In the previous module we looked at various kinds of randomized experiments,

the assignment mechanism.

And we looked at how randomization based inference allows you to test hypotheses,

and estimate, as well, the sample average treatment effect.

So, we assume some things then that get us back on the same page.

The ith of the experimental units had two potential outcomes, yi(0) if you

were selling treatment 0, and y1 if assigned to treatment 1.

However, we only looked at one of these two because, if you get one,

we can't see the other.

So, could think of it as missing data problem with half the data missing.

The potential outcomes in that module are regarded as fixed constants.

And randomness got introduced through the random assignment of treatments to

subjects.

Now, in this lesson, we're going to extend the randomization based inference

to the finite population average treatment effects.

And, in this case,

we look at the n units as a random sample from a population of capital N units.

The potential outcomes are still fixed constants, okay?

But, in addition to mission half the potential outcomes for

the N units that we do look at, we don't observe either potential outcome for

the remaining units, the ones that aren't in our sample.

So now the randomness has two sources,

the random assignment of treatments to the n subjects, little n subjects and

the random selection of the little n subjects from the population of capital N.

So the estimator the difference in the sample averages is y-bar

1 minus y-bar 0 from the previous module.

That remains unbiased for the finite average treatment effect,

but the randomization based approach to hypothesis testing obviously isn't going

to work anymore.

So let's talk about estimating the population treatment effect.

The n units are simple random sample from the population of capital N units.

So that means that each possible sample size N,

taken without replacement from the capital N the units is equally likely.

So, lets let Ti denote the random variable that takes the value of 1,

if unit i appears in the sample, and 0, otherwise.

Now, because of the way we're taking simple random sample,

each Ti has an expectation of little n over N.

Due to random sampling, the sampler average treatment effect from

before can be rewritten as n inverse times this in the second line.

And then now, you notice in the next equality following that,

we're summing over all the units but multiplying by Ti.

So that way, it's equivalent to the other.

And now, this is a random variable because the presence of Ti.

Okay, now the sample average treatment effect is on bias for

the finite average population treatment effect.

Okay, and so we going to take this and just show you immediately why that's true.

And down below we see that the first equality,

the second equality all we done is put the n outside and

put the expectation inside, which we can do as you know.

And then also now remember that little ys are constant and so

the Ti is only random variable but the expectation of constant times the random

variable is the constant times expectation of the random variable,

and that what we get it's give us a constant line.

Okay so now we want to talk about estimating the finite

population average treatment effect.

So for every sample, the difference between the sample means is unbiased for

the sample average treatment effect.

And the sample average treatment effect is unbiased for

the expected value of Y1- Y0, then over the distribution induced by the sampling.

So we can write this as an equation too.

So first we're taking the expectation over the randomization distribution and

then we're taking the expectation over the sampling distribution.

So as in the case of the sample average treatment effect, the variance, or

the estimator is tedious and depends on the unknown unit effects.

The result given earlier for

the sample average treatment effect that we saw is just a special case.

So here now the variance of the difference between means has these three components

and the first one you can see has to do with the difference of the y0s

from their true value the second from the difference of the y1s from the true value.

And the third, the differences of the unit effects from their values.

Okay, now again, if the treatment effects are constant,

you can see that the last term is going to be 0.

But otherwise,

we can get conservative estimate of the variance when we ignore this last term.

So now we can do hypothesis test and confidence intervals for the population,

finite population average treatment affect.

And we can use a normal approximation for

the distribution of the difference between sample means.

And we get the familiar kind of result that expression there under the null

hypothesis of no treatment effect, no average treatment effect will be

approximately in normal 0,1 random variable.

All right, I want to switch gears a little bit now.

Now I'm sure you're more familiar with the model based approach inference in which

potential outcomes are random variables.

And throughout much of the remainder of the course,

that's the approach we're going to take.