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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

Modeling Interdependence: Toward a General Framework

Richard Gonzalez, U of Michigan

Dale Griffin, U of British Columbia

slide2
The

Nested

Individual

underlying premises
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
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
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
mcdougall 1920 p 23
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.
statistical framework should mimic theoretical framework
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
menu of techniques
Menu of Techniques
  • Repeated measures ANOVA
  • Intraclass correlation
  • Hierarchical linear models (HLM)
  • Structural equations models (SEM)
common beliefs about interdependence in dyadic data
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)
slide11
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.

dyadic designs three major categories
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
intraclass correlation building block
Intraclass Correlation: Building Block
  • Structural Univariate Models:
    • Exchangeable
    • Distinguishable
intraclass correlation hlm language
Intraclass Correlation:HLM Language
  • Two level model:
  • Intraclass correlation is given by
pairwise coding
Pairwise Coding

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

interdependence
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)
pairwise generalization
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
example stinson ickes 1992
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)
some formulae
Some formulae
  • Actor regression coefficient
  • V(Actor reg coeff)

Partner coef replaces Y with Y’

interdependence example
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
slide31
a

Xm

Ym

b

rx

r

c

Yc

Xc

d

slide32
Not on Welfare

Xm

Ym

-.1

.2

.3

Yc

Xc

slide33
Welfare

.09

Xm

Ym

.1

Yc

Xc

simple actor partner model pre post death of spouse
V-post

V-pre

S-pre

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

No interdependence problem on the dependent variable

slide35
Return to Original Model: Special Case

a

Xm

Ym

b

rx

r

c

Yc

Xc

d

Set a=d and b=d

slide36
ri

ri

Eym

Eyc

Exc

Exm

Yc

Ym

Xm

Xc

Y

X

rd

latent variable model
Latent Variable Model
  • ri = individual level correlation
  • rd = dyad level correlation
  • The square root of intraclass correlations are the paths
using path analysis rules
Using Path Analysis Rules

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

slide40
.4

.4

Eym

Eyc

Exc

Exm

Yc

Ym

Xm

Xc

.47

.45

.47

.45

Y

X

-.8

Not on Welfare

slide41
Eym

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
What does the correlation of two dyads means?

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

multivariate model hlm lingo
Multivariate Model: HLM Lingo
  • Three-level model: one level for each variable, one level for individual effect, and one level for group effect
difference scores
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
difference scores46
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
solutions
“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”
model based approach
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)
model based approach50
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
patterns of regression coefficients
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
patterns of regression coef
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.
conclusion
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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