contextual effects in information integration

Journal ol Experimental Psychology1971, Vol. 88, No. 2, 158-170

CONTEXTUAL EFFECTS IN INFORMATION INTEGRATION

MICHAEL H. BIRNBAUM,* ALLEN PARDUCCI, AND ROBERT K. GIFFORD

University of California, Los Angeles

Category judgments of the average lengths of sets of lines were inconsistentwith context-independent models of information integration: the effects of anyparticular line upon the judgment of average length varied inversely with thelengths of the other lines within the same set. This interaction, obtained infive separate experiments, was similar to that previously reported for auditoryintensities. The judgments reflect two kinds of contextual effects: (a) within-set effects, in which the judgment of the set varies directly with the range ofvalues within the set, and (b) between-set effects, in which the apparent inter-action between the stimuli within a set depends upon the context provided bythe different sets. A simple range model provides a method for separating thetwo types of contextual effects. The context between sets is postulated toaffect only the response scale; when the responses are rescaled to allow for thebetween-set context, the integrated impression is dependent upon both themean and the range of components within the set.

The term "information integration"refers to the process whereby the psycho-logical values of several stimuli are com-bined to produce a single impression. Forexample, 5s have been asked to judge theoverall loudness of four bursts of noise ofvaried intensity (Parducci, Thaler, &Anderson, 1968). Additive models of in-formation integration assume that the inte-grated impression is simply a weighted sum(or average) of the values associated witheach of the component stimuli comprisingthe set. In the usual application of suchmodels, the effect of each componentstimulus is assumed to be independent ofthe other stimuli. The term "additivemodels" will be used here to represent thisassumption of independence from context.

In apparent contradiction to additivemodels, the loudness study demonstratedthat the effect of the intensity at any oneserial position within the set was greaterwhen the intensities in the other three serialpositions were lower. This finding was also

1 This research was supported by United StatesPublic Health Service Grant HD-00923, with thecomputing assistance of the University of Cali-fornia, Los Angeles Health Sciences ComputingFacility, sponsored by National Institutes ofHealth Special Resources Grant RR-3. The authorswish to thank Norman H. Anderson, who made manyhelpful comments and suggestions on an earlier draft.

2 Requests for reprints should be sent to MichaelH. Birnbaum, Psychology Department, Universityof California, Los Angeles, California, 90024.

inconsistent with the results of information-integration studies using verbal descrip-tions of personality (Anderson, 1968b).Consequently, the first aim of the presentresearch was to determine whether thesewithin-set effects are peculiar to judgmentsof loudness.

EXPERIMENT I: REPLICATION WITHVISUAL LENGTH

This experiment followed the design ofthe loudness study as closely as possible;however, the stimuli composing each setwere lines of varying length presentedsimultaneously.

Method

Stimuli.-—Sets of four black lines were centered 13mm. apart and parallel to the longer side of 100X 150 mm. white cards.8 The length of the line ineach position was either 25, 38, 51, or 64 mm., witha separate card for each of the 256 permutations ofthese four lengths.

Subjects.—Eight University of California, LosAngeles, undergraduates with no prior experience inthis task were paid to serve as 5s.

Procedure.-—-The 5s were instructed to judge theaverage length of the four lines on each card, in com-parison with the averages for the sets of lines on theother cards. Judgments were made on a 6-pointscale from "1—Very Short to 6—Very Long." Eachsuccessive card was placed on the table before a

8 Stimulus values for all of these experiments havebeen converted from the original English units andconsequently are subject to rounding errors.

158

CONTEXT IN INFORMATION INTEGRATION 159

and removed as soon as he reported hisjudgment, usually within 5 sec. Each block of 64sets in the regular series was as representative aspossible of the total series. Order within blocks wasrandom. A warm-up of 18 sets was representativeof the total series with respect to physical averages,the different lengths appearing with equal frequency.Each 5 went through all 256 cards twice, once oneach of 2 successive days. Individual sessions took amaximum of 40 min.

Results and Discussion

Each panel of Fig. 1 plots the judgmentstabulated with respect to the lengths of thelines in two of the four positions. Position 1refers to the top line on a card, Position 4to the bottom line. Each point is a mean of256 judgments: one for each of the 16permutations of lengths for two of the posi-tions, on each of 2 days, by each of eight 5s.The upward trend of each curve representsthe effects of the length of the line in thelower of the two positions; the separationbetween the curves represents the effects ofthe length of the line in the upper position.

Deviations from parallelism representwithin-set contextual effects. These devia-tions are systematic and similar to theinteractions obtained in the previous ex-periment on judgments of loudness. Again,

the curves converge toward the right. Thegreater the length in one position, the lessthe effects of the length in the other posi-tion. This convergence characterizes theindividual data (graphed as in Fig. 1) foreach of the eight 5s.

The significance of these interactions wastested by an overall analysis of variance inwhich the positions were the independentfactors, each position having four levels(lengths) i n a 4 X 4 X 4 X 4 design. Themean squares for each two-way interactionand its error term are given in each of thepanels in Fig. 1. Since the critical valueof F (9, 63) at the .001 level is approxi-mately 3.8, the interactions between ad-jacent lines (Positions 1 and 2, 2 and 3, 3and 4) are in each case significant. Theother three two-way interactions are notsignificant (p > .05); of the five higherorder interactions, only the 1 X 2 X 3interaction is statistically significant(p < .01).

The similarity between the results of lineaveraging and those of the previous experi-ment with auditory intensities strengthensthe case against additive models. Thesemodels imply that the interactions in theanalysis of variance are zero (Anderson,

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LOWER POSITION STIMULUS

FIG. 1. Two-way data tables for each pair of positions. (Each data point is averagedover the 16 permutations of lengths at the other two positions.) (Exp. I)

160 M. H. BIRNBAUM, A. PARDUCCI, AND R. K. GIFFORD

1968b). Consequently, they are contra-dicted by the convergence found in thepresent experiment as well as in the pre-vious experiment with auditory stimuli.

EXPERIMENT II : THE WITHIN-SETCONTEXT

Several alternative interpretations havebeen suggested to explain the within-seteffects found for judgments of loudness(Parducci et al., 1968). One of theseassumes that implicit judgments of the com-ponents of each set are determined by thesame contextual conditions as overt judg-ments of single stimuli. In particular, theimplicit judgment of each component of aset is assumed to vary inversely with thelengths of the other components in the set.This is the contrast effect typically found injudgments of single stimuli. If the overtjudgment of the set were simply a weightedaverage of these implicit judgments, con-vergence interactions would be obtained,particularly if contrast were greater forlonger lines.

To test this interpretation directly, thepresent experiment required some 5s tomake separate judgments of each of thecomponents of the sets while other 5sjudged the average length of the set, as inExp. I. The component judgments shouldreflect the postulated contrast effectsdirectly, and the mean of the judgments ofthe components of a set should predict thejudgment of the average length of the set.

The number of components was reducedfrom four to two. This permitted an in-crease in the number of lengths for eachposition from four to eight without makingthe series too long for replication within asingle experimental session. Furthermore,the use of only two lines in a set eliminatesa number of complex interpretations, suchas the possibility that 5 ignores shorterlines in Positions 2 and 3 when longer linesare presented in Positions 1 and 4.

Method

Stimuli.—Sets of two black lines were centered 13mm. apart and parallel to the longer sides of 130

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A PREDICTEDRANGE MODEL

B

I 2 3 4 5 6 7 8 I 2 3 4 5 6 7 8

LOWER STIMULUS

FIG. 2. (a) Mean judgment of average length, plotted as joint function oflengths of lower and upper lines in the set; additive models predict parallelismfor these curves, and (b) predictions from the range model described in theDiscussion section, for the same conditions. (Exp. II)


X 200 mm. white cards. The eight lengths variedfrom 25 to 70 mm., in 6-mm. steps, with a separatecard for each of the 64 permutations. The cardswere exposed manually for approximately 5 sec.each, at a distance of from 2 to 4 m.

Subjects.—Eighty University of California, LosAngeles, undergraduates were run for single sessionsin eight groups of from 8 to 15 5s each. As in eachof the following experiments, 5s were fulfilling a re-quirement of the introductory course in psychology.No 5 served in more than one experiment or in morethan one experimental condition.

Procedure.—Four of the groups, the 44 5s judgingaverage length, were instructed to compare theaverage length of the two lines on each card with theaverages of the lengths on the other cards, as inExp. I. These 5s made a single overall judgmentfor each card. The rest, the 36 5s judging the indi-vidual components, were instructed to "judge thelength of each line compared to all the other lines,not just relative to the other line on the same card";they were thus required to make two separate judg-ments for each card, judging the upper line first,then the lower line. In both conditions, 5s recordednumerical judgments for categories running from"1—Very Very Short" to "9—Very Very Long."After a warm-up of 18 representative presentations,the 64 cards were presented in random order andthen again in reverse order. A different order wasused for each of the four groups in each condition.


Figure 2a plots the mean judgments ofthe "average length" for each set, tabu-lated as in Fig. 1. Again, there is a signifi-cant convergence to the right, F (49, 2107)= 8.02, p < .001. Most of the variance ofthe interaction (71%) is in the LinearX Linear trend. As in Exp. I, additiveand averaging models predict no inter-actions, i.e., parallel curves. This predic-tion is again contradicted by the converg-ence in Fig. 2a. Figure 2b represents theo-retical predictions from a range model thatwill be developed in the final discussionsection.

Figure 3 plots the separate judgment ofthe lower line as a function of the lengthof the upper line on the same card. Con-trary to hypothesis, the linear trend ofthese functions is positive rather than nega-tive ; judgments of the individual lines varydirectly rather than inversely with thelength of the other line on the same card,F (1, 35) = 39.42, p < .001. Figure 3suggests that this overall assimilation iscomplicated by a contrast for pairs of

UPPER STIMULUS

FIG. 3. Mean judgment of lower line as functionof length of upper line in same set. (Exp. II)

similar lengths. Thus, the most prominentdips in these functions occur where thelength of the lower stimulus is one stepbelow the length of the upper stimulus.The interaction is statistically significant,F (49, 1715) = 3.15, p < .001, with muchof the variance in the quadratic trends.Highly similar effects were found for judg-ments of the upper lines.

Figure 4 plots the mean of the two sepa-ate judgments for each pair. Although theinteraction is statistically significant,F (49, 2445) = 1.97, p < .001, it is smalland without the marked convergence ofFig. 2a. Convergence would have beenobtained if the component judgments, asrepresented in Fig. 3, had shown both con-trast and convergence to the right, i.e., acontrast that was greater for the longerlines. Neither the separate judgments northeir mean seem consistent with the con-trast interpretation. The difference be-tween the curves shown in Fig. 2a and 4 is

162 M. H. BIRNBAUM, A. PARDUCCI, AND R. If. GIFFORD

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LOWER STIMULUS

FIG. 4. Mean of the separate judgments of com-ponent lengths, plotted for each permutation as inFig. 2. (Exp. II)

inconsistent with the hypothesis that theaverage of component ratings would explainthe rating of the average.

It is possible that the request for compo-nent judgments reduces the importance ofthe other line in the same set. Anotherpossibility, consistent with previous data(Anderson, 1966; Anderson & Lampel,1965; Wyer & Dernier, 1968), is that overtjudgments of the components reflect a con-fusion by 5 between the component he issupposed to be judging and his overallimpression of the set. Although either ofthese possibilities might have obscured areal within-set contrast operating on im-plicit judgments, the fact that differences incomponent ratings are generally in thedirection of assimilation rather than con-trast encourages consideration of alterna-tive interpretations.

Figure 5 plots the judgments of the setsin Exp. II as a function of within-set range,i.e., the difference between the lengths of

the two components of each set. Thephysical lengths average to the same valuefor each point on the abscissa so that therange and mean of the lengths are not con-founded in Fig. 5. The linear componentof this function is statistically significant,F (1, 43) = 28.62, p < .001, judgmentsvarying directly with range. Similar resultswere obtained for judgments of the four-component sets of Exp. I. This suggeststhat the integrated impression dependsupon the within-set range.

EXPERIMENT III: THE WITHIN-SETRANGE EFFECT IN COMPARA-

TIVE JUDGMENT

In this experiment, the hypothesis of awithin-set range effect is tested directly byhaving 5s compare the average length of apair of lines with the length of a single line.

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5.8

5.7

5.6

5.5

5.4

5.3

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WITHIN-SET RANGE (CM)

FIG. S. Mean of judgments of those sets havingthe same range of lengths. (Each vertical extends± 1 SE.) (Exp. II)


The psychological average of each pair isthen the length of the single line to whichit is equated. The experimental designvaries the within-set range for differentpairs of lines while controlling for meanphysical length.

Method

Stimuli.—Nine standards and 12 comparisonstimuli were used. The standards were single linesdrawn lengthwise and centered on 130 X 200 mm.white cards; their lengths varied from 41 to 54 mm.,in 1.6-mm. steps. The comparison pairs werechosen from the sets used in Exp. II, with each pairagain on a separate card. For each of three meanlengths (44, 48, and 51 mm.), there were four com-parison pairs differing in within-set range: (44, 44),(38,50), (31,57), (25,63); (45,51), (38,58),(32,64), (26,70); (51,51), (45,57), (38,64),(32, 70).

Subjects.—AM 108 permutations of the standardand comparison stimuli were judged by each of 18 5s.

Procedure.—The task on each trial was to com-pare the average length of the two lines on the com-parison card with the length of the single line on thestandard card. The comparison categories on the6-point scale were; 1. Shorter; 2. Slightly Shorter;3, Very Slightly Shorter; 4. Very Slightly Longer;5. Slightly Longer; and 6. Longer. Each standardremained on a stand 2 to 4 m. from 5s, while the 12comparison sets were placed next to it, 1 at a time,for approximately 5 sec. each. Different randomorders of standards and comparisons were used foreach squad of nine 5s.

Results

Figure 6 plots the mean judgments ofeach of the 12 comparison pairs as a functionof their difference in length (within-setrange), but averaged over the nine stand-ards. Each curve represents the judgmentsof 4 comparison pairs with the samephysical mean. It is clear that these com-parative judgments increase directly withdifference in length, even when meanphysical length is held constant. Thelinear trend of these functions is significant,F (1, 17) = 6.64, p < .05, which providesdirect evidence for the within-set rangeeffect inferred from the data of Exp. Iand II.

The direction of the within-set rangeeffect appears inconsistent with the averag-ing process described by the basic adapta-tion-level equations (Helson, 1964). Theseequate adaptation level or the psychologi-

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FIG. 6. Mean judgment of comparison set asjoint function of range of lengths within set andmean length of components. (Exp. Ill)

cal average to the geometric mean of thestimulus values. But since geometric meanvaries inversely with range when arithmeticmean is held constant, the adaptation-levelapproach incorrectly implies that the func-tions in Fig. 6 should have negative sloperather than positive slope.

EXPERIMENT IV: TEST OF THE WITHIN-SET RANGE EFFECT WITH

CATEGORY RATINGS

The comparative judgments of Exp. Illcan also be analyzed with respect to thetraditional psychophysical measure of cen-tral tendency, the point of subjectiveequality (PSE). This is the value of thestandard to which each comparison setwould have been equated (with the inter-polated standard value corresponding to3.5 on the 6-point scale of comparativejudgment). The following equation de-scribes the relationship of PSE to the physi-


cal values and range of the comparison set:

PSE = .5(*L + *B) + .25 (*L - *„), [1]

where $L and $s are the lengths of thelonger and shorter of the two lines in thecomparison set, and .25 is an empiricallyfitted constant reflecting the importance ofthe within-set range effect.

Insofar as Equation 1 provides a usefulcharacterization of the contextual effectswithin sets, then category ratings should bethe same for all line pairs having the samePSE. In particular, for each pair of lineswith unequal lengths, there is a pair oflines of equal length which should receivethe same judgment. To test this implica-tion, two different series of stimuli are em-ployed in the present experiment: (a) a5 X 5 factorial series and (6) an equal-component series in which the length of eachpair corresponds to the PSE for one of thesets in the factorial series. Insofar as thewithin-set effects are accounted for by thePSE equation from Exp. Ill, both seriesshould have the same context between sets.Consequently, the judgments should be thesame for corresponding sets in the twoseries.

Method

Stimuli.-—The factorial series was composed ofpairs of lines, arranged as in Exp. II but with eachline taking only five values (in 10-mm. steps from25 to 65 mm.). In the equal-component series, bothlines in the pair had the same length. This lengthwas the value of PSE from Equation 1 for the cor-responding set in the factorial series.

Subjects.—Four groups (29 5s) judged the fac-torial series first, then the equal-component series;the remaining three groups (19 5s) judged the equalcomponent series first, then the factorial series.

Procedure.—Each series was preceded by a warm-up of 12 presentations. After the 25 sets in eachseries had been presented in random order and againin reverse order, 5s were told,

Now there is another series of cards. Your taskremains the same, i.e., to judge the average lengthof each set compared with the average length ofthe other sets; but now the two lines on each cardwill always be of equal length (or "may have dif-ferent lengths").

The first replicate of each stimulus type wastreated as additional warm-up, the regular warm-upbeing insufficient to prevent strong ordinal effectsfor some of the stimuli presented near the beginning

of the series. Since tests for transfer between thetwo series proved insignificant, a within-5s designwas employed in analyzing the results.


Comparison of Fig. 7a and 7b showsobvious similarity of form, verified by thenonsignificance of the Series X UpperX Lower interaction, F (16, 576) = 1.50,p > .10. Deviations from the best fit forthe additive models can be represented bythe sum of squares associated with theUpper X Lower interaction, which is al-most six times as great as the sum of squaresfor the Series X Upper X Lower inter-action. The nonsignificance of the othersources of variance associated with series isconsistent with the prediction that theequal-component sets would receive thesame judgments as the corresponding fac-torial sets. The results thus support theinterpretation that apart from between-seteffects, which were presumably the samefor both series, interactions are determinedby the within-set range as described byEquation 1.

EXPERIMENT V: THE CONTEXTBETWEEN SETS

In addition to the contextual effects asso-ciated with the range of stimuli within sets,interactions may also depend upon thecontext between sets. Previous researchhas shown that sets are assigned highercategories when the frequency distributionof their physical means is positivelyskewed, i.e., when the sets with lowerphysical means are presented more fre-quently (Parducci et al., 1968). In thisrespect, the judgments of sets are like thoseof single stimuli. A general finding withsingle stimuli is that when the frequencydistribution of the stimuli is negativelyskewed, the judgment function is positivelyaccelerated; when the frequency distribu-tion is positively skewed, the function isnegatively accelerated (Parducci, 1965).

This suggests that experimental manipu-lation of the frequency distribution of setswould affect the form of the plot of thejudgments of the sets against their physical


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FIG. 7. Mean judgment of average length of each set. Coordinates forequal-component sets represent upper and lower lines for the factorial setwhose PSE (calculated from Equation 1) corresponds to the lengths of theequal-component set. (Exp. IV)

means. Systematic nonlinearity in thisplot implies interactions in the analysis ofvariance of the responses, assuming homo-geneity of response to sets with the samephysical mean. For example, if the judg-ment of a set with Physical Values 1 and 5were closer to the judgment of a set bothof whose physical values were 5 than to aset whose values were both 1, the judgmentswould show the convergent interaction ob-tained in Exp. I and II.

Apart from within-set'contextual effects,any nonlinear change in the function re-lating the judgment of the set to the physi-cal mean of the stimuli in the set must beparalleled by a change in this interaction.This would ordinarily be interpreted as evi-dence against the additive models. Thechange would be described as a "between-set effect" since it would be produced bymanipulation of the frequency distributionof sets.

Experiment V manipulates the contextby adding extra sets to those whose judg-

ments are analyzed for interactions. Fillersets have been used for various purposes(e.g., Anderson & Jacobson, 1965), includ-ing an effort to break up whatever patternmay be present in the experimental sets.It is possible that the use of filler sets mayaffect or even obscure the interactions thatmight otherwise be obtained in research oninformation integration.

Method

Stimuli.—The sets from the 5 X 5 factorial seriesof Exp. IV were used as the regular stimuli. Inaddition to these 25 sets, 25 filler pairs were addedfor each of two contextual conditions. The meansof the lengths of the filler pairs in the Low- and High-Filler conditions were approximately 38 and 52 mm.,respectively. In the Low-Filler condition, thelengths in each filler pair (inside parentheses) andtheir frequencies of presentation were as follows:(35, 35)3, (25, 55)6, (35, 45)6, and (25, 45)10. Inthe High-Filler condition, these filler values were:(55, 55)3, (35, 65)6, (45, 55)6, and (45, 65)10.

Subjects.—The 30 5s in the High-Filler conditionand the 22 in the Low-Filler condition were run ingroups of from 8 to 11 5s.


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LOWER STIMULUS

Mean judgment of average length for Low-Filler andHigh-Filler contexts. (Exp. V)

Procedure.—Instructions and method of presenta-tion followed the procedure used for the condition ofExp. II that required judgments of the averagelength of each set. The entire series was presentedin random sequence, and then in reverse, with adifferent sequence for each group.


The mean judgments of the 25 sets com-mon to both conditions are shown in Fig. 8.Convergence is much less for the High-Filler condition (Fig. 8&), than for theLow-Filler condition (Fig. 80), and theFiller Context X Length of Upper Line XLength of Lower Line interaction is sta-tistically significant, F (16, 672) = 2.37,p < .01. Although the Upper X Lowerinteraction is also significant for the High-Filler data analyzed separately, F (16, 464)= 2.92, p < .001, the shape of the inter-action is clearly dependent upon the contextestablished by the filler sets.

The two functions in Fig. 9 have differentforms, and the Stimulus Mean X FillerContext interaction is highly significant,F (8, 336) = 5.79, p < .001, with about

half of the variance in the quadratic trend.The Low-Filler condition has a negativelyaccelerated function with a highly signifi-cant quadratic trend, F (1, 21) = 31.83,p < .001, whereas the High-Filler functionexhibits a slight but statistically insignifi-cant positive acceleration. It is this non-linearity that implies the interactionsshown in Fig. 8.

When the data are segregated by within-set range (as for Fig. 5), the range variableis again significant, F (4, 42) = 18.52,p < .001. Its interaction with filler con-text is not significant, F (4, 42) = 1.56,p > .10. There is thus no evidence thatthe within-set range effect itself dependsupon the context between sets. This sug-gests that the between-set context has anindependent influence upon the interactions.

The major finding of Exp. V is that thestatistical interactions ordinarily inter-preted as within-set contextual effects arealso determined by the context betweensets. Although only the filler sets weremanipulated in this experiment, it seems


likely that the interactions in Fig. 8 werealso influenced by the particular valuesselected for the regular factorial series.Thus, the context between sets must betaken into account in any interpretation ofeither the presence or absence of this typeof interaction.

DISCUSSION

Figure 10 provides a general framework fordiscussing different interpretations of the data.In this framework, the physical values of thecomponent stimuli, $*, are transformed topsychological values, st, by the psychophysicalfunction, H. These psychological values arecombined by the integration rule, 7, to formthe overall impression, &i . . . * . . . n, which isthen transformed into the overt response,RI . . . * . . . «, by the judgment function, /.

Additive models.—Additive models of infor-mation integration assume that the psycho-logical values are added or averaged to form anintegrated impression. The best supported ofthese models (Anderson, 1968b) asserts thefollowing function for I:

[2]

in which MI& is the weight of whatever stimulusis in Position k, and Sk is the psychologicalvalue of the stimulus in that position. In theusual application, it is assumed that theweights are independent of the stimulus valuesin the set, and the effects of each stimulus areassumed to be independent of the other stimuliwith which it is combined. Although it is theovert response, R, rather than the psychologi-cal impression, ty, that is actually observed inexperiments on information integration, the

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MEAN LENGTH (CM)

FIG. 9. Mean judgment of average length as func-tion of mean length of components. (Exp. V)

relationship, /, between these two variables isusually assumed to be linear. These assump-tions permit the use of analysis of variance on

Stimuli H —»- Psychological—I -*- IntegratedValues Impression

JudgmentalResponse

FIG. 10. Framework outline of information integration analysis.


the overt responses to test the additive models(Anderson, 1968b).

The convergence interaction obtained ineach of the present experiments constitutesevidence against this class of additive models.The simplest interpretation is that the inte-gration of information, at least in the presentsituations, is not simply an additive or aver-aging process. The effect of any line in the seton the overall impression depends on the otherlines in the same set.

A simple additive or averaging rule might bepreserved for / at the expense of some compli-cation of other features of Fig. 10. It is un-necessary, for example, to assume that J is alinear function. Indeed, if it were the rawphysical values that were averaged (or if Hwere a linear function), J might be expectedto take the logarithmic form that describes thetypical empirical relationship between categoryjudgments and physical magnitudes of singlestimuli (Stevens & Galanter, 1957). With theactual integration still conceived as an addi-tive process, an antilog transform of the re-sponses could account for part of the con-vergence interactions (Parducci et al., 1968).However, this possibility is contradicted bythe finding (Exp. I and II) that judgments ofsets with the same physical average varydirectly with the within-set range. The within-set effect is also found when the task requiresdirect comparison of each set with a standard(Exp. III).

The possibility of a nonlinear function re-lating the integrated impression to the overtresponse need not be tied to physical averag-ing. If it were assumed that the psychologicalvalues, Sk, were averaged in accordance withthe weighted-averaging model (see Equa-tion 2), the elimination of interactions wouldbe the appropriate criterion for evaluatingalternative candidates for / (Anderson, 1970).However, the demonstration (Exp. V) thatchanging the filler sets also changes the inter-action indicates that J would itself be subjectto contextual effects. Even if there were nofiller sets, J might be expected to depend onthe other sets in the series presented for judg-ment, just as the form of the judgment func-tion for single stimuli depends on the otherstimuli in the series (Parducci, 1965).

Contextual effects might also precede theintegration process. Thus, the psychologicalscale value of a component stimulus, Sk, mightdepend not just on its own physical value, $t,but also on the other components, $1, $2, . . .,$„, of the same set. The integrated impression

might then be a weighted average of the con-textually determined scale values. However,the judgments of the components in Exp. IIdid not average to the overall judgment of theset.

Another possibility, one that has seemedattractive for interpreting certain interactionswith stimuli that are more complex (e.g.,Lampel & Anderson, 1968), attributes theinteraction to changes in the weights ratherthan to changes in the scale values: the valueof wic in Equation 2 might depend upon thevalue of Sk. For example, the weight or rela-tive influence of a particular line in the setmight be directly proportional to its length.Preliminary attempts to fit the present databy permitting the weights to vary in thismanner have not been encouraging.

A range model.—The data from these ex-periments can be described more simply byassuming a nonadditive integration of physicalvalues. The integration function, 7, is ex-pressed in terms of the physical values, by-passing the need for H; and the integratedimpression, ^f, is assumed to vary directly withthe range of the set. Thus, the within-setrange effect is ascribed to the integrationprocess.

The effect of the context between sets isattributed by this model to the final transfor-mation, /. Once ̂ is achieved for a particularset, it is compared with the values of "9 for theother sets in the series. These different ¥values constitute a frequency distribution inwhich sets with greater within-set ranges haverelatively higher values. The judgmentassigned to each value in this distributionpresumably follows the principles describingjudgments of single stimuli. The general trendof the differences between the Low- and High-Filler conditions of Exp. V are consistent withthe contextual effects found with single stimuli(Parducci, 1965), as were the between-seteffects in the experiment with auditory stimuli.

Equation 1, which was fitted to the compara-tive judgments of Exp. Ill, was a preliminarycandidate for the integration function, I. Thecross-validation of Equation 1 in Exp. IVdemonstrated that the within-set range couldaccount for the interaction. However, themuch more extensive explorations of the otherexperiments suggest a modification of Equa-tion 1. In particular, a post hoc analysis ofExp. II (by connecting all points in Fig. la,representing judgments of sets with the samephysical mean) suggests that the effect of


within-set range is inversely proportional tothe mean of the components.

Accordingly, Equation 1 can be modified asfollows to provide a better account of thewithin-set range effects :

), [3]

in which M^ is the psychological average of thelonger and shorter of the component lengths,$L and $s respectively, and o> is an empiricalconstant representing the magnitude of thewithin-set range effect.

The value of co and also the relationshipbetween ^ and the actual judgment are esti-mated from empirical data. First, the judg-ments of those pairs with equal components(and thus no within-set range) are fitted usingthe least-squares criterion for the cubicfunction :

R = 8.

The inverse function for Equation 4 is thenapplied to transform the judgment of each setinto its corresponding ¥ value. The purposeof this is to allow for between-set effects.These ¥ values and the physical lengths foreach pair are then substituted in Equation 3to obtain the best least-squares estimate of co.This fit to the data requires the estimation ofonly the four parameters of Equation 4, inaddition to co. For the data of Fig. la, thesefive parameters (with co = .823) yield the pre-dictions shown in Fig. 2b. The correspondencebetween Fig. 2a and 2b represents a markedimprovement over the fit obtained for the addi-tive models, even when the judgments are firstrescaled using Equation 4 and the usualanalysis of variance procedures are employedto fit these same data (i.e., with parallel lines).The variance left unexplained by the additivemodels is closely associated with the rangeterm from Equation 3. The range model doessignificantly better than additive models,F (1, 61) = 60.37, p < .001, reducing the un-explained variance to less than one-third of thevariance left unexplained by the additivemodels. An even greater improvement of fitis obtained for the Low-Filler condition ofExp. V. Additive models do a better job onlyon the High-Filler condition where their fit isvery good. However, when co is estimatedseparately for each physical mean, the rangemodel reduces the error variance of the High-Filler condition to one-half of that from theadditive models, still using fewer parameters.

It should be emphasized that the effect ofbetween-set context in the range model is esti-mated solely from sets with equal components.

Equation 4 is assumed to be the function,/, that transforms the psychological impres-sions to the responses. Application of 7"1 canbe considered a method for rescaling the re-sponses. The fact that significant interactionsremained after this rescaling suggests that thewithin-set range effects are "real" and not anartifact of the response scale.

Concluding comments.—There are numerousdifferences in procedure between the presentexperiments and those that are more con-sistent with the weighted-averaging or additivemodels. For example, in order to isolate theintegration process from the problems ofshifting scales of judgment, the between-setcontext has sometimes been anchored by end-sets that are more extreme than the sets whosejudgments are analyzed in tests of thesemodels. Before judging the regular series, Smay be shown two extreme sets and told theyrepresent "1" and "20" on a 20-point scale.There is some evidence that such anchoringreduces the interactions (Anderson, 1967b).But insofar as variability of judgment for eachstimulus is proportional to the range of all thestimuli, the power of the test of the inter-actions is also reduced.

A recent experiment (Weiss & Anderson,1969) used the method of reproduction tostudy averaging of serially presented lines. Theinteractions were described as slight. Thismethod may reduce the effects of the between-set context, but in preliminary work4 we haveobtained large convergent interactions usingthe method of reproduction and stimuli similarto those of Exp. II. There were other differ-ences in procedure, including the use of onlytwo alternative lengths in each position in theWeiss and Anderson study. Earlier psycho-physical support for the weighted-averagingmodel used lifted weights, just two differentvalues for the component stimuli, and a 20-point rating scale (Anderson, 1967a). Al-though it would be difficult to pin down theeffects of the procedural differences, the presentexperiments give substantial evidence that theaveraging and additive models may hold onlyunder very restricted conditions.

Interactions reflecting either the between-set or within-set contexts do not appear to berestricted to judgments of lines and noisebursts. They are prominent in our unpub-

4 Birnbaum, M. Studies in information integra-tion, unpublished manuscript, 1967.


lished work on lifted weight, (see Footnote 4)on the favorableness of adjectives, and on themorality of acts of behavior. Evidence con-sistent with the within-set range effect is alsofound in published studies of information inte-gration (Anderson, 1965, 1968a; Lampel &Anderson, 1968; Weiss, 1963; Willis, 1960).Although some of the interactions reportedin these studies have been interpreted as shiftsin the weighting of different classes of stimuli,they may also be interpreted as within-setrange effects.

The present research demonstrates thatthere are conditions in which one must expectstrong contextual effects upon informationintegration. The effect of a particular compo-nent stimulus upon the judgment of a setdepends upon the range of component values.The judgment of the set also reflects the con-text established by the differences between sets.

REFERENCES

ANDERSON, N. H. Averaging versus adding as astimulus-combination rule in impression forma-tion. Journal of Experimental Psychology, 1965,70, 394-400.

ANDERSON, N. H. Component ratings in impressionformation. Psychonomic Science, 1966, 6,279-280.

ANDERSON, N. H. Application of a weightedaverage model to a psychophysical averaging task.Psychonomic Science, 1967, 8, 227-228. (a)

ANDERSON, N. H. Averaging model analysis of set-size effect in impression formation. Journal ofExperimental Psychology, 1967, 75, 158-165. (b)

ANDERSON, N. H. Application of a linear-serialmodel to a personality-impression task using serialpresentation. Journal of Personality and SocialPsychology, 1968, 10, 352-362. (a)

ANDERSON, N. H. A simple model for informationintegration. In R. P. Abelson et al., Theories of

cognitive consistency: A source book. Chicago:Rand McNally 1968. (b)

ANDERSON, N. H. Functional measurement andthe psychophysical law. Psychological Review,1970, 77, 153-170.

ANDERSON, N. H., & JACOBSON, A. Effect of stimu-lus inconsistency and discounting instructions inpersonality impression formation. Journal ofPersonality and Social Psychology, 1965, 2, 531-539.

ANDERSON, N. H., & LAMPEL, A. K. Effect ofcontext on ratings of personality traits. Psycho-nomic Science, 1965, 3, 433^34.

HELSON, H. Adaptation-level theory. New York:Harper & Row, 1964.

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. Category judgment: A range-fre-quency model. Psychological Review, 1965, 72,407-418.

PARDUCCI, A., THALER, H., & ANDERSON, N. H.Stimulus averaging and the context for judgment.Perception & Psychophysics, 1968, 3, 145-150.

STEVENS, S. S., & Galanter, E. H. Ratio scales andcategory scales for a dozen perceptual continua.Journal of Experimental Psychology, 1957, 54,377-411.

WEISS, D. J., & ANDERSON, N. H. Subjective aver-aging of length with serial presentation. Journalof Experimental Psychology, 1969, 82, 52-63.

WEISS, W. Scale judgments of triplets of opinionstatements. Journal of Abnormal and Social Psy-chology, 1963, 66, 471-479.

WILLIS, R. H. Stimulus pooling and social precep-tion. Journal of Abnormal and Social Psychology,1960, 60, 365-373.

WYER, R. S., & DERMER, M. Effect of context andinstructional set upon evaluations of personality,trait adjectives. Journal of Personality andSocial Psychology, 1968, 9, 7-14.

(Received June 15, 1970)

contextual effects in information integration

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