just for simplicity I'm going to assume there are no always
takers in the population.
That we only have never takers and compliers, that means that over
here in the control group we don't have this first group.
There are no always takers. Everybody is either a never taker or a complier, and
over here in the treatment group in this part we also don't have
any always takers.
So that means that we know these people are compliers, and we know
these people are never takers.
But the problem is over here in the control group we don't know
whether these people are nevertakers or compliers.
So let's see how we get the local average treatment effect.
Well remember how it's defined.
The local average treatment effect is the difference between the
average potential outcome of compliers when they do receive treatment
minus
Average potential outcome for complyers when they don't receive
treatment, which I've written here.
It's ust a difference of those two averages for compliers.
The potential outcome when they receive treatment minus the potential
outcome when they don't.
Now this first number, the average potential outcome for compliers
when they do receive treatment.
We learn directly from the data.
We just go over here to our treatment group and because there are
no always takers, we know that these people are compliers and they
received treatment.
So we look at the average outcome, say in the organ healthcare experiment,
the average cholesterol
for the people in the treatment group who got treated.
That's gonna tell us the average potential outcome for compliers
when they get treated.
That's the first part that we need to compute.
The next part is a little more complicated, if we went over
here to the control group and we looked at the average outcome,
the average cholesterol
for these people if there were no never takers we'll
be done, that will be that second thing we needed to know.
That's exactly what happens in a randomized experiment when there's
perfect compliance, when there are no never takers.
The problem is the average cholesterol for this group of people
is a mixture of two averages.
It's the mixture of the outcomes for compliers and the outcomes
for the never takers.
But we only care about the compliers here, so these never takers
are a problem.
How are we going to deal with that?
Well over here I've written out an equation that describes that
mixture.
So on the left-hand side of this equation, I wrote the average outcome,
say the average cholesterol in the control group, that means the
average of these people.
Remember we're assuming there're no always takers, so these
people aren't here.
This is everybody in the control group.
So the average of this outcome is going to be an average of the
never takers and the compliers.
So the average outcome in the control group equals the average potential
outcome for the compilers when they don't receive treatment,
because nobody here receives treatment, that's what we want to know
multiplied by the proportion of the compliers in our population
plus the average outcome for the never takers when they don't receive
treatment multiplied by the proportion of never takers.
So, this is the number we want to know, if we can get that we're
done.
This number here, on the left-hand side we get directly from our
data, we just take the average outcome in the control group.
This number here, the proportion of compilers that we can get directly
from our data in the treatment group.
Because over here, we see some people are treated and we know that
they're compliers.
Some people are not treated and we know that they're never takers.
So we just look at what percentage of people
in our assigned to treatment group or actually getting treatment.
That's the percentage of compliers in the data assuming that there
are no always takers.
And remember it's really important that this is a random sample
of people so that percentage reflects the true proportion of compliers
in our population and also reflects the proportion of compliers
over here in the control group because of random sampling and random
assignment to treatment and random assignment to control.
So, we know the proportion of compliers, similarly we also know
the proportion of never takers because assuming that only people
are either compliers or never takers.
If I know how many compliers there are, I also know how many never
takers there are.
So finally what about this number? The average potential outcome
for never takers when they don't receive treatment.
Well that I can get by looking at my scientific
And then looking at the average outcome for these people, the people
who don't get treated.
Now why is that?
Well this is a group of never takers, it's a random sample from
the population and they don't get treated so the outcome I see,
their cholesterol is equal to their potential outcome when they don't
get treated.
So this average is exactly the average potential outcome for never
takers when they are not treated, and that number is precisely this
number here.
So remember the local average effect is this number the average outcome
for compliance when they do receive treatment minus this number
the average potential out come for compliance when they don't receive
treatment, and I've got this equation here where I've just shown
that this number here on the left hand side is known from my data 106 00:05:57,910 --> 00:00:00,000 this number here is known from our data, this number here is known from our data
and this number is not from our data, only this number is unknown.
But now we just have one equation and one unknown and we can solve
for this number here and therefore we solve this equation to get
this subtract it from that and that gives us the local average
treatment effect.
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