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One-with-Many Design: Estimation

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David A. Kenny

- Introduction to the One-with-Many Design

- People
- Focal person (the one)
- Partners (the many)

- Source of Data
- Focal persons (1PMT)
- Partners (MP1T)
- Both (reciprocal design: 1PMT & MP1T)

- Multilevel analysis
- Indistinguishable partners
- Many partners
- Different numbers of partners per focal person

- Confirmatory factor analysis
- Distinguishable partners
- Few partners
- Same number of partners per focal person

- Each record a partner
- Levels
- Lower level: partner
- Upper level: focal person

- Random intercepts model (nonindependence)
- Lower level effects can be random

Estimate a basic multilevel model in which

There are no fixed effects with a random intercept.

Yij = b0j + eij

b0j = a0 + dj

Note the focal person is Level 2 and partners Level 1.

MIXED

outcome

/FIXED =

/PRINT = SOLUTION TESTCOV

/RANDOM INTERCEPT | SUBJECT(focalid) COVTYPE(VC) .

Could add predictors here.

Fixed Effects

Covariance Parameters

So the actor variance is .791, and ICC is .791/(.791+1.212) = .395

- Can add to the model
- Focal person characteristics
- Would be actor if 1PMT design
- Would be partner if MP1T design

- Partner characteristics
- Would be partner if 1PMT design
- Would be actor if MP1T design
- Can be random: The coefficient may vary by focal person

- Important to make zero interpretable

- Focal person characteristics

Sources of nonindependence

- More complex…

- Individual-level effects for the focal person:
- Actor & Partner variances
- Actor-Partner correlation

- Relationship effects
- Dyadic reciprocity corelation

Data Structure: Two records for each dyad; outcome is the same variable for focal person and partner.

Variables to be created:

role = 1 if data from focal person; -1 if from partner

focalcode = 1 if data from focal person; 0 if from partner

partcode = 1 if data from partner; 0 if from the focal person

A fairly complex multilevel model…

MIXED

outcome BY role WITH focalcodepartcode

/FIXED = focalcodepartcode | NOINT

/PRINT = SOLUTION TESTCOV

/RANDOM focalcodepartcode | SUBJECT(focalid) covtype(UNR)

/REPEATED = role | SUBJECT(focalid*dyadid) COVTYPE(UNR).

- Taken from Chapter 10 of Kenny, Kashy, & Cook (2006).
- Focal person: mothers
- Partners: father and two children
- Outcome: how anxious the person feels with the other
- Distinguishability of partners is ignored.

.

The estimates show the intercept is the mean of the ratings made by the mother (focalcode estimate is 1.808). The partcode estimate indicates the average outcome score across partners of the mother which is smaller than mothers’ anxiety. This difference is statistically significant.

- The relationship variance for the partners is .549. (Role = -1) and for mothers (Role = 1) is .423.
- The correlation of the two relationship effects is .24: If the mother is particularly anxious with a particular family member, that member is particularly anxious with the mother.
- Var(1) (focalcode is the first listed random variable) is the actor variance of mothers and is .208.
- Var(2) is the partner variance for mothers (how much anxiety she tends to elicit across family members) and is .061. (p = .012; p values for variances in SPSS are cut in half).

- The ICC for actor is .208/(.208+.423) = .330 and the ICC for partner is .061/(.061+.549) = .100.
- The actor partner correlation is .699, so if mothers are anxious with family members, they are anxious with her.

- Two ways to think about fixed effects
- Standard way
- Focal person characteristics (fx)
- Partner characteristics (px)

- APIM way (the same variable must be measured for the focal person and partners)
- Actor characteristics (ax)
- Partner characteristics (ptx)

- Standard way

/FIXED = focalcodepartcodefX*focalcode

fX*partcodepX*focalcodepX* partcode| NOINT

or

/FIXED = focalcodepartcodeaX*focalcode

aX*partcodeptX*focalcodeptX*partcode| NOINT

Note: fX*focalcode = aX*focalcode

fX*partcode = ptX*partcode

pX*focalcode = ptX*focalcode

pX*partcode = aX*partcode

http://davidakenny.net/doc/onewithmanyrecip.pdf

Thanks to Deborah Kashy

Reading: Chapter 10 in Dyadic Data Analysis by Kenny, Kashy, and Cook