Using k to Estimate and Test Patterns in the APIM

1. Using k to Estimate and Test Patterns in the APIM David A. Kenny

2. You need to know the Actor Partner Interdependence Model and APIM patterns! APIM APIM Patterns

3. 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

4. The Parameter k • Suggested by Kenny and Ledermann (2010) • kis 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

5. -1 0 +1 Contrast Actor Only Couple a = -p p = 0 a = p k

6. But k might equal 0.5. -1 0 +1 Contrast Actor Only Couple a = -p p = 0 a = p k

7. Phantom Variables • One way to estimate k is using a phantom variable. • Phantom variable • No conceptual meaning • Forces a constraint • Latent variable • No disturbance

8. Standard APIM X1 a1 Y1 1 E1 p21 p12 Y2 X2 1 E2 a2

9. Phantom Variables to Estimate k • Now the indirect effect from X2 to Y1, p12 equals a1k1 • Thus, k1 = and k2= and X1 a1 Y1 1 E1 k1 a2 P1 P2 a1 k2 Y2 X2 1 E2 a2

10. 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.

11. 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.

12. 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 = =

14. 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)

15. Example Setups Amos and Mplus (and soon laavan) setups can be downloaded at davidakenny.net/papers/k_apim/k_apim.htm

16. Defining k in Terms of X or kX • 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.

17. X1 a1 Y1 1 E1 p21 p12 Y2 X2 1 E2 a2

18. X1 a1 Y1 1 E1 p21 p12 Y2 X2 1 E2 a2

19. 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 kʹ =

20. 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.