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November 1, 2011
Web Appendix for
Using the Actor-Partner Interdependence Model to Study the Effects of Group Composition
David A. Kenny and Randi L. Garcia
University of Connecticut
We welcome questions and points of clarifications. This is a working document that we plan to periodically edit.
Table of Contents (hyperlinked)
Measurement of Heterogeneity and Person Fit or Actor Simiarity
Triadic and Higher Order Interactions
Details Concerning the Submodels
GAPIM-I
GAPIM-D
Additional Data Analyses
GAPIM-D: Prior Opinion on Likeability
GAPIM-D: Ethnicity on Likeability
GAPIM-D: Gender on Persuasiveness
Solo Status
Distinguishable Members: A Leader and Followers
Random Effects in the GAPIM
GAPIM and the SUTVA Assumption
Details about Estimation and Computer Syntax
Estimating the GAPIM-I in SPSS
Macro for the estimation of the GAPIM-I
Estimating the GAPIM-D in SAS
Tables
References
Measurement of Heterogeneity and Person Fit or Actor Simiarity
Diversity has been conceptualized in many different ways. Recently, Harrison and Klein (2007) discussed three different ways diversity can be conceptualized: separation, variety, and disparity. We consider diversity in terms of separation in which diversity is at its maximum when the group has “half of unit members at the highest and lowest endpoints” (Harrison & Klein, 2007, p. 1203).
In this section of the technical appendix we consider some technical issues in the measurement of interaction effects. Using formal equations we first consider person fit or actor similarity and then diversity. We initially presume the group composition variable or X is a dichotomy, like being female or being male. Later in this section, we extend our discussion to consider non-dichotomous variables.
One common way to measure how a person fits in the group is to compute the average absolute difference between the person and the others in group. Let Xi be a characteristic of the person for whom we are predicting outcomes, and we can define person fit as Sj|Xi – Xj|/(n – 1) where j ≠ i, and n is the group size, and the summation is across the other n – 1 members of the group. In words, this measure of person fit is the average absolute difference between the person and all other persons in the group. Alternatively, we might compute the interaction between Xi and Xj by computing their product or XiXj and average these products across the n – 1 other members of the group SjXiXj/(n – 1) where j ≠ i. Note that when X is a dichotomy that is effects coded (1 and –1), XiXj equals 1 – |Xi – Xj|. Moreover, because SXiXj/(n – 1) has a perfect negative correlation with the S|Xi – Xj|/(n – 1), it can be used as a measure of person fit. We note that absolute difference measure times minus one plus one exactly equals the average of product terms.
In the diversity literature, there are many measures of group diversity and most are based on a variable’s variance or a measure that is very similar to the variance (e.g., the standard deviation). For instance, in sociology a common measure of diversity is the Blau’s index of heterogeneity (Blau, 1977) which equals 1 – ∑pi2 where pi is the proportion of the groups’ members who are of the demographic group i. For example, if i is female, then pi is the proportion of females in the group. Note that when there are just two levels of the variable (i goes from 1 to 2), as in the case of gender, the Blau index can be shown to equal the (n – 1)s2/(2n). Thus, although the Blau index does not look like a variance, it can be reexpressed as such.
Variance is conventionally defined as the sum of squared deviations of scores from the grand mean divided by group size less one. An alternative and mathematically equivalent definition is to define the variance as twice the average of the sum of squared differences between all pairs of members in the group (see Kenny & Judd, 1996). That is, it can be shown that the variance or s2 equals
(1)
where the summation is between all possible pairs of observations (i < j). Thus, a variance can be viewed as nothing more than an aggregate measure of the differences between all possible pairs of persons. Note that when X is an effect-coded dichotomous variable (+1 and –1), (Xi – Xj)2 is equal to 2(1 – XiXj) which results in Formula 1 equaling
(2)
Thus, the variance of the scores can be viewed the interaction between all possible pairs of persons in the groups.
Joseph Olsen of Brigham Young University has made the following suggestion to extend this approach to continuous variables. We thank Joe for his excellent suggestion and present it here: Traditional methods of centering predictors at their grand means or scale midpoints could be used to maintain meaningful interpretability of intercept and main effect parameters. It is possible to provide a general alternative to Equation 1 above which can handle continuous variables:
(3)
Combining this with Equation 2 above, we can also derive a general simplified expression for others’ similarity:
where and . The necessary information to carry out the analysis consists of the individual predictor scores (Xik) and the group size (), predictor mean (), and predictor variance (sk2) for group k. This reduces to the following in the special case of effect-coded dichotomous predictors:
Triadic and Higher-Order Interactions
The APIM is originally a dyadic model, and so it considers only dyadic interactions. However, in groups with more than two members, higher-order interactions are possible. We can add to the GAPIM-I or the GAPIM-G equation a triadic term of XikXjkXlk where the X variables are the three members’ scores on the group composition variable. Consider the case where X measures a skill that is present or absent. Note that if we use dummy coding with a dichotomy (1 = skill present; 0 = skill absent), then the triadic interaction term would be one only when all three members have the skill. Note if we reverse dummy code the variable (0 = skill present; 1 = skill absent), then the interaction only equals one, when none of the members had the skill. In this way we can allow for effects for conjunctive and disjunctive task (Steiner, 1972).
As discussed in Bond and Kenny (2002), the number of these interactions can become large. For a group of size 6, there are 15 dyadic interactions, 20 triadic interactions, 15 quartic interactions, 6 quintic interactions, and 1 septic interaction. Moreover for GAPIM-I and GAPIM-D, these would have to be apportioned in different ways. For instance for dyadic outcomes, there are 4 triadic interactions involving actor and partner, 6 involving actor and others, 6 involving partner and others, and 4 involving others. In practice most of these effects can be assumed to be zero.
We do plan to test the triadic, quartic, and quintic interactions for the GAPIM-G example in the paper and we shall present those interactions here very soon.
Detail concerning Submodels
The material in this and the next section is very similar to what is in the text, but here we include symbols and formulas. What is different is that we compare models using a deviance difference test which results in chi square tests. To conduct these tests we estimate the models using maximum likelihood and not restricted maximum likelihood. There are more submodels presented here than there are in the text.
GAPIM-I
We consider three submodels involving the two interaction effects. The first is the Diversity Model in which we include the main effects and a measure the overall diversity of the group. Diversity contains 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 similarity into a measure of diversity. The diversity model implies that b3 = b4 ≠ 0. Second, there is the Person Fit Model in which we assume that what matters is how similar the actor is to the others in the group, and not how similar the other members are to each other (i.e., from Equation 1 in the paper, b3 ≠ 0 and b4 = 0). The third model is the Contrast Model of interaction effects. In this model, a participant’s similarity is measured relative to the similarity of the others in the group. This model implies that the two interaction effects are of opposite sign: b3 – b4 ≠ 0.
We should note that all four variables in the Complete Model are all level-one or person-level variables. However, as group size increases, group members would have more similar scores on Xik' and Iik'. That is for large groups, Xik' and Iik' varies primarily between the groups. We think that here should be relationship between the random group effect, b0, and the two “group” effects in the model, Xik' and Iik'. If we find effects due to others or effects due to others similarity, we should also find group variance, because these effects are group level variables. Conversely, if there is little or no group variance, we might expect little or no effect of Xik' and Iik'. In a parallel fashion, we would likely not find evidence for either the Group or Diversity Models if the variance in the group intercepts was small.
There are five main effects models, all of which set both the interaction terms to zero:
I: Main Effects: b1 ≠ 0, b2 ≠ 0,
II: Actor Only: b1 ≠ 0, and b2 = 0,
III: Others Only: b2 ≠ 0, and b1 = 0,
IV: Group: b1 = b2, and
V: Contrast b1 − b2 ≠ 0.
While the first three models are relatively straight-forward, the fourth and fifth models require some additional explanation. 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 outcome. That is, if X is ethnicity, what affects the person is the ethnicity of the group, and not the ethnicity of the respondent alone. For the Contrast Model, Model V, the group member implicitly or explicitly compares his or her standing on X to the other group members. There is a strong boundary between self and other in this model. That is, the person determines how different his or her ethnicity is from the other group members’ ethnicity.
Additionally, there are four models for the interaction effects, all of which contain the two main effects, Xik and Xik’:
VI: Complete: b3 ≠ 0 and b4 ≠ 0,
VII: Person Fit: b3 ≠ 0 and b4 = 0,
VIII: Diversity: b3 = b4, and
IX: Contrast: b3 – b4 ≠ 0.
In the Complete Model, there is no constraint on the two interaction effects. For the Person Fit Model, what matters is only how similar the actor is to the others in the group, and the others’ similarity does not affect the person’s outcome. The Diversity Model focuses on how different all the members of the group are, including the actor. For the last model, the Contrast Model, the group member compares how similar he or she is to the others with how similar the others are to each other. For instance, a person who is of a different ethnicity from the other group members, while these other members are similar in ethnicity, might not identify with the group. In the Contrast Model, the actor similarity effect and others similarity effect have opposite signs.
GAPIM-D
Although the GAPIM-D has seven terms in the model, normally much simpler and theoretically more plausible models will fit better than the full model. In this section, we consider 15 different submodels. Having these different submodels helps the researcher find a simpler and more conceptually relevant model than the complete model with all seven terms. We first consider six submodels of the main effects. The first is the saturated main effects model and the next five are submodels:
Model I: The Main Effects Model includes all three main effects, the actor effect, the partner effect, and the others effect (b1 ≠ 0, b2 ≠ 0, and b3 ≠ 0).
Model II: In the Actor Only Model, there is only an actor effect, b1 ≠ 0 and b2 = b3 = 0.
Model III: In the Partner Only Model, there is only a partner effect, b2 ≠ 0, and b1 = b3 = 0.
Model IV: The Climate Model has as the effect of the weighted average of the three main effects, b1 = b2 = b3; the predictor variable is [Xi + Xj + (n – 2)Xij′ ]/n.
The next two main effects submodels can be viewed as Contrast models.
Model V: In the Self vs. Partner Model, the actor and partner effects are compared, b1 – b2 ≠ 0, and b3 = 0.
Model VI: In the Self vs. Others Model, the actor compares him or herself to all the others in the group, b1 – (b2 + b3).
The next nine models focus on the interaction terms.
Model VII: The Complete Model has all four interaction terms. It is the model to which the next eight models are compared.
Model VIII: The first submodel is the Relational Model where b4 ≠ 0 and b5 = b6 = b7 = 0: The actor likes (or dislikes) the partner based on the others’ similarity to the actor.
Model IX: The Diversity Model sets all for interaction effects 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.
Model X: 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’s 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 where person is the actor.
Model XI: Analogously, what might be most important in liking one’s partner might be how similar the partner is to the others in the group and not the actor’s similarity as in Model X. This model, which we refer to as the Partner Similarity Model, implies that b4 = b6 and b5 = b7 = 0. This model is the person fit model where person is the partner.