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Communication between Fixed and Random Effects: Examples

from Dyadic Data

David A. Kenny

University of Connecticut

davidakenny.net\kenny.htm

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Overview

I. Introduction: Dyadic Designs and Models

II. Specification Error

III. Respecifying Fixed Effects Based on the Random Effects

IV. Resolving an Inconsistency

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Caveat

Linear Models

Normally Distributed Random Variables

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Three Major Dyadic Designs Standard

– Each person has one partner.– Married couples

Social Relations Model (SRM) Designs– Each person has many partners, and each

partner is paired with many persons.– Group members state liking of each other.

One-with-Many– Each person has many partners, but each

partner is paired with only one person.– Patients rate satisfaction with the physician.

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Standard Design

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SRM Designs

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One-with-Many Design

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Model for the Standard Design

Two scores for dyad j:

Y1j = b01 + b11X11j + 1j

Y2j = b02 + b12X12j + 2j

where C(1j, 1j) may be nonzero

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Members Indistinguishable

Y1j = b0 + b1X11j + 1j

Y2j = b0 + b1X12j + 2j

where V(1j) = V(2j) and C(1j,2j) may be nonzero

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Partner Effect with Members Indistinguishable

Y1j = b0 + b1X11j + b2X12j + 1j

Y2j = b0 + b1X12j + b2X11j + 2j

where V(1j) = V(2j) and C(1j,2j) may be nonzero

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Partner Effects with Members Distinguishable

Y1j = b01 + b11X1j + b21X2j + 1j

Y2j = b02 + b12X2j + b22X1j + 2j

where C(1j, 1j) may be nonzero

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Social Relations Model

• model of dyadic relations embedded in groups

• Xijk: actor i with partner j in group k

• round-robin structures: everyone paired with everyone else

• other structures possible

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Round Robin Design

1 2 3 4 5 61 - x x x x x

2 x - x x x x

3 x x - x x x

4 x x x - x x

5 x x x x - x

6 x x x x x -

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The Statistical Model: Random Effects

Xijk = k + ik + jk + ijk

variances (4): 2,

2, 2,

2

covariances (2): ,

(fixed effects discussed later)

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Liking: How much Dave likes Paul

VariancesGroup (

2): How much liking there is in the group.Actor (

2): How much Dave likes others in general.Partner (

2): How much Paul is liked by others in general.Relationship (

2): How much Dave particularly likes Paul.

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Liking: How much Dave likes Paul

Covariances:

Actor-Partner (): If Dave likes

others, is Dave liked by others?

Relationship (): If Dave

particularly likes Paul, does Paul particularly like Dave?

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EstimationANOVA Expected Mean Squares (Warner,

Kenny, & Stoto, JPSP, 1979)Maximum likelihood (Wong, JASA, 1982)Bayesian estimation and fixed effects (Gill &

Swartz, The Canadian Journal of Statistics, 2001)

Estimation with triadic effects (Hoff, JASA, 2005)

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Consensus and Acquaintance

• We would think that we would agree more about targets the longer we know them.

• Evidence (Kenny et al., 1994) does not support this hypothesis.

• Measure: s2/(s

2 + s2 + s

2)

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0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.1 0.2 0.3 0.4 0.5 0.6

Time 1

Tim

e 2

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Model for the One-with-Many Design

Person i (the one of nj persons) is paired with person j (the many):

Yij = b0j + b1X1ij + jj

where V(b0j) may be nonzero

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Model for the One-with-Many Design

Person i (the one of nj persons) is paired with person j (the many):

Yij = b0j + b1jX1ij + ij

where V(b0j), V(b1j), and C(b0j, b1j) may be nonzero

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II. Specification Error

How does specifying the wrong model in one part affect the other part?

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Specification Error in the Random Part

• Unbiased estimates of fixed effects.

• Bias in standard errors• under-estimation

• over-estimation

• little or none

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Example of Bias in Standard Errors

Consider a standard dyad design and a simple model for person i (i =1, 2) in dyad j

Yij = b0 + b1Xij + ij

where C(X1j,X2j)/V(X) = x and C(1j, 2j)/V() = .

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Bias in Standard Error for b1 If e Is Falsely Assumed To Be Zero

• under-estimation: x > 0

• over-estimation: x < 0

• none: x = 0

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Specification Error in Fixed Part

• can dramatically bias random effects

• example: roommate effects and liking

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Kenny & Lowe Study

• 19 round robins of 4• 2 pairs of roommates• Roommates like one another: Mean

difference between roommates and non-roommates

• What if the roommate effect were ignored?

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Kenny & Lowe Study: Results Using ANOVA

Component Estimate

Actor -.754

Partner -.794

Relationship 3.693

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Kenny & Lowe Study: Results Using ANOVA

Component Estimate Revised

Actor -.754 .444

Partner -.794 .597

Relationship 3.693 1.139

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III. Respecifying Fixed Effects Based on Random Effects

• A fixed effect often corresponds to certain random effect.– e.g., fixed effect at a given level

• What if those random effects have zero variance?

• May need to rethink the fixed effects.

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Example• People like being in homogeneous groups.

• Demographic homogeneity

• Same ethnicity

• Same gender

• Same age

• Opinion homogeneity

• People do not like being in diverse groups.

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Conclusion

People do not like being in diverse groups.

But is there group variance in liking?

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Group Variance in Liking?

• Liking tends not to show group effects– SRM analyses of lab groups– SRM studies of families– Rather the dominant component is

relationship.• Group (?) diversity as a relationship effect?

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The Measurement of Diversity

• The usual measure of diversity is a variance or some related measure.

• Not well known is that the variance can be expressed as the sum of squared differences:

s2 = i(Xi – M)2/(n – 1)

= ij(Xi – Xj)2/[n(n - 1)]

i > j

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The Disaggregation of Group Diversity

• Instead of thinking about diversity as a property of the group (i.e., a variance), we can view diversity as the set of relationships.

• It then becomes an empirical question whether it makes sense to sum across different components.

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Dyad Group Data

• Group of Dave, Paul, Bengt, and Thomas

• Dave states how much he likes Paul.

• Dave: actor

• Paul: partner

• Bengt and Thomas: others

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Model for Dyad Group Data

Dyadic Similarity: How similar am I to Paul?

Actor Similarity: How similar am I to others in the group?

Partner Similarity: How similar is Paul to others in the group?

Group: How similar are others to each other?

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What Does Group Diversity Presume?

• Presumes the four effects are of equal magnitude.

• Predicts group similarity has an effect.

• Presence of group variance.

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Example• Harmon Hosch’s Data• Gathered in El Paso, Texas• 134 6-person juries from the jury pool• Mock jury case• Jurors rate how likeable the other

jurors are.• Diversity in terms of initial verdict

preference

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Results: Random EffectsTerm Estimate SE

2 .000 ----

2 .165 .016

2 .045 .013

2 .477 .018

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Results: Fixed EffectsTerm Estimate SE

Diversity -.007 .009

More diversity, others seem less likeable.

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Results: Fixed EffectsTerm Estimate SEDyad .129 .015Actor -.029 .041Partner .006 .041Group .005 .058

You see someone as likeable if they have the same opinion as you.

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What Did We Learn?

At least for the data set under consideration, it is not that group diversity leads to lower liking, but rather being similar to the other, a relationship effect, that leads to perceptions of likeability.

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IV. Resolving an Inconsistency

• It has just been argued that a fixed effect at one level should be “accompanied” by a random effect at that level.

• Blind following of this approach can be problematic.

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Statistical Logic Is Not Necessarily Logical

• We might conclude that groups of various types of persons are different, even though groups may not be different.

• For example, we often conclude women are better than men at understanding others (a fixed effect) while at the same time we conclude that people do not differ in understanding others (a random effect).

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Model• two-level multilevel model• n people each judge m different targets, nm targets

total • two types of judges: n/2 men and n/2 women

Yij = b0 + b1Xj + j + ij

j is person, ij is target, and

Xj(0, 1) is gender]

(Simulation performed by Randi Garcia.)

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Parameter ValuesYij = b0 + b1Xj + j + ij

b0 = 0 and 2 = 1

Fixed effect is a medium effect

size: d = .5.

The fixed and random effects explain

the “same” amount of variance.

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Power of the Tests of Fixed and Random Effects (nm =200)

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.70

0.80

0.90

1.00

20 (10) 10 (20) 5 (40) 2 (100)

m (n)

Po

we

r

Fixed Power

Random Power

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What we have here is a

failure to communicate.

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Conclusion• In most of the social and

behavioral sciences, there is relatively little attention paid to random effects.

• A parallel examination of both fixed and random effects would be beneficial.

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