[MUSIC] Remember CATE, the Conditional Average Treatment Effect?
This is just the average effect for a subset of units.
Here I'm going to introduce a new kind of average effect, and to do that I've got a
table with eight data points.
I've got eight people: one, two, three, four, five, six, seven, eight.
Four of these people are male, and four of them are female.
And over here, I've got a column telling me in my data was this person treated - yes - or
not treated - no.
So the first three people are treated, the second - the fourth person is not, and for
the females, the first person is treated and the other three people are not.
Now over here, I have my columns of potential outcomes.
So the first column is what this person would get, what outcome would they have, if they
were treated.
So the first person was 80, 75, 85, 70.
And then similarly for the females.
The next column tells me the potential outcome of what would happen to them if they weren't
treated: 60, 70, 80, 60, etc.
Now we're never ever, ever going to see both of these numbers simultaneously for the same
person in a dataset.
That's the fundamental problem of causal inference: it’s that you can't observe the counterfactual
outcome, what didn't happen.
So in real data, for say these people, we will only ever observe this number - what
happened to them when they were treated - because they were treated.
We would not observe these numbers.
For this person, this person was not treated, so we were to observe this number - what happens
to them when they are not treated - but not this number.
But for the purpose of defining what effect we want to go for, I've just written out the
full potential outcomes even though we can't actually observe them in the data.
Now once I got these outcomes that you get with the policy and without, I can compute
the unit level causal effect, which is just the difference in these two numbers.
So for the first person, the unit level causal effect is 20: 80 minus 60.
So this is the effect that the person would get if they went in the policy, a 20 point
increase in their outcome, and similarly for everybody else.
So remember back from earlier?
The average treatment effect is just the average of these eight numbers, it’s the average
unit causal effect for everybody in the population, for all of these people.
The conditional average treatment effect for men is just the average of these four numbers:
just the average unit level causal effects only for the men.
And similarly the conditional average treatment effect for females is just the average of
these four numbers: the average of the unit level causal effects only for females.
So that's what we did before.
Now what's new?
We can think about the treated people as a particular subset of people: the people who
were treated.
And we can define the average treatment effect for those people.
So the Average Treatment on the Treated, ATT, is just going to be the average unit causal
effects for the people who are treated.
So this person, this person, this person, that's 20 plus 5, plus 5 and then this person
here - this one female who is treated - plus 5, divided by 4 people.
So that's the average treatment on the treated, the average of these three people and this
person.
We are just thinking of this variable telling us whether someone was treated or not is just
another variable on our dataset, just like sex is here, male or female,
So that's Average Treatment on on the Treated.
Now under the unconfoundedness assumption or other assumptions, even though you don't
observe both of these columns, you're still going to be able to compute this number ATT
from your data, and as well as ATE, or CATE, or other parameters of interest.
[MUSIC].