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Modeling Interdependence: Toward a General Framework

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Modeling Interdependence: Toward a General Framework

Richard Gonzalez, U of Michigan

Dale Griffin, U of British Columbia

The

Nested

Individual

- Nonindependence
- provides useful information
- is not a nuisance
- is a critical component in the study of interpersonal behavior
- but may not be required in all analyses

- Explanatory priority placed on the group
- Meade-- individual in context of group
- Durkheim
- Comte—family as primary social unit

- Explanatory priority placed on the individual
- Allport--individual is primary (“babble of tongues”)

- Homogeneity: similarity in thoughts, behavior or affect of interacting individuals
- E.g., group-level, emergent processes, norms, cohesiveness

- Interdependence: individuals influencing each other
- E.g., actor-partner effects

- The essential conditions of a collective mental action are, then, a common object of mental activity, a common mode of feeling in regard to it, and some degree of reciprocal influence between the members of the group.

- Make concepts concrete
- Avoid Allport’s “babble” critique
- Make the model easy to implement
TODAY’s Talk

- One time point; dyads
- Two or three variables
- Normally distributed data; additive models

- Repeated measures ANOVA
- Intraclass correlation
- Hierarchical linear models (HLM)
- Structural equations models (SEM)

- If you don’t correct for interdependence, your Type I errors will be inflated
- If you don’t correct for interdependence, your results will be ambiguous
- An HLM program will eliminate all nonindependence problems
- If you have dyadic data, you must run HLM (or else your paper won’t be published)

- These beliefs miss what we believe to be the fundamental issue:
There is useful psychological information lurking in the “nonindependence”

Interdependence is the “very stuff” of relationships.

- Subjects nested within groups
- Exchangeable (e.g., same sex siblings)
- Distinguishable (e.g., different sex siblings, mother-child interaction)
- Mixed exch & dist (e.g., same sex & different sex dyads in same design)

- Univariate versus multivariate
- Homogeneity versus interdependence

- Structural Univariate Models:
- Exchangeable
- Distinguishable

- Two level model:
- Intraclass correlation is given by

The Pearson corr of X and X’ is the ML estimator of the intraclass correlation.

- The degree to which one individual influences another
- Need not be face to face
- We have a good time together, even when we’re not together (Yogi Berra)

- Predictor X represents the actor’s influence on actor’s Y
- Predictor X’ represents the partner’s influence on actor’s Y
- Predictor XX’ represents the mutual influence of both on actor’s Y

- ActorS = ActorV + PartnerV
- Strangers: an effect of the partner’s verb frequency on the actor’s laughter (in ordinal language, the more my partner talks, the more I smile/laugh)
- Friends: an effect of the actor’s verb frequency on the actor’s laughter (the more I talk, the more I smile/laugh)

- Actor regression coefficient
- V(Actor reg coeff)

Partner coef replaces Y with Y’

- Mother and child witness victimization (WV) related to each individual’s fear of crime (FC).
- Does child’s WV predict child’s FC?
- Does mother’s WV predict child’s FC?
- etc

a

Xm

Ym

b

rx

r

c

Yc

Xc

d

Not on Welfare

Xm

Ym

-.1

.2

.3

Yc

Xc

Welfare

.09

Xm

Ym

.1

Yc

Xc

V-post

V-pre

S-pre

No interdependence problem on the dependent variable

Return to Original Model: Special Case

a

Xm

Ym

b

rx

r

c

Yc

Xc

d

Set a=d and b=d

ri

ri

Eym

Eyc

Exc

Exm

Yc

Ym

Xm

Xc

Y

X

rd

- ri = individual level correlation
- rd = dyad level correlation
- The square root of intraclass correlations are the paths

Two equations in two unknowns; reason why rxy may be uninterpretable

Solving those two equations….

.4

.4

Eym

Eyc

Exc

Exm

Yc

Ym

Xm

Xc

.47

.45

.47

.45

Y

X

-.8

Not on Welfare

Eym

Eyc

Exc

Exm

Yc

Ym

Xm

Xc

.2

.3

.2

.3

Y

X

1.6

Welfare: latent variable model doesn’t hold

So, there are multiple components to the correlation of dyad means making it uninterpretable….

- Three-level model: one level for each variable, one level for individual effect, and one level for group effect

- Frequently, a question of similarity (or congruence) comes up in dyadic research
- Diff of husband and wife salary as a predictor of wife’s relationship satisfaction
- Diff of husband and wife self-esteem as a predictor of husband’s coping

- Correlations with difference scores can show various patterns depending on their component correlations
- The numerator is a weighted sum of the correlations: (rX1Y SX1 – rX2YSX2)Sy
- Toy Examples
- One variable is a constant
- One variable is random

- One can use multiple regression, entering the two variables as two predictors (rather than one difference score).
- Y = 0 + 1X1 + 2 X2
- Problem: doesn’t test specific hypotheses such as “similar is better” or “self-enhancing is better”

- Questions
- Discrepancy model (woman’s sat is greatest the more she earns, the less her husband earns)
- Similarity model (woman’s sat is greatest the smaller the absolute diff in salary)
- Superiority model (woman’s sat is greatest when she earns more than her husband)

- Run separate regressions for subjects below and above the “equality line” (or use dummy codes and include an interaction term)
- The three different models imply different patterns on the coefficients

- Discrepancy model:
- Both regressions should yield a negative coef for the husband and a positive coef for the wife (maximizing the difference)

- Similarity model:
- For dyads where salary W>H, positive coef for husband and neg coef for wife because in this region higher husband salary identifies couples closer to equality
- For dyads where salary W<H, neg coef for husband and pos coef for wife because in this region higher wives’ salary identifies couples closer to equality

- Superiority model
- For couples where W>H on salary, a larger positive coef for wive’s salary

- The main point is that each model implies a qualitatively different pattern of regression weights across the two regressions.

- The take home message is that nonindependence due to interaction does not require a “statistical cure”
- Nonindependence provides an opportunity to measure and model social interaction
- Follow your conceptual models and your research questions
- There is still much room for careful design in correlational research with couples