using the actor-partner interdependence model to study the effects of group composition david a....
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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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