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So our notation for

the variable, that's the outcome is Y and I'll give a few examples.

So most of the time in this course,

we're going to think about situations where there's just two treatment options.

So you can think of a generically as treated versus untreated,

drug versus placebo, exposed versus unexposed.

But the field of causal inference is really more general than that where there

could be multiple levels of treatment.

So, you could have several different drugs that you could take.

You could have levels of exposure to some environmental toxin, for example.

But to keep things simple,

we're going to think about just two possible treatments that you could have.

So, one example is influenza vaccine.

So remember, our variable A is treatment.

So here, A could take two possible values.

So A could equal 1 if you receive influenza vaccine or

A could equal 0, if you don't or another example is statins.

So A would equal 1 if you take statins and A would equal 0, otherwise.

So when we are defining these variables in terms of a number,

we are assigning a number to them.

Because ultimately, we are going to be analysing data and

we need them to be numeric.

Another example is slightly more generic example is maybe you receive an active

drug versus a placebo.

So A=1 is active drug, A=0 is placebo.

What are some out coming examples?

So I work mostly in a medical study, in environmental research.

So I tend to think of outcomes as biomedical outcomes.

So, one example would be cardiovascular disease.

So Y=1 if you develop cardiovascular disease within two years and Y=0,

otherwise.

So again, here's a binary outcome.

Meaning, there's just two possible options, but

it doesn't have to be a binary outcome.

Why could be time until death?

So in this case, the outcome is continuous.

So, what they actually be some period of time.

How long do you survive?

Or how long until you die?

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Treated versus untreated and so on.

In this case, your outcome Y is an actual time.

Those are just sort of hypothetical examples.

But now, we're going to move on to what are known as potential outcomes.

So i the previous slide, what I called Y would ultimately be an observed outcome,

something we would see.

But potential outcomes, we're going to think a little more hypothetically.

So going to think of, potential outcomes is the outcomes we would observe

under each possible treatment option.

So you could think of this as before the study takes place, what could happen?

So, here is our notation and

we're going to use superscript notation to indicate potential outcomes.

So on the previous slide, I just had a Y by itself.

So, that was just an observed outcome.

Here, we're using superscript notation to indicate a potential outcome.

So Y, superscript little a is the outcome that would be observed if

treatment was set to A equals little a.

So you'll notice that remember, capital A represented treatment.

So that's our variable, but we could set it to a particular value.

In this case, we'll set it to value little a.

If we're just in binary sort of treatment world,

then little a could equal either 0 or 1.

Then each person we could think of as having two potential outcomes, Y0 and

Y1 and what I think it will become more clear what we really mean by potential

outcomes when we look at examples.

So, I'm going to walk through a few examples here.

So, suppose the treatment we're interested in is influenza vaccine.

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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.