Estimation: Confirmatory Factor Analysis
Social Relations Model:
Estimation Distinguishable Dyads
David A. Kenny
Background
Social Relations Model
Confirmatory Factor
Analysis
Data Structure
Members of the groups are distinguishable.
Each member has a different role.
Prototypical example a family mother, father, & child
Other examples work teams laboratory teams with roles or types
Four-Person Family
In the four-person family, there are twelve possible relationships: mother-father (MF) father-mother (FM) mother-older child (MO) father-older child (FO) mother-younger child (MY) father-younger child (FY) older child-mother (OM) younger child-mother (YM) older child-father (OF) younger child-father (YF) older child-younger c. (OY) younger child-older c. (YO)
The first letter corresponds to the actor and the second letter corresponds to the partner.
Strategy
Create a variance-covariance matrix of the 12 variables
(MF, MO, MY, FM … YO).
Analyze by Confirmatory
Factor Analysis.
Factors
Each measure loads on a group, actor, and partner factor.
Separate actor and partner variances can be estimated for each member of the group.
All loading fixed at 1.
Relationship effects are treated as
“errors.”
OF: Older Child with
Father
Loadings
Actor Factor: Older Child
Partner Factor: Father
Group or Family Factor
Correlations
Generalized reciprocity: Actorpartner correlation, one for role
Dyadic reciprocity: Correlation of errors, one for each pair of roles
Identification
Need at least 4 members of the group to estimate all the SRM variances and correlations.
With 3 members, an identifying assumptions must be made, e.g., no group variance.
Degrees of Freedom
CFA with 4 members: df = 47
CFA with 3 members and no group variance: df = 3
Diagram for
3-Person
Family
Model the Means
We can estimate factor means for each of the factors.
To be identified, we nee to make constraints.
One idea is ANOVA constraints: actor and partner effects sum to zero; relationship effects sum to zero by row and column.
Separating Error from Relationship
Need multiple measures.
xxx
What To Do If the
Model Does Not Fit?
Generally the model does fit.
For families, if it does not, can estimate correlations for intragenerational effects. See Kenny et al.
(2006) for details.
Variance Partitioning
For a four-person, each of 12 scores has four different sources of variance.
Except for the family variance, the other three sources explain a different amount.
Different profile of proportion of variance explained for each score.
Reference
Reading: Chapter 9 of Dyadic Data
Analysis by Kenny,
Kashy, and Cook.
