Using the Actor-Partner Interdependence Model to Study the Effects of Group Composition

David A. Kenny & Randi Garcia

University of Connecticuthttp://davidakenny.net/doc/gapim.ppthttp://davidakenny.net/doc/gapim.doc

Example Question Jill is a member of a six-person

group. Jill is female. We measure how influential Jill is in

the group. The research question: How does a

person’s gender and the genders of the other group members affect how influential a person is seen?

Denote gender as X and presume X is a dichotomy.

Multilevel Data The answer to the research question

requires a multilevel data set. Two levels

– The lower level or level 1: Person– The upper level or level 2: Group

To have unbiased estimates of standard errors, we must allow for nonindependence due to groups.

Variables and NotationYij = the outcome of person i in

group j (How influential is Jill seen?)

Xij = gender of person i in group j (Jill is -1 and a male would be +1)

Mj = the average X scores for group j (if greater than zero, there would be more males in the group)

Traditional Multilevel Modeling of Groups

Variables X (level 1) and Mj (level 2) to predict Y.

Or X – Mj (X “group mean centered”) and Mj to predict Y.

Problems with the Traditional MLM Formulation

Part-whole problem.Can be difficult to interpret.Linkage to theory unclear.What about other effects of X,

especially diversity in the Xs (or the similarity of the Xs)?

Actor-Partner Interdependence Model

The “group effect,” called “Others,” is the effect due to OTHER members of the group, denoted as Mj’.

The individual’s score is removed from the group mean.

Others is a level 1 variable but most of its variance is between groups.

Y1

Y2

X1

X2

X3

X4

a

p/(n-1)

p/(n-1)

p/(n-1)

p/(n-1)p/(n-1)

p/(n-1)

a

Y1

Y2

X1

X2

X3

X4

a

M1'

M2'

p

p

1/((n-1)

1/(n-1)

a

1/(n-1)

1/(n-1) 1/(n-1)

1/(n-1)

Main Effects for the ExampleActor: Are men (or women) more likely to be seen as influential?

Others: If most of the partners are men (or women), is the person seen as influential?

InteractionsActor x Others: If the person is similar to others, is the person seen as influential?

Other x Other: If the other members of the group are similar to each other, is the person seen as influential?

Re-conceptualization of Diversity

Instead of thinking about diversity as a property of the group (i.e., a variance), we can view diversity as the set of relationships.

Variance as the Measure of Diversity

s2 = i(Xi – M)2/(n – 1)

s2 = ij(Xi – Xj)2/[n(n - 1)] i > j

s2 = 1 - ij(XiXj)/[n(n - 1)/2] i > j

Thus, diversity can be viewed as a summary of the similarity of all the possible relationships in the group.

Group Diversity as the Sum of All Possible Relationships

Group Diversity = Actor Similarity + Others Similarity

The Two Types of Similarity• Actor Similarity

• How well the person fits into the group.• “Relational Demography” of Elfenbein

and O’Reilly• Others Similarity

• Combined with actor similarity becomes diversity

• If Actor and Others Similarity have the same coefficients, there is a pure diversity effect.

Example Data Set• PI: Harmon Hosch• Gathered in El Paso, Texas• 134 6-person juries from the jury

pool– The sample was 54.7% Female, 58.7%

Hispanic, 31.5% White, 3.9% Black, and 2.2% Asian American or Native American.

• Mock jury case: theft• We have a measure of influence (1

to 5; to be discussed later).

SPSS SyntaxMIXED influential WITH gender other_gender

actor_sim others_sim /FIXED = gender other_gender

actor_sim others_sim /PRINT = SOLUTION TESTCOV /REPEATED = memnum |

SUBJECT(group) COVTYPE(CSR) .

Results: Main Effects

Effect Coefficient SE pActor 0.093 0.025 >.001Partners -0.077 0.073 .291

Men seen as persuasive.

Results: Interactions

Effect Coefficient SE pActor Similarity -0.050 0.062 .422Others Similarity 0.257 0.106 .016

A person is seen as more persuasive if others in the group are similar.

Conclusions• Men are seen as more influential

than women.• If others are similar, a person is

seen as influential.

What was the measure of “Influential”?

• Based on a relational measure.• Each person asked (round-robin

design): “How persuasive is each other person in the group.”

• We need to extend the model, both fixed and random, to a dyadic outcome.

Group: How much influence in the group?

Individual

– Actor: How much influence Jill sees others?

– Partner: How influential is Jill seen by others (may be correlated with Actor)?

Dyad: If Jill sees Sally as influential, does Sally see Jill as influential?

(The Social Relations Model)

Levels or Random Effects

Three Main Effects

Actor

Partner

Others

Main EffectsActor: Are men (or women) more likely to see others as influential?

Partner: Are men (or women) more likely to be seen by others as influential?

Others: If the most of the partners are men (or women), is the person seen as influential?

Results: Main Effects

Effect Coefficient SE pActor -0.007 0.024 .776Partner 0.086 0.026 .001Others -0.092 0.062 .142

Men seen as more influential.

Interactions

Instead of thinking about diversity (or homogeneity) as a property of the group (i.e., a variance), we can view diversity as the set of relationships.

Four Types of Similarity

Actor

Partner

Others

Four Types of Similarity

Group similarity equals the sum of these components.

Dyadic SimilarityActor Similarity

Partner Similarity

Others Similarity

The Four APIM Interactions

Dyadic: Actor-PartnerActor: Actor-OthersPartner: Partner-OthersOthers: Other-Other

Interaction Results

Similarity Effect SE p Dyadic 0.018 0.200 .368Actor 0.148 0.056 .009Partner -0.102 0.058 .080Others 0.076 0.074 .306

If the partner is different from others (partner similarity) and you are similar to others (actor similarity), you see the partner as influential.

Partner Seen Relatively Low on Influential

Actor

Partner

Others

Partner Seen Relatively High on Influential

Actor

Partner

Others

SAS Syntax

PROC MIXED COVTEST;

CLASS dyad group;

MODEL influential = actor partner other dsim asim psim osim / S DDFM=SATTERTH;

RANDOM a1 a2 a3 a4 a5 a6 p1 p2 p3 p4 p5 p6 INTERCEPT / G SUB=group TYPE = LIN(4) LDATA=g;

REPEATED /TYPE=CS SUB=dyad (group);

Extensions Some people may have a bigger partner

effect (e.g., leaders). Non-dichotomous X variables:

– Interval variables– Nominal variables with more than two

levels Multiple X variables Solo effects

Limitations Requires

– Interval outcomes – At least four-person groups– a large number of groups– considerable variation in diversity

Does not provide an account dynamic factors of group interaction.

Conclusions The model presented offers some

unique opportunities for the study of groups.

Approach combines state-of-the-art statistical methods with theories of groups.

Thank You!

http://davidakenny.net/doc/gapim.ppthttp://davidakenny.net/doc/gapim.doc

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