of whom was described by 2 adjectivesTwo adjectives were printed on the left side ofthe page and 2 on the right side of the page The5s were instructed to form impressions of thepersonalities of the 2 persons before judging thedifference in likableness The 2 adjectives de-scribing each person were described as being ofequal accuracy and impoi tance, S's task was toimagine that they were contributed by different,but equally reliable, sources Judgments were interms of a 9-point scale, labeled as in the differencetask of Experiment 111

Subjects The & were 195 University of Cali-fornia, Los Angeles, undergraduates, 65 in eachof 3 stimulus replicates

Stimuh The stimulus for each judgment con-sisted of 4 adjectives, 2 printed on the left sideof the page and 2 on the right The adjectivepairs describing the "person on the left" wereproduced from a 3 X 3, A X B, factorial design,the adjective pairs on the right were constructedfrom a 2 X 2 , C X D, factorial design Eachperson on the left was combined with each personon the right producing a 3 X 3 X 2 X 2 , ( A X B ) X(C X D), factorial design


-2

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LEVEL OF ADJECTIVE A

,

FIGURE 6 Transformed impression values for the person on the leftderived from MONANOVA applied to difference ratings Experiment IV(The 3 levels of subscripts refer to low, medium, and high hkableness adjec-tives of Experiment I For example, 01 = PHONY, 02 = BLUNT, and 03= SINCERE, for Replicate 1 To assess metric information in mean ratings,compare this figure with Figure 2 )

The basic idea of the stimulus construction canbe understood by consideration of Figure 2 Forthe third replicate, the adjectives were Oi = MALI-CIOUS, a» = CHANGEABLE, aa - LOYAL, 61 = OB-NOXIOUS, 63 = SHY, b3 = UNDERSTANDING, Ci = MA-LICIOUS, ca = LOYAL, di = OBNOXIOUS, and di =UNDERSTANDING With respect to Figure SC, thelast entry in the first row (9) would represent therating of the difference in hkableness betweensomeone who is LOYAL & UNDERSTANDING andsomeone who is MALICIOUS & OBNOXIOUS Nineadditional cells in the design were constructed foreach replicate by pairing each person on the leftwith an additional person on the right who wasINFORMAL & LIGHTHEARTED Therefore, the entiredesign represents a 9 X 5 factorial design with thefactors representing the person on the left and theperson on the right (see Birnbaum, 1972b)

Procedure The 45 trials were printed in 1 of 3different random orders in booklets with differentpage orderings for different 5s The 5s readthrough the entire list before beginning to recordtheir judgments


The mean judgments of difference inHkableness are presented in Table 1 forthat part of the design illustrated inFigure 5 Comparison of the rank orderof the ratings with the direction of thedifferences for the circled values showsthat the pattern resembles that of Figure5C representing an interaction. Differences

in rating from Experiments I, II, and IIIpredict the ratings of differences in thepresent experiment

The crucial comparisons (circled entriesin Figure 5D), which must be equal forthe additive or constant-weight averagingmodel, are significantly in the directionpredicted by the interactions in Figures2-4 for all 3 replicates, Fs (1, 64) = 71 13,115.88, and 89 54, respectively. The indi-vidual ratings by single 5s show the samepattern of rank orders for these crucialcomparisons as the mean judgments in thetable Of the 195 5s, 81 were in perfectordinal agreement with the interactionobtained in the first 3 experiments for all4 comparisons. Only 12 5s showed agreater number of rankings in the direc-tion opposite from that predicted by theinteraction, compared with 170 5s whoserankings were in the direction predictedby the interactions obtained in Experi-ment I. These results are clearly incon-sistent with the additive model whichwould predict an equal split of theserankings between the 2 directions.

The larger, 9 X 5 design was used totest the subtractive model for preferencesbetween combinations. The subtractive


model also predicts parallelism, and thedata appeared roughly parallel when plot-ted However, statistically significant in-teractions were obtained for each stimulusreplicate, Fs (32, 2048) = 3 72, 4 98, and5 10 for Replicates 1, 2, and 3, respectivelyApplication of MONANOVA to theseratings of preference reduces the inter-action to less than 1% of the total vari-ance, so it is apparent that the preferenceratings are at least ordmally consistentwith the subtractive model

Following the application of MONA-NOVA to the 9 X 5 subtractive modeldesign, the derived impression valuesfor the combinations were inconsistentwith the additive and constant-weightaveraging models Figure 6 plots the im-pression values derived from the MONA-NOVA transformation for the 9 personson the left Since these were producedfrom a 3 X 3 factorial design of adjectives,the nonparallelism of these scale-free valuesrefutes the additive models of impressionformation. Similar results were obtainedfor the 2 X 2 design for persons on theright

In summary, the minimal assumptionthat the preference ratings are monotom-cally related to subjective differences leadsto the conclusion that the additive andconstant-weight averaging models must berejected Comparison of scale-free valuesin Figure 6 with the scale-dependentratings in Figures 2, 3, and 4 shows thatratings contain metric information that cor-rectly describes the divergent interaction

Krantz et al (1971, pp 445-447) haveargued that if an interaction can be re-moved by monotonic transformation, thenit should be attributed to the rating scaleand should not be given a psychologicalinterpretation However, Experiment IVshows that this procedure could lead toerroneous conclusions Although the in-teraction obtained in Experiment I couldbe removed by monotonic transformation,ratings of differences in impression arepredictable from the differences in ratingsThis indicates that the interactions are ofpsychological significance Rescahng thosedata to parallelism would be inappropriate,

as demonstrated by the qualitative con-tradictions of Experiment IV

However, the ratings also appear tocontain some nonlmeanty that reduces theapparent magnitude of the interaction fornonnegative items It would be improperreasoning to conclude that nonnegativeitems combine additively based on thelack of interaction in the ratings Thescale-free values in Figure 6 appear toshow a continued divergence Nonlmear-ity in the response scale may actuallymake nonadditive data appear additiveThe procedures of Experiment IV avoidthe rescahng problems and provide a trulyscale-free test of the additive models

GENERAL DISCUSSIONThis research demonstrates that impressions

of hkableness cannot be represented as simplesums or constant-weight averages of the valuesof the adjectives Instead, one bad traitresults in an unfavorable overall impressionwith the other trait having less influenceThis effect reflects a real psychological inter-action and cannot be attributed to the re-sponse scale

Some of the eaily research on impressionformation concluded that the constant-weightaveraging model could give a reasonable fitto ratings of adjective combinations Theparallelism prediction appeared to be roughlysatisfied, and this finding was interpreted tovalidate both the model and the rating scaleSystematic deviations did occur in this re-search, but were often attributed to experi-mental difficulties or nonhneanty of theresponse scale However, it is now clear thatthe constant-weight averaging model is notan appropriate general description, but maygive a reasonable approximation when thestimulus range is restricted and the experi-mental design lacks power 3

3 An alternative view might maintain that theconstant-weight model is descriptive of impressionformation, but only under limited experimentalcircumstances The E would be advised to care-ful ly select adjectives, instructions, response pro-cedures, and other conditions to minimize thenonparallelism When the data appear parallel,this view contends that -E would have the rightto conclude that the parallelism jointly supportsthe constant-weight model and the response scaleHowever, this approach seems unsatisfactory forseveral reasons First, the present research showsthat different response procedures, "equal impor-


Theoretical Implications

It is useful to consider a set of conditionsthat yield the parallelism prediction to con-sider what nonparallelism might mean. Thefollowing conditions underlie the parallelismprediction of the averaging model (a) theintegration function is an averaging process;(6) the adjectives within each factor have equalabsolute weight, (c) the adjectives do notchange value nonlmearly in combination,nor do the weights depend upon the partic-ular stimulus configuration, and (d) the/ function is linear Based on a single ex-periment such as Experiment I, nonparallelismcould be interpreted as evidence to disproveat least one of the premises Without furtherconstraints, it would be impossible to specifywhether nonparallelism was due to a non-additive / process (Conditions a-c) or a non-linear J function (Condition d) These ex-periments provide the leverage to indicate

-that the nonparallelism is not attributable tothe / function, There are several remainingpossibilities The integration function may notbe an averaging process, the stimulus pa-rameters may depend upon the configuration,or the weights may not be equal for theadjectives

The differential-weight averaging model(Anderson, 1971, 1972, 1974) can account forthe interactions by allowing the weight of anitem to depend upon its scale value Dif-ferential weighting requires many more pa-rameters and seems unnecessarily complicatedto fit the simple divergence interactions Themodel can account for a wide variety ofinteractions, making it difficult to disprove,However, with large enough designs, thereare degrees of freedom left over to permit

tance and accuracy" instructions, and different-selections of adjectives all yield similar divergentinteractions If the model is to be deemed cor-rect, then it must apply under very special con-ditions indeed Second, since it is possible tomanipulate the form of /, it follows that selectionof experimental procedures could lead to an "ex-perimental rescahng" of the data Thus, it should

_ be possible to select end anchors and filler stimulito reduce the interaction m the analysis of varianceTherefore, it would be inappropriate to select ex-perimental procedures that yield parallel data andthen conclude that the parallelism is joint supportfor model and response scale That parallelismcould result from a combination of nonadditive Iand nonlinear / is not merely an untestable philo-sophical possibility but represents a plausible hy-pothesis that can be tested by the procedures ofExperiment IV.

tests ( e g , Anderson, 1972; Birnbaum, 1973)Configural weighting (Birnbaum, 1972b), inwhich the weight of an item depends in parton its rank within the set, requires fewerparameters and gives a slightly better fit tothe data.4

The configural-weight model predicts steadydivergence for Figures 2, 3, 4, and 6. Thedifferential-weight model is more flexible andcould account for reconvergence, among otherpatterns It is thus of theoretical interest toask if the curves show any evidence of re-convergence The scale-dependent ratings ofFigure 2 appear nearly parallel for non-negative traits However, the scale-free valuesin Figure 6 show steady divergence for Repli-cates 1 and 3 The ratings may contain asmall scale-end effect that reduces the ap-parent interaction The bulge in Figure 2,Replicate 2 is apparently due to IMPULSIVEmultiplying the effect of the other adjective,rather than having less weight on its ownSince there is no evidence for more than asimple divergence, the existing data cannottest between differential and configural-weightmodels Methods for distinguishing thesemodels have been suggested by Birnbaum(1973)

Methodological Implications

Previous conceptualizations of the stimulusconcatenation problem have not separated the

4 An averaging mechanism can be analogized toa balance plank and fulcrum In the differential-weight model, each adjective corresponds to aweight (w) placed at a certain location (s) on theplank The integrated impression (*) is the loca-tion where the fulcrum must be placed so thatthe plank will balance This location is the weightedaverage, * = £w.ii/£)«>, The configural-weightaveraging model assumes that the weight of astimulus depends upon its rank within the set tobe judged For 2 stimuli, the simple range model(Birnbaum et al , 1971), *„ = ,5(s, + s,) +u Si — Sj |, can be rewritten as a configural-weightmodel, *„ = ( S + u)s, + (5 — u)sj, when s, > s,This model can also be represented by a balanceplank mechanism, however, the configural-weightmodel does not require each location on the plankto have its own weight For 2 stimuli in the set,there would be only 2 weights, one for the lowervalued item, and one for the higher valued item.The same weight can be placed at any locationon the plank, if it holds the same rank Thismodel becomes a minimum model when a = — 5,a constant-weight model when w = 0, and a maxi-mum model when w = 5, and can describe a familyof simple convergent or divergent interactions.

560 MICHAEL H B1RNBAUM

/ and / functions of Figuie 1 This articleoffers a conceptualization that separates theseproblems The propriety of rescalmg thedata has been uncertain in previous workFunctional and conjoint measurement (Ander-son, 1970, Krantz et al , 1971) both allowfor monotone transformation, but differ intheir outlook about when transformation isappropriate

Krantz et al (1971, pp 445-447) reanalyzedthe data of Sidowski and Anderson (1967) andconcluded that since the interaction could beeliminated by a monotonic transformation ofthe mean ratings, the original authors wereincorrect in attributing psychological signifi-cance to their findings An investigator fol-lowing this rescalmg procedure with the dataof Experiment I would have erroneously con-cluded that the additive model was an ap-propriate description of impression formationThe present research, by demonstrating thatdata transformation can lead to erroneousconclusions, provides a warning against thispractice

Anderson6 originally attempted to rescalethe data for ratings of the seventy of dis-turbed behaviors and to attribute the non-additivity to the rating scale, but he hasrecently reinterpreted the same data as evi-dence for the configurahty of clinical judg-ment by fitting these data to an averagingmodel with differential weights (Anderson,1972) The latter interpretation assumed thelinearity of the J function

The present research supports this attitudetoward rating scales, but it also providesscale-free constraints that make the inter-pretations of Krantz et al (1971) and Ander-son (1972, see, also, Footnote 5) matters forexperimentation rather than assumption

In Experiment IV it would be possible todistinguish multiplicative from additive mod-els on the basis of ordinal information (seeFigure 5) The simple view of conjoint scalingwould not allow this distinction, since a mono-tonic (logarithmic) transformation of a posi-tive product yields a sum By assuming thatratings of differences are monotomcally relatedto subjective differences, the / function can beseparated from the J function Experiment IVallows rescalmg, but rescalmg does not pre-clude the possibility of testing the / function

Although the latmg scale contains metric

5 N H Anderson Application of an additivemodel to impression formation Paper presentedat the third annual meeting of the PsychonomicSociety, St Louis, August 1962

information, it does not seem appropriate toassume that J is linear in every experiment,The / function can be predictably nonlinearIt depends upon contextual effects that areexplainable (Birnbaum et al , 1971, Parducci& Perrett, 1971) A nonlinear J functioncould result in 2 possible errors (a) non-additive data when the underlying impressionsare additive, or (6) additive data when theimpressions are nonadditive Thus, the deci-sion to rescale or not to rescale cannot beconvincingly settled without further con-straints, such as those applied in ExperimentsIII and IV

Conclusions

Impressions of likableness cannot be rep-resented as simple sums or averages of singlevalues of the adjectives They appear to bea predictable, but nonadditive function of thecomponent values The data show consistent,regular deviations from additivity that aresimilar for different selections of adjectivesWhen one adjective is dishkable, the personis rated as dishkable, and variation of theother trait has less effect

Differential or configural weighting of themore dishkable traits can account for theinteractions Representation of the stimuliby distributions could explain why the ad-jectives are integrated by a nonadditive func-tion It is less likely for a person possessinga dishkable trait to be likable than for aperson with a likable trait to be dishkable

Finally, this research illustrates that ratingscales contain metric information that shouldnot be uncritically scaled away. The proce-dures employed in Experiments III and IVprovide constraints that allow a model to betested without having to assume the validityof the rating scale

REFERENCES

ANDERSON, N H Application of an additive modelto impression formation Science, 1962, 138,817-818

ANDERSON, N H Likableness ratings of SSS per-sonality-trait words Journal of Personality andSocial Psychology, 1968, 9, 272-279 (a)

ANDERSON, N H A simple model for informationintegration In R P Abelson, E Aronson,W J McGuire, T M Newcomb, M J Rosen-berg, & P H Tannenbaum (Eds), Theories ofcognitive consistency A sourcebook ChicagoRand McNally, 1968 (b)

ANDERSON, N H Functional measurement andpsychophysical judgment Psychological Review,1970, 77, 153-170


ANDERSON, N H, Integration theory and attitudechange. Psychological Renew, 1971, 78, 171-206

ANDERSON, N H Looking for configurahty inclinical judgment Psychological Bulletin, 1972,78, 93-102

ANDERSON, N H Cognitive algebra Integrationtheory applied to social attribution In LBerkowitz (Ed ), Advances in experimental socialpsychology Vol 7 New York AcademicPress, 1974

ANDERSON, N H Information integration theoryA brief survey In D H Krantz, R C Atkin-son, R D. Luce, & P Suppes (Eds), Contem-porary developments in mathematical psychologyVol. 2 New York' Academic Press, in press.

ASCH, S E Forming impressions of personalityJournal of Abnormal and Social Psychology, 1946,41, 258-290

BIRNBAUM, M H Morality judgments Tests ofan averaging model. Journal of ExperimentalPsychology, 1972, 93, 35-42 (a)

JBIRNBAUM, M H The nonadditivity of impres-sions Unpublished doctoral dissertation, Uni-versity of California, Los Angeles, 1972 (b)

BIRNBAUM, M H Morality judgment Tests ofan averaging model with differential weightsJournal of Experimental Psychology, 1973, 99,395-399

BIRNBAUM, M H , PARDUCCI, A , & GIFFORD, R KContextual effects in information integrationJournal of Experimental Psychology, 1971, 88,158-170

BIRNBAUM, M H , & VEIT, C T Scale conver-gence as a criterion for rescaling Information

integration with difference, ratio, and averagingtasks. Perception & Psychophysics, 1974, IS, 1-9

GARNER, W R , HAKE, H W , & ERIKSEN, C WOperatiomsm and the concept of perception.Psychological Renew, 1956, 63, 149-159

KANOUSE, D E , & HANSON, L R , JR Negativityin evaluations Mornstown, N J . General Learn-ing Press, 1972,

KRANTZ, D H , LUCE, R D,, SUPPES, P., &TVERSKY, A Foundations of measurementNew York Academic Press, 1971

KRANTZ, D H , & TVERSKY, A Conjoint mea-surement analysis of composition rules in psy-chology Psychological Review, 1971, 78, 151-169

KRUSKAL, J B Analysis of factorial experimentsby estimating monotone transformations of thedata Journal of the Royal Statistical Society(Series B), 1965, 27, 251-263

KRUSKAL, J B , & CARMONE, F J MONANOVA •A FORTRAN-IV program for monotone analysisof variance Behavioral Science, 1969,14, 165-166

LAMPEL, A K , & ANDERSON, N H Combiningvisual and verbal information in an impression-formation task Journal of Personality and SocialPsychology, 1968, 9, 1-6

PARDUCCI, A , & PERRETT, L F Category ratingscales Effects of relative spacing and frequencyof stimulus values Journal of ExperimentalPsychology Monograph, 1971, 89, 427-452

SIDOWSKI, J B , & ANDERSON, N H Judgmentsof city-occupation combinations PsychonomicScience, 1967, 7, 279-280

TORGERSON, W S Theory and methods of scalingNew York Wiley, 1958

(Received November 16, 1972)

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