Tengiat Loi posted on Tuesday, April 17, 2018 - 4:52 pm
Dear Prof. Muthen,
For 1-1-1 multilevel mediation, the indirect effect is sum of the product of the parameter estimates of the two paths and the covariances of their slopes.
For 1-1-1-1 (x-m1-m2-y) model, the indirect effect would be the product of three paths, plus three components of (covariance of a1 with a2 TIMES the coefficient of a3 ), (covariance of a2 with a3 TIMES the coefficient of a1), and (covariance of a1 with a3 TIMES the coefficient of a2 ) is this correct?
Also, do you have reference for it so that we can cite in our papers.
Tengiat Loi posted on Wednesday, April 18, 2018 - 8:02 am
Also, if there is a moderator which moderates the first path, then the product of paths would be [a1+a1w*(+/-1 SD of W)]*a2*a3], what about the covariance component?
Mediation with random slopes for the 1-1-1-1 case has the indirect effect that I show below. I don't know a reference for this but it can be derived from basic statistical principles assuming normal errors for the random slopes. With random slope means b1, b2, b3 corresponding to the regressions of y on m1, m1 on m2, and m2 on x, respectively, and their corresponding errors e1, e2, e3, the indirect effect boils down to the expectation
because E[e]=0 and E[e1e2e3]=0 for normal variables. An expressions such as E[e2e3] is the same as Cov(e2e3), the covariance of the corresponding two random slopes.
If the regression of m2 on x is moderated by w and the x*w interaction also has a random slope (with mean b4 and error e4), the indirect effect is
b1*b2*(b3+b4*W)+ the above 3 terms involving the covariances + b4*W*Cov(e1e2)+W*Cov(e1e4)+W*Cov(e2e4),