Author: Aboozar Hadavand, Minerva Schools at KGI

Let's get our mind set on our goal. We are trying to find the treatment effect in a randomized experiment in which not every subject that is assigned to the treatment takes the treatment and not every subject that is assigned to the control takes the control. In other words, we have what we previously called non-compliance. In randomized experiments with non-compliances, treatment assigned isn't the same as treatment received.

Even if we're talking about randomized experiments, the treatment effect isn't simply the difference between the potential outcomes under treatment and no treatment. Because treatment received isn't randomized.

A little bit of warning. This short writing has a lot of math notations but we're pretty sure you'll survive 🥴

One thing is easy to estimate. We can easily estimate the causal effect of treatment assignment or what we called the intention-to-treat (ITT) causal effect:

$$ \text{ITT } = E(Y^{Z=1} - Y^{Z=0}) = E(Y|Z=1) - E(Y|Z=0) $$

Let's see how we can go from ITT to treatment effect (effect of treatment received). Let's use the following acronyms:

• Always-taker (AT)
• Complier (CM)
• Never-taker (NT)
• Defier (DF)

Because we have four compliance groups, we can break down $E(Y|Z=1)$ into the effect for each of the groups:

$$ E(Y|Z=1) = E(Y|Z=1, AT) \Pr(AT) + E(Y|Z=1, CM) \Pr(CM) + E(Y|Z=1, NT) \Pr(NT) + E(Y|Z=1, DF) \Pr(DF)  $$

Because we assumed there is no defiers, then the last term is equal to zero:

$$ E(Y|Z=1) = E(Y|Z=1, AT) \Pr(AT) + E(Y|Z=1, CM) \Pr(CM) + E(Y|Z=1, NT) \Pr(NT)  $$

Because for always-takers and never-takers, the assignment, $Z$, doesn't matter and $E(Y|Z=1, AT) = E(Y|AT)$ and $E(Y|Z=1, NT) = E(Y|NT)$. Thus, we can simplify the expression further like below

$$ E(Y|Z=1) = E(Y|AT) \Pr(AT) + E(Y|Z=1, CM) \Pr(CM) + E(Y|NT) \Pr(NT)  $$

Similarly, for $Z=0$ we have:

$$ E(Y|Z=0) = E(Y|AT) \Pr(AT) + E(Y|Z=0, CM) \Pr(CM) + E(Y|NT) \Pr(NT)  $$