specification issues in relational models david a. kenny university of connecticut talk can be...
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Specification Issues in Relational Models
David A. Kenny
University of Connecticut Talk can be downloaded at:
http://davidakenny.net/talks/nd.ppt
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OverviewPreliminaries
Group Effects: Univariate
X Y Effects with Group Data
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What Is a Group?• dyads
– husband-wife– teacher-student– siblings
• more than two people– families– work groups – classrooms
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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.
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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.
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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
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Example Data
Acitelli Study148 married heterosexual couplesY (outcome): satisfactionX: how positively the partner is
viewedWill use SPSS to illustrate some of the
computations
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Univariate Case
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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
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Y11 Y12 Y13 Y14
Group Y
Person 2 in Group 1
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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(
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Interpretation
The intraclass correlation can be viewed as the proportion of variance due to the group.
s + s
s = rEG
GI 22
2
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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.
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Example
Error Variance (sE2) .094
Group Variance (sG2) .153
rI = .153/(.094 + .153) = .621
Husbands and wives similar in satisfaction.
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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?
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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.
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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.
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Group Processes• Make members similar:
Solidarity
• Differentiate members: Status
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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.
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II. X Y Effects with Group Data
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Y11 Y12 Y13 Y14
Group Y
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Y11 Y12 Y13 Y14
Group Y
X11 X12 X13 X14
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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)
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X Y as a Random Variable
• The effect of X Y varies across groups.
• Requires groups of size 3 or more.
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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.
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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.
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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)
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Y11 Y12 Y13 Y14
Group Y
X11 X12 X13 X14
Mean X
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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) .
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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.
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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.
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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.
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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.
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Y11 Y12 Y13 Y14
Group Y
X11 X12 X13 X14
Group X
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Estimation• Not so easy to estimate the model with
multilevel modeling
• Can use the Olsen & Kenny procedure (Psychological Methods, June issue).
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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
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Example: Latent Group
CI
Variable Effect Lower Upper
Individual .112 .000 .224
Latent Couple 1.532 .574 2.490
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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
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Y11 Y12 Y13
Group Y
X11 X12 X13
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Y11 Y12 Y13
Group Y
X11 X12 X13
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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′
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Example: Partner Effects
CI
Effect b Lower Upper
Actor or X .400 .307 .494
Partner (XMEAN′) .288 .195 .381
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Four Answers
Effect Individual Couple
X & Mean .112 .576
X′ & Mean .112 .689
X & Latent .112 1.532
X & Mean′ .400 .288
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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)
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Which Is Right?
All four are right!
Each has advantages and disadvantages.
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X & Mean
Long history: contextual analysis
Easily embedded within multilevel modeling
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X′ & Mean (Between-Within)
Statistical advantage: two effects orthogonal
Easily embedded within multilevel modeling as group centered
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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.
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X & Mean′ (APIM)
Has a simple interpretation
Interaction can be meaningful
Very popular in dyadic analysis
Not used frequently in group research
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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.
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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!!!
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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