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1 Communication between Fixed and Random Effects: Examples from Dyadic Data David A. Kenny University of Connecticut davidakenny.net\kenny.htm

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2 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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3 Caveat Linear Models Normally Distributed Random Variables

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5 Three Major Dyadic Designs l Standard –Each person has one partner. –Married couples l 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. l 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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6 Standard Design

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7 SRM Designs

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8 One-with-Many Design

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9 Model for the Standard Design Two scores for dyad j: Y 1j = b 01 + b 11 X 11j + 1j Y 2j = b 02 + b 12 X 12j + 2j where C( 1j, 1j ) may be nonzero

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10 Members Indistinguishable Y 1j = b 0 + b 1 X 11j + 1j Y 2j = b 0 + b 1 X 12j + 2j where V( 1j ) = V( 2j ) and C( 1j, 2j ) may be nonzero

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11 Partner Effect with Members Indistinguishable Y 1j = b 0 + b 1 X 11j + b 2 X 12j + 1j Y 2j = b 0 + b 1 X 12j + b 2 X 11j + 2j where V( 1j ) = V( 2j ) and C( 1j, 2j ) may be nonzero

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12 Partner Effects with Members Distinguishable Y 1j = b 01 + b 11 X 1j + b 21 X 2j + 1j Y 2j = b 02 + b 12 X 2j + b 22 X 1j + 2j where C( 1j, 1j ) may be nonzero

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13 Social Relations Model model of dyadic relations embedded in groups X ijk : actor i with partner j in group k round-robin structures: everyone paired with everyone else other structures possible

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14 Round Robin Design 1 2 3 4 5 6 1 - 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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15 The Statistical Model: Random Effects X ijk = k + ik + jk + ijk variances (4): 2, 2, 2, 2 covariances (2):, (fixed effects discussed later)

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16 Liking: How much Dave likes Paul Variances Group ( 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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17 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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18 Estimation ANOVA 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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19 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: s 2 /(s 2 + s 2 + s 2 )

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21 Model for the One-with-Many Design Person i (the one of n j persons) is paired with person j (the many): Y ij = b 0j + b 1 X 1ij + jj where V(b 0j ) may be nonzero

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22 Model for the One-with-Many Design Person i (the one of n j persons) is paired with person j (the many): Y ij = b 0j + b 1j X 1ij + ij where V(b 0j ), V(b 1j ), and C(b 0j, b 1j ) may be nonzero

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23 II. Specification Error How does specifying the wrong model in one part affect the other part?

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24 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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25 Example of Bias in Standard Errors Consider a standard dyad design and a simple model for person i (i =1, 2) in dyad j Y ij = b 0 + b 1 X ij + ij where C(X 1j,X 2j )/V(X) = x and C( 1j, 2j )/V( ) =.

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26 Bias in Standard Error for b 1 If e Is Falsely Assumed To Be Zero under-estimation: x > 0 over-estimation: x < 0 none: x = 0

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27 Specification Error in Fixed Part can dramatically bias random effects example: roommate effects and liking

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28 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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29 Kenny & Lowe Study: Results Using ANOVA Component Estimate Actor-.754 Partner-.794 Relationship 3.693

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30 Kenny & Lowe Study: Results Using ANOVA Component EstimateRevised Actor-.754.444 Partner-.794.597 Relationship 3.693 1.139

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31 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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32 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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33 Conclusion People do not like being in diverse groups. But is there group variance in liking?

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34 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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35 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: s 2 = i (X i – M) 2 /(n – 1) = i j (X i – X j ) 2 /[n(n - 1)] i > j

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36 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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37 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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38 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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39 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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40 Example Harmon Hoschs 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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41 Results: Random Effects Term Estimate SE 2.000 ---- 2.165.016 2.045.013 2.477.018

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42 Results: Fixed Effects Term Estimate SE Diversity -.007.009 More diversity, others seem less likeable.

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43 Results: Fixed Effects Term Estimate SE Dyad.129.015 Actor -.029.041 Partner.006.041 Group.005.058 You see someone as likeable if they have the same opinion as you.

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44 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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45 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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46 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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47 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 Y ij = b 0 + b 1 X j + j + ij j is person, ij is target, and X j (0, 1) is gender] (Simulation performed by Randi Garcia.)

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48 Parameter Values Y ij = b 0 + b 1 X j + j + ij b 0 = 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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50 What we have here is a failure to communicate.

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51 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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