This video is on causal effects.

Here we're going to more formally define what we mean by causal effects.

And in particular, we'll discuss two types of causal effects.

Average causal effects, and the causal effect of treatment on the treated.

So we're going to formally define them using statistical notation,

potential outcomes.

So this video is going to be a little more technical.

And then we're also going to spend some time focusing on the difference between

conditioning and a variable or variables versus manipulating or setting variables.

So first we're going to talk about hypothetical worlds and

average causal effects.

So, let's begin by thinking of some population of interest,

which we're depicting just with a circle.

So this circle is everybody you're interested in.

It's representing a whole population of people that you're interested in.

It's sort of your target of interest.

So if you are interested in people who have diabetes and what

treatment is better, then your population would be the population of diabetics.

This circle represents the whole population that you're interested in.

And now, we're going to think about hypothetical worlds.

So remember with potential outcomes, we were thinking about hypothetical worlds

and hypothetical interventions, so we are still thinking about that now.

So we're not really thinking about data yet.

This is, we're just imagining what we would ideally like to see.

So what we would ideally like to see is two worlds, two hypothetical worlds.

So World 1, which is depicted by this grayish circle,

is that everyone in our population gets treatment A=0.

So treatment A=0 could be, it actually could be no treatment,

it could be a placebo, whatever you imagine.

So we're picturing now, this is a world where our entire population,

every single person, got treatment A=0.

Versus, some other hypothetical world where everyone received

the other treatment, A=1, so depicting that with this light blue circle.

But the most important thing here is that World 1 and

World 2 have the exact same people, it's the same population of people.

But in one case, we do one thing to them, and in another case, we do another.

And then, if we were able to observe both of these worlds simultaneously,

we could collect the outcome data from everyone in the populations, and

then we could take the average value.

So I say mean of Y in World 1.

And mean of Y in World 2.

And then that difference would be the average causal effect.

And so this is what we mean by an average causal effect.

So, it's an average in the sense that it's a mean and

it's a population sort of level average causal effect.

It's over the whole population.

We're saying, what would the average outcome be if everybody got one treatment,

versus if everybody got another treatment?

So, of course, in reality, we're not going to see both of these worlds.

But this is what we want, this is what we define as the average causal effect.

This is what we would like to see, and this is what we're hoping to estimate.

And we can define that more formally using statistical notation.

So here, the E refers to expected value and that also means that's the mean.

And here we're then taking the average of

difference of these two potential outcomes.

So, remember, Y^1 is a potential outcome if treated with A=1,

and Y^0 is the potential outcome if treated with A=0.

And so then,

we could take the average difference of that to get an average causal effect.

So this quantity,

this average causal effect is the average value of Y if everybody was treated with

A=1 minus the average value of Y if everyone was treated with A=0.

And that's exactly what I showed you on the previous slide.

Mean of Y for World 1 versus mean of Y for World 2.

So in the case where Y is binary for example,

this would just be a risk difference.

In fact, it would be a causal risk difference.