A new round robin analysis of variance for social interaction data

Journal of Personality and Social Psychology1979, Vol. 37, No. 10, 1742-1757

A New Round Robin Analysis of Variancefor Social Interaction Data

Rebecca M. WarnerUniversity of Miami, Coral Gables

Michael StotoHarvard University

David A. KennyUniversity of Connecticut

Experimental social psychology has dealt primarily with situations that are nottrue social interactions; in a typical study, a subject responds to a fixed, arti-ficial social stimulus such as a photograph, written description, or performanceby a confederate. Although these artificial social stimuli provide experimentalcontrol over independent variables and can be analyzed using the types ofstatistical models originally developed for nonsocial experimental research, theyprovide little or no information about the interactive aspects of social behav-ior—the reciprocity or mutual contingency of the behavior of interaction part-ners. This paper describes a nonexperimental design specifically tailored tosocial interaction data that provides more information about individual differ-ences and social influence in social interactions: a round robin design in whicheach person interacts with every other person. After a brief review of availablemodels, a new and more general model for the analysis of social interactiondata is presented, with an empirical demonstration using vocal activity data.

Experimental social psychology has oftenbeen limited to the study of artificial, one-sidedsocial situations in which the subject respondsto some fixed social stimulus created by theexperimenter; this has been the case even inresearch on person perception and interpersonalattraction, where it is especially apparentthat the behaviors and attitudes of personsare mutually contingent (i.e., A's liking forB affects B's liking for A, and vice versa).In research on subject reactions to fixedstimuli (rather than the reactions to otherpersons in the context of naturally occurringsocial interactions), information about thereciprocity of social behaviors is lost. More-over, the use of standardized, artificial stimulioften precludes having stimuli that arerepresentative of typical interactions. Thus,

Development of this model was supported in partby National Science Foundation Grant BNS 77-03271,David A. Kenny, principal investigator. Additionalfunds were provided by the Livingston Fund Award(Harvard Medical School).

Requests for reprints should be sent to Rebecca M.Warner, Department of Psychology, University ofMiami, Coral Gables, Florida 33124.

the generalizability of such studies may berather limited.

A round robin design is one in which allpossible pairs of subjects from some set ofsubjects interact (like the pairings formed in atennis tournament). This term was introducedby Gleason and Halperin (1975). For each per-son paired with every other person, an observa-tion is made of some social behavior (speechpattern, rating, or degree of attraction). Thusthe "treatments" to which each subject re-acts are the behaviors of other subjects.This design provides two kinds of informationabout social behavior—first, information aboutindividual differences among subjects (inspeech patterns, ratings, or other behaviors);second, information about the mutual influencethat interaction partners have on each other'sbehaviors (for instance, the tendency toreciprocate positive feelings or to match thedurations of certain kinds of pauses in speech).

Before introducing a new round robinanalysis of variance, two simpler designs forsocial interaction data will be briefly reviewed.Each of these designs provides partial informa-tion about social behavior, and the roundrobin design can be understood as a model

Copyright 1979 by the American Psychological Association, Inc. 0022-3514/79/3710-1742$00.75

1742

ROUND ROBIN ANOVA 1743

that integrates and elaborates on these twosimpler designs.

Intraclass Correlation

A common nonexperimental design forsocial interaction research is one in which anumber of different dyads are formed andeach subject's behavior toward one randomlyassigned partner is observed. An example isa study reported by Welkowitz, Cariffe, andFeldstein (1976) on the congruence of vocalactivity (i.e., the tendency for conversationpartners to match the mean durations of theirswitching pauses, the pauses that occur afterone person has stopped speaking and beforethe other person begins to speak). For eachsubject there is one observation, the meanswitching pause duration in seconds. Theobservations are arranged by dyads, asillustrated by the hypothetical data displayedin Table 1.

To determine the strength of the tendencyfor partners to match pause durations, it isnecessary to look at the correlation betweenthe two columns of data, but an ordinarycorrelation coefficient is inappropriate herebecause the data do not consist of orderedpairs, that is, the assignment of any particularobservation to the first or second column isarbitrary. This type of layout also occurs inexamining twin data, as in research on IQ;here also there is no basis for labeling onetwin X and the other twin X''. For this typeof data layout an intraclass correlation isrequired (Snedecor & Cochran, 1967, 294-296).The r intraclass is calculated slightly differentlyfrom the ordinary product-moment r; each

Table 1Hypothetical Data Layout for IntraclassCorrelation of Switching Pause Durations

Dyad X X'

.71.21.5

.91.1

Note. Each entry is a hypothetical mean switchingpause duration (in seconds). The sequence canof course be extended for Dyads 4, 5, 6, and soforth. X = switching pause duration for one of thespeakers; X' = switching pause duration for theother speaker.

Table 2Hypothetical Data Layout for an A nalysis ofVariance Design: Mean Proportion of TimeSpent Speaking by Doctors

Patients

Doctor

123

Quiet,depressed

23,436.645.9

Talkative,anxious

15.724.126.3

Note. Design used by Goldman-Eisler (1952).

pair of observations is counted twice whencomputing the covariance term in the numera-tor—once as X, X' and again as X', X.Instead of calculating the variances of the Xand X' groups separately, the denominatorconsists of the variance of all the observations.The r intraclass is just the ratio of these twoterms. The intraclass correlation providesinformation about the matching or reciprocityof behaviors between partners, but it ignoresindividual differences among persons.

Ordinary Two-Way Analysis of Variance

A second approach to the study of socialinteraction involves the partitioning of var-iances to look for individual subject differences.Individual persons (or groups of persons) aretreated as levels of subject factors in anordinary analysis of variance. When there issome asymmetry of roles (such as doctor/patient or interviewer/interviewee), the dataare easily translated into a two-way analysisof variance. An example of this design is astudy by Goldman-Eisler (1952); the hypo-thetical data in Table 2 represent the meanproportion of time each doctor spent talkingwhen paired with individual patients of twodifferent types. This design can be treated asa two-way analysis of variance, and compar-isons can be made of the following: individualdifferences among doctors in' talkativeness;differences in the activity levels elicited fromdoctors by the two different types of patients;and interactions between doctor and patienttype.

This design does deal with individualdifferences (both in the types of social behavior

1744 R. WARNER, D. KENNY, AND M. STOTO

Table 3Round Robin Data Layout

Subject

Subject

1 -^121

Xiw

— *m

2 JTsn —XmA 213

3 --^311 ^321"V VA 3 12 A 322V V^-A 313 " 323

4 ^4U ^42,

-A-412 -^422

-^413 -^423

A-U,•^132

^133

^231

A232

^233

^431

^432

^433

xlX ,X,X,X,X.

X3X,X3

—

—

12

3

12

3

12

3

Note. Xijk = a behavior of Person i toward Personj on Day k.

that individuals exhibit and in the types ofsocial behavior that groups of patients elicit);but it is less well suited to the study of socialinteractions among peers (i.e., social interac-tions that do not have a role asymmetrythat makes it easy to assign some individualsto the row factor and others to the columnfactor); furthermore, it provides no informa-tion about mutual influence of social behaviors.

Round Robin Analysis of Variance

Model Specification

Round robin designs make it possible toobtain both kinds of information about socialinteractions—individual differences and mu-tual contingency. Several round robin-typedesigns have been developed (Bechtel, 1971;Gleason & Halperin, 1975; Lev & Kinder,1957); all of these designs are applicable todata layouts in which each subject is pairedwith every other subject, but they differ withrespect to the assumptions that they have madeabout the kinds of nonindependence amongobservations. Since these models have devel-oped as a special application of the paired-comparisons design in psychophysical research,some of the assumptions of these modelsare rather restrictive or in other respects notideally suited for some of the special difficultiesthat arise in social psychological research.

The reason for developing a new model is totailor the round robin design more specificallyto the problems of social interaction and tomake available a model with less restrictiveassumptions about the type of dependenceamong observations. Depending on the researchproblem, however, one of these previouslysuggested models may be more appropriate;accordingly, the specifications of these othermodels will be described briefly later in thispaper to assist readers in choosing the bestmodel.

Round robin designs arise frequently insmall-groups research, where records are keptof the frequency or duration of various typesof acts that each group member addresses toeach other group member; in studies of personperception, where each person rates everyother person in a group; and such data maybe generated by creating all possible pairs froma small subject pool and studying theirbehavior in isolation or by collecting data ingroup settings.

Any social psychological dependent variablecan be used: summary information on speech,gaze, or body movements; frequency countsof the number of acts (aggressive, altruistic,or other); ratings, perceptions, or self-reportmeasures of utility—the amount of someresource (money, time, materials, or whatever)that each person gives to each other person.1

A round robin data layout is illustrated inTable 3, where X,,k is an observation of somesocial behavior of Person i toward Person jon Day or Time k.

The diagonal cells are empty, since ordinarilya person cannot be paired with himself orherself, and the matrix is asymmetric, sincethe behavior of Person i toward _;' is ordinarilydifferent from that of j to i, even though theymay be highly correlated. With some modifica-tions, this will be treated as an n X n X ranalysis of variance, where n is the number ofsubjects and r is the number of observationsmade on each dvad. This is a random effects

1 Note two limitations on appropriate data for theround robin analysis of variance—there should not bemany cells with zero entries (apart from the diagonalcells), and neither the row nor column sums should beconstant, as would occur if each subject allocatedlimited resources among other subjects (such asrank-order preferences or limited amounts of money).


model, that is, the set of n subjects is con-sidered to be a very small random sample froma very large population of subjects; subjectswere not selected to represent levels of somedimension (such as dominance). By treatingthis as a random effects model, we achievegreater generality. Formally stated, the modelis as follows :

a2 IB el

= M + «i j +

The a, term represents the contribution ofPerson i as an actor, a source of behaviors;for instance, in a study of the proportion oftime spent speaking in conversations, a;represents Person i's talkativeness. The jSyterm represents Person j's effect as a partner(i.e., the amount of talk that Person _;' tends toelicit from people when he/she is their partner).Note that these effects can be logicallydistinguished, although in the case of vocalactivity there is a negative correlation betweenthe proportion of time a person tends to talk,and the proportion of time he/she allows otherpersons to talk when he/she is their partner.For some other social variables, there mightconceivably be a positive or zero correlationbetween the subject-as-actor and subject-as-partner effects. The 7,-,- term is an interactioneffect, representing the special adjustmentwhich Person i makes in level of talkativenesswhen paired with Person j. As usual, tijkrepresents the error term, which picks upvariability in behavior at different times.

Looking at the layout in Table 3 may clarifythe distinction made between actor and partnereffect. The tth row mean represents the averagebehavior of Person i toward n — 1 differentpartners; thus the row variance indicateswhether there are clear-cut and consistentindividual differences among persons whentheir behavior is observed with a number ofdifferent partners. The jth column meanrepresents the average of behaviors of personswho have Person j as a partner; thus thecolumn variance indicates whether there aresignificant differences among individuals aspartners (that is, as social stimuli to whichother persons react). These two main effects,row (actor) and column (partner) are bothsubject variables, and they are in fact based

on the same set of subjects ; in developing ournew round robin model, we will distinguishthe actor and partner factors and allow thesetwo factors to be either positively or negativelycorrelated. Among the other round robindesigns, different approaches are used: Bechtel(1971) treats actor and partner factorsseparately and allows them to be positivelycorrelated; Lev and Kinder (1957) treatactor and partner as separate factors andrequire that they be independent; Bechtel(1967) and Gleason and Halperin (1975) treatactor and partner as equivalent and combinethem into a single subject factor.

This nonindependence between the row andcolumn factor is just one of the special consider-ations that necessitate the development of aspecial analysis of variance model; anotherspecial problem is the nonindependence ofpartner behaviors that comes about becausesocial behaviors are mutually contingent.This mutual contingency means that thebehavior of _;' to i is correlated with thebehavior of i to j; or, to put it another way,in the layout in Table 3, the cells across thediagonal from each other (the i,j and j,ipairs of cells) are correlated, either positivelyor negatively. Since this violates the ordinaryanalysis of variance assumptions, it requiresspecial adaptations in the specification of themodel and the derivation of expected meansquares.

Many of the basic assumptions aboutexpectations are unchanged : The expectationsof row, column, and interaction effects andthe expectation of the error are still zero.Formally, £(«,) = 0, Efa) = 0, £(7y) = 0,and E(tijk) — 0, for all i, j, and k (j ^ i),where E is the expectation operator. Theusual variance specifications are also made :

= <rv2;

and

£(«y*2) = o-,2, for all i, j, and k, (j ^ i).

The sources of nonindependence outlinedearlier (the correlation between row andcolumn factors and between pairs of cross-diagonal cells) must also be incorporated intothe model specifications. The covariancebetween Person i's effect as an actor («<) andPerson i's effect as a partner (/3,-) is formally


Table 4Summary of Round Robin Model Specifications

£(«<) = 0, £(ai2) = <ra* for all »

£(/3,) = 0, E03y2) = ,? for all jE(ya) = 0 for all i and j, j ^ iE(tijk) = 0 for all i, j, and k

*(«,/»-{&"•" n*iE(ya, yt'i') =

0

if t = »', j = j'if i' = j, j' - iotherwise

.2 if i = «', j = j', k = k'sff,2 if »' = j, j' = i, k = k1

O otherwise

stated as £(«,-, &) = p\aav$, hereafter referredto as the row and column covariance.

The covariance between pairs of cross-diagonal cells is more complex, since it consistsof two parts. First, it is reasonable to assumethat there is some correlation between 7,7and ?,-,- (that is, between i's special adjustmentto j and j's special adjustment to i). Thisnonzero covariance between the interactioneffects within dyads is represented as p2<TT

2;formally, the nonindependence assumptionrequires that we specify E(jij, yy.) = p2<rT

2.This term will be referred to as the cross-diagonal interaction covariance, and it willbe interpreted as evidence for some kind oflasting reciprocity of behaviors within dyads.Furthermore, it is likely that the error termsfor the behavior of social interaction partnerson a particular day or time will also be corre-lated; formally, E ( f i i k , tya) = p3<r<2; this willbe interpreted as evidence for a situation-specific reciprocity in partner behaviors. Allother covariances are assumed to be zero;the model specifications outlined in thissection are summarized in Table 4.

For some types of social interaction data,there may be a further source of nonindepen-dence among observations; in person percep-tion data, if Persons h and _;' discuss theirperceptions of Person i, then the h, i andj, i ratings may be correlated. In the vocalactivity data presented as an illustration inthis paper, this additional source of nonin-dependence is assumed not to occur, and somecare should be taken to minimize this type ofnonindependence in designing studies that areto be analyzed using round robin designs.

Before outlining the calculations and param-eter estimation procedures, we should notethe alternative model specifications brieflydescribed in Table 5; depending on the natureof the data, one of the previously developedmodels may provide more powerful means ofanalyzing some data. This new model has theleast restrictive assumptions of any of thesedesigns.

Calculation of Means and Mean Squares

The strategy adopted for the purpose ofdeveloping the new model was as follows:First, we set up formulae for the sample meansand mean squares; next, taking into accountthe empty diagonal cells and the nonindepen-dence assumptions included in the modelspecifications, we worked out the expectedmean squares; finally, we used the expectedmean square equations to solve for the variancecomponents of the model (e.g., o-a

2, a f , aj)in terms of the sample mean squares. Signif-icance testing on these variance componentswas done by jackknifing, since ordinary Fratios are not easily set up. After these pro-cedures have been outlined in detail, the modelwill be demonstrated using vocal activity data.

The computation of sample means isstraightforward; the only special considerationis the presence of empty cells in the diagonal.

Grand mean:

M... =

Row mean:

Mi.. =

Column mean:

^ y y v-» y

( 4 \ L—4 2m* * --• •"• ijk •n — 1) i j^ k

1r(n - 1) £

E E Xnk.

Cell mean:r(n —

-T Xr k

As usual, the estimate of the grand mean isM... . Because of the empty diagonal cell,Mi.. — M... is not an unbiased estimate ofthe row effect («<); the following argumentwill explain why the empty diagonal cellsproduce bias in the row means of this layout.


When you sum across row i, for instance, youget an average of Person i's behavior withn — 1 partners, not including self. This meansthat the estimate of Person i's talkativenessrelative to the talkativeness of the otherpersons in the sample is biased, since Person iis not observed with the same set of partnersthat all the other persons had. Since we canget an estimate of Person i's effect as a partner(j8,-), however, it is possible to correct for this"missing partner" bias. For instance, in thedata on proportion of time spent speaking inconversations, any particular observation (X,;)depends upon the talkativeness of i, theperson being observed, and the amount oftalk elicited by j, the partner. Typically, theamount of talk elicited by j depends on j'stalkativeness. If Person i is the most talkativeperson in the sample, the the fact that i lackshimself or herself as a partner means theestimate of his/her talkativeness is slightlyinflated, since he/she had the last talkativeset of partners of any individual in the sample.An analogous argument holds for bias in thecolumn means. The estimates for the roweffects and the column effects are as follows:

(n - I)2

n(n - 2)

(n - 1)'

(n - 1)«(n - 2)

(» ~ 1)w(w — 2)

( H - l )M... ,

(n-2)M... .

These equations make sense in light of theprevious discussion; to get an estimate ofPerson i's true row effect, it is necessary tocorrect for the "missing partner" bias byadjusting by some fraction of Person i'scolumn mean. As usual, once the row andcolumn effects are known, estimates of theinteraction effects can be obtained by subtrac-tion:

7,v = Ma. - M... - & - ft.

The calculations for mean square row andmean square column are straightforward, butthe coefficients are slightly different; notethat the total number of observations in the

design is rn(n — 1) and the number of observa-tions in a row or column is r(n — 1).

MS.column

(«-1) it(w--l)v

(n - 1) i

',-.. - M...)2.

'.,-. - M...)2.

Before going further, notice how the emptycells and the special nonindependence assump-tions affect the variance components thatare estimated by the row mean square. ForMi., and M... , we have:

M,.. = ju + «i —

M... = u + a. +(n- 1)8. + ?..

When these expressions are used to derivethe expected mean square for row effect, thespecial assumptions incorporated into themodel specifications mean that many cross-multiplication terms whose expected valueswould be zero in an ordinary analysis ofvariance now have nonzero expectations; forinstance, since on is correlated with /J., therewill be a nonzero covariance term picked upwhen the on term in the equation for Mi., ismultiplied by the /3. term in the equation forM... and their expectation is taken. Theexpected mean square for row in the roundrobin model includes the following terms.

EMSrr = r(n- l)«ra2 + T a? + raj

(n — 1;

— 2rpio-a<r0 —\n —

1

M- 1PsoY

where pio-aa0 is the Row X Column covariance,P2CTY

2 is the cross-diagonal interaction covar-iance, and p3<re

2 is the cross-diagonal errorcovariance.

The complete derivation for the expectedmean squares will not be presented here sinceit is rather lengthy; a copy of the derivationis available on request (Stoto, Kenny, &Warner, Note 1).

The number of untraditional variancecomponents included in this expected meansquare suggests that it will not be easy to findan appropriate error term that will make it


Table 5Summary of Round Robin Design Specifications

Study

Lev &Kinder(19S7)

Bechtel(1971)

Gleason &Halperin(1975)

Actor/partnerfactors

Separate andindependent

Separate andpositivelycorrelated

Combinedintosingle factor

Relationshipof crossdiagonal

cells

Independent

Positivelycorrelated


Relationshipof cells

within samerow or column

Independent

Independent


Significancetesting

F

F

Pseudo F

Special features

—

Tests for identity of row andcolumn factors, and symmetryof interaction effects

Split plot, with all subjectswithin each condition pairedround robin style; mainemphasis on treatment ratherthan subject variables

Warner,Kenny& Stoto(1979)

Separate, Positively Independent / —eitherpositivelyor negativelycorrelated

ornegativelycorrelated

(jackknifing)

possible to set up an F ratio to evaluate thesignificance of the row variance. Instead, wedecided to solve directly for estimates of thevariance components themselves; this requiresseven equations in seven unknowns. Someadditional sample mean squares are thereforeneeded, and the nature of these mean squarescan be guessed from the set of unknownsappearing in EMSrow. The new mean squaresare a mean square for row and column covar-iance (denoted MSBowXCol)»

a mean squarefor the covariance of cross-diagonal interactioneffects (denoted MSc^xceu) > an(^ a meansquare for the covariance of cross-diagonalerrors (denoted .ÊrrorXError)- Also, in settingup the sample mean squares, we have chosento calculate a mean square for cells that isnot the usual mean square for interaction;this was done because it is easier to evaluatethe expected mean square for cells. The setof mean squares to be computed for the roundrobin layout are as follows:

MS,cell

.n(n - 1) - 1 i &iE L (Ma. - M...Y,

MS,CellXCell

£ (Mi,: ~ M...)

X (Ma. - M...),

and

,n(n-X (XJilt - Ma.).

MS,column

The coefficients for the variance componentsin the seven expected mean square equationscorresponding to these sample mean squaresare summarized in Table 6.

We now have the information necessary tosolve for the estimates of the seven variancecomponents. Let M be the (7 X 1) vector of


CO|, M5cell,sample mean squares: (MSTOW, M5^•^RowXCoD •^'^CellXCelli -êrrori •Let V be the (7 X 1) vector of variancecomponents to be estimated :

Let C be the (7 X 7) matrix of coefficientsfrom the expected mean squares equations,as in Table 6.

By definition, M = CV. Therefore we cansolve for estimates of the variance componentsby multiplying both sides by the inverse ofthe coefficient matrix, to get V = C~' M.

One can of course consider various specialcases of the model as a a = <rf and pi = 1 ;P2 = 0;p2 = I;p2 = — I;p8 = 1; or pa = — 1.For any of the above assumptions one cansimply drop the relevant mean square andchange the coefficient matrix accordingly.In the case of a£ = af and pi = 1, a sensiblestrategy would be to combine MSnw and-^column. and dr°P ^5RowXColumn-

In order to ensure that all the parametersin the model are identified, it is necessary tohave a minimum of four subjects. The modelmay be applied where r = 1 (each dyad isobserved only once), but in this case it willnot be possible to distinguish the interactionvariance (o-7

2) from the error variance (<rf),or the cross-diagonal interaction covariance(p2<ry

2) from the cross-diagonal error covariance

some means of evaluating whether they differsignificantly from zero.2 A useful strategy isjackknifing' (Hosteller & Tukey, 1977). Theidea is to estimate the variance of a parameterby examining the empirical distribution ofestimates of that parameter (such as <7«2),since there is no simple way to determine thevariance of its theoretical distribution. For<7«2, for instance, this can be done as follows:First, create n different subsets of the data,each subset consisting of the full data matrix,omitting the data for one subject (i.e., onerow and the corresponding column). For eachof these leave-out-one subsets, we can estimatea value for a<?. Now, these n different estimatesof <ra

2 are not independent, since the subsetsoverlap in membership; but Hosteller andTukey have devised a way around thisproblem. A set of pseudoestimates is generatedfrom the leave-out-one estimates, and thepseudoestimates can be treated as if they wereindependent. Let Fall be the value of o-a

2 forthe whole group of n subjects; let FO> be thevalue of <ra

2 for the subset of n — 1 subjectscreated by omitting person j. Then thepseudoestimates are given by F*y = «(Fall)- (n - 1)F0), for j = 1, 2, . . . , « . The F,,pseudoestimate is a "fake" estimate of <ra

2,as if we had been able to estimate a,,2 basedon only subject _/ ; using this set of pseudo-estimates, we can estimate the standard error

Significance Testing! To see why F ratios are difficult to set up, consider

the expected mean squares associated with the ratioof mean square rows and mean square error. It is clear

Once estimates of the variance components that this ratio would not be an unbiased estimate ofhave been obtained, it is necessary to have the row effect variance.

Table 6Expected Mean Square Coefficients

MS

Row

Column

Cell

RowXColumn

Cell XCell

Error

Error XError

if ».«

n— I r(n — I)

n-l -L-n — 1

n-l -r

( 1) 1 '("-11

.(.-iXr-i) "<"~1)~ I

«(n-l)(r-l) 0

**

rn-l

r(n-l)

r(n-l) 'n(n-l)-l

— r

'(n-l)

0

0

,•,*

r

r

r

rn-1— r

0

0

„...,

-2r

-2r

-2r(n-l)n(«-l)-l

r[(M_l)i+l]n —I

2r(n-l)s

0

0

rHffy* <r«s pafff1

_ r j L_

' i 'n-l n-l

-' . I«(»-!)-! «(n-l)-l

-iÎ

r 1 ,

0 1 0

0 0 1


of aj- by the variance of the 7#, values anduse this to set up a / test (with n — 1 degreesof freedom) to evaluate whether a* = 0.Let y# be the mean of the F*j pseudovalues,and let s2 be the variance of the F*y pseudo-values. Then (nly^/s is distributed as /with n — 1 degrees of freedom if aa

2 = 0.This jackknifing procedure can be used to

test the significance of each of the varianceand covariance components estimated for themodel. For justification of the procedure andfurther examples of its applications, see Hos-teller and Tukey, 1977 (chap. 8).

Interpretation of the Variance and CovarianceComponents

Since the components of this model differfrom those of an ordinary analysis of variance,some comments on the meaning that thevariance components have for various typesof social interaction data may be helpful.Notice first that three variance components(actor, partner, and interaction) correspondto the components of an ordinary two-wayanalysis of variance and that three covariancecomponents (row and column, cross-diagonalinteraction, and cross-diagonal error) provideinformation somewhat similar to that of anintraclass correlation design.

A ctor effect. This factor is related to individ-ual differences among persons as sources ofsocial behaviors; depending on the dependentvariable, this factor can be renamed speaker,rater, perceiver, and so forth. Since eachindividual's behavior is observed over a numberof different partners, the round robin analysisof variance provides a fairly strict test forstability; the actor effect cannot be largeunless individuals are fairly consistent intheir behavior, regardless of the person withwhom they are paired.

Partner effect. This factor indicates whetherthere is a strong tendency for individuals toelicit particular types of behavior from otherpersons—for instance, high or low trait ratings,high or low level of attraction, high or lowamounts of speech activity, and so forth.

Notice that it is possible to have a strongactor effect and a weak partner effect (thisoccurs in the vocal activity data presented inthis paper as an illustration of the model)

or a strong partner effect and a weak actoreffect (as we have seen in preliminary workwith person perception data, where thesubjects are being trained to apply an objectiverating system; there are small differencesamong ratings given by different raters andlarge differences in the ratings received bydifferent individuals). Thus, although theyare often highly correlated, it is importantto stress again the logical distinction betweenthese factors, and in some situations, partic-ularly in person perception research, it isinteresting to compare the magnitude ofthese two main effects.

Interaction effect. This component relatesto the particular adjustment that each personmakes to each particular partner. In vocalactivity research, for instance, interactioneffect indicates whether Person i paired withPerson j consistently talks a little more (or alittle less) than would be expected on thebasis of i's talkativeness and j's tendency toelicit talk. This interaction effect also hasmeaning in person perception and attractionresearch—does Person i tend to like Person jmore than would be expected on the basis ofi's average liking and j's average tendencyto inspire liking? The interaction effect picksup any tendency to make adjustments thatare unique to each partner and hold up acrossrepeated pairings with that partner. In asense the interaction term measures what isunique to the interaction between partners.

Row and column covariance. If this covar-iance is very large relative to the row andcolumn covariances, and if it is positive, itsuggests that the row and column factors arenearly indistinguishable, in which case asingle factor design (one that combines actorand partner into a single subject factor) maybe more appropriate. In some situations, therow and column covariance may have asubstantive interpretation; if the dependentvariable is related to utility (for instance, thefavorableness of a rating or the amount ofsome resource given by one person to another),then the relationship between the row factor(amount Person i gives out, on the average)and the column factor (amount Person ireceives, on the average) may be viewed as akind of "equity" factor. Another theoreticalconstruct that comes close to describing this


factor in person perception is projection; ahigh row and column covariance could betaken to mean that the ratings a person givesto others are closely tied to his or her owntraits (as perceived by others). For the vocalactivity data, there is no simple interpretationof this covariance.

Cross-diagonal interaction covariance. Thisis one of the two dyadic reciprocity factors inthe round robin design; it should be distin-guished from the cross-diagonal error covar-iance, which picks up a different type ofreciprocity. A large cross-diagonal interactioncovariance means that for each of the possible(i, j] dyads, the systematic and enduringadjustment that i makes to _;' is correlatedwith the enduring adjustment that / makesto i. For interpersonal attraction data, itseems likely that this could be a strongrelationship; that is, if i always tends to like_;' more than expected, it seems probable thatin return j always tends to like i more thanexpected. The reciprocity picked up by thiscovariance is an enduring reciprocity, that is,a relationship between partner behaviors thatholds up across different times or situations.

Cross-diagonal error covariance. This is asituation-specific reciprocity effect. If thiscovariance is large, then the behaviors of thepartners are highly correlated in any particularsituation, although they may not necessarilybe related in the same way across differentsituations. For instance, in the vocal activitydata, we will find a large cross-diagonalerror covariance—since the proportion oftime available for speaking in any particularconversation is approximately zero sum, thevocal activity of partners in specific conversa-tions is highly negatively correlated. However,the reciprocity turns out not to be enduring—that is, although the sum of activity is approx-imately 1 in any particular conversation, theallocation of time between partners may beworked out differently in different conversa-tions—in one, Person i may do more thanhis or her share of the talking; in the next,Person j may do more than his or her share.A fanciful example illustrating the indepen-dence of the situation-specific and the enduringreciprocity factors might be found in a soapopera, where the attractions and antipathiesamong the characters may be highly correlated

and mutual in any one episode, but the patternof attractions may shift from one week to thenext—indicating strong situation-specific reci-procity and a lack of enduring reciprocity.

Summary

Clearly, the interpretation of the varianceand covariance components in the round robinmodel depends upon the nature of the depen-dent variable and the type of substantive ortheoretical questions that arise about thereciprocity of behaviors. The goal of thissection is to show the flexibility of the newround robin model; there are some readilyunderstandable parallels between well-knowntheoretical constructs and the factors in theround robin design. The following section willdemonstrate the application of the roundrobin design to vocal activity data.

Empirical Illustration

Various aspects of speech activity in con-versations and interviews have been singledout for study, for example, mean durationof vocalizations or pauses, probability ofinitiating or maintaining speech, and propor-tion of time spent speaking; these vocalactivity parameters are interrelated, andcertain aspects of vocal activity are highlyconsistent for individual speakers (Jaffe &Feldstein, 1970). Of these parameters, thesimplest one to study is the proportion of timespent speaking, or speaker activity level. Pastresearch on individual speaker consistency hasgenerally involved at most two or threedifferent partners; the round robin designprovides a natural method of examiningindividual differences across many differentpartners in order to evaluate parametricallywhether the reliable individual speaker differ-ences claimed on the basis of this earlierresearch actually exist.

A round robin study of proportion of timespent speaking was conducted by Warner(Note 2). Eight participants (four male, fourfemale) were enlisted for a study of conversa-tion. Each of the 28 possible pairs conversedprivately on 3 separate days for about 12 toIS minutes each time. Each speaker's voicewas recorded onto a separate channel of a


Table 7Proportion of Time Spent Speaking inConversations

Subject

Subject 1 2 3 4 5 7 8RowM

1 55 65 59 62 87 69 81 6758 58 54 68 68 76 7645 85 50 79 76 56 71

2 22 21 34 25 35 39 58 3850 42 27 38 35 36 5336 35 38 34 49 53 28

3 31 56 34 31 24 34 36 3545 45 40 33 25 48 1612 54 24 37 42 37 30

46 31 58 72 5525 89 43 8365 33 59 65

36 58 73 6154 53 5637 71 71

676561

50

55

4 26 62 6830 62 5545 60 81

5 66 56 70 6770 56 89 6952 58 64 44

6 27 30 74 45 49 70 40 4849 53 81 11 35 78 6928 41 47 48 37 46 41

7 52 40 77 32 52 4945 66 54 40 48 2267 34 60 48 23 49

8 39 47 75 47 37 45 3759 80 90 47 51 46 6745 70 73 49 37 51 51

Col urn n(

M 43 54 65 43 43 47 54 58 51

Note. X i j t = the proportion of time spent speakingby Person i to Person j on Day k (in percent).

stereo tape. The participants wore headsetswith noise-canceling microphones close tothe mouth; this type of microphone arrange-ment, which was somewhat intrusive, wasnecessary because ordinary microphones allowtoo much spillage of voices between channels.The proportion of time spent speaking by eachperson was determined by using a computervoice-operated relay to detect the presenceor absence of speech in each of the two channelsof the tape-recorded conversations. The voice-operated relay was calibrated by means of anindicator light that showed when the systemwas detecting speech activity; thus a humanlistener could manipulate the threshold andother settings on the voice-operated relayuntil the pattern shown by the light matchedthe perceived on-off pattern of speech activity.

The on-off vocal activity judgment wasmade twice per second, and this informationwas used to determine the proportion of timespent speaking by each person over the timeof the whole conversation.

Subjects were instructed that the study wasabout the process of becoming acquaintedand were told that they could talk aboutwhatever they liked. Pilot tests had indicatedthat this was probably much more naturalthan imposing some kind of task; most dyadshad little difficulty in carrying on a conversa-tion, and a number of conversations werequite animated and contained information ofa rather personal nature. Subjects spent theirfree time between recording sessions in alounge where they were free to talk.

An effort was made to promote a feelingof ease in the situation, at the cost of experi-mental control; for example, several personsin the study were previously acquainted, someof the conversations that took place for thetape recorder were continuations of conversa-tions that had begun in the lounge, and manyconversations continued after the tape recorderwas turned off. Partly because of the marathonnature of the scheduling (the eight subjectsspent three 8-hour days participating in thestudy), a certain amount of group cohesivenessdeveloped. For all these reasons, it seemslikely that the conversations reported in thisstudy are more natural than those that havebeen elicited from persons who come into thelaboratory "cold" for just one session orwho converse via intercom without visualcontact.

The proportion of time spent speaking wastabulated for all 84 conversations, and theresults are displayed in the 8 X 8 X 3 datamatrix shown in Table 7.8

The O-Bird round robin analysis of varianceprogram, which carries out all the calculations,parameter estimation, and significance testingas outlined in this paper, was run on thesedata.4 The results are displayed in a modifiedsource table, with / tests in place of the usualF ratios. Since the true variance components

3 No transformation for proportional data (such asarcsine) was used, since none of the proportions werevery near zero or one.

4 The O-Bird program (in Fortran) is available fromthe second author.


(row, column, interaction) must be nonnega-tive, a one-tailed t was used for these threetests; the covariances (row and column,cross-diagonal interaction, and cross-diagonalerror) can be either positive or negative, andso a two-tailed / is needed for these three tests.5

Discussion

There were two significant effects: (a)clear-cut individual speaker differences inactivity level, as shown by the significant /value for row effect and (b) strong situationalreciprocity between the speech of partners inparticular conversations, as indicated by thesignificant t value for the cross-diagonal errorcovariance. No other effects were statisticallysignificant (although the sizes and signs of allother effects were generally consistent withour expectations, which lends some additionalplausibility to the model). Our results confirmearlier findings of individual differences; wealso have information about the cross-diagonalerror convariance, which can be interpretedas evidence of a situation-specific reciprocityof speech activity level. This was anticipated,since it is a common sense observation that thetime allocation within any particular conversa-tion is approximately zero sum, that is, themore time taken up by one speaker, the lesstime available to the other speaker. Thus, theproportions of time spent speaking by thetwo persons in a particular conversation(Xijk, Xjik) are negatively correlated.

The partner effect was smaller than thespeaker effect, which would suggest that theimpact of partner activity level on an individ-ual's speech production is not as great as theeffect of his or her own "preferred" activitylevel. As anticipated, there was a negativecovariance between the activity level of aperson and that person's effect on otherprople's activity level when he or she wastheir partner; however, this covariance wasnot significant, so it seems appropriate toconclude that the relationship between anindividual's activity level and his or hertendency to elicit a high or low activity levelfrom others is not so strong that these twofactors should be considered equivalent.

There was a relatively modest and non-significant interaction effect. This can be

rationalized as follows: although it is true thatin any particular conversation between personsi and j they adjust their activity levels toeach other to achieve a total proportion oftime active of about 1.0, this adjustment canbe made in a number of different ways, and theadjustment that they make in any one partic-ular conversation is unique to that conversa-tion and does not carry over to later conversa-tions (at least, not in our study, which con-siders only three conversations per dyad).That is, in one conversation, i may talk alittle more than would be expected (based oni's talkativeness and j's tendency to elicittalk); in another conversation, i may talkless than would be expected. The smallinteraction variance indicates that there isnot a fixed adjustment for each dyad, sothat when i talks to j, he or she always talksa little more (or less) than usual.

The cross-diagonal interaction covariancefor these data is virtually zero. This is reason-able; if i's adjustment to j is not stable orfixed across different situations, it is notreasonable to expect a strong relationshipbetween i's average adjustment to _;', andj's average adjustment to i. Furthermore,even if these partner adjustments wereconsistent, as would be indicated if there werea large interaction effect, the adjustments inactivity level might not necessarily be corre-lated for partners. Any combination is possible—depending upon the "preferred" activitylevels of i and j, both partners could talkmore, both partners could talk less, * couldtalk more and j could talk less, or j couldtalk more and i could talk less; any of thesetypes of adjustment could conceivably beneeded in order to work out the time allocationbetween partners. Since all these combinationscan and do occur, on the average the covariancebetween i's adjustment to j and j's adjustment

6 When testing whether a variance component iszero in the population, a one-tailed / is appropriate,since the true variance component in the populationcannot be negative. Since the pseudovalues can benegative (and will be negative about half the time if thetrue variance component being estimated is zero),there is no reason why these pseudoestimates couldnot be / distributed. Of course, the covariance termscan be either positive or negative, so two-tailed <sshould be used for the covariance terms.


Table 8Source Table for Round Robin Analysis ofVariance

Variance or

Source

Row(speaker)

Column(partner)

InteractionRow and

columnCell X CellError

X ErrorError

* p < .05,'(7).

MS

2438.3167

1388.1423622.8201

-1380.7346-354.1238

-95.5282146.0462

one-tailed t ( 7 ) .

covarianceestimate

91.9680

40.917829.7534

-40.37494.0811

-95.5282142.0462

**/> < .05,

«(7)

2.0144*

1.26691.0262

-1.15470.0997

-2.4500**—

two- tailed

to i will be nearly zero. The small (virtuallyzero) covariance obtained for this cross-diagonal interaction covariance indicates alack of enduring reciprocity in speech activitylevel—there is no adjustment that i alwaysmakes to j that is correlated with an adjust-ment that j always makes to i. It seems likelythat certain other vocal activity parameters(such as mean switching pause duration) andother variables such as attraction, might showenduring reciprocity; but there seems to beno such effect for activity level of speakers.

The significant individual speaker differ-ences, taken together with the nonsignificantinteraction effect, suggest that the appropriateunit of analysis for proportion of time speakingis the individual person rather than the dyad.The strong situational reciprocity, togetherwith the lack of enduring reciprocity, suggeststhat the reciprocity of speech activity levelsdepends heavily on situational factors—per-haps topic of conversation, time of day,moods of the participants, and so forth,rather than on some lasting "agreement"between partners as to their relative dominanceor right to claim speaking time. At presentthe model is not set up to handle order effects,but it could be extended to account for predict-able shifts in the time allocation betweenpartners over time (if these occur).

Although the main focus of this discussion isinterpretation of the significant effects, thereare additional aspects of the results in Table 8

that bear further examination. It is possibleto estimate correlations from the meansquares:

= -.7504,

and

f-, =MS,CellXCell = -.5686.

The f i is the correlation estimate for therelationship between the row and columnfactors that would be obtained directly fromthe raw data. It is a biased estimate of thetrue relationship between row and columnfactors; this can be seen by inspecting theset of variance components belonging to£M5RowXColumn, EMSrow, and £M5oolumn. Theratio of these raw mean squares clearly doesnot provide an estimate of the true correlationpi. The same problem applies to the estimatefor r2, which is also biased; however, no suchproblem arises in estimating p3, which is just(M5ErrorXError)/Af5error.

An alternative means of estimating thecorrelations is to take ratios of the varianceand covariance components derived from theround robin model, as follows:

A PI a p ACOOPi - li. 2i 2 \J - ~ -tooA

P2 = = .1372.

Although the components of these new correla-tion estimates are unbiased, our initialexperiences with them indicate that they maybe unreliable. Also, just as estimates for thevariance components of the round robinmodel can be less than zero due to samplingerror, correlation estimates derived from thevariance and covariance components mayfall outside the range of plus or minus one,especially if the denominators are small.Thus some caution is in order when using orinterpreting these correlation estimates. Itshould be kept in mind that correlationsbased on the raw data in a round robin-typelayout do not estimate what they seem to beestimating. Such raw correlations are com-monly reported in person perception literature;their interpretation is highly problematic.


Another useful strategy is comparing themagnitude of various effects. A simple instancewhere this may be substantively interestingis in person perception research, where itmay be useful to know whether there isgreater variability among the ratings thatpeople give (rater or actor effect) or theratings that people receive (partner effect);that is, is the trait in question "in the eye ofthe beholder" (influenced mainly by theperson doing the rating)? Or is it a trait thatdifferent observers can agree upon, influencedmainly by characteristics of the person beingrated? This comparison can be made bylooking at the ratio (or the difference) betweenthe row and column variance components,and the significance of the ratio or differencecan be evaluated by jackknifing. This compar-ison of actor and partner effects was notsignificant for the vocal activity data, /(7) =.635. That is, even though the row variancewas significantly greater than zero and thecolumn variance was not significantly greaterthan zero, the row variance was not signif-icantly greater than the column variance.

More elaborate comparisons among thevariances can also be constructed. For example,for some applications it might be instructiveto ask to what extent the variability of socialbehavior is accounted for by characteristicsof the individuals (actor and partner effects)as opposed to characteristics of the dyad orthe situation (interaction and error). To theextent that actor and partner factors pre-dominate, social behavior can be viewed asadditive; social interaction can be conceivedof as a linear system, that is, a system withbehavior that can be predicted from thebehavior of its components. Such a test ofadditivity might take the following form:

A = . 2 '

For our data, this ratio was significantlygreater than zero, t(1) — 2.44, p < .025 one-tailed, based on the jackknifing of this ratioover different subsets of the data; that is tosay, some nonnegligible amount of the variabil-ity in vocal activity level is accounted for bythe components of the conversational system,the speakers themselves. Depending upon thenature of the research problem, other compar-isons among variances can be set up.

Special Applications

As with any statistical model, the decisionto use the round robin model depends on twoconsiderations—first, whether the assumptionsof the model are satisfied, and second, whetherthe model provides the desired information.It is possible to use the round robin model toanalyze data on unconstrained social interac-tions in natural settings or to make groupcomparisons; however, it is important torealize that these data may violate certainassumptions of the round robin model.

Consider the type of study in which socialinteractions are observed in natural settingssuch as classrooms. One might count thenumber of aggressive or attention-getting actsby each child. Since such data violate theindependence assumptions of ordinary chi-square and analysis of variance, one shouldnot use these tests. Clearly each act within agiven classroom is dependent to some extenton the other acts that occur within thatclassroom. This might suggest that theround robin model is suitable for these data;in addition to reciprocity effects, however,there are likely to be other forms of socialinfluence such as modeling or shared attitudestoward particular class members. In postulat-ing the round robin model, we assume that(except for the reciprocity covariances) theobservations are independent, for example, h'sbehavior toward Person * is assumed to beuncorrelated with j's behavior toward i.Clearly if Person j imitates h's behavior orshares h's attitudes toward other individuals,the behaviors of h and j will be correlated.This means that the independence assumptionsof the round robin model are violated. It maybe possible to minimize these other forms ofsocial influence by preventing the participantsfrom observing the interactions of other dyadsand preventing participants from discussingtheir perceptions and attitudes. In manynatural social environments, however, thiskind of control is not feasible, and the use ofsuch stringent controls may sacrifice too muchexternal validity.

We know of no statistical test of the covar-iance structure of the data to evaluate whetherthe independence assumptions of the roundrobin model are satisfied. Such a test couldconceivably be developed along the lines of


work done by Huynh and Feldt (1970). Inthe absence of statistical criteria, the investiga-tor must realize that modeling and sharedexpectations will bias the estimates of theround robin model parameters.

Another difficulty in the analysis of interac-tion data from unconstrained social situationsis that participants self-select their partnersand may not interact with everyone else inthe group. This results in large numbers ofmissing observations or zero frequencies. Atpresent the analysis we propose cannot handlemissing data. The presence or absence of tiesbetween persons reveals network structure,however, so block models or network analysisare viable alternatives (White & Breiger,1975).

Another possible application involves groupcomparisons: either comparisons of two ormore round robin layouts or comparisons ofsubsets of subjects within a round robin layout.For instance, the vocal activity study reportedearlier included four male and four femaleparticipants. One might wish to examinedifferences in talkativeness between malesand females or between same sex and cross-sexdyads. This might be done by taking the meandifference in the row effect estimates (a;)for males versus females in such of the leave-out-one subsets of data and jackknifing toget a significance test. One might also wishto correlate the actor effect (a;), talkativeness,with a personality scale such as dominance.

Another consideration in applications ofthe round robin model is sample size. Theround robin analysis requires a minimum offour persons, and the jackknife procedureraises this to five. In fact, we believe that theanalysis will provide unstable estimates forsample sizes less than eight. Recall that ifthere are only seven members of the group,there are only six degrees of freedom in the/ test of the actor and partner main effects.To increase the efficiency in estimation, onecan replicate the round robin design ondifferent sets of persons. The parameterestimates could then be pooled across thedifferent groups, resulting in more stableestimates.

Summary

The new round robin analysis of varianceprovides a tool for dealing with social interac-

tion data that allows assessment of bothindividual differences and reciprocity in socialbehaviors. It readily lends itself to the sub-stantive problems encountered in social psy-chological research on person perception,attraction, vocal activity and other mutuallycontingent behaviors. This means that insteadof regarding the mutual contingency of socialbehaviors as a problematic deviation fromthe independence assumptions of statisticalmodels traditionally used in the nonsocialsciences, it is possible to explicitly incorporatemutual contingency into the design and totreat it as an interesting effect in its own right.By tailoring statistical models to the questionsthat arise in social psychological research, wecan leave behind some of the limitations thatare inherent in the use of statistical modelsthat were created to study nonsocial phenom-ena.

The round robin data layout is not new;this type of design has been employed in anumber of classic social psychological studies(Campbell, Miller, Lubetsky, & O'Connell,1964; Cronbach, 1955; Newcomb, 1961). Itcontinues to be used in small group and personperception research (e.g., Bales, 1970). Whatis new is the statistical model specificiallytailored to this design, which makes it possibleto test the significance of individual differences,interaction effects, and various types ofreciprocity. This new round robin model isideally suited to the kind of research Tagiuricalled for in his 1969 article on person percep-tion in the Handbook of Social Psychology. Herecommended more study of person perceptionin the context of ordinary transactions thatoccur in the natural environment, where thepersons are interacting with each other andeach person is simultaneously judge andobject.

The round robin design deserves seriousconsideration as an alternate research strategythat provides information complementary tothat from traditional experimental designs.We hope that the availability of techniquesthat facilitate analysis of data from naturallyoccurring social interactions will stimulateinterest in the interactive, mutually contingentaspects of social behavior and the relationshipbetween individual differences and socialinteraction.


Reference Notes

1. Stoto, M., Kenny, D. & Warner, R. Derivation of theexpected mean squares for the round robin analysisof variance. Unpublished manuscript, HarvardUniversity, 1978.

2. Warner, R. Temporal patterns in dialogue. Unpub-lished doctoral dissertation, Harvard University,1978.

References

Bales, R. F. Personality and interpersonal behavior.New York: Holt, Rinehart & Winston, 1970.

Bechtel, G. G. The analysis of variance and pairwisescaling. Psychometrika, 1967, 32, 47-65.

Bechtel, G. G. The analysis of variance for pairedcomparisons. Psychometrika, 1971, 36, 135-154.

Campbell, D. T., Miller, N., Lubetsky, J., Si O'Connell,E. J. Varieties of projection in trait attribution.Psychological Monographs, 1964, 78 (15, Whole No.592).

Cronbach, L. J. Processes affecting scores on "under-standing of others" and "assumed similarity."Psychological Bulletin, 1955, 52, 177-193.

Gleason, J. R., & Halperin, S. A paired comparisonsmodel for round robin experiments. Psychometrika,1975, 40, 43-54.

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interviewers and their effect on interviewee's speechbehavior. Journal of Mental Science, 1952, 98,660-671.

Huynh, H., & Feldt, L. S. Conditions under whichmean square ratios in repeated measurementsdesigns have exact ^-distributions. Journal of theAmerican Statistical Association, 1970, 65, 1582-1589.

Jaffe, J., & Feldstein, S. Rhythms of dialogue. New York:Academic Press, 1970.

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Hosteller, F., & Tukey, J. Data analysis and regression:a second course in statistics. Reading, Mass.; Addison-Wesley, 1977.

Newcomb, T. M. The acquaintance process. New York:Holt, Rinehart & Winston, 1961.

Snedecor, G. E., & Cochran, W. G. Statistical methods.Ames: Iowa University Press, 1967.

Tagiuri, R. Person perception. In G. Lindzey & E.Aronson (Eds.), Handbook of social psychology (Vol.3, 2nd ed.). Reading, Mass.: Addison-Wesley, 1969.

Welkowitz, J., Cariffe, G., & Feldstein, S. Conversa-tional congruence as a criterion of socialization inchildren. Child Development, 1976, 47, 269-272.

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Received November 2, 1978 •

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A new round robin analysis of variance for social interaction data