APIM with Indistinguishable Dyads: SEM Estimation David A. Kenny March 22, 2013

You Need to Know APIM (click for webinar) Structural Equation Modeling Amos APIM with Distinguishable Dyads Using SEM (click for webinar)

Example Dataset: Acitelli dyad dataset 148 married couples Outcome Satisfaction (Wife and Husband) Predictor Variable Other-Positivity (Wife and Husband) How positive the Wife views her Husband, and how positive the Husband views his Wife

Constraints Set six parameters equal: Two actor effects Two partner effects Two error variances Two Y intercepts Two X variances Two X means

Actor Effect: The effect of own X on Y 5

Partner Effect: The effect of partner’s X on Y 6

Intercept: The expected value for Y when both X1 and X2 equal zero 7

Error Variance: Unexplained variance in Y 8

Means: Mean for X 9

Variances: Variance of X 10

(pretending indistinguishable) Acitelli Example (pretending indistinguishable)

Meaning of the c2 c2(6) = 9.192, p = .163 What does it mean? If members were randomly assigned to 1 and 2, the value would be totally arbitrary and merely reflects the non-random assignment of persons to be a “1” and a “2.” If there was a different assignment pattern, the c2 would change, but the df and all of the estimates and their standard errors would not. Thus, if dyad members were indistinguishable, the c2 would not at all meaningful and be totally arbitrary.

I-Sat Model Because the c2 changes and is fundamentally arbitrary, it should not be interpreted. For the basic APIM, the c2 for the indistinguishable model with 6 df evaluates the non-random sorting of dyad members to the roles of 1 and 2. Because this is a saturated model, this model is designated as the I-Sat (interchangeable and saturated) model. For any non-saturated model, the I-Sat c2 (e.g., 9.192) and its df (e.g., 6) must be subtracted off. For example, for the model in which we have equal actor and partner effects, the c2(7) = 12.995, but when we adjust for the I-Sat model, we have c2(1) = 3.803.

Fit Indices: RMSEA To compute the RMSEA, we adjust the c2 and the df using the I-Sat model values. These adjusted values are then used to compute the RMSEA.

Fit Indices: TLI and CFI These indices need a null or independence model. The usual null model: variances free but correlations fixed to zero. For indistinguishable dyads Means and variances for the members from the same dyad equal. Adjust the df and c2 using the I-SAT model.

Strategy Use o&k.xls Need c2 and df for the following models Your model I-SAT model (may the same as above) Null model with mean and variances set equal for the two dyad members

o&k.xks Example

Fit Statistics Model c2 df c2’ df ’ RMSEA CFI ISAT 9.192 6 0 Null 157.863 10 p = 0 44.061 7 34.869 1 .480 .771 a = p 12.995 7 3.803 1 .138 .981

The ISAT Model The df are kq + q(q + 1) where q is the number of mixed variables in the model, k the number of between-dyads variables, and df are the total number of constraints. Check to see the “implied moment matrix” is symmetric for persons 1 and 2. If not, the equality constraints are not correct.

Standardization The standardized solution in Amos and other SEM programs are correct if dyad members are treated as completely indistinguishable.

Summary Estimation with of the APIM with indistinguishable dyads using SEM is tricky. Alternatives Multilevel Modeling Olsen & Kenny discuss using pairwise data and weighting cases by 0.5.

Additional Readings Olsen, J. A., & Kenny, D. A. (2006). Structural equation modeling with interchangeable dyads. Psychological Methods, 11, 127-141. Kenny, D. A., Kashy, D. A., & Cook, W. L. Dyadic data analysis. New York: Guilford Press, Chapter 5 (especially pp. 111-112) and Chapter 7, pp. 168-169.