Dyadic designs to model Dyadic designs to model relations in social interaction relations in social interaction

datadata

Todd D. Little

Yale University

OutlineOutline

•Why have such a symposium

•Dyadic Designs and Analyses

•Thoughts on Future Directions

Some Bad MethodsSome Bad Methods

•Dyad-level Setups (Ignore individuals)•Target-Partner Setups

• Arbitrary assignment of target vs partner•Loss of power•Often underestimates relations•Ignores dyadic impact

•Target with multiple-Partner• Take average of partners to reduce dyad-

level influences•Doesn't really do it•Ignores dyadic impact

Intraclass SetupsIntraclass Setups

•Represents target with partner & partner with target in same data structure

•Exchangeable case (target/partner arbitrary)•Distinguishable case (something systematic)

• Keeps dyadic influence• Contains dependencies• Requires adjustments for accurate statistical

inferences (see e.g., Gonzalez & Griffin)

Between-Friend Correlations

NN EE OO AA CC

Child-Rated

.05 .10 .06 .06 .06

Parent-Rated

.07 .02 .06 -.04 .17

Teacher-Rated

.35 .21 .29 .36 .30

Canonical Correlations

All .16 .26 .485 .31 .59 .536 .34 .66 .637 .17 .42 .738 .16 .34 .739 .28 .44 .65

10 .26 .40 .61

Child-Rated

Parent-Rated

Teacher-Rated

Grade

Social Relations Model (Kenny et al.)Social Relations Model (Kenny et al.)

•Xijk = mk + ai + bj + gij + eijk

Where Xijk is the actor i's behavior with partner j at occasion kmk is a grand mean or intercept ai is variance unique to the actor ibj is variance unique to the partner jgij is variance unique to the ij-dyadeijk is error variance

•Round-Robin designs: (n * (n-1) / 2)• Sample from all possible interactions

•Block designs: p persons interact with q persons• Checker-board: multiple p's and q's of 2 or more

Development

Gender

Persistence

Tenure

RelativeAbility to Compete

Onlooking

Directives

Imitation.12

.39

-25

.68

.51

-.26

-.27

From Hawley & Little, 1999

SEM of a Block Design SEM of a Block Design

Multilevel ApproachesMultilevel Approaches

• Distinguish HLM (a specific program) from hierarchical linear modeling, the technique– A generic term for a type of analysis

• Probably best to discuss MRC(M) Modeling– Multilevel Random Coefficient Modeling

• Different program implementations– HLM, MLn, SAS, BMDP, LISREL, and others

"Once you know that "Once you know that hierarchies exist, hierarchies exist,

you see them you see them everywhere."everywhere."

-Kreft and de Leeuw (1998)

Logic of MRCMLogic of MRCM

• Coefficients describing level 1 phenomena are estimated within each level 2 unit (e.g., individual-level effects)– Intercepts—means

– Slopes—covariance/regression coefficients

• Level 1 coefficients are also analyzed at level 2 (e.g., dyad-level effects)– Intercepts: mean effect of dyad

– Slopes: effects of dyad-level predictors

Negative Individual, Positive GroupNegative Individual, Positive Group

Positive Individual, Negative GroupPositive Individual, Negative Group

No Individual, Positive GroupNo Individual, Positive Group

No Group, Mixed IndividualNo Group, Mixed Individual

A Contrived ExampleA Contrived Example

• Yij = Friendship Closeness ratings of each

individual i within each dyad j.

• Level 1 Measures: Age & Social Skill of

the individual participants

• Level 2 Measures: Length of Friendship

& Gender Composition of Friendship

The EquationsThe Equations

yij = 0j + 1jAge + 2jSocSkill + 3jAge*Skill + rij

The Level 1 Equation:

0j = 00 + 01(Time) + 02(Gnd) + 03(Time*Gnd) + u0j

1j = 10 + 11(Time) + 12(Gnd) + 13(Time*Gnd) + u1j

2j = 20 + 21(Time) + 22(Gnd) + 23(Time*Gnd) + u2j

3j = 30 + 31(Time) + 32(Gnd) + 33(Time*Gnd) + u3j

The Level 2 Equations:

Future DirectionsFuture Directions

•OLS vs. ML estimator and bias

•Individual-oriented data vs. dyad-oriented data

•Thoughts on Future Directions

Level 1 Equations: Level 1 Equations: Meaning of Intercepts Meaning of Intercepts

• Y = Friendship Closeness Ratings– i individuals– across j dyads– rij individual level error

• Intercept (Dyad-mean Closeness)– Yij = 0j + rij

Level 2 Equations:Level 2 Equations:Meaning of Intercepts Meaning of Intercepts

• Do Dyad Means Differ?

• Mean Closeness across Dyads– 0j = 00 + u0j

• Mean Closeness and dyad-level variables (time together and gender composition)– 0j = 00 + 01 (TIME) + 02 (Gen) + u0j

Level 1 Equations: Level 1 Equations: Meaning of SlopeMeaning of Slope

• E.g., Relationship between Closeness and Social Skill within each dyad– Yij = 0j + 2j (SocSkil) + rij

• Intercept for each dyad:0j

• Social Skill slope for each dyad:2j

Level 2 Equations: Level 2 Equations: Meaning of SlopesMeaning of Slopes

• Mean Social Skill-Closeness relationship across all dyads

– j = 10 + u1j

• Does SocSkill-Closeness relationship vary as a function of how long the dyad has been together?– 1j = 10 + 11(TIME) + u1j