using k to estimate and test patterns in the apim david a. kenny february 17, 2013
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Using k to Estimate and Test Patterns in the APIM
David A. Kenny
February 17, 2013
You need to know the Actor Partner Interdependence Model and APIM patterns!
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APIM APIM Patterns
APIM Patterns
• Couple Model– Equal Actor and Partner Effects: a = p
• Contrast Model– Actor plus partner sums to zero: a – p = 0
• Actor Only Model– Partner effect is zero: p = 0
• Partner Only Model– Actor effect is zero: a = 0
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• Suggested by Kenny and Ledermann (2010) • k is the ratio of the partner effect to the actor
effect or p/a• k is named after Larry Kurdek, a pioneer in
the study of dyadic data• Special cases of k:
–k is 1, couple model–k equal to −1, contrast model–k equal to zero, actor-only model
The Parameter k
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-1 0 +1
Contrast Actor Only Couple a = -p p = 0 a = p
k
5
-1 0 +1
Contrast Actor Only Couple a = -p p = 0 a = p
But k might equal 0.5.
k6
Phantom Variables
• One way to estimate k is using a phantom variable.
• Phantom variable– No conceptual meaning– Forces a constraint– Latent variable– No disturbance
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Standard APIM
X1
X2
Y1
Y2
E1
E2
1
1
a1
p21
p 12
a2
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Phantom Variables to Estimate k
• Now the indirect effect from X2 to Y1, p12 equals a1k1
• Thus, k1 = and k2 = and
X1
X2
Y1
Y2
E1
E2
1
1
a1
a2
P1
a1
k1
P2
a2
k2
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Estimates and Confidence Interval
• Use bootstrapping to obtain the asymmetric confidence interval (CI).
• Check to see if 1, -1, or 0 are in the CI of k.
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• Note that k is not defined when the actor effect is zero.
• Thus, k and its confidence interval should not be computed if the actor effect is small.
Caution in Computing the Parameter k
Distinguishability and k
• For distinguishable dyads, k may differ for the two members which might be theoretically interesting: e.g., wives couple model and husbands contrast model.
• Need to test to see if k varies across the distinguishing variable.
• Note that k may not vary, even if a and p vary by the distinguishing variable:
k = = 12
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ResultsDistinguishable
Wives: kW = 0.851 (0.223 to 2.038 )
Husbands: kH = 0.616 (0.294 to 1.187)
Equal values of k
kW = kH = 0.710 (0.489 to 0.989 )
c2(1) = 0.320, p = .571
Indistinguishable: k = 0.719 (0.484 to 1.027)14
CI
Example SetupsAmos and Mplus (and soon laavan) setups can be downloaded at
davidakenny.net/papers/k_apim/k_apim.htm
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• When dyads are distinguishable, we previously took the two paths leading into Y to define k: k1X = and k2X =
• Alternatively k can be defined by the two paths coming from X:
k1X = and k2X =
• For instance if one person is more “influential” than the other, that person would have kX of 1 and the partner may have a kX of zero.
Defining k in Terms of X or kX
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X1
X2
Y1
Y2
E1
E2
1
1
a1
p21
p 12
a2
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X1
X2
Y1
Y2
E1
E2
1
1
a1
p21
p 12
a2
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• In some contexts the partner effect is larger than the actor effect, i.e., partner-only models.
• Note if a = 0, k = ∞! • In this case, it may make more sense
to define k as the ratio of the actor to the partner effect or kʹ =
Defining k in as Actor Effect Divided by Partner Effect
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ConclusionUsing k can simplify the model and link the model to theory.
Reading
Kenny & Ledermann (2010), Journal of Family Psychology, 24, pp. 359-366.
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