Hi everyone,
I have been reading through VanderWeele, T. (2015) and Hayes, A. F. (2017) to learn about mediation analysis, especially when there are multiple mediators. In their book, they both mentioned estimating a set of equations. For example, if there are 3 mediators, the equations would look like:
M1 = a0+a1*X+e1
M2 = b0+b1*X+e2
M3 = c0+c1*X+e3
Y = d0+d1*X+d2*M1+d3*M2+d4*M3+e4
No mediator causally influences another. But Mediators M1, M2 and M3 are likely to be correlated. Error terms e1, e2 and e3 could be correlated because of unmeasured confounding factors of mediators.
Question: I do not understand why estimating these equations allows the correlation among mediators. Are the distribution of error terms be measured jointly?
Thank you in advance for your kind help.
Best wishes,
Jingya
References:
Hayes, A. F. (2017). Introduction to mediation, moderation, and conditional process analysis: A regression-based approach. Guilford publications.
VanderWeele, T. (2015). Explanation in causal inference: methods for mediation and interaction. Oxford University Press.
It is the basic concept in multiple regression, that the intercorrelations of the predictor variables are taken into account. Otherwise it would make no difference if you use a set of bivariate correlations/regressions or multiple regression. The outcome (in the sense of the overall explained variance R^2) of MR as compared to bivariate correlations/regressions can be very different. If the predictors would be perfectly uncorrelated to each other, the sum of all r^2 from bivariate correlations would be indentical to R^2 of the MR. In case of redundancies, i.e. the predictors are somehow correlated, R^2 of the MR would be smaller than the sum of all r^2. And in some special cases (e.g. suppression effects), the R^2 can get even larger than the sum of all r^2. Regression weights "b" are typically called "partial regression coefficients" which makes it a little bit clearer (see Cohen, Cohen, West, and Aiken 2003)
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