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I am trying to calculate direct, indirect, and total effects with an SEM model. If I have a model like this one: enter image description here Then I understand from Wright's path tracing rules that the total effect of $X_1$ on $X_4$ is $ab+cd$. And for instance the total effect of $X_2$ on $X_4$ is $b + acd$.

But what if the two mediators $X_2$ and $X_3$ are allowed to be correlated? i.e. enter image description here

My questions are:

  1. In calculating the effect of $X_1$ on $X_4$, should I now include $ard$ and $crb$?
  2. In calculating the effect of $X_2$ on $X_4$, should I now include $rd$?
  3. In calculating the effect of $X_2$ on $X_4$, I think Wright's ray tracing rules forbid other pathways e.g. $rcab$ which would have me enter $X_2$ twice. Is that correct?

I don't know R so I am using semopy in Python, which does not calculate these automatically as best I can tell.

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the total effect of $𝑋_2$ on $𝑋_4$ is $𝑏+π‘Žπ‘π‘‘$

No, it is just $b$. The implied covariance Cov($𝑋_2,𝑋_4$) includes $𝑏+π‘Žπ‘π‘‘$, but also all other paths between those 2 variables that can be traced (1) up any directed arrow from one variable, (2) around a double-headed arrow once, to go (3) down any directed arrow(s) to the other variable.

  1. should I now include $π‘Žπ‘Ÿπ‘‘$ and $π‘π‘Ÿπ‘$?

To calculate the implied Cov($𝑋_2,𝑋_4$), yes. But that is not part of the implied causal effect, which only involves any path traced from $X_1$ by traveling down directed arrows until arriving at $X_4$.

Same answer as for 1.

  1. I think Wright's tracing rules forbid pathways which would have me enter $𝑋_2$ twice. Is that correct?

I think the relevant rule is that you can only switch direction (by going around a double-headed arrow) once. Your diagram does not include variances for any variables, which are implied by the $π‘π‘Žπ‘$ path: You would have to go around the double-headed arrow from $X_1$ pointing back to itself, which allows you to change direction from up to down. But the $π‘Ÿπ‘π‘Žπ‘$ path is already not allowed because by starting from $X_2$ and going up $r$, you already change directions. So you can't go "up" from $X_3$ to $X_1$; you can only go down to $X_4$.

Here is the book I learned about tracing-rules from, along with the covariance algebra they represent. A later edition is coauthored by Alexander Beaujean.

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  • $\begingroup$ The distinction between (direct, indirect, and total) "causal effect" and (total) "correlation" is one I hadn't thought about previously. Thank you! $\endgroup$ Commented Mar 25 at 2:28
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Good question! Part of the confusion may lie in how you created your parallel mediation model. The usual notation would denote your variables as:
X for the Independent (predictor) variable;
M1 and M2 to denote the Mediator variables;
Y for the Dependent (outcome) variable.

Next, your parameters (a, b, c, d) should be instead, respectively: a1, b1, a2, b2. This allows us to denote the direct effect of X on Y as c' (c prime), which you would include as an arrow going directly from X to Y. I'm not familiar with semopy, though c' should be automatically calculated, just as the other parameters are.

Now, to answer your questions:

  1. This is the total effect (usually denoted by c) of X on Y. It is calculated as $c' + (a1 * b1) + (a2 * b2)$. You may be thinking of the total indirect or mediated effect, which is $(a1 * b1) + (a2 * b2)$.
  2. This is the first indirect effect, calculated as just $a1 * b1$.
  3. This does not apply, as r is not used in any calculations.

TLDR: Correlation between the mediators has no effect on calculating indirect and total effects. These are calculated the same as if there were no correlation between mediators.

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    $\begingroup$ Thank you for taking the time to write, but your answer confuses me quite a bit β€” the variable renaming is unnecessarily confusing, and I think some of the answers you gave don't make any sense. For instance in #2, why is $a$ (what you called $a_1$) involved in calculating the effect of $X_2$ on $X_4$ (what you called $M_1$ on $Y$)? I'm not trying to appeal to a parallel mediator modelβ€”my SEM model is actually much more complex than what I've drawn, I'm just trying to give a minimum working example. $\endgroup$ Commented Mar 23 at 23:20
  • $\begingroup$ Maybe aside from this, here's the real thing I am stuck on: why doesn't correlation between the mediators matter when calculating effects? If $M_1$ and $M_2$ have some correlation, doesn't it imply that e.g. $M_1$ has an indirect effect on $Y$ through $M_2$, because if $M_1$ varies a little, $M_2$ does too? $\endgroup$ Commented Mar 23 at 23:24

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