In ideal experiments, everyone follows instructions. In the real world, they don't. This gap between who is **assigned** a treatment and who actually **receives** it is called noncompliance. Here’s how we draw valid conclusions anyway.
To understand noncompliance, we classify people into four hidden groups based on how they would behave. This is **Principal Stratification**.
✓
Take the treatment if, and only if, they are assigned to it. They follow the rules.
✗
Will not take the treatment, no matter what group they're in.
!
Will get the treatment, no matter what group they're in.
?
Do the opposite of their assignment. These are rare and usually assumed away.
The Key Simplification: In **one-sided noncompliance**, the control group can't get the treatment. This means Always-Takers and Defiers are impossible, leaving us with just Compliers and Never-Takers!
With noncompliance, we can't just compare treated vs. untreated. We must choose the right causal question.
The effect of OFFERING the treatment.
This compares everyone assigned to treatment vs. everyone assigned to control. It's a real-world policy effect, but it's "diluted" because many people in the treatment group don't actually get treated.
The effect of RECEIVING the treatment.
This measures the true effect of the treatment, but only for the sub-group of Compliers. It adjusts for the dilution to estimate the treatment's true efficacy on those who use it.
A classic study in Indonesia assigned villages to receive Vitamin A supplements. Not everyone took them, creating a one-sided noncompliance problem. Let's see the data.
Comparing the two randomly assigned groups shows the raw effect of the program offer (ITT).
The difference in who actually received the supplement tells us the share of the population who are Compliers.
The logic to find the true effect on Compliers (CACE) is surprisingly simple. We just "inflate" the diluted ITT effect by the compliance rate.
Effect of Program Offer (ITT)
(Difference in mortality rates)
÷ Compliance Rate
(Difference in take-up rates)
= Effect for Compliers (CACE)
(True effect of the treatment)
Plugging in the numbers from the Vitamin A study reveals the program's impact.
Intention-to-Treat (ITT) Effect
-0.26%
Offering Vitamin A reduced mortality by 0.26 percentage points across the whole group.
Complier Average Causal Effect (CACE)
-0.32%
For those who took it, Vitamin A reduced mortality by 0.32 percentage points. The effect is larger once "dilution" from non-compliers is removed.