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:

kkkk eIbXbbY __

2

__

10 (2)

The variable kX is the average gender of the n members of the group k and kI 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

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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

Main Effects Interactions Fit

Gender Others Gender Actor Similarity Others Similarity SABICb

Empty 0a 0

a 0

a 0

a 214.94

Main Effects Only -0.082 0.157 0a 0

a 217.32

Complete 0.013 0.973+ 0.053 0.746

+ 213.47

Others Only 0a 0.918* 0

a 0.728* 210.47

aFixed to zero.

bSmaller SABIC means a better fitting model.

+ p < .10

* p < .05

** p < .01

35

Table 2

Effect Coefficient Estimates of Gender (Female = -1, Male = 1) on Dyadic Perceptions of

Persuasiveness

Model

Main Effects Interactions Fit

Actor

Gender

Partner

Gender

Others

Gender

Dyad

Similarity

Actor

Similarity

Partner

Similarity

Others

Similarity SABIC

b

Empty 0a 0

a 0

a 0

a 0

a 0

a 0

a 11477.79

Main

Effects -0.023 0.102** -0.098 0a 0

a 0

a 0

a 11456.27

Complete

Model -0.007 0.086** -0.092 0.018 0.148* -0.102+ 0.076 11449.98

Contrast:

Actor vs.

Partner -0.007 0.086** -0.097 0a 0.125** -0.125** 0

a 11446.90

aFixed to zero.

bSmaller SABIC means a better fitting model.

+ p < .10

* p < .05

** p < .01

36

Figure 1

Understanding the Different Ties in the GAPIM-D

Dyadic Similarity:

Actor Similarity:

Partner Similarity:

Other Similarity:

Actor:

Partner:

Other:

37

Figure 2

Gender Composition Effects on Ratings of Persuasiveness

2.6

2.8

3

3.2

3.4

3.6

3.8

Male Female

Gender of Partner

Pe

rsu

as

ive

ne

ss

Actor and Partner Similar to Others

Actor Similar and Partner Dissimilar to Others

Actor Dissimilar and Partner Similar to Others

Actor and Partner Dissimilar to Others

38

Footnotes

1In 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.

2The group and group diversity effects, which are estimated in some models, are at the group level.

3The 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.‖

4Note 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.

5We 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.