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In the Book Of Why by Judea Pearl there is the mention of the dependence of weights of guinea pig pups on gestation period, as explored by Sewall Wright. The following is the causal diagram provided -

Causal diagram depicting impact of gestation period on weight

In the above diagram,

X = weight of pup in grams

P = Gestation period in days

Q = Utero growth rate

L = litter size

A and C = exogenous factors

We are interested in finding the impact of changing P on X in terms of grams/day, depicting the additional weight gain for each extra day of gestation.

The following are the path coefficients for each causal relation -

$L{\rightarrow}Q=l$

$L{\rightarrow}P=l'$

$Q{\rightarrow}X=q$

$P{\rightarrow}X=p$

Then the book says that, the influence of $P$ on $X$ can be found by $p+(l{\times}l'{\times}q)={\,}\text{rate of change of }X\text{ per day in grams/day}$.............$(i)$

My query is, what do the path coefficients denote exactly, in terms of their mathematical formulation? How are changes $\Delta P$ and $\Delta L$ related in terms of $l'$? Based on the definition of the coefficients, how is equation $(i)$ derived?

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The coefficients are slopes in a data-generating (structural) model corresponding to a linear regression model. That is, $X := qQ + pP + u$, where $u$ corresponds to other (unobserved) factors (not included in the graph). This is the context of path analysis as described by Sewall Wright. See also Pearl's excellent paper "Linear Models: A Useful "Microscope" for Causal Analysis".

$(i)$ is derived using the rules of path analysis, which state that when variables are standardized to have a variance of 1, the relationship between two variables is equal to the sum of the open paths between them, where open paths are unique chains of arrows from one variable to the other that do not involve two arrows pointing at the same node (i.e., an unconditioned-upon collider). The two open paths from $P$ to $X$ are $P \rightarrow X$ and $ P \leftarrow L \rightarrow Q \rightarrow X$, and their magnitudes are $p$ and $l \times l' \times q$, respectively.

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    $\begingroup$ The asker talk about “impact of changing” $P$ on $X$, so the causal effect of $P$ on $X$. It is $p$, while the covariance between $X$ and $P$ is $p+ll’q$. It is right? $\endgroup$
    – markowitz
    Jan 19, 2022 at 11:24
  • $\begingroup$ Yes, that is correct, because in the "mutilated" graph associated with $do(P)$, you remove all arrows pointing at $P$. $\endgroup$
    – Noah
    Jan 19, 2022 at 17:01
  • $\begingroup$ @Noah How come $l'$ is entering in multiplicative form given that the causal relationship is $L{\rightarrow} P$? Do I need to learn "do-calculus" to understand this? $\endgroup$
    – Anirban Chakraborty
    Jan 19, 2022 at 17:55
  • $\begingroup$ Sorry for my initial comment. I now understand from @markowitz's comment that $ll'p$ accounts for the correlation between $P$ and $X$ as rising from the path $P{\leftarrow}L{\rightarrow}Q{\rightarrow}X$. Correct? $\endgroup$
    – Anirban Chakraborty
    Jan 19, 2022 at 18:21
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    $\begingroup$ Yes, that is correct and is what the last sentence of my answer meant to convey. You do not need to know do-calculus to understand the path tracing rules, but to understand the concept of "impact of changing" you do. $\endgroup$
    – Noah
    Jan 19, 2022 at 18:35

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