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---
title: "Understanding Structural Coefficients and Causal Effect in Linear Systems 6/6"
author: "AI Engineer"
date: "July 29, 2023"
output:
  beamer_presentation: default
  slidy_presentation: default
---

## Table of Contents

1.  Total Effect in Linear Systems
2.  Identifying Structural Coefficients and Causal Effect
3.  Mediation in Linear Systems
4.  Summary

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## Total Effect in Linear Systems

-   In a linear system, the total effect of $X$ on $Y$ is the sum of the products of the coefficients of the edges on every nonbackdoor path from $X$ to $Y$.
-   To find the total effect of $X$ on $Y$, follow these steps:
    -   Find every nonbackdoor path from $X$ to $Y$.
    -   For each path, multiply all coefficients on the path together.
    -   Add up all the products.
-   This identity is a consequence of the nature of Structural Causal Models (SCMs).

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## Total Effect in Linear Systems (continued)

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![](https://cdn.mathpix.com/cropped/2023_07_28_a103dc94c1ec860738c6g-31.jpg?height=443&width=345&top_left_y=1857&top_left_x=867) Figure 3.13: Graphical representation of a linear system
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-   Consider the graph in Figure 3.13.
-   To find the total effect of $Z$ on $Y$, intervene on $Z$, removing all arrows going into $Z$, then express $Y$ in terms of $Z$ in the remaining model.
-   This can be done with a little algebra, resulting in the equation $Y=\tau Z+U$, where $\tau=d+e c$ and $U$ contains only terms that do not depend on $Z$ in the modified model.
-   An increase of a single unit in $Z$ will increase $Y$ by $\tau$-the definition of the total effect.
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## Identifying Structural Coefficients and Causal Effect

-   The problem of estimating total and direct effects from nonexperimental data is known as "identifiability".
-   It involves expressing the path coefficients associated with the total and direct effects in terms of the covariances $\sigma_{X Y}$ or regression coefficients $R_{Y X \cdot Z}$.
-   In many cases, to identify direct and total effects, we do not need to identify each and every structural parameter in the model.

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## Identifying Structural Coefficients and Causal Effect (continued)

-   To determine the causal effect of $X$ on $Y$, we can use the backdoor criterion to find a set of variables $Z$ to adjust for.
-   Once we obtain the set, $Z$, we can estimate the conditional expectation of $Y$ given $X$ and $Z$.
-   Averaging over $Z$, we can use the resultant dependence between $Y$ and $X$ to measure the effect of $X$ on $Y$.
-   This procedure can be translated to the language of regression.

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## Identifying Structural Coefficients and Causal Effect (continued)

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![](https://cdn.mathpix.com/cropped/2023_07_28_a103dc94c1ec860738c6g-32.jpg?height=471&width=343&top_left_y=958&top_left_x=845) Figure 3.15: A graphical model in which $X$ has direct effect $\alpha$ on $Y$
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-   To find the direct effect of $X$ on $Y$, we can use a similar procedure to the backdoor criterion, but now we need to block not only backdoor paths but also indirect paths going from $X$ to $Y$.
-   First, we remove the edge from $X$ to $Y$ (if such an edge exists), and call the resulting graph $G_{\alpha}$.
-   If, in $G_{\alpha}$, there is a set of variables $Z$ that $d$-separates $X$ and $Y$, then we can simply regress $Y$ on $X$ and $Z$.
-   The coefficient of $X$ in the resulting equation will equal the structural coefficient $\alpha$.
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## Identifying Structural Coefficients and Causal Effect (continued)

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![](https://cdn.mathpix.com/cropped/2023_07_28_a103dc94c1ec860738c6g-33.jpg?height=477&width=345&top_left_y=490&top_left_x=867) Figure 3.16: By removing the direct edge from $X$ to $Y$ and finding the set of variables $\{W\}$ that $d$-separate them, we find the variables we need to adjust for to determine the direct effect of $X$ on $Y$
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-   In the linear model of Figure 3.15, we can find the direct effect of $X$ on $Y$ by this method.
-   First, we remove the edge between $X$ and $Y$ and get the graph $G_{\alpha}$ shown in Figure 3.16.
-   In this new graph, $W d$-separates $X$ and $Y$.
-   So we regress $Y$ on $X$ and $W$, using the regression equation $Y=r_{X} X+r_{W} W+\epsilon$.
-   The coefficient $r_{X}$ is the direct effect of $X$ on $Y$.
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## Mediation in Linear Systems

-   When we can assume linear relationships between variables, mediation analysis becomes much simpler than the analysis conducted in nonlinear or nonparametric systems.
-   Estimating the direct effect of $X$ on $Y$ amounts to estimating the path coefficient between the two variables, and this reduces to estimating correlation coefficients.
-   The indirect effect is computed via the difference $I E=\tau-D E$, where $\tau$, the total effect, can be estimated by regression.

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## Summary

-   In linear systems, the total effect of $X$ on $Y$ is the sum of the products of the coefficients of the edges on every nonbackdoor path from $X$ to $Y$.
-   The backdoor criterion can be used to identify the set of variables to adjust for in order to determine the causal effect of $X$ on $Y$.
-   The direct effect of $X$ on $Y$ can be found by removing the edge from $X$ to $Y$ and finding a set of variables that $d$-separates $X$ and $Y$.
-   In linear systems, mediation analysis is simplified, with the direct effect of $X$ on $Y$ estimated by the path coefficient between the two variables, and the indirect effect computed via the difference between the total effect and the direct effect.