So the potential value of treatments are in this case, A0 and
A1, where A0 is the treatment that you would receive if you were
randomized to the control condition.
Or in other words, if the instrument was Z = 0, or
if you were randomized to not receive encouragement, essentially.
So we're imagining that everyone has this potential value, A0.
So it's the treatment that you would receive
if you had been randomized to the Z = 0 group.
Whereas A1 is the value of treatment that you would receive
if you were randomized to the Z = 1 condition.
So if you had been assigned treatment.
Right, so these are potential values of treatment.
And we imagine this exists for everybody, we just don't necessarily see them,
and we certainly don't see them all for everybody.
But what we can do is we can take this pair A0 and A1, and
then classify people or label people based on their pair values, okay?
So if we look at this first row for example.
So the first row is people who,
if they were randomized to the control condition, they would not take treatment.
So that's A0 equals 0, means if they were randomized to the control condition,
they would not take treatment.
But for these individuals also, if they were randomized to receive the treatment,
so if they were in the Z = 1 arm, they still would not take the treatment.
So we'll call these never-takers, and
that just of course means that the never take the treatment.
So no matter what you assign them, whether you assign them to the control group or
the treatment group, they're just never going to take the treatment.
So they could be just people who just are not interested in that treatment for
whatever reasons.
And you can do the same thing with other possible contrast.
So you have people who, if they were assigned the control condition,
they don't take treatment.
But if they were assigned to treatment, they do take it,
and we'll call them compliers.
So these are people who are doing what they're assigned to do.
And then we also have two other groups.
So there's the defiers who do the opposite of what they're told.
So if you assign them the control condition, they take the treatment.
And if you assign them the treatment condition, then they don't take treatment.
And finally, there's the always-takers and
they just always take treatment no matter what they're assigned.
So this sort of layout is also what's known as part of the Rubin causal model.
And it's also a general kind of approach that's known as principal stratification.
So we'll briefly talk about each of these subpopulations.
But one thing to note is to really think of these as subpopulations of people.
So think of never-takers as one group of people.
Again, so these are people who no matter what they're assigned,
they are not going to take treatment.
So encouragement for this group does not work.
So one thing to note about this population is that,
if we knew who this population was, we wouldn't be able to
learn anything about the causal effect for treatment for them, right?
Because there would be no actual variation in treatment received.
So in this population, they never take treatment, so
we would never observe an outcome under treatment for anyone in this group.
All right, so there's no way from data that we could
learn about the causal effect of treatment for this population,
at least without making some other strong assumptions.
So in general, we don't have variability in treatment in this group.
So we don't have much hope of learning
about the causal effect of treatment for that group.
Then we have this population of compliers.
So they take treatment when they're encouraged to, and they don't otherwise.
So treatment received for this group is always equal to treatment assigned.
So in this group, we get variation in treatment received.
So in this population, some people will take the treatment and
some people won't, and it's entirely based on this coin flip.
It's based on randomization.
So this is a population we have a lot of hope for learning about a causal effective
treatment since we're directly randomizing Z treatment assignment.
And they do what they're told.
So we actually are directly randomizing treatment received in this population.
So this is a population that we hope we can learn something about.
Then we have defiers, and they do the opposite of what they're encouraged to do.
So in this group you could think of treatment received as randomized, but
just sort of in the opposite way than you intend.
So there's still sort of randomization happening here but
they're just doing the opposite of what they were told.
So in principle,
we also could hope to learn about the causal effect of treatment in this group.
As we'll see later,
that this is a group that we tend to think would either be very small or not exist.