Modeling Interdependence: Modeling Interdependence: Toward a General FrameworkToward a General Framework

Richard Gonzalez, U of MichiganRichard Gonzalez, U of Michigan

Dale Griffin, U of British ColumbiaDale Griffin, U of British Columbia

The

Nested

Individual

Underlying PremisesUnderlying Premises

Nonindependence Nonindependence – provides useful informationprovides useful information– is not a nuisanceis not a nuisance– is a critical component in the study of is a critical component in the study of

interpersonal behaviorinterpersonal behavior– but may not be required in all analysesbut may not be required in all analyses

Historical AnalysisHistorical Analysis

Explanatory priority placed on the groupExplanatory priority placed on the group– Meade-- individual in context of groupMeade-- individual in context of group– DurkheimDurkheim– Comte—family as primary social unitComte—family as primary social unit

Explanatory priority placed on the individualExplanatory priority placed on the individual

– Allport--individual is primary (“babble of tongues”)Allport--individual is primary (“babble of tongues”)

Necessary ConditionsNecessary Conditions

Homogeneity: similarity in thoughts, Homogeneity: similarity in thoughts, behavior or affect of interacting behavior or affect of interacting individualsindividuals– E.g., group-level, emergent processes, E.g., group-level, emergent processes,

norms, cohesivenessnorms, cohesiveness Interdependence: individuals influencing Interdependence: individuals influencing

each othereach other– E.g., actor-partner effectsE.g., actor-partner effects

McDougall, 1920, p. 23McDougall, 1920, p. 23

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

Statistical Framework Should Mimic Statistical Framework Should Mimic Theoretical FrameworkTheoretical Framework

Make concepts concreteMake concepts concrete Avoid Allport’s “babble” critiqueAvoid Allport’s “babble” critique Make the model easy to implementMake the model easy to implement

TODAY’s TalkTODAY’s Talk– One time point; dyadsOne time point; dyads– Two or three variablesTwo or three variables– Normally distributed data; additive modelsNormally distributed data; additive models

Menu of TechniquesMenu of Techniques

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

Common Beliefs about Common Beliefs about Interdependence in Dyadic DataInterdependence in Dyadic Data

If you don’t correct for interdependence, your If you don’t correct for interdependence, your Type I errors will be inflatedType I errors will be inflated

If you don’t correct for interdependence, your If you don’t correct for interdependence, your results will be ambiguousresults will be ambiguous

An HLM program will eliminate all An HLM program will eliminate all nonindependence problemsnonindependence problems

If you have dyadic data, you must run HLM If you have dyadic data, you must run HLM (or else your paper won’t be published)(or else your paper won’t be published)

These beliefs miss what we believe to These beliefs miss what we believe to be the fundamental issue:be the fundamental issue:

There is useful psychological There is useful psychological information lurking in the information lurking in the “nonindependence”“nonindependence”

Interdependence is the “very stuff” of Interdependence is the “very stuff” of relationships.relationships.

Dyadic Designs:Dyadic Designs:Three Major CategoriesThree Major Categories

Subjects nested within groupsSubjects nested within groups– Exchangeable (e.g., same sex siblings)Exchangeable (e.g., same sex siblings)– Distinguishable (e.g., different sex siblings, Distinguishable (e.g., different sex siblings,

mother-child interaction)mother-child interaction)– Mixed exch & dist (e.g., same sex & Mixed exch & dist (e.g., same sex &

different sex dyads in same design)different sex dyads in same design) Univariate versus multivariateUnivariate versus multivariate Homogeneity versus interdependenceHomogeneity versus interdependence

Intraclass Correlation: Intraclass Correlation: Building BlockBuilding Block

• Structural Univariate Models:Structural Univariate Models:

• ExchangeableExchangeable

• DistinguishableDistinguishable

ij i ijY

jij i ijY

ANOVA Intraclass (& REML)ANOVA Intraclass (& REML)DyadsDyads

MSB MSWICC

MSB MSW

Intraclass Correlation:Intraclass Correlation:HLM LanguageHLM Language

Two level model:Two level model:

Intraclass correlation is given by Intraclass correlation is given by

1:

2 :

ij i ij

i i

Y

2

2 2

Pairwise CodingPairwise CodingDyad #Dyad # XX X’X’

11 11 22

11 22 11

22 55 11

22 11 55

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

Pairwise Intraclass CorrelationPairwise Intraclass Correlation

'xxr

Example: Personal VictimizationExample: Personal VictimizationCeballo et al, 2001Ceballo et al, 2001

Not WelfareNot Welfare WelfareWelfare

Mom Mom meanmean

1.61.6 2.82.8

ChildChild

meanmean

2.62.6 3.73.7

NN 2828 7272

rr 0.290.29 0.340.34

Pairwise Intraclass (ML):Pairwise Intraclass (ML):DyadsDyads

ML

SSB SSWICC

SSB SSW

ANOVA

MSB MSWICC

MSB MSW

InterdependenceInterdependence

The degree to which one individual The degree to which one individual influences anotherinfluences another

Need not be face to faceNeed not be face to face– We have a good time together, even when We have a good time together, even when

we’re not together (Yogi Berra)we’re not together (Yogi Berra)

Pairwise GeneralizationPairwise Generalization' '

0 1 2 3Y X X XX

•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

ExampleExample(Stinson & Ickes, 1992)(Stinson & Ickes, 1992)

ActorS = ActorV + PartnerVActorS = ActorV + PartnerV Strangers:Strangers: an effect of the partner’s an effect of the partner’s

verb frequency on the actor’s laughter verb frequency on the actor’s laughter (in ordinal language, the more my (in ordinal language, the more my partner talks, the more I smile/laugh)partner talks, the more I smile/laugh)

Friends:Friends: an effect of the actor’s verb an effect of the actor’s verb frequency on the actor’s laughter (the frequency on the actor’s laughter (the more I talk, the more I smile/laugh)more I talk, the more I smile/laugh)

Some formulaeSome formulae

Actor regression Actor regression coefficientcoefficient

V(Actor reg coeff)V(Actor reg coeff)

' '1 2

'

( )

(1 )y xy xy xx

x xx

s r r r

s r

2 2' ' ' ' '

2 2'

(1 )

2 (1 )y xy xx xx yy xy

x xx

s r r r r r

Ns r

Partner coef replaces Y with Y’

Interdependence ExampleInterdependence Example

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

Xm Ym

XcYc

a

b

c

d

rx r

Xm Ym

XcYc

-.1

.2 .3

Not on Welfare

Xm Ym

XcYc

.09

.1

Welfare

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

V-pre

S-pre

V-post

No interdependence problem on the dependent variable

Xm Ym

XcYc

a

b

c

d

rx r

Return to Original Model: Special Case

Set a=d and b=d

X Y

Xm XcYm Yc

ExmExc Eym Eyc

rd

ri ri

'xxr 'xxr 'yyr 'yyr

Latent Variable ModelLatent Variable Model

rrii = individual level correlation = individual level correlation

rrdd = dyad level correlation = dyad level correlation

The square root of intraclass The square root of intraclass correlations are the pathscorrelations are the paths

Using Path Analysis RulesUsing Path Analysis Rules

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

Solving those two equations….

X Y

Xm XcYm Yc

ExmExc Eym Eyc

-.8

.4 .4

.45 .45 .47 .47

Not on Welfare

X Y

Xm XcYm Yc

ExmExc Eym Eyc

1.6

.3 .3 .2 .2

Welfare: latent variable model doesn’t hold

What does the correlation of two What does the correlation of two dyads means?dyads means?

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

Multivariate Model: HLM LingoMultivariate Model: HLM Lingo Three-level model: one level for each Three-level model: one level for each

variable, one level for individual effect, variable, one level for individual effect, and one level for group effectand one level for group effect

0 0 1 1

0 0 0 0

1 1 1 1

ijkY X X

0 0 1

0 1 1

0,V C

NC V

0 0 1

0 1 1

0,V C

NC V

Difference scoresDifference scores

Frequently, a question of similarity (or Frequently, a question of similarity (or congruence) comes up in dyadic congruence) comes up in dyadic researchresearch– Diff of husband and wife salary as a Diff of husband and wife salary as a

predictor of wife’s relationship satisfactionpredictor of wife’s relationship satisfaction– Diff of husband and wife self-esteem as a Diff of husband and wife self-esteem as a

predictor of husband’s coping predictor of husband’s coping

Difference ScoresDifference Scores

Correlations with difference scores can Correlations with difference scores can show various patterns depending on show various patterns depending on their component correlationstheir component correlations– The numerator is a weighted sum of the The numerator is a weighted sum of the

correlations: (rcorrelations: (rX1YX1Y S SX1X1 – r – rX2YX2YSSX2X2)S)Syy

– Toy ExamplesToy Examples

» One variable is a constantOne variable is a constant

» One variable is randomOne variable is random

““Solutions” Solutions”

One can use One can use multiple regressionmultiple regression, , entering the two variables as two entering the two variables as two predictors (rather than one difference predictors (rather than one difference score).score).

Y = Y = 00 + + 11X1 + X1 + 22 X2 X2

– Problem: doesn’t test specific hypotheses Problem: doesn’t test specific hypotheses such as “similar is better” or “self-such as “similar is better” or “self-enhancing is better”enhancing is better”

Model-Based ApproachModel-Based Approach

QuestionsQuestions– Discrepancy model (woman’s sat is Discrepancy model (woman’s sat is

greatest the more she earns, the less her greatest the more she earns, the less her husband earns)husband earns)

– Similarity model (woman’s sat is greatest Similarity model (woman’s sat is greatest the smaller the absolute diff in salary)the smaller the absolute diff in salary)

– Superiority model (woman’s sat is greatest Superiority model (woman’s sat is greatest when she earns more than her husband)when she earns more than her husband)

Model-Based ApproachModel-Based Approach

Run separate regressions for subjects Run separate regressions for subjects below and above the “equality line” (or below and above the “equality line” (or use dummy codes and include an use dummy codes and include an interaction term)interaction term)

The three different models imply The three different models imply different patterns on the coefficientsdifferent patterns on the coefficients

Patterns of Regression CoefficientsPatterns of Regression Coefficients Discrepancy model:Discrepancy model:

– Both regressions should yield a negative coef for the Both regressions should yield a negative coef for the husband and a positive coef for the wife (maximizing husband and a positive coef for the wife (maximizing the difference)the difference)

Similarity model:Similarity model:– For dyads where salary W>H, positive coef for For dyads where salary W>H, positive coef for

husband and neg coef for wife because in this region husband and neg coef for wife because in this region higher husband salary identifies couples closer to higher husband salary identifies couples closer to equalityequality

– For dyads where salary W<H, neg coef for husband For dyads where salary W<H, neg coef for husband and pos coef for wife because in this region higher and pos coef for wife because in this region higher wives’ salary identifies couples closer to equalitywives’ salary identifies couples closer to equality

Patterns of Regression CoefPatterns of Regression Coef

Superiority modelSuperiority model– For couples where W>H on salary, a larger For couples where W>H on salary, a larger

positive coef for wive’s salarypositive coef for wive’s salary

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

ConclusionConclusion

The take home message is that The take home message is that nonindependence due to interaction does not nonindependence due to interaction does not require a “statistical cure”require a “statistical cure”

Nonindependence provides an opportunity to Nonindependence provides an opportunity to measure and model social interactionmeasure and model social interaction

Follow your conceptual models and your Follow your conceptual models and your research questionsresearch questions

There is still much room for careful design in There is still much room for careful design in correlational research with couplescorrelational research with couples