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
– is a critical component in the study of
interpersonal behavior
– but may not be required in all analyses
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

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

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
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
Repeated measures ANOVA
 Intraclass correlation
 Hierarchical linear models (HLM)
 Structural equations models (SEM)

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)

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

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
•
Structural Univariate Models:
• Exchangeable
Yij     i   ij
• Distinguishable
Yij     i   j   ij
ANOVA Intraclass (& REML)
Dyads
MSB  MSW
ICC 
MSB  MSW
Intraclass Correlation:
HLM Language

Two level model:
1: Yij   i   ij
2 : i     i

Intraclass correlation is given by

2
 
2
2
Pairwise Coding
Dyad #
X
X’
1
1
2
1
2
1
2
5
1
2
1
5
The Pearson corr of X and X’ is the ML
estimator of the intraclass correlation.
Pairwise Intraclass Correlation
rxx '
Example: Personal Victimization
Ceballo et al, 2001
Not Welfare
Welfare
Mom
mean
Child
mean
N
1.6
2.8
2.6
3.7
28
72
r
0.29
0.34
Pairwise Intraclass (ML):
Dyads
ICCML
SSB  SSW

SSB  SSW
ICC ANOVA
MSB  MSW

MSB  MSW
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
Y   0  1 X   2 X   3 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
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


Actor regression
coefficient
1 
s y (rxy  rxy ' rxx ' )
sx (1  r )
2
xx '
V(Actor reg coeff)
s (1  rxy ' rxx '  rxx ' ryy '  r )
2
y
2
xy '
2 Ns (1  r )
2
x
Partner coef replaces Y with Y’
2
xx '
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
a
Xm
Ym
b
rx
r
c
Xc
d
Yc
Not on Welfare
Xm
Ym
-.1
.2
Xc
.3
Yc
Welfare
Xm
.09
Ym
.1
Xc
Yc
Simple Actor-Partner Model:
Pre-post death of spouse
V-pre
V-post
S-pre
No interdependence problem on the
dependent variable
Return to Original Model: Special Case
a
Xm
Ym
b
rx
r
c
Xc
d
Yc
Set a=d and b=d
ri
ri
Exm
Exc
Eym
Eyc
Xm
Xc
Ym
Yc
ryy '
ryy '
rxx '
rxx '
Y
X
rd
Latent Variable Model
 ri
= individual level correlation
 rd = dyad level correlation

The square root of intraclass
correlations are the paths
Using Path Analysis Rules
Two equations in two unknowns; reason
why rxy may be uninterpretable
Solving those two equations….
.4
.4
Exm
Exc
Eym
Eyc
Xm
Xc
Ym
Yc
.45
.45
Y
X
Not on Welfare
.47
.47
-.8
Exm
Exc
Eym
Eyc
Xm
Xc
Ym
Yc
.3
.3
.2
.2
Y
X
1.6
Welfare: latent variable model doesn’t hold
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

Three-level model: one level for each
variable, one level for individual effect,
and one level for group effect
Yijk   0 X 0  1 X 1
 0  0   0   0
1  1   1  1

  V 0
N  0, 
 C 
  01
C 01  
 
V1  

  V 0
N  0, 
 C 
  01
C 01  


V1  
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 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”
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 “selfenhancing is better”
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 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

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

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




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