Using the Actor-Partner Interdependence Model to Study the Effects of Group Composition
David A. Kenny and Randi L. Garcia
University of Connecticut
We especially wish to thank Harmon Hosch, John Mathieu, and Susan Mohammed who generously
provided data for the analyses presented in this paper. We also thank Keli Holloway who kindly
assisted us in the preparation of El Paso jury study. Additionally, we also thank Hillary Anger
Elfenbein who shared with us her data and Joseph Olsen and Thomas Ledermann who read a prior
draft of this paper.
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Using the Actor-Partner Interdependence Model to Study the Effects of Group Composition
Abstract
We extend the Actor-Partner Interdependence Model (Kenny, Kashy & Cook, 2006), a model
originally proposed for the analysis of dyadic data, to the study of groups. We call the model the
Group Actor-Partner Interdependence Model or GAPIM. For individual outcomes (e.g., satisfaction
with the group), we propose four effects of group composition; for group-level outcomes (e.g., group
productivity), we propose two effects; and for dyad-level outcomes (e.g., liking of each of the other
members of the group), we propose a model with three main effects and four interaction effects. For
these different models we discuss the ways in which different submodels map onto different social
psychological processes. We illustrate the GAPIM with two datasets.
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Using the Actor-Partner Interdependence Model to Study the Effects of Group Composition
The study of group composition is one of the oldest topics in the study of small groups
(Haythorn, 1968; Levine & Moreland, 1998). Understanding how a group member thinks, feels, and
behaves as a function of who he or she is and who are the others in the group is obviously a critical
question. There are a multitude of questions that fall under the general rubric of “group composition.”
The most basic question is does one’s experience of the group depend on one’s own characteristics;
another key question is the effect of diversity in group membership (Tsui, Egan & O'Reilly, 1992; Tsui
& O’Reilly, 1989); an additional key question is the effect of a person’s fit into the group or the
person’s similarity to the other members (Elfenbein & O’Reilly, 2007); still another is the effect of
being the only member of a certain type, i.e., a solo (Kanter, 1977; Sekaquaptewa & Thompson,
2002); there is also the effect of the group’s average, sometimes called group climate (Florin,
Giamartino, Kenny, & Wandersman, 1990). These are just some of the ways that group composition
can affect member and group outcomes.
Moreland and Levine with their collaborators have made important contributions to the
conceptualization of the effects of group composition. For example, Moreland, Levine, and Wingert
(1996) have proposed a general theoretical model for the study group composition that describes how
individual characteristics combine to affect group-level outcomes (e.g., productivity). One important
question highlighted by this work is whether group member’s characteristics have an additive or
interactive composition effect. The method we propose offers one straightforward way to answer this
question for not only group-level outcomes, but also for individual and dyadic outcomes. Furthermore,
Levine and Moreland (1998) have usefully organized group composition research in three basic
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categories: composition as consequence, composition as context and as a cause. This paper focuses on
the later two categories.
Building on these past contributions, we propose in this paper a new and very general method
for studying all the effects of group composition on group members. We shall see that the approach
allows us to measure and test each of these different group phenomena. Moreover, we apply this
model when one outcome is obtained for each member of the group – an individual outcome – when
there is a single outcome for the entire group – group outcome – and when an outcome is obtained for
each person with all other members in the group – a dyadic outcome.
In what follows, we consider the effects of group composition for different levels of
measurement: individual, group, and dyad. By far the most common type of outcomes in group
research is individual outcomes, and so we first extensively discuss it. To simplify the discussion, we
use gender as the group composition variable throughout the paper. Technical details and computer
setups are contained in an online website: davidakenny.net/doc/gapim_tech.doc.
Individual Outcomes
The Actor-Partner Interdependence Model (APIM; Kenny et al., 2006) was developed primarily
for dyadic data, but has been extended to the study of groups (Gonzalez & Griffin, 2001; Kenny,
Mannetti, Pierro, Livi, & Kashy, 2002). In its use as a dyadic model, one member of the dyad’s
response depends on his or her own characteristics, the actor effect, and also on the characteristics of
the person’s interaction partner, the partner effect. These two effects can interact which can be
interpreted as a similarity effect: When an actor is similar to his or her partner, is the actor more
satisfied in the relationship than when an actor is dissimilar to his or her partner?
For groups, we refer to the individual who provides the data point as the actor and the
remaining n − 1 members of the group as the others.1 For instance, we might ask Sally how satisfied
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she feels about being in her group. Consider work teams in a given company that vary in gender
composition. We start with a dichotomy, gender, and later consider other types of variables. Some of
the groups might be of the same gender, either all male or all female, and other groups may contain a
mixture of men and women. A particular person’s feelings of group satisfaction might depend on his
or her own gender (women designated as −1 and men as +1), denoted as Xik, and depend on the average
gender of the other n − 1 members in the group, denoted as Xik'. For example, if the group is all
women, then the Xik' would be −1, and if all men, then average gender of others would be 1. If the
others in the group were half of each gender, the average gender of others would equal zero. These two
variables are the main effects of gender and others gender on group satisfaction.
There are potentially two interactions due to gender composition:
1. the actor effect by others effect interaction or Iik, which measures how similar the
person’s gender is to the other n – 1 members of the group – actor similarity, and
2. the interactions of all possible pairs of others or Iik' which measures how similar the
other n – 1 members’ genders are to each other, a total of (n − 1)(n − 2)/2 dyads – others
similarity.
Each of these interactions is formed as the mean of product terms of all pairs of gender terms, i.e.,
XikXjk.
More formally, we denote the individual outcome as Yik for person i in group k and the full
group APIM for individuals or GAPIM-I model is:
Yik = b0k + b1Xik + b2Xik' + b3Iik + b4Iik' + eik
(1)
where, for the example, the effects are defined as follows:
b1: the effect of a person’s own gender, the actor effect;
b2: the effect of the average gender of the other n – 1 members of the group, the others effect;
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b3: the average similarity of person i’s gender to the gender of the other n – 1 members of the
group, the actor similarity; and
b4: the average similarity of the genders of all possible pairs of others or (n – 1)(n – 2)/2 dyads
in the group, the others similarity.
The Iik term, or actor similarity, equals XikXik' and was called the actor-partner interaction in Kenny et
al. (2002). The Iik' or others similarity is the average of the partner-partner product terms. This second
interaction was not considered in Kenny et al. (2002). Presuming the X variable is effects coded (+1,
-1), these “I” variables equal one when there is exact similarity and –1 when there is complete
difference.
As explained in Kenny et al. (2002), the APIM does not use the usual multilevel formulation of
entering the group mean of X to predict Y. Because a person’s X is part of the group mean of X, (a
person’s own gender is part of the group’s gender) the two effects are correlated. Additionally, it
might be argued that the key conceptual contrast in groups is between self and others and not between
self and group.
Kenny and Cook (1999) considered four submodels of the APIM, which parallel models of
interdependence theory (Kelley, Holmes, Kerr, Reis, Rusbult, & Van Lange, 2003). It is sometimes
forgotten that interdependence theory began as a theory of groups, the title of Thibaut and Kelley’s
(1959) classic book being “The Social Psychology of Groups.” It is these submodels that evaluate
social psychological theories of group processes. Using member ability as example of the X variable
and member satisfaction as the Y or outcome variable, the four submodels are: 1) the Actor Only
Model, where only the actor’s ability has an effect on the outcome (b1 ≠ 0 and b2 = 0 in Equation 1); 2)
the Others Only Model, where only the average ability of the other n − 1 persons in the group has an
effect on the outcome (b1 = 0 and b2 ≠ 0); 3); the Group Model in which the mean of the group’s
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abilities has an effect on the outcome (b1 = b2 ≠ 0); and 4) the Contrast Model is which a person
compares his or her own ability to the average ability of the others in the group (b1 – b2 = 0).
In the Actor Only Model, the abilities of the others in the group have no effect on the
individual’s satisfaction whereas in the Others Only Model it is the others who matter. These models
parallel individual and altruistic motives of Interdependence Theory. In the Group Model, it is the
overall composition of the group that matters and the individual’s own X, or personal value on some
characteristic, plays no special role in affecting one’s satisfaction. In some sense, in this model there is
no boundary between self and other in a way akin to the inclusion of self perspective of Aron, Aron,
and Tudor (1991) and the equality motive dominates. However, in the Contrast Model, there is sharp
boundary between the individual and the others in the group. What matters is how different the person
is from others, and the direction of the difference matters. In some contexts, this effect has been
dubbed the frogpond effect in which Davis (1966) found that college students had higher career
aspirations if they were smarter than most of the others students at their college. Hence the name, it is
better to be a big fish (or frog) in a small pond, than a bigger fish in larger pond.
We can also consider three submodels involving the two interaction effects that also test
various social psychological theories. We change the example to consider the X variable as ethnicity,
members of the majority group and members of a minority group. The first is the Diversity Model
which contains the two main effects and a measure of the overall diversity of the group. We can
measure group diversity as a weighted average of the two interaction terms: –[2(n – 1)Iik + (n – 1)(n –
2)Iik']/[n(n – 1)]. The negative sign changes the measure of group similarity into a measure of
diversity. The diversity model implies that b3 = b4 ≠ 0. Several different investigators (Antonio,
Chang, Hakuta, Kenny, & Levin, 2004; Harrison, & Klein, 2007; Harrison, Price, & Bell, 1998;
Jackson et al., 1991; Sommers, 2006; Tsui, & Gutek, 1999) have made predictions that group diversity
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has an effect on individual, as well as group and dyadic outcomes. For instance, we might expect
members of ethnic minority groups to feel less identified with the group. Second, the Person Fit Model
assumes that what matters is how similar the actor is to the others in the group (i.e., from Equation 1,
b3 ≠ 0 and b4 = 0), and not how similar the other members are to each other. For instance, Elfenbein
and O’Reilly (2007) found if a group member “fitted into the group,” he or she had better performance
ratings. The third model is the Contrast Model of interaction effects. In this model, actor similarity is
measured relative to the similarity of the others in the group. This model implies that the two
interaction effects are of equal but opposite sign: b3 – b4 = 0. For the Interaction Contrast Model, the
group members look to see how different they are from the others and how different the others are
from each other. This model taps into the idea that individuals who are solos experience the group
differently from individuals who are not solos.
The last two submodels have both main effects and interaction effects of the GAPIM-I. First is
the Actor Only Model in which we would include only effects involving the actor—the actor effect and
the actor similarity effect. Second, for the Partner Only Model, we would include the others effect and
the other similarity effect.
Estimation and Testing
We note that there are two random variables in the model contained in Equation 1:
b0k: the extent to which some groups have more or less satisfaction than other groups which can
be viewed as the group effect, and
eik: error, or the extent to which person i is satisfied with the group more than others.
Thus, the model in Equation 1 has two levels, and so multilevel modeling estimation must be used.
Individual is at level one and group is at level two with all of the group composition predictors
in Equation 1 at level one.2 There might be additional predictors at level one (e.g., how long one has
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been in the group) and at level 2 (e.g., the type of group). The variance of the intercepts or b0k
represents the group level effects and the variance of ik represents the combination of error and person
variance. We used SPSS in our analyses, but any multilevel modeling program (e.g., HLM, SAS, or
MLwiN) could have been used.
We computed a measure of fit for each model that we estimated. We used the sample-size
adjusted Bayesian information criterion (SABIC) which equals D + qln[(N + 2)/24] where D is the
model’s deviance, N is the number of groups, and q is the number of parameters in the model. To get a
baseline value, we estimated the SABIC for a model in which all four coefficients are set to zero, an
Empty Model. Our model testing strategy was as follows. We first estimated a Main Effect Model,
and we tested the two main effects for statistical significance, and we compared its fit to the Empty
Model. We then estimated the Complete Model, the model with all four terms. We compared the fit of
this model to the Main Effects Model. Finally, based on these analyses we chose a submodel that had
better fit than any of the prior models and whose terms were statistically significant.
Example
As an example dataset to illustrate the use of the GAPIM-I, we used data originally collected by
Mohammed, Mathieu, and Bartlett (2002). At the beginning of a 15-week semester the students were
randomly separated into teams distributed evenly by gender. There are 25 teams of 4 to 5 students (5
groups of 4 and 20 groups of 5, a total of 120 participants) earning extra credit for a Hotel, Restaurant,
and Institutional Management (HRIM) course. Six participants did not have a gender recorded, so these
6 groups were not included in the analyses. Of the remaining 91 participants, 2 had missing group
satisfaction scores—the outcome variable that we investigate— and so the final analyses included 89
participants, 69 females and 20 males, in 19 groups. The participants’ ages ranged from 19 to 33 with a
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mean age of 21.37 and standard deviation of 1.55. They were 94.5% White, 2.2% Hispanic, and 3.3%
Asian.
At two points in the semester, separated by five weeks, each group planned and managed an
entire cafeteria style meal with the help of only two staff members. Roles, which changed from one
meal day to the next, were assigned to the group members. At the end of the semester, the participants
completed a team effectiveness survey in which they reported how satisfied they were with this group
experience (i.e., “I am satisfied with my experience in HRIM 330 [the class]” from 1—strongly
disagree or definitely false, to 5—strongly agree or definitely true) which is the outcome variable of
our analyses.
Results
We have effect coded gender, +1 for males and −1 for females. Table 1 presents the results
from our analysis of the Mohammed et al. (2002) dataset. First, the Main Effects Only model was
estimated and neither of the effects, actor’s gender or others’ gender, was statistically significant and in
fact this model fits worse than the model with no effects. We next estimated the Complete Model and
found that the fit improved. We found marginally statistically significant effects of others’ gender (b =
0.973, p = .099), and others’ similarity (b = 0.746, p = .056).3 There were no statistically significant
effects of actor gender or actor similarity. It appears as though group members’ satisfaction is
influenced by the gender of the other members of the group, not the members’ own gender. To test
this, the Others Only Model was estimated—actor gender and actor similarity were fixed to zero. This
was the best fitting model of all the models that we estimated. As seen in Table 1, there were
statistically significant effects of both others’ gender (b = 0.918, p = .019) and others similarity (b =
0.728, p = .023). The positive others’ gender effect implies that the person would be more satisfied if
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the other group members are male. In addition, the person would be more satisfied if the others were
more similar to each other.
In summary, we find that people were more satisfied when the others in the group (and not the
group itself) are more homogeneous. Additionally, people were more satisfied when there were more
males in the group. Ironically, the model predicts that satisfaction would be maximal when the group
was all male, something which did not occur in the study.
Group-Level Outcomes
There is often interest in the effect of group composition on group-level outcomes such as
productivity. For example, a question of some interest is the effect of group diversity on group
productivity. We can adapt the GAPIM to model group outcomes (GAPIM-G). With such a model,
we would have only group-level predictors:
__
__
Yk  b0  b1 X k  b2 I k  ek
(2)
The variable X k is the average gender of the n members of the group k and I k is the average similarity
of all n(n – 1)/2 pairs of members of group k. Thus, a group’s productivity is determined by the
proportion of males in the group and the extent to which the group has an equal number of males and
females. We note that the interaction captures in part (see web appendix) the non-additive effect of
group composition, chemistry or synergistic effects.
For illustration purposes only, we took the Mohammed et al. (2002) data and aggregated the
satisfaction to the group level. We found that b1 = .958 (p = .184) and b2 = .768 (p = .193). We find
that groups with more males and more homogeneity have more a positive outcome, a result that
parallels our individual analysis. However, neither of these effects are statistically significant.
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Dyad-Level Outcomes
For most applications of the GAPIM, we would have individual outcomes – a single measure
for each person. In this section, we consider the less frequent, but very interesting case of a dyadic
outcome.
Consider n-person groups in which for each person there is a measure of some characteristic of
the person (e.g., ethnicity, gender, or opinion). We again denote this characteristic as Xik for person i in
group k. We assume here that it a dichotomy and we use effects coding, 1 and –1. (We later discuss
how to handle non-dichotomous categorical variables, as well as continuous variables.) Again we
presume that we have males (+1) and females (-1). Here we consider an outcome variable that is
dyadic, for example how persuasive person i thinks person j is in group k, or Yijk. We refer to i as the
actor and to j as the partner, and subscript k refers to group. The main effects model of the GAPIM for
dyad data (GAPIM-D) is:
Yijk = b0 + b1Xik + b2Xjk + b3Xijk' + eijk
(3)
where Xijk' is the mean of all Xs in group k besides persons i and j. Thus, if person i is a man, person j
is woman, and the other members of the group were all women, then Xik would equal +1, Xjk would
equal –1, and Xijk' would equal –1. We would interpret the coefficients for these three variables a
follows:
b1: actor effect: If positive, then men tend to think that others are more persuasive than
women think others are persuasive,
b2: partner effect: If positive, male partners are thought to be more persuasive than female
partners, and
b3: others effect: If positive, an actor tends to think the partner is persuasive if the others in
the group are male.
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These three main effects can also interact in four ways:
1. Actor by partner interaction or how similar actor i is to partner j; we refer to this
interaction as dyad similarity; note that it equals 1 when i and j are both men or women
and –1 when one is a man and the other a woman; we denote this variable as Iijk which
equals XikXjk. The variable measures whether the actor and partner are same gendered
or not.
2. Actor by others interaction or how similar person i is to the other n – 2 members of the
group; we refer to this as actor similarity; this variable equals 1 when the person has the
same gender as the other n – 2 persons in the group and –1 when the person is of a
different gender; we denote this variable as Ii.k and it measures how similar gender is
between the actor and the others.
3. Partner by others interaction or how similar person j is to the other n – 2 members of the
group; we refer to this as partner similarity; note that this variable equals 1 when the
partner has the same gender as the other n – 2 persons in the group and –1 when the
partner is of a different gender; we denote this variable as I.jk and it measures how
similar the partner is to the others.
4. Interaction of all pairs of the others’ effects or how similar the other n – 2 are to each
other, a total of (n − 2)(n − 3)/2 dyads; we call this variable the others similarity and
denote it as Iijk' and it measures how similar the others are to each other.
Figure 1 can help in the understanding of these four interaction terms. There is a six-person group, one
member we designate at the actor (the white face), one as the partner (the grey face), and four as the
others (the black faces). There are a total of 15 different ties between members in the group, and for
each of these ties it can be determined if the two people are similar or not. The tie would be +1 if the
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two persons were similar and −1 if they were dissimilar. The 15 ties are allocated to the four
interaction terms. One of those ties is between the actor and partner; four between the actor and the
others; four between the partner and the others; and six between the four others. These ties represent
dyad, actor, partner, and others similarity, respectively. The full equation with the three main effects
and four interactions is:
Yijk = b0 + b1Xik + b2Xjk + b3Xijk' + b4Iijk + b5Ii.k + b6I.jk + b7Iijk' + eijk
(4)
Submodels
Although the GAPIM-D has seven terms in the model, normally a much simpler and more
theoretically more plausible model would fit better than the Complete Model. Having these different
submodels helps the researcher find a simpler and ideally more conceptually appropriate model than
the Complete Model with all seven terms. By estimating and testing the submodels, we can much
more clearly begin to understand the social psychological process that affects group members. Again,
we borrow from the Kenny and Cook (1999) formulation which is based on interdependence theory.
We first consider six submodels of the main effects.
The Main Effects Model includes all three main effects, the actor effect, the partner effect, and
the others effect, but no interactions. In the Actor Only Model, there is only an actor effect. Such a
model is analogous to an individual motive in Interdependence Theory. In the Partner Only Model,
there is only a partner effect: Persuasiveness depends on the gender of the partner. The Group Model
has as the effect the weighted average of the three main effects and the outcome variable is determined
by the average of the group members’ compositional variable (e.g., gender). The next two main effects
submodels can be viewed as contrast models. In the Self vs. Partner Model, the actor and partner
effects are compared, and so the operative variable is Xi – Xj. In the Self vs. Others Model, the actor
compares him or herself to all the others in the group (including the partner) and this comparison is
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what affects his or her evaluation of persuasiveness. As we discussed with individual outcomes, this
pattern of results is sometimes called a frogpond effect.
We can also consider seven of submodels of the interaction terms. In the Diversity Model, all
four interaction effects would be equal, b4 = b5 = b6 = b7. This model implies that if we aggregated the
four interaction terms into a measure of group diversity, the fit of the model would not suffer. The
operative variable is how similar (or different) everyone in the group is. The Actor Similarity Model
posits that the important similarity variables are actor similarity, the effect of the actor’s similarity to
the others (b5) and dyad similarity (b4). Together these variables represent the actor similarity to
everyone else in the group. This model implies that b4 = b5 and b6 = b7 = 0 and can be considered to be
the person fit model (Elfenbein & O’Reilly, 2007), where person is the actor. Analogously, what
might be most important in seeing how persuasive one’s partner might be is how similar the partner is
to the others in the group: If the partner is similar (or different from) to others in the group, he or she is
seen as more persuasive. This model is referred to as the Partner Similarity Model and implies that b4
= b6 and b5 = b7 = 0.
At least four different contrast models of the interaction effects can be estimated and tested. A
contrast model implies that interaction effects have opposite signs, but equal magnitudes. We might
test that an actor sees a partner as persuasive if the partner is similar to the actor but the actor is
different from others. In this model, the dyad similarity and actor similarity effects are of opposite
sign, and the remaining interaction effects equal zero. Alternatively, what might matter is how similar
the partner is to the others in the group relative to how similar the actor is to the partner. The actor
might see the partner as persuasive when the partner is similar to the actor but different from others.
We might also examine whether an actor versus partner contrast model in which the actor sees the
partner as persuasive if the actor was similar to others but the partner was different from others. In the
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final contrast model, we test an “us against them” effect: The actor and partner are similar to each
other but different from the others in the group who are all similar to each other. When this effect is at
its maximum, the actor and partner are the only two persons in the group with the same identity.
Estimating and Testing
The GAPIM-D is a complicated multilevel model with several types of nonindependent terms.
They are the usual Social Relations Model’s (Kenny & La Voie, 1984) variances and covariances that
are detailed in the web appendix (davidakenny.net/doc/gapim_tech.doc). Because the major focus in
this paper is the effects of group composition, these SRM effects are not discussed here. Nonetheless,
we need to compute these variances and covariances in order to obtain the proper standard errors for
the seven effects of group composition in the GAPIM-D. We used a method of estimation presented
by Kenny and Livi (2009).
Again we used the SABIC to compare the relative fit of the models. To get a baseline value,
we estimated the SABIC for a model in which all seven terms of the model are set to zero, an Empty
Model. Our model testing strategy was as follows. We first estimated a Main Effect Model, and we
tested the two main effects for statistical significance, and we compared its fit to the Empty Model.
We then estimated the Complete Model, the model with all seven terms. We compared the fit of this
model to the Main Effects Model. Finally, based on these analyses we chose a submodel that had
better fit than any of the prior models and most of whose terms were statistically significant.
Example
As an example data set to test the GAPIM-D, we used the data from Culhane, Hosch, and
Weaver (2004). Six-person juries were assigned to one of six videotaped versions of a mock criminal
burglary trial. The mock trials were videotaped in a local courtroom with a judge, attorneys, and a
police officer playing their actual roles. The language and race of defendant varied between juries; the
17
victim was always female and the perpetrator was always male. There were 804 mock trial participants
forming 134 6-person juries from El Paso County, Texas. The median age was 45, ranging from 18 to
89. The sample was 54.6% Female, 58.5% Hispanic, 31.3% White, 3.9% Black, and 2.2% Asian
American or Native American. After viewing their assigned mock trial, jurors deliberated for up to
three hours to reach a unanimous verdict or were declared hung. After deliberation, each juror gave
round-robin ratings of persuasiveness of the other five jury members which were on a 1 to 5 scale.
The composition predictor variable is gender. There was only one participant with a missing
value for gender so we dropped his or her entire group from the analyses resulting in 133 six-person
juries and 798 jurors. Each juror rated the other 5 members, resulting in 3990 dyadic observations.
However, there were two missing cases on the persuasiveness measure and so the resultant sample size
is 3988.
Results
We first consider the Main Effects Model. Recall that we have effect coded gender, +1 for
males and −1 for females. In Table 2 we see that the one statistically significant effect is the main
effect of gender of partner. This effect is positive, which indicates that males were seen as more
persuasive than females, a result found in the Eagly and Carli (1981) meta-analysis. We do note that
the effect of others gender is larger than the actor gender effect, but it is not statistically significant (p =
.118). However, it should be realized that there is less power for the test of the others gender than there
is for the test of actor gender because the former varies mostly by group and the latter varies mostly by
individual within group. We note also that these two effects are of opposite sign. It is the contrast
between partner’s gender and the gender of others that leads to perceptions of persuasiveness.
When we added the four interactions and estimated the Complete Model, we see that the fit of
the model improves. Although the dyad and others similarity effects are not statistically significant,
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the actor similarity effect is statistically significant and the partner similarity effect is marginally
significant. Note that the actor similarity effect is positive whereas the partner similarity effect is
negative, suggesting that if the actor is similar to others in gender and the partner is different from
others, then the partner is seen as more persuasive. When we consider the submodels, the one that fits
best is the Actor vs. Partner Model. A female is seen as relatively more persuasive by a male, if the
other members of the group are mostly male; likewise, a male is seen as relatively more persuasive by
a female, if the other members of the group are mostly female. These results suggest that persons who
are different from others are seen as more persuasive, a result previously found in other studies (e.g.,
Antonio et al., 2004).
In Figure 2, we have graphed the results from the Complete Model for Gender of Partner, Actor
Similarity, and Partner Similarity. In the graph, similarity means that the actor or partner is similar to
all of the others and dissimilarity means dissimilar to all of the others in the group. Note also that the
combination of partner similarity and partner gender completely determines the gender of the others.
For instance, if the partner is similar to others and the partner is male, then the others must be male.
Examining Figure 2, there is the effect of actor similarity: the dark lines tend to be above the light
lines. Also the effect of partner gender occurs only when the partner and others are dissimilar. Finally,
there is an effect of partner similarity only when the partner is male. This result might be interpreted
as an interaction between gender of partner with partner similarity, but that “interaction” is really the
main effect of partner gender.4
In summary, the effects in the model can be viewed in terms of contrast effects: A partner is
viewed as relatively more persuasive if the partner’s gender is different from the others in the group
and the actor’s gender is similar to the others in group. Additionally, it is found that a male partner is
seen as more persuasive if the others in the group are female. Moreover, these two main effects are
19
roughly equal which means that when the partner and others are the same gender, male partners more
than female partners are seen as persuasive, but when the partner and others are of a different gender,
there is no effect of gender of partner. Note that we find that the well-established finding that men are
seen as more persuasive should be qualified by the composition of the group. Certainly, these results
require replication, but they do illustrate the potential benefits of the GAPIM.
Summary of the Two Examples
For the individual and dyad examples, we found evidence for main effects and interactions and
for both we found that a submodel was the best fitting model. However, we do caution the reader that
we in no way claim the results here and earlier that we obtained are universal of gender effects.
Interestingly, we find evidence in both cases of contrast effects, but we suspect that often group
composition effects would appear as group effects, actor gender and other gender effects being the
same sign.
Extensions
In this section, we discuss aspects of the model that we have thus far ignored. They are solo
status, unequal group sizes, missing data, and continuous variables. Moreover, in the last part of this
section we discuss design issues and consider the important question of the number of groups that are
needed to estimate the model.
Solo Status
We have not yet explicitly discussed the effect of solo or token status in groups. Solo members,
especially persons of minority demographic groups (i.e., tokens), may experience the group very
differently from the rest of the members of the group (Sekaquaptewa & Thompson, 2002). Much of
the research on solo and token status has focused on issues of power, for example identifying group
20
decision makers, measuring participation, and whose ideas are most persuasive (Benenson & Heath,
2006; Hewstone et al., 2006; Kanter, 1977; Saenz, 1994).
The GAPIM would predict that solos should see the group very differently from other members
of the group because they have relatively extreme scores on many of the variables of the model. In
terms of the GAPIM-I, if one is a solo then the person is different from everyone else in the group (Iik =
–1) and everyone else in the group is the same (Iik' = 1). For example, if there was a large effect of
being solo then the model would forecast that b4 – b3 would be large in absolute value. Note that if b4 –
b3 were large, there would be what we called a contrast effect at the level of the two interaction effects.
Usually, there is more interest in solo status for a person who is member of a low status group.
Although not obvious, if there was a token effect (i.e., solo group member of low status), we would see
a contrast effect of the main effects as well as contrast effect of interactions.
However, it is possible that the reaction of someone who is a solo or a token may be more
extreme than predicted by the GAPIM which would imply that the additive model does not explain all
of the results. To see if there was a pure solo effect, we would create an additional variable Sik which
would equal one if the person was the only person of that type in the group and would be zero
otherwise. The test of this variable would evaluate if there was anything special about being a solo
above the additive effect of our composition variables. Moreover, we can have this variable interact
with Xik to determine if there were token effects, that is, if being a solo and a member of a subordinate
social status (e.g., female) has an effect on the outcome.
Unequal Group Sizes
Unequal group sizes present various complications for the analysis that we have developed in
this paper. Note that several of the terms, both main effects and interactions, are defined in terms of
means across others or relationships. We would then use the group size for that particular group to
21
compute those means. To be able to compute these means, there must be at least three persons in the
group for the individual model and four persons for the dyadic model.
Missing Data
Missing data can be especially problematic for the GAPIM. Even if there are missing data on
the outcome variable, ideally there would not be missing data on the characteristic that would be used
to compute the composition variables. For example, if we are studying the effects of gender
composition on liking, we might still have information about the gender of the participants who did not
provide a rating of liking. This is a better scenario than if the converse pattern were present: if we have
liking scores, but no gender. Note that if one participant in a group has a missing Xik score, gender, then
the scores on the other three variables of the GAPIM-I are missing for all the other respondents in the
group; moreover, the scores for at least five of the six variables would be missing in the GAPIM-D
case. In this situation the researcher is faced with a choice on how to handle the missing data: Either
the entire group is treated as if it were missing, or the participant who has a missing gender is ignored
and it is presumed that the group has one less member. While neither of these solutions is ideal, if the
group size is fairly large or if the sample consists of a small number of groups, the latter strategy may
be preferable.
Non-dichotomous Variables
If X, the composition variable, were a continuous variable, like age, and not a dichotomy, we
might modify the strategy as follows. We determine the largest score in the sample or XL and the
smallest score in the sample or XS. For variables that have upper and lower limits, these limits would
be used to determine XL and XS. We would then recode Xik as 2(Xik − XS)/(XL − XS). This will change
the largest score to +1 and the smallest score to −1. In this way the interaction variables can be
interpreted as equaling 1 when people have exactly the same score on X and equaling −1 when persons
22
have the maximally different scores. A score of zero would indicate a pair of scores that fell half way
between no difference and the largest possible difference in the sample.
Additionally, the variable X might be categorical, but not a dichotomy. For instance X might be
ethnicity, with five different ethnicities possible. For the main effect composition variables Xik and Xik',
we suggest using multiple dummy variables to code for those variables. That is, if we had five
ethnicities, we might create four dummy variables, using one ethnicity as the standard group. We can
retain the same coding for the interaction terms, Iik and Iik'. For example, if Iik' equals 1, this score
would imply the others in the group are all members of the same ethnic group. However if it is equal
to −1, then that would imply that all the other members of the group are members of different ethnic
groups.
We should also make one point about the lower limit of others similarity in the GAPIM-I (Iik')
and in the GAPIM-D (Iijk'), when Xik is a dichotomy. Recall that these variables are scaled such that a
value of one means complete similarity and minus one means complete dissimilarity. When the
variable takes on only two values (e.g., male and female), the lower limit of this variable is –1/m if m is
odd and –1/(m – 1) if m is even, where m is the number of others in the group besides the actor or the
actor and partner in the dyadic case. For instance, if m is 10, the lower limit is not −1, but rather
−0.111. If, however, Xik can take on as many values as n, group size, as in the case of a six ethnicities
in a group of n = 6, then both Iik' and Iijk' can equal their theoretical lower limit of –1. That is, a person
can be completely dissimilar to the others in the group and they can also be completely dissimilar to
each other. In this case, diversity would measure variety, as defined by Harrison and Klein (2009).
Multiple Xs
We have assumed so far that there is just a single X variable. As is the case in the Culhane et
al. (2004) dataset that we previously analyzed, there may be several different composition variables
23
simultaneously affecting one outcome, and we may wish to combine these variables. For instance, we
might be interested in the combined effects of gender and ethnic composition on liking. Thus, we
would want to combine, in one analysis, information about the group members’ genders and
ethnicities. This problem is closely related to the idea of demographic faultlines — subgroups may
form within groups based on differences in group members’ characteristics (Lau & Murnighan, 1998;
Lau & Murnighan, 2005). Furthermore, these faultlines increase in strength as subgroup differences
based on one characteristic are reinforced by differences on other characteristics. For example, half of
the group may be Hispanic and female while the other half of the group may be White and male.
Clearly, testing hypotheses about multiple variables can become complicated.
One idea might be, create the various GAPIM variables and have them work through a
composite (Bollen & Lennox, 1991). That is, we would create all of the relevant composition variables
for gender and ethnicity separately as we have discussed. Then, each pair of variables is assumed to
cause a composite variable. For instance, others ethnicity and others gender, from the APIM-I, would
cause others ethnicity-gender. The weighting of the composite would depend on the data. Thus, if for
the variable of interest, say liking, gender effects were stronger than ethnicity, then gender would
receive the greater weight. These weights would be fixed to be same for the different variables in the
GAPIM model.
Number and Sizes of Groups
In this section, we discuss the design of a GAPIM study. We first discuss the decision about the
number of groups to study. We then discuss what happens with the model when the X variable is
skewed or when there is only a single group.
Number of groups. The reader likely has noticed that we have 133 groups in the dyadic
example dataset. Obvious questions are how many groups are needed to perform the analysis and are
24
over 100 groups necessary. Although we cannot provide definitive answers, we can provide some
guidance.
If the interest is in estimating the full GAPIM, the number of groups cannot be small for at
least two reasons. First, as we have emphasized, several different terms of the model (the others and
diversity effects) vary primarily at the group level. The effective sample size for such effects is not the
total number of persons in the sample, but rather the number of groups.
Second, the terms of the model are not independent and can be correlated. The presence of
co-linearity of effects increases standard errors and reduces power. In the next section, we discuss
skewed X variables which can result in very strong correlations between the predictor variables in the
model.
It is difficult to give a rule of thumb about the minimum group size as it would vary depending on
effect sizes and what terms were present in the model. We do note that we did find effects for the
Mohammed et al. (2002) dataset that contained 19 groups.
One way of increasing power is to estimate one of the submodels. For instance, if the interaction
terms are correlated, we might estimate the Diversity and Contrast Models. Because these models have
fewer terms and co-linearity is less of an issue.
Skewed X variables. In some studies, especially naturalistic ones, X might be highly skewed.
For instance, a company might have 100 working groups of size 5, or 500 workers, but only 50 of these
workers might be male. By chance, it is very unlikely there would be a group that had a majority of
male workers. In this case, actor gender and actor similarity as well as others gender and others
similarity become highly correlated. That is, if a person is male, likely he is dissimilar to others in the
group and if the others in the group are homogenous, the group is very likely to be mostly female. For
instance, if we take the Culhane et al. (2004) data and omit those groups in which males are in the
25
majority—in effect creating a minority male sample, and thus a skewed X variable—the correlation
between gender and actor similarity is –.612 between others gender and others similarity is –.799,
which is quite a change from r = –.240 and –.143, respectively, in the full sample. That is, when the
group is not diverse, it is almost certainly because most of the other group members are female and not
mostly males. We suggest that in cases where there are only a few groups in which one type of person
is the majority, it might be best to estimate only the Main Effects Model. In such an analysis, the effect
of Xik is confounded with Iik and that the effect of Xik' (or the average of X of the others in the group) is
confounded with Iik' (the diversity of others in the group).
Single group. One question that we might ask is what happens to our model when we have just
one very large group? For instance, we might ask members of a large organization how much they like
being in the organization and we seek to compare men’s and women’s response to that item. We
assume in the following discussion that women are in minority in this particular organization. We first
note that the variables Xik' and Iik' are constants. That is, their effect is contained in the mean of all
responses. Also note that the effects of Xik and Iik have a perfect negative correlation, making their
effects confounded. Thus, if we found that women liked being in the organization less than did men,
we would not know if that lowered level of liking was due to gender (men liked and women disliked
the organization) or due to actor similarity (men liked the organization because there were mostly men
in the organization and women disliked the organization because there were not many people in it
whom shared their gender). Typically, such effect are interpreted as an actor effect (e.g., women
relative to men do not like being in the organization) when in fact the effect may actually be one of
actor similarity (e.g., people do not like being in organization when there are very few members who
are similar to them).
26
Summary. It can easily happen that the researcher is unable to estimate the full GAPIM, either
because effects are highly correlated or because there are too few groups. Even if the full GAPIM
cannot be estimated, it is still valuable to realize that the researcher is estimating a special case of the
general model and is assuming that some effects are zero. By using the GAPIM, the research can
better understand his or her results, even if all the GAPIM parameters cannot be estimated.
Conclusion
The reader may have noticed that the terms in the model involve individuals − the main effects
− or dyads − the interaction effects. Moreover, our extended examples examined individual and dyadic
effects. As one of the reviewers asked, “Where is the group in the GAPIM?” We admit that the
Complete Models decompose group effects into individual and dyadic group processes. But if there
are group effects such as diversity and group composition effects, then we would find that a relevant
submodel would fit better than the Complete Model. Thus, the GAPIM allows group researchers to
show that effects occur that are not at the level of the individual or the dyad but are at the group level.
Additionally the reader should not confuse the level of measurement of the outcome measure with level
at which the phenomenon occurs within groups. For instance, we might measure liking dyadically by
asking each member how much he or she likes the other members of the group. Such a measurement
process does not imply that the process is dyadic. Using the GAPIM-D, we might find that the
phenomenon is at the group level (i.e., cohesiveness), at the individual level (popularity effects), or the
dyadic level (the match between perceiver and partner). By measuring the phenomenon at the lowest
possible level, we can empirically evaluate the level at which the phenomenon operates (Bond &
Kenny, 2002) which might well be a level higher than the unit of measurement.5
How group composition affects groups, people and relationships is a complex process. We
have shown that there is not one effect of group composition, but several. Moreover, these effects
27
occur at multiple levels of analysis, the individual and the group, and effects can be main effects and
interactions. These effects can work in combination. Following Brown (1965), there is a dichotomy of
processes in groups. The first is solidarity. The boundaries between persons break down and the
person and the group merge. The second is status. Here the person sees how he or she is different
from the others in the group. With the development of the GAPIM, we can now estimate and test these
processes more carefully.
We focused on the topic of demographic group composition because it interests us and it has
been a classic question in the study of groups (Levine & Moreland, 1998). However, the variable X
need not be a demographic compositional variable. For instance, consider the hypothesis that
perceptions of fairness lead to feelings of being satisfied in the group (Tyler & Blader, 2003). We
could perform a GAPIM-I analysis treating perceptions of fairness as the X variable. That is, the group
composition variable of interest may be surface or deep characteristic (Harrison et al., 1998). Perhaps
the most classic questions of group composition are the effects of member ability on member’s
perceptions and group performance (Jones & Regan, 1974; Tziner & Eden, 1985) as well as the effects
of newcomers and personnel turnover (Jackson et al., 1991; Levine, Choi, & Moreland, 2003). The
GAPIM can be used to address these questions. Alternatively, the X variable might be an
experimentally manipulated variable. For instance, the experimenter might manipulate group
identification and determine how hard the individual works in the group, or randomly assign
participants to artificially created groups, such as a red team or blue team, and investigate group
cooperation under varying red-blue compositions.
The study of group composition is much more complicated than might be thought. There are
main effects and interactions and there is the group and individual level. Moreover, these effects can
be combined to form a group and a contrast model. With individual outcomes, there are 4 fixed
28
effects, 2 random effects, and 11 models that we recommend estimating. With dyadic outcomes, there
are 7 fixed effects, many random effects, and 15 models. It is our view that the full complexities must
initially be considered if we are to understand the many different ways group composition affects the
behavior and perceptions of persons in groups. In one model, we can simultaneously measure
individual differences, person fit, climate, diversity, contrast, and solo effects. Only by considering all
of these effects simultaneously can the group researcher fully understand the complications and
interesting nuances of group composition. With the GAPIM, researchers interested in main effects will
also consider interactions and vice versa; researchers interested in individual-level effects would also
consider group-level effects and vice versa; and researchers interested in both individual and group
effects will also consider contrast effects. Armed with the GAPIM, we anticipate significant advances
in the understanding of group and intergroup processes.
29
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34
Table 1
Effect Coefficient Estimates of Gender (Female = -1, Male = 1) on Individual Satisfaction with Being
in the Group
Model
Empty
Main Effects Only
Complete
Others Only
a
Main Effects
Gender
Others Gender
0a
0a
-0.082
0.157
0.013
0.973+
a
0
0.918*
Fixed to zero.
b
Smaller SABIC means a better fitting model.
+ p < .10
* p < .05
** p < .01
Interactions
Actor Similarity
Others Similarity
0a
0a
0a
0a
0.053
0.746+
a
0
0.728*
Fit
SABICb
214.94
217.32
213.47
210.47
35
Table 2
Effect Coefficient Estimates of Gender (Female = -1, Male = 1) on Dyadic Perceptions of
Persuasiveness
Model
Empty
Main
Effects
Complete
Model
Contrast:
Actor vs.
Partner
a
Main Effects
Actor
Partner Others
Gender Gender Gender
0a
0a
0a
Interactions
Dyad
Actor
Partner
Others
Similarity Similarity Similarity Similarity
0a
0a
0a
0a
* p < .05
** p < .01
11477.79
0.102**
-0.098
0a
0a
0a
0a
11456.27
-0.007
0.086**
-0.092
0.018
0.148*
-0.102+
0.076
11449.98
-0.007
0.086**
-0.097
0a
0.125**
-0.125**
0a
11446.90
Smaller SABIC means a better fitting model.
+ p < .10
SABICb
-0.023
Fixed to zero.
b
Fit
36
Figure 1
Understanding the Different Ties in the GAPIM-D
Actor:
Dyadic Similarity:
Actor Similarity:
Partner:
Partner Similarity:
Other Similarity:
Other:
37
Figure 2
Gender Composition Effects on Ratings of Persuasiveness
3.8
Actor and Partner Similar to Others
Actor Similar and Partner Dissimilar to Others
Persuasiveness
3.6
Actor Dissimilar and Partner Similar to Others
Actor and Partner Dissimilar to Others
3.4
3.2
3
2.8
2.6
Male
Female
Gender of Partner
38
Footnotes
1
In Kenny et al. (2002), the term partners is used. Because in the dyad model, there is a term called
partner, we use others here to reduce confusion.
2
The group and group diversity effects, which are estimated in some models, are at the group level.
3
The reader may have noticed that the coefficient for others gender dramatically increased when the
interaction terms were added to the model. We have here a case of suppression in that others gender
and others similarity is correlated -.878. We return to this issue when we discuss “skewed X
variables.”
4
Note that the interaction of partner similarity with partner gender can be expressed as (Partner Gender
X Others Gender) X Partner Gender. Following Schaffer’s (1977) discussion of aliasing, the two
Partner Gender terms cancel each other out leaving just Others Gender.
5
We could further complicate the GAPIM to allow for interactions at triadic and higher levels. Models
to evaluate conjunctive and disjunctive process (Steiner, 1972) can be viewed as including an
interaction effect at the group level containing the product of n terms.