0:33

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.

6:24

And just as a reminder, if there is unmeasured confounding,

then we can't average or marginalize overall the confounders, right?

because some of the confounders are unobserved, all right?

So we can't sort of condition or match on these confounders and

then average over it, because we don't observe them all.

So then I'm emphasizing this here because the causal methods that we focused on in

other videos really does focus on causal effects in the whole population, right.

But if you have unmeasured confounding, it would be very difficult to

actually obtain causal effects for the whole population,

considering really the way to do that is to average over all confounders.

So instrumental variable methods are not going to focus on

the average casual effect for the whole population.

So remember instrumental variable methods, we're hoping that they can still

be used to estimate a valid casual effect even if there's unmeasured confounding.

But as I just mentioned, if you have unmeasured confounding it's really, you

shouldn't expect to be able to estimate a causal effect for the whole population.

So instrumental variable methods are not going to do that.

Instrumental variable methods are going to try to estimate a causal effect locally.

So what we'll call local average treatment effect, and

I'll define what we mean by that.

8:31

So now what we need to do though is think about who is the subpopulation of people?

So I'll just clear this off a little bit so you can see better.

So what you'll notice is that A0 = 0, and A1 = 1.

That's actually just the population of compliers, right?

So we're looking at the subpopulation of people who,

if they were prescribed treatment or if assigned treatment, they would take it.

So that's A1 = 1.

But if they were assigned to the control condition, then they wouldn't take it,

that's A0 = 0.

We're conditioning on that.

So we're saying,

restrict to the subpopulation of people that if assigned treatment, take it.

If not assigned treatment, don't take it.

Well that's a subpopulation of compliers.

So what we're really looking at here is the average causal

effect of treatment assigned on the population of compliers.

So this is what we mean by local.

It's local in the sense that it's an inference about a subpopulation.

So that's what we mean by local as a subpopulation.

The subpopulation happens to be compliers.

And it actually turns out, if you restrict to the subpopulation of compliers,

I can do this little thing here where I go from indexing by Z to indexing by A.

So now I'm actually comparing potential outcomes based

on treatment received, as opposed to treatment assigned.

And so as a thought experiment, or just something to think about, I'm asking why?

Why was I able to just swap what I'm indexing the potential outcomes by?

So I went from Z = 1 to A = 1, and from Z = 0 to A = 0.

So now, this would be a good place to pause the video and

think about it for a minute or two and have your answer.

I'll assume you've done that.

Well so now,

remember that we're restricting to the subpopulation of compliers.

So compliers are people who do what they're told.

So Z = 1 for compliers is always going to correspond exactly to A = 1.

And for a complier Z = 0 is always going to correspond exactly to A = 0,

meaning Z = 0 implies A = 0 for compliers.

Z = 1 implies A = 1 for compliers.

So as long as I'm restricting the subpopulation of compliers,

I can use Z and A interchangeably.

So as I mentioned, this is a causal effect.

It's contrasting counterfactuals in a common population, and in fact,

it's a causal effect of treatment received, right?

Because you see that I have potential outcomes index by treatment received, so

it is a causal effect of treatment received.

But it's in a subpopulation so its local.

It's only looking at compliers.

And this is sometimes refered to as the complier average causal effect.

So sometimes this is refered to a local average treatment effect.

That's kind of the general term for causal effects in subpopulations.

But in this specific situation where you are in a randomized trial with

noncompliance, then this is referred to as the complier average causal effect or

sometimes abbreviated CACE.

So you'll notice that what I'm saying in an instrumental variable analysis,

the target of inference is the cause and

effect of treatment received among compliers.

So you'll notice that we're not talking about defiers, always-takers, or

never-takers.

So we're not actually, in instrumental variable analysis,

going to make inference about those populations.

And we'll discuss that more in other videos.

But hopefully it's especially

clear why we won't be making inference about always-takers and never-takers.

And that's because for those groups,

there's no variation in treatment received.

And so we shouldn't expect to learn anything about

the causal effect in those subpopulations.

13:27

And then the other one is a potential treatment if assigned to treatment.

And so you'll notice if you're in the Z = 0,

A = 0 group, we observe this one, right?

So this is somebody who was actually assigned to the control condition.

And in this case, they did what they were told, they didn't actually take treatment.

So we know that A0 is equal to 0.

But we don't know what they would have done, had they been assigned treatment 1.

So what all we know about this group of people is that they're either never-

takers, right?

They might be people who just would never take the treatment regardless of what they

were assigned.

Or they might be compliers, they might always do what they're told.

So we've narrowed it down to two choices, but we don't know for

sure which one they are.

So you can do to have a similar exercise with these other cases.

So somebody who was assigned the control condition but

actually took treatment, right?

Well, we would know that if assigned Z = 0, they take treatment,

which means they're either always-takers or defiers.

So if they're always-takers that would mean A1 would equal 1,

if they were defiers that would mean A1 equals 0.

But we don't know which one they are.