k for APIM Patterns

Report
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!
APIM
APIM Patterns
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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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The Parameter k
• 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
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-1
Contrast
a = -p
0
Actor Only
p= 0
+1
Couple
a=p
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But k might
equal 0.5.
-1
Contrast
a = -p
0
Actor Only
p= 0
+1
Couple
a=p
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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
a1
a2
Y1
1
Y2
1
E1
E2
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Phantom Variables to Estimate k
a1
X1
1
Y2
1
E1
k1
a2
P1
P2
a1
X2
Y1
a2
k2
E2
• Now the indirect effect from X2 to Y1, p12 equals a1k1
p
p
• Thus, k1 = 12 and k2 = 21 and
a1
a2
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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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Caution in Computing
the Parameter k
• 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.
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:
p12 p21
k=
=
12
a1
a2
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Results
CI
Distinguishable
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)
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Example Setups
Amos and Mplus (and soon laavan) setups
can be downloaded at
davidakenny.net/papers/k_apim/k_apim.htm
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Defining k in Terms of X or kX
• When dyads are distinguishable, we previously
took the two paths leading into Y to define k:
p12
p21
k1X =
and
k2X =
a1
a2
• Alternatively k can be defined by the two paths
coming from X:
p21
p12
k1X =
and
k2X =
a1
a2
• 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.
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X1
X2
a1
a2
Y1
1
Y2
1
E1
E2
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X1
X2
a1
a2
Y1
1
Y2
1
E1
E2
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Defining k in as Actor Effect
Divided by Partner Effect
• 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
a
kʹ =
p
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Conclusion
Using 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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