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

k5

-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.

ReadingKenny & Ledermann (2010), Journal of Family Psychology, 24, pp. 359-366.

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