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Modeling Interdependence: Toward a General Framework. Richard Gonzalez, U of Michigan Dale Griffin, U of British Columbia. The Nested Individual. Underlying Premises. Nonindependence provides useful information is not a nuisance

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Modeling interdependence toward a general framework l.jpg

Modeling Interdependence: Toward a General Framework

Richard Gonzalez, U of Michigan

Dale Griffin, U of British Columbia


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The

Nested

Individual


Underlying premises l.jpg

Underlying Premises

  • 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


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Historical Analysis

  • 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”)


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Necessary Conditions

  • 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


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McDougall, 1920, p. 23

  • 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.


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Statistical Framework Should Mimic Theoretical Framework

  • 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


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Menu of Techniques

  • Repeated measures ANOVA

  • Intraclass correlation

  • Hierarchical linear models (HLM)

  • Structural equations models (SEM)


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Common Beliefs about Interdependence in Dyadic Data

  • 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)


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  • 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.


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Dyadic Designs:Three Major Categories

  • 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


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Intraclass Correlation: Building Block

  • Structural Univariate Models:

    • Exchangeable

    • Distinguishable


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ANOVA Intraclass (& REML)Dyads


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Intraclass Correlation:HLM Language

  • Two level model:

  • Intraclass correlation is given by


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Pairwise Coding

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


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Pairwise Intraclass Correlation


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Example: Personal VictimizationCeballo et al, 2001


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Pairwise Intraclass (ML):Dyads


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Interdependence

  • 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)


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Pairwise Generalization

  • 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


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Example(Stinson & Ickes, 1992)

  • 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)


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Some formulae

  • Actor regression coefficient

  • V(Actor reg coeff)

Partner coef replaces Y with Y’


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Interdependence Example

  • 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


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a

Xm

Ym

b

rx

r

c

Yc

Xc

d


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Not on Welfare

Xm

Ym

-.1

.2

.3

Yc

Xc


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Welfare

.09

Xm

Ym

.1

Yc

Xc


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V-post

V-pre

S-pre

Simple Actor-Partner Model:Pre-post death of spouse

No interdependence problem on the dependent variable


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Return to Original Model: Special Case

a

Xm

Ym

b

rx

r

c

Yc

Xc

d

Set a=d and b=d


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ri

ri

Eym

Eyc

Exc

Exm

Yc

Ym

Xm

Xc

Y

X

rd


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Latent Variable Model

  • ri = individual level correlation

  • rd = dyad level correlation

  • The square root of intraclass correlations are the paths


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Using Path Analysis Rules

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


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Solving those two equations….


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.4

.4

Eym

Eyc

Exc

Exm

Yc

Ym

Xm

Xc

.47

.45

.47

.45

Y

X

-.8

Not on Welfare


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Eym

Eyc

Exc

Exm

Yc

Ym

Xm

Xc

.2

.3

.2

.3

Y

X

1.6

Welfare: latent variable model doesn’t hold


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What does the correlation of two dyads means?

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


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Multivariate Model: HLM Lingo

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


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Difference scores

  • 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


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Difference Scores

  • 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


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“Solutions”

  • 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”


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Model-Based Approach

  • 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)


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Model-Based Approach

  • 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


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Patterns of Regression 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


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Patterns of Regression Coef

  • 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.


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Conclusion

  • 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


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