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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! APIM APIM Patterns 2 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 3 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 4 -1 Contrast a = -p 0 Actor Only p= 0 +1 Couple a=p 5 But k might equal 0.5. -1 Contrast a = -p 0 Actor Only p= 0 +1 Couple a=p 6 Phantom Variables • One way to estimate k is using a phantom variable. • Phantom variable – No conceptual meaning – Forces a constraint – Latent variable – No disturbance 7 Standard APIM X1 X2 a1 a2 Y1 1 Y2 1 E1 E2 8 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 9 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. 10 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 13 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) 14 Example Setups Amos and Mplus (and soon laavan) setups can be downloaded at davidakenny.net/papers/k_apim/k_apim.htm 15 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. 16 X1 X2 a1 a2 Y1 1 Y2 1 E1 E2 17 X1 X2 a1 a2 Y1 1 Y2 1 E1 E2 18 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 19 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. 20