One-with-Many Design:EstimationDavid A. Kenny

June 22, 2013

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What You Should Know Introduction to the One-with-Many

Design

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The One-with-Many Provider-Patient Data

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Terminology People

Focal person (the one) Partners (the many)

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

MP1T)

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Analysis Strategies• 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

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Multilevel Analyses: Nonreciprocal Design Each record a partner Levels

Lower level: partnerUpper level: focal person

Random intercepts model (nonindependence)

Lower level effects can be random

Data Analytic Approach for the Non-Reciprocal One-with-Many Design

FocalID PartID DV1 1 61 2 51 3 52 1 32 2 22 3 42 4 33 1 73 2 8

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.

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SPSS Output

Covariance Parameters

Fixed EffectsEstimates of Fixed Effectsa

6.934020 .228724 21.066 30.316 .000 6.458453 7.409587ParameterIntercept

Estimate Std. Error df t Sig. Lower Bound Upper Bound95% Confidence Interval

Dependent Variable: DV.a.

Estimates of Covariance Parametersa

1.212359 .189978 6.382 .000 .891758 1.648222.790917 .336679 2.349 .019 .343391 1.821681

ParameterResidual

VarianceIntercept [subject= FOCALID]

Estimate Std. Error Wald Z Sig. Lower Bound Upper Bound95% Confidence Interval

Dependent Variable: DVa.

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

Fixed Effects: Nonreciprocal Design 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

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Reciprocal One-with-Many DesignSources of nonindependence More complex…

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Sources of Nonindependence in the Reciprocal Design Individual-level effects for the focal

person: Actor & Partner variances Actor-Partner correlation

Relationship effects Dyadic reciprocity corelation

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Data Analytic Approach for Estimating Variances & Covariances: The Reciprocal Design

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

partnerpartcode = 1 if data from partner; 0 if from the

focal person

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Data Analytic Approach for Estimating Variances & Covariances: The Reciprocal Design

A fairly complex multilevel model…

MIXED outcome BY role WITH focalcode partcode /FIXED = focalcode partcode | NOINT /PRINT = SOLUTION TESTCOV /RANDOM focalcode partcode |

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

COVTYPE(UNR).

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Example 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.

.

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Output: Fixed Effects

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.

Estimates of Fixed Effectsa

ParameterEstimate Std. Error df t Sig.

95% Confidence IntervalLower Bound Upper Bound

focalcode 1.807695 .040989 207.000 44.102 .000 1.726886 1.888505partcode 1.698269 .034249 207.000 49.587 .000 1.630748 1.765790

a. Dependent Variable: outcome.

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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).

Estimates of Covariance Parametersa

ParameterEstimate Std. Error Wald Z Sig.

95% Confidence IntervalLower Bound Upper Bound

Repeated Measures Var(1) .549234 .038083 14.422 .000 .479444 .629184Var(2) .423155 .029341 14.422 .000 .369385 .484753Corr(2,1) .239029 .046228 5.171 .000 .146585 .327334

focalcode + partcode [subject = focalid]

Var(1) .208409 .035715 5.835 .000 .148952 .291601Var(2) .060898 .027134 2.244 .025 .025430 .145838Corr(2,1) .698818 .170996 4.087 .000 .206931 .908699

a. Dependent Variable: outcome.

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Output: Nonindependence 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.

Fixed Effects: Reciprocal Design 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)

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Fixed Effects: Reciprocal Design

/FIXED = focalcode partcode fX*focalcode fX*partcode pX*focalcode pX* partcode| NOINT

or/FIXED = focalcode partcode aX*focalcode aX*partcode ptX*focalcode ptX*partcode| NOINTNote: fX*focalcode = aX*focalcode fX*partcode = ptX*partcode pX*focalcode = ptX*focalcode pX*partcode = aX*partcode

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

Thanks to Deborah Kashy

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

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