In many experiments, we can't force people to act. We can only randomize an *encouragement*. Some who are encouraged won't act, and some who *aren't* encouraged will act anyway. This is **Two-Sided Noncompliance**, and it makes finding the true effect of a treatment tricky.
To solve this, we imagine everyone belongs to one of four hidden groups (**Principal Strata**) based on how they'd react to being encouraged or not.
✓
Take the treatment *if and only if* encouraged. They follow directions.
→
Take the treatment *no matter what*. They are self-motivated.
✗
Will *never* take the treatment, even if encouraged.
?
Do the *opposite* of what they are encouraged to do. (Assumed to be rare).
With noncompliance, we must choose our question carefully. We can measure the effect of the *offer* or the effect of the *treatment itself*—but only for one group.
The effect of the ENCOURAGEMENT.
This compares everyone assigned to the encouragement group vs. everyone assigned to control. It's a real-world policy effect, but it's "diluted" by people who don't follow their assignment.
The effect of the TREATMENT for Compliers.
This estimates the true effect of the treatment itself, but only for the Compliers—the group whose behavior was actually changed by the encouragement.
A classic example used a random lottery number as an "encouragement" (instrument) for military service. This allows us to estimate the causal effect of serving in the military on later-life mortality.
This is the Intention-to-Treat effect on the outcome ($ITT_Y$). It shows the raw effect of the encouragement.
This is the Intention-to-Treat effect on participation ($ITT_D$). It shows how much the encouragement changed behavior.
The logic to find the true effect on Compliers (CACE) is to simply "inflate" the diluted ITT effect by the compliance rate. This is called the Wald Estimator.
Effect on Mortality ($ITT_Y$)
Effect on Service Rate ($ITT_D$)
Effect for Compliers (CACE)
Plugging in the numbers from the draft lottery reveals the true impact of military service for those induced to serve.
Intention-to-Treat (ITT) Effect
+0.6%
Being draft-eligible increased mortality by 0.6 percentage points. This is the diluted policy effect.
Complier Average Causal Effect (CACE)
+4.0%
For Compliers (men who served *because* of the draft), military service increased mortality by 4.0 percentage points.