Where does this formula for the effect of the mediator on the outcome comes from?

Question

I am reading the paper

Caron, P.-O., & Valois, P. 2018. A computational description of simple mediation analysis. The Quantitative Methods for Psychology, 14(2): 147–158.

There there is the following model for mediation:

Where:

• $a_{xm}$ is the coefficient obtained by the following regression: $$ M = a_{xm} X + \epsilon $$

• $c_{xy}$, also called total effect, is the coefficient obtained by the following regression: $$ Y = c_{xy} X + \epsilon $$

• $c_{xy|m}$ and $b_{my|x}$, are obtained by the following regression: $$ Y = c_{xy|m} X + b_{my|x} M + \epsilon $$

• $b_{my}$ (unreported in the picture) is the coefficient obtained by: $$ Y = b_{my} M + \epsilon $$

The paper says that:

$$ b_{my} = b_{my|x} (1-a_{xm}^2) + a_{xm} c_{xy} $$

This is equation (1) in page 148.

But there is no proof for that.

Why is that? Where does this formula comes from?

Answer 1

This comes from path analysis. We'll assume these variables are standardized, which is the only case in which these equations make sense.

Path analysis says that the unconditional correlation between two variables, which in this case is just the standardized regression slope of one on the other, is the sum of all the open pathways between the two variables. The contribution of a pathway is the product of all the regression slopes along that pathway.

In this case, we want $b_{my}$, which is the unconditional correlation between $M$ and $Y$. There are two pathways from $M$ to $Y$: the direct pathway and the indirect pathway through $X$. So, $b_{my} = Path_{MY} + Path_{MXY}$. We can compute those paths in a straightforward way using the rules above:

\begin{align} Path_{MY} &= b_{my|x} \\ Path_{MXY} & = a_{xm} c_{xy|m} \end{align}

That gives us $b_{my} = b_{my|x} + a_{xm} c_{xy|m}$. We're halfway there.

Next, realize that the total effect, $c_{xy}$, is the sum of the paths from $X$ to $Y$, which include the one through $M$ (the indirect path) and the direct path; the following equation gives the standard mediation decomposition of the total effect into the indirect and direct effects: $$ c_{xy} = a_{xm} b_{my|x} + c_{xy|m} $$ which can be rearranged as $$ c_{xy|m} = c_{xy} - a_{xm} b_{my|x} $$ If we substitute this formula into our formula for $b_{my}$, we get the result: \begin{align} b_{my} &= b_{my|x} + a_{xm} c_{xy|m} \\ &= b_{my|x} + a_{xm} (c_{xy} - a_{xm} b_{my|x}) \\ &= b_{my|x} - a_{xm}^2 b_{my|x} + a_{xm} c_{xy} \\ &= b_{my|x}(1 - a_{xm}^2) + a_{xm} c_{xy} \\ \end{align}