So then in this case, Y superscript 1 is a potential outcome.

And in fact,

it's a potential on outcome under treatment which here is influenza vaccine.

So Y1 is a time until the individual would get the flu,

if they received the flu vaccine.

So, you'll notice we say would get the fluids.

It's hypothetical.

So this is before any sort of data are collected,

before even treatment's been assigned.

Just what would happen, if this person were to receive the influenza vaccine.

Alternatively, Y0 is the time until the individual

would get the flu if they did not receive the flu vaccine.

So Y superscript 0 is a potential outcome,

it's an output we would see if the person does not get the vaccine.

So, this is what we mean by potential outcomes and

this is what we mean with the superscript notation.

I'll give another example to hopefully help clarify.

So imagine that treatment is regional anesthesia, A=1 versus general anesthesia,

A=0 and this is among people who are getting hip fracture surgery.

So, our population is the population of people who are getting hip fracture

surgery and they could either be given regional or general anesthesia.

Our outcome is major pulmonary complications and

the outcome again is Y, is denoted by Y.

So, we would be interested in does regional

anesthesia lead to either higher or lower

risk of major pulmonary complications compared to general anesthesia.

In terms of potential outcomes then, we could think of

Y1 as a potential outcome if you were given, in this case, regional anesthesia.

Because remember, we defined A=1 as regional.

So Y1 then, we could define that as equal to 1 if major pulmonary

complications occurred, if given regional anesthesia.

And we could say that the potential outcome Y superscript 1 would equal 0,

if pulmonary complications did not occur.

So in this case, the outcome is binary.

You can take on two values.

Remember in the previous example, Y was time, time until the event.

Here, Y is a binary kind of outcome.

It could either be one, if pulmonary complications and zero, otherwise.

And then the potential outcome Y superscript 1 is referring to pulmonary

complications or not, if they personal was given regional anesthesia.

Alternatively, if the person would be given general anesthesia,

they would have an outcome Y superscript 0.

And again, that would take value 1 if major pulmonary complications occurred and

it would equal to 0, otherwise.

So the Y1 and Y0, those are the potential outcomes in this setting.

So now, we're going to move on to counterfactuals.

And counterfactuals as we'll see are very related to potential outcomes and you'll

see these terms used interchangeably, but they're slightly different and

we'll talk about what they are and how they're different.

So counterfactual outcomes, we could think of as outcomes that would have

been observed had treatment been different.

So now we're thinking of data have already been collected,

things have already taken place.

But what would have happened under some hypothetical alternative scenario?

So if my treatment actually was A=1,

then my counterfactual outcome is Y0, is Y superscript for 0.

Because that's the outcome that what have occurred had my treatment been A=0.

But my treatment wasn't actually A=0, it was A=1.

Alternatively, if my actual observed treatment was A=0,

then my counterfactual outcome would be Y1, Y superscript 1.

It would be the outcome I would see had my treatment been different,

had my treatment been A=1.

So, let's get back to an example.

Did influenza vaccine prevent me from getting the flu?

So imagine that and you can see by the way, I worded that question that I'm

implying that you did receive the influenza vaccine.

So imagine that that's already known, A=1.

A has a value of one for you.

You did receive the influenza vaccine, but

did it actually prevent you from getting the flu.

So what actually happened, you received the vaccine and you did not get sick.

Your actual exposure was A=1.

Your observed outcome in this case, Y is equal to Y superscript 1.

That's the outcome that would have occurred, if you were given the vaccine.

And since you were given the vaccine,

we can think of that as your observed outcome.

So, that's your observed outcome.

It's the outcome that you would've seen if you were given the vaccine.

And since you were given the vaccine, it is your observed outcome.

But, what would have happened is contrary to fact.

What would have happened contrary to fact?

And we can think of that in this case,

as had I not gotten the vaccine, would I have gotten sick?

So your counterfactual exposure, in this case is A=0 and

your counterfactual outcome is Y0.

So hopefully here, it's clear that the difference between an observed outcome and

a counterfactual outcome.

So an observed outcome is what we actually do see,

given whatever your actual treatment is, and your counterfactual outcome

is the one we would have seen under some hypothetical alternative scenario.

So now, we can think about the link between Potential Outcomes and

Counterfactuals.

So before the treatment decision is made, any outcome is a potential outcome.

So, Y0 and Y1.

So, and that's where the word potential is coming from.

You have the potential for Y0 and you have the potential for Y1, and

we don't see either of those until what treatment decision is actually made.

So before treatment decision's made, before you're given a influenza vaccine,

you had these two hypothetical outcomes that we imagine could occur in the future.

But after the study, there's an observed outcome which is the one that corresponds

with the treatment you did receive and there's a counterfactual outcome.

And you'll notice here, I'm using this notation Y superscript A capital for

the one we observed and that's because we observed treatment capital A and

I'm using for counterfactual outcome Y superscript 1-A.

And that's because here, we're in binary treatment world is what we're thinking of.

So if capital A is equal 1, then 1 minus capital A is equal to 0.

Whereas if capital A is equal to the other value, then you would get the opposite.

So why superscript capital A is the one you would always observe and

Y superscript 1-A is the one you don't observe, but

we can think of it in a counterfactual sense.

What would have happened?

So, counterfactual outcomes are assumed to be the same as potential outcomes.

So they sort of have this different motivation behind them and

different sort of reasoning about them, but the ones that we hypothetically

imagine occur prior to any treatment decisions that potential outcomes.

We assume that these line up with what we would observe counterfactually.

So the potential outcomes Y0 and Y1, we assume are the same

sort of values as Y0 and Y1 that are counterfactuals.

So, people use these terms interchangeably.