- [[potential outcomes framework example]], [[bias]] # Idea **Association (or correlation)** is measured by the following (**observed** outcomes): $E[Y \mid T=1]-E[Y \mid T=0]$ which is identical to the following (**observed difference in outcome between treated and control**): $E\left[Y_{1} \mid T=1\right]-E\left[Y_{0} \mid T=0\right]$ which is equivalent to saying the mean of the treated ($T = 1$) minus the mean of the untreated ($T = 0$). ## [[potential outcomes|Potential outcomes]] (not *observed* outcomes!) But **causation** is measured/expressed via [[potential outcomes]]: $E[Y_1 - Y_0]$ ## The counterfactual $E[Y_0|T = 1]$ relates causation and bias What is [[bias]]? We can derive "[[bias]]" via a simple algebraic manipulation: Subtract and add $E\left[Y_{0} \mid T=1\right]$, which is a very important **counterfactual (unobserved) outcome—what would have been the outcome of the treated, had they not received treatment**. $ E\left[Y_{1} \mid T=1\right]-\underbrace{\left(E\left[Y_{0} \mid T=1\right]+E\left[Y_{0} \mid T=1\right]\right)}_{\text {subtract and add the counterfactual }}-E\left[Y_{0} \mid T=0\right] $ Then regroup the terms, merge some expectations, and we get [[average effect of treatment on the treated]] and [[bias]]: $ \underbrace{E\left[Y_{1} \mid T=1\right] - E\left[Y_{0} \mid T=1\right]}_\text{merge expectations} + \underbrace{E\left[Y_{0} \mid T=1\right]}_\text{counterfactual outcome for treated} -\underbrace{E\left[Y_{0} \mid T=0\right]}_\text{observed outcome for control} $ $ \underbrace{E\left[Y_{1}-Y_{0} \mid T=1\right]}_{ATT}+\underbrace{\left\{E\left[Y_{0} \mid T=1\right]-E\left[Y_{0} \mid T=0\right]\right\}}_{bias} $ If there is no [[bias]], then [[when is association equals to causation|association equals causation]]. # References - [01 - Introduction To Causality — Causal Inference for the Brave and True](https://matheusfacure.github.io/python-causality-handbook/01-Introduction-To-Causality.html#bias) - [[potential outcomes framework example]] - [[Angrist 2008 mostly harmless econometrics]] (p13-15)