Specification Issues in Relational Models

David A. Kenny

University of Connecticut Talk can be downloaded at:

http://davidakenny.net/talks/nd.ppt

OverviewPreliminaries

Group Effects: Univariate

X Y Effects with Group Data

What Is a Group?• dyads

– husband-wife– teacher-student– siblings

• more than two people– families– work groups – classrooms

A. Distinguishability• In some groups, members can be

distinguished by the role: e.g., heterosexual couples are usually distinguished by gender.

• In other groups, e.g., some work groups, members are indistinguishable. That is, members of the group cannot be ordered.

B. Distinguishability• Both a theoretical and empirical issue.

• Differences by variable.

• Partial distinguishability.

• Will assume in the rest of the talk that members are indistinguishable.

DesignPresume that each person in the group

measured once. Alternative designs

one measure per groupeach dyad in the group is measured

(Social Relations Model)one informant or target in the group

Example Data

Acitelli Study148 married heterosexual couplesY (outcome): satisfactionX: how positively the partner is

viewedWill use SPSS to illustrate some of the

computations

Univariate Case

Nonindependence

Definition: the degree of greater similarity (or dissimilarity) between two observations from members of the same group than between two scores from members of different groups

How to model: a group effect

Y11 Y12 Y13 Y14

Group Y

Person 2 in Group 1

Intraclass CorrelationGroup is treated as the independent variable in a one-way, between-subjects ANOVA:

where: MSB is the mean square between groups, MSW is the mean square within groups, and k is the group size.

WB

WBI MSkMS

MSMSr

)1(

Interpretation

The intraclass correlation can be viewed as the proportion of variance due to the group.

s + s

s = rEG

GI 22

2

Computing Group Variance by SPSS

MIXED Y /FIXED = /PRINT = SOLUTION TESTCOV /RANDOM INTERCEPT | SUBJECT(GROUP) COVTYPE(VC) .

Person is the unit of analysis. “GROUP” is a variable that codes what group each person is in.

Example

Error Variance (sE2) .094

Group Variance (sG2) .153

rI = .153/(.094 + .153) = .621

Husbands and wives similar in satisfaction.

What if Negative?• Nonindependence is a correlation.

• A correlation can be negative, but the proportion of group variance cannot be.

• Why would nonindependence be a negative intraclass correlation?

A. How Negative CorrelationsMight Arise?

• Compensation: If one person has a large score, the other person lowers his or her score. For example, if one person acts very friendly, the partner may distance him or herself,

• Social comparison: The members of the dyad use the relative difference on some measure to determine some other variable. For instance, satisfaction after a tennis match is determined by the score of that match.

B. How Negative CorrelationsMight Arise?

• Zero-sum: The sum of two scores is the same for each dyad. For instance, the two members divide a reward that is the same for all dyads.

• Division of labor: Dyad members assign one member to do one task and the other member to do another. For instance, the amount of housework done in the household may be negatively correlated.

Group Processes• Make members similar:

Solidarity

• Differentiate members: Status

Negative Intraclass Correlations Using SPSS

MIXED Y /FIXED = /PRINT = SOLUTION TESTCOV /REPEATED = MEMBER | SUBJECT(GROUP) COVTYPE(CS).

“MEMBER” is a variable that codes the different person in the group; e.g., it is “1,” “2,” and “3” in a three-person group.

Not going to consider this any more.

II. X Y Effects with Group Data

Y11 Y12 Y13 Y14

Group Y

Y11 Y12 Y13 Y14

Group Y

X11 X12 X13 X14

Computing X Y Effects in SPSS

MIXED

Y WITH X

/FIXED = X

/PRINT = SOLUTION TESTCOV

/RANDOM INTERCEPT |

SUBJECT(GROUP) COVTYPE(VC) .

X for example = .314 (CI of .219 to .408)

X Y as a Random Variable

• The effect of X Y varies across groups.

• Requires groups of size 3 or more.

Random X Y Effects in SPSS

MIXED Y WITH X /FIXED = X /PRINT = SOLUTION TESTCOV /RANDOM INTERCEPT X | SUBJECT(GROUP) COVTYPE(IN) . “IN” allows for intercept and X effects to be

correlatedNot going to consider this any more.

X Y Effect May Occur at the Group Level

Just because X is measured at the individual level does not mean that the effect of X on Y occurs only at that level.

Need to model the effect of X on Y at more than the individual level.

A simple idea but not so simple to do.

Consider Four Ways To Do So

Group Mean (Contextual Analysis)

Group Mean with Group Centering

(Between-Within Analysis)

Group Effect as a Latent Variable

Group Effect as “Everyone Else” (Actor-Partner Interdependence Model)

Y11 Y12 Y13 Y14

Group Y

X11 X12 X13 X14

Mean X

Computing X Y Effects at Two Levels by SPSS

MIXED

Y WITH X XMEAN

/FIXED = X XMEAN

/PRINT = SOLUTION TESTCOV

/RANDOM INTERCEPT |

SUBJECT(GROUP) COVTYPE(VC) .

Example: Group Mean

X .112 (CI: -.001 to .226)

XMEAN .576 (CI: .390 to .762)

Suggests that when couples idealize, the couples are more satisfied.

Centering

Group centering: Subtract from X the mean of X for the group in which the person is in.

SPSS syntax is the same but now X become X′ or X minus the mean of X for the group.

Example: Group Centering

X′ .112 (CI: -.001 to .226)

XMEAN .689 (CI: .539 to .837)

Suggests that when couples view partner more favorably, the couples are more satisfied.

Group X as a Random Variable

Group Mean may be an imperfect measure of the couple score.

Treat X11 and X12 as indicators of a latent variable.

Proposed by Kenny & La Voie in 1984 and a modified version by Griffin & Gonzalez used here.

Y11 Y12 Y13 Y14

Group Y

X11 X12 X13 X14

Group X

Estimation• Not so easy to estimate the model with

multilevel modeling

• Can use the Olsen & Kenny procedure (Psychological Methods, June issue).

4.26

Male Perceptionof the Partner

3.13

MaleSatisfaction

4.26

Female Perceptionof the Partner

3.13

FemaleSatisfaction

0, .06

CouplePerception

0

CoupleSatisfaction

1.00

1.00

1.00

1.00

0, .19

e11

0, .19

e2

1

0, .09

f1

1

0, .09

f2

1

0, .00

U11.53

.11

.11

Example: Latent Group

CI

Variable Effect Lower Upper

Individual .112 .000 .224

Latent Couple 1.532 .574 2.490

Partner Effects• Actor Effect or X

– Member A’s X affects the member A’s Y

• Partner Effect or XMEAN′

– Member A’s X affects the member B’s Y

Y11 Y12 Y13

Group Y

X11 X12 X13

Y11 Y12 Y13

Group Y

X11 X12 X13

Estimating Partner Effects by SPSS

MIXED

Y WITH X XPART

/FIXED = X XPART

/PRINT = SOLUTION TESTCOV

/RANDOM INTERCEPT |

SUBJECT(GROUP) COVTYPE(VC) .

XPART is the mean of X of the other members in the group or XMEAN′

Example: Partner Effects

CI

Effect b Lower Upper

Actor or X .400 .307 .494

Partner (XMEAN′) .288 .195 .381

Four Answers

Effect Individual Couple

X & Mean .112 .576

X′ & Mean .112 .689

X & Latent .112 1.532

X & Mean′ .400 .288

Four Ways

Group Mean (Contextual Analysis)

Group Mean with Group Centering

(Between-Within Analysis)

Group Effect as a Latent Variable

Group Effect as “Everyone Else” (Actor-Partner Interdependence Model)

Which Is Right?

All four are right!

Each has advantages and disadvantages.

X & Mean

Long history: contextual analysis

Easily embedded within multilevel modeling

X′ & Mean (Between-Within)

Statistical advantage: two effects orthogonal

Easily embedded within multilevel modeling as group centered

X & Latent

Cannot work if the intraclass for X is not positive and estimates are unstable when intraclass is small

Latent variable must make sense

Not easily estimated

Can lead to anomalous results

Not frequently adopted by practitioners.

X & Mean′ (APIM)

Has a simple interpretation

Interaction can be meaningful

Very popular in dyadic analysis

Not used frequently in group research

Translation of EffectsWe use the X and XMEAN analysis as the basic

analysis.Denote i as the effect of X and g as the effect of

XMEAN and k as group size:within= i and between = g + iactor = i + g/k and partner = (k – 1)g/k

For the latent variable model, the X effect is again i, and the group effect equals p[1/(k – 1) + rx]/rx where p is the partner effect and rx is the intraclass correlation for X.

Concluding Comments• In studying groups you need to give careful

thought as to what type of effects might occur.

• No one “right” way to model effects.

• Be open to alternative ways to estimate effects.

• Beware of over-simplification

• Beware of over-complexity

THINK!!!

Kenny, D. A., Mannetti, L., Pierro, A., Livi, S., & Kashy, D. A. (2002). The statistical analysis of data from small groups. Journal of Personality and Social Psychology, 83, 126-137.

Kenny, D. A., Kashy, D. A., & Cook, W. L. (2007) Dyadic data analysis. New York: The Guilford Press.

Talk can be downloaded at:

http://davidakenny.net/talks/nd.ppt