In an experiment testing how soil treatments affect plant height, the presence of fungus can act as a post-treatment variable that negatively affects growth. Including fungus in the analysis can mask the soil treatment's true effect, because the treatment may improve plant growth primarily by reducing the fungus.

University of Connecticut

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Omitted Variable Bias - missing variables that could affect the experiment
Included Variable Bias - including variables that could affect the experiment
Post-treatment Bias - example of included variable bias, pg 171 example

Post-treatment bias

- Fungus is a post-treatment variable
- Shows there is a flaw in our experiment because soil has an effect on the growth through reducing fungus. Better to omit the post-treatment variable fungus 

Affects of a Post-Treatment Variable

Plant Growth Example of Post-Treatment Bias

**Directional separation (d-separation)** is a graphical criterion used to determine whether a set of variables is conditionally independent of another set in a directed acyclic graph (DAG). In a directed graph, two variables $$X$$ and $$Y$$ are d-separated by a set of conditioning variables $$Z$$ if there is no active path connecting them, meaning they are conditionally independent given $$Z$$. Understanding d-separation is essential for proper experimental design and observational analysis to prevent biases, such as post-treatment bias, which occurs when conditioning on an intermediate variable (like a mediator) blocks the true causal pathway or introduces spurious associations.

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