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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


Slide2 l.jpg

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


Historical analysis l.jpg
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”)


Necessary conditions l.jpg
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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Dyadic Designs: issue: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: issue:Building Block

  • Structural Univariate Models:

    • Exchangeable

    • Distinguishable



Intraclass correlation hlm language l.jpg
Intraclass Correlation: issue:HLM Language

  • Two level model:

  • Intraclass correlation is given by


Pairwise coding l.jpg
Pairwise Coding issue:

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



Example personal victimization ceballo et al 2001 l.jpg
Example: Personal Victimization issue:Ceballo et al, 2001



Interdependence l.jpg
Interdependence issue:

  • 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 issue:

  • 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


Example stinson ickes 1992 l.jpg
Example issue:(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)


Some formulae l.jpg
Some formulae issue:

  • Actor regression coefficient

  • V(Actor reg coeff)

Partner coef replaces Y with Y’


Interdependence example l.jpg
Interdependence Example issue:

  • 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


Slide31 l.jpg

a issue:

Xm

Ym

b

rx

r

c

Yc

Xc

d


Slide32 l.jpg

Not on Welfare issue:

Xm

Ym

-.1

.2

.3

Yc

Xc


Slide33 l.jpg

Welfare issue:

.09

Xm

Ym

.1

Yc

Xc


Simple actor partner model pre post death of spouse l.jpg

V-post issue:

V-pre

S-pre

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

No interdependence problem on the dependent variable


Slide35 l.jpg

Return to Original Model: Special Case issue:

a

Xm

Ym

b

rx

r

c

Yc

Xc

d

Set a=d and b=d


Slide36 l.jpg

r issue:i

ri

Eym

Eyc

Exc

Exm

Yc

Ym

Xm

Xc

Y

X

rd


Latent variable model l.jpg
Latent Variable Model issue:

  • 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 issue:

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



Slide40 l.jpg

.4 issue:

.4

Eym

Eyc

Exc

Exm

Yc

Ym

Xm

Xc

.47

.45

.47

.45

Y

X

-.8

Not on Welfare


Slide41 l.jpg

E issue:ym

Eyc

Exc

Exm

Yc

Ym

Xm

Xc

.2

.3

.2

.3

Y

X

1.6

Welfare: latent variable model doesn’t hold


What does the correlation of two dyads means l.jpg
What does the correlation of two dyads means? issue:

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


Multivariate model hlm lingo l.jpg
Multivariate Model: HLM Lingo issue:

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


Difference scores l.jpg
Difference scores issue:

  • 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


Difference scores46 l.jpg
Difference Scores issue:

  • 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


Solutions l.jpg
“Solutions” issue:

  • 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”


Model based approach l.jpg
Model-Based Approach issue:

  • 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)


Model based approach50 l.jpg
Model-Based Approach issue:

  • 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


Patterns of regression coefficients l.jpg
Patterns of Regression Coefficients issue:

  • 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


Patterns of regression coef l.jpg
Patterns of Regression Coef issue:

  • 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 issue:

  • 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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