Post-Treatment Variable

Post-treatment variables are variables that confound the “controlling for endogenous factors by including them in the regression” technique. They can cause problems is two ways:

Mediator §

Motivation. Suppose regressing Earnings vs Tutoring:

$${\text{Earnings}}_i={\beta }_0+{\beta }_1\text{isTutored}+{ϵ}_i$$

But secretly, there was a correlation where: Therefore, $\mathbb{E}[\widehat{{\beta }_0}]={\gamma }_1+\alpha {\gamma }_2$ Now, simply controlling for reading scores like this:

$${\text{Earnings}}_i={\beta }_0+{\beta }_1\text{isTutored}+{\beta }_2\text{readingScores}+{ϵ}_i$$

Is not effective, because there’s a multicollinearity between $\text{isTutored}$ and $\text{readingScores}$. In this case, $\mathbb{E}[\widehat{{\beta }_1}]={\gamma }_1$, and this doesn’t capture the $\alpha {\gamma }_2$ portion.

Confounder §

Collider Bias §

Collider bias

A stupid model:

$$Y_i={\beta }_0+{\beta }_1{X}_{1i}+{\beta }_2{X}_{2i}$$

will have biases:

• $\mathbb{E}[\widehat{{\beta }_1}]={\gamma }_1+\alpha \frac{{\rho }_2}{{\rho }_1}$
• $\mathbb{E}[\widehat{{\beta }_2}]={\gamma }_2+\frac{{\rho }_2}{{\rho }_1}$

Example. Suppose we are regressing: Flu vs. Car accident. We also try to control for being being infected while being hospitalized:

$${\text{Flu}}_i={\beta }_0+{\beta }_1\text{hadAccident}+{\beta }_2\text{isHospitalized}+{ϵ}_i$$

Suppose, however, reality went like this:

1. Car accidents lead to hospitalization, but don’t cause the flu
2. Fevers lead to both hospitalization and cause flu testing Then:

• $\mathbb{E}[\widehat{{\beta }_1}]={\gamma }_1-\alpha \frac{{\rho }_2}{{\rho }_1}$ i.e. accident and flu is
• $\mathbb{E}[\widehat{{\beta }_2}]={\gamma }_2+\frac{{\rho }_2}{{\rho }_1}$