[MUSIC] Remember that each person has a unit level treatment effect or unit level causal
effect, and that's the difference between their potential outcome if they have the treatment
minus their potential outcome if they didn't have the treatment, and the average treatment
effect is just the average of all of these unit level causal effects.
Take each person, look at their unit causal effect, and you take the average.
Now, we can also define the Conditional Average Treatment Effect as the average for just some
of those people.
For example, the conditional average treatment effect for men, where you just look at the
average unit causal effects for men, or the conditional average treatment effect for women,
where you just look at the average for women.
So over here, I have an example in this population there are four people: person one, two, three
and four.
There are two men and two women.
And here I've got their potential outcomes.
So in this column is their outcome if they got treated.
This guy would have a 40, this guy a 20, this woman a 10 and this person a 30.
Over here is their potential outcome if they don't get treated.
30, 20, 15, 30.
So this first person, if he is treated, he gets 40.
If he isn't treated, he gets 30.
So his unit causal effect is 10, 40 minus 30.
The second person gets 20 with treatment and 20 without treatment - it actually doesn't
matter whether they are treated or not, they have the same outcome.
So their unit causal effect is 0, 20 minus 20.
The third person is 10 with treatment, 15 without, so their unit causal effect is -5:
10 minus 15.
And finally our last person has 30 with treatment, and 30 without.
So she also has a unit causal effect of 0.
Now remember, later on, we are going to deal with the problem that in data we never see
both of these numbers for the same person, because each person can only either get treated
or not get treated.
So one of these numbers will always be hidden for us, depending on whether these people
are treated or not.
That's called the fundamental problem of causal inference that I discussed earlier.
And that's going to make it very difficult because we're not going to be able to see
the unit causal effects from the data, and we're going to try to figure out how can we
learn about these things when we have data that only shows us one of the numbers from
these columns.
But before we get to that, we just want to understand the kinds of numbers we can learn
if we observed both of these columns.
So one of those numbers is the average treatment effect, which is just the average of all these
unit causal effects.
So in this example, the average treatment effect is equal to the sum of these: 10 plus
0 plus -5 plus 0, divided by four, because there are four people in the population that
we're averaging over.
So it's going to equal 10 minus 5 over 4.
5/4, that's our average treatment effect.
Now let's look at the conditional average treatment effect for men.
This is the same thing as the average treatment effect, but only looking at the men and the
population.
So we throw out the women’s data.
So that's going to be 10 plus 0, divided by two because we have two people that we're
averaging over, and that's going to be 10/2, or 5.
So finally let's look at the conditional average treatment effect for women, and that's going
to be just like with the men, we're going to look at the average only just the women
and throw out the men’s data.
So that's going to be -5, plus 0, divided by two, because there are two people.
So that's going to be -5/2.
So we see the average treatment effect is 5/4 - positive - so usually we conclude treatment
must be good.
We look at men, we see it's even bigger, so the treatment is very good for men, but we
would say for women the conditional average treatment effect is negative, so we would
say actually the treatment is bad for women.
So what we see here, and also from unit causal effects, is treatment effect heterogeneity.
In this example, the treatment affects different people differently.
This person benefits, this person is hurt, and for both of these people it doesn't matter
whether they are treated or not.
Now when we go to the data, and later on in these models, we're going to see how we learn
these numbers from real data.
Here I computed it, but that's because I assumed I knew both columns, and we're not going to
know both columns.
So how do we learn these numbers?
And there are batch of other kinds of averages you can take, and we will talk about later
on.
For example, there's something called LATE, a local treatment effect.
We will talk about this later — no pun intended.
So those are Average Treatment Effects and Conditional Average Treatment Effects.
[MUSIC].