# Idea The average causal effect of **treatment assignment** (not actual treatment received; see [[randomized trials with non-compliance]]) on the outcome. $E\left(Y^{z=1}-Y^{z=0}\right)$ The average values of the outcome if everyone had been **assigned** to received treatment, minus the average outcome if no one had been **assigned** to receive the treatment. It is based on ==treatment assignment==, not ==treatment received==, which are two very different constructs that is related through [[non-compliance]] (or compliance). We can usually estimate the intent-to-treat effect from the observed data, where $E\left(Y^{z=1}\right)=E(Y \mid z=1)$ $E\left(Y^{z=0}\right)=E(Y \mid z=0)$ If we have perfect [[non-compliance|compliance]], this would be equal to the causal effect of treatment ([[average treatment effect|average causal effect]]). # References