Modeling Interdependence: Toward a General Framework

1. Modeling Interdependence: Toward a General Framework Richard Gonzalez, U of Michigan Dale Griffin, U of British Columbia

2. The Nested Individual

3. 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

4. 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”)

5. 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

6. 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.

8. 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

9. Menu of Techniques • Repeated measures ANOVA • Intraclass correlation • Hierarchical linear models (HLM) • Structural equations models (SEM)

10. 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)

11. 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.

12. 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

13. Intraclass Correlation: Building Block • Structural Univariate Models: • Exchangeable • Distinguishable

14. ANOVA Intraclass (& REML)Dyads

15. Intraclass Correlation:HLM Language • Two level model: • Intraclass correlation is given by

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

21. Pairwise Intraclass Correlation

23. Example: Personal VictimizationCeballo et al, 2001

25. Pairwise Intraclass (ML):Dyads

26. 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)

27. 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

28. 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)

29. Some formulae • Actor regression coefficient • V(Actor reg coeff) Partner coef replaces Y with Y’

30. 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

31. a Xm Ym b rx r c Yc Xc d

32. Not on Welfare Xm Ym -.1 .2 .3 Yc Xc

33. Welfare .09 Xm Ym .1 Yc Xc

34. V-post V-pre S-pre Simple Actor-Partner Model:Pre-post death of spouse No interdependence problem on the dependent variable

35. Return to Original Model: Special Case a Xm Ym b rx r c Yc Xc d Set a=d and b=d

36. ri ri Eym Eyc Exc Exm Yc Ym Xm Xc Y X rd

37. Latent Variable Model • ri = individual level correlation • rd = dyad level correlation • The square root of intraclass correlations are the paths

38. Using Path Analysis Rules Two equations in two unknowns; reason why rxy may be uninterpretable

39. Solving those two equations….

40. .4 .4 Eym Eyc Exc Exm Yc Ym Xm Xc .47 .45 .47 .45 Y X -.8 Not on Welfare

41. Eym Eyc Exc Exm Yc Ym Xm Xc .2 .3 .2 .3 Y X 1.6 Welfare: latent variable model doesn’t hold

42. What does the correlation of two dyads means? So, there are multiple components to the correlation of dyad means making it uninterpretable….

43. Multivariate Model: HLM Lingo • Three-level model: one level for each variable, one level for individual effect, and one level for group effect

44. 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

46. 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

48. “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”

49. 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)

50. 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