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Hi, inverse probability of treatment weighting is a method for

estimating causal effects.

In this video, our objective will be to gain an intuitive

understanding of inverse weighting by relating it to matching.

As a motivating example, we'll focus on the situation where there's just

a single binary confounded that we'll call X.

So, there's just a single variable that we need to control for.

Now, let's also imagine that among people who have this confounder value equal to 1.

So, X = 1.

The probability of treatment is equal to 0.1.

So among people with X = 1, only 10% of them would receive treatment.

What this means is that the value of the propensity score for

people with X = 1 is equal to 0.1.

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So everybody who has a value of X = 1 has a propensity score, 0.1.

Now let's look at the other group of people,

those are individuals who have X = 0.

Here are the probabilities that A = 1 which is a probability of treatment,

given that X = 0 is equal to 0.8.

So in other words, the subpopulation of people who have X = 0,

their propensity score is 0.8 which means that 80% of

people who have X = 0 will receive the treatment.

So people with X = 1 will unlikely to receive the treatment

where is people with X = 0 are very likely to receive it.

We could depict this in a picture where we separate the treated and

controls via this vertical line, and

then we're using just blue, and red to indicate the values of X.

So, blue is for X = 1 and red is for X = 0.

And this is just a hypothetical kind of situation where, but

that's corresponding to what we saw with the propensity score where for

the people with X = 1, the large majority of them.

And in fact, 90% of them are in the control group.

And for X = 0, we see that 4 out of 5 of them are in the treatment group which

again, corresponds to what we saw for our assumption about the propensity score.

So, let's just focus on one of these groups for now.

So, this is the X = 1 group.

So, this is a subpopulation.

And in that subpopulation, for everyone one treated subject,

you would expect to have nine control subjects.

Or in other words, one out of every ten people with X = 1 is treated.

So, that's what we'd expect on average.

So out of 10 total people with X = 1, we'd expect for one of them to be treated.

Now, let's imagine we're going to do propensity score matching.

So you see we have this imbalance here in the sense that among people with X = 1,

they are way more likely to be in the control group.

Whereas if you had a randomized trial and you were strictly randomizing people

to treat it in control groups, you would expect an equal number of treated and

controlled subjects in the subpopulation.

So, what propensity score matching would do then is remember that.

Because there's only one covariant here,

everybody with that is equal to 1 has the same value of a propensity score.

So all of these ten blue dots here, they all have the same propensity score.

So if we were going to do propensity score matching, what we would do is we would

match one treated subject to one here randomly selected control subject.

So with propensity score matching,

the data that we would end up using is right here.

We'd just use all of that and

what we see then is that there's one person in the treatment group now.

There's also one person in the control group, but

there was originally one person in the treated group.

So this one person in the treated group.

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So, this particular treated person should end up having nine times more weight

than any of these individuals from the control group.

So we saw that from the previous slide when we did that one-to-one matching,

we saw that the treated person counted the same as nine people in the control group.

So what you can do then is just weight these observations to make that happen.

So that's what inverse probability of treatment weighting is going to do.

Inverse probability treatment weighting or

weight based on treatment actually received.

So for treated subjects,

we were weighed by the inverse of the probability of treatment.

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So that's a propensity of score.

So for treated subjects, we weight by the inverse probability of treatment.

But for control subjects, we would actually weight by the inverse

of the probability of not getting the treatment.

So in other words, you are always waiting by the inverse of the probability of

whatever it is they actually received.

So treated subjects will see treatment.

So we had by the inverse of that probability,

control subjects we see the control.

So we weigh by the inverse of that probability.

So this is what known as inverse probability

of treatment weighting or IPTW.

So we could go back to this example where we have one treated subject and

nine control subjects.

And now, we can weight by the inverse of the probability

of their particular treatment.

So in this treated group,

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We want to weight by the inverse of the propensity score and

recall from an earlier slide that the propensity score for

treated subjects with X = 1 was equal to 0.1.

So here, we end up taking 1 and dividing by 0.1 which is the propensity score.

Or in other words, they get a weight of 10.

So, this one treated subject will have a weight of 10.

Whereas for the control subjects,

we weight by 1 over the probability of getting the control treatment.

So, the difference here is we're looking for the probability that A = 0.

Well, the probability that A = 0 is just 1 minus the propensity score.

So this 0.9, that comes 1- 0.1.

Because the probabilities have to add up to 1.

So if you have 10% chance of getting the treatment,

you have a 90% chance of getting the control.

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Now, imagine we want to do a propensity score matching.

So we would just takes, there's only one person in the control group.

We would find a match for them at the treated group.

In this case, they all have the same value of the propensity score.

So, we will just randomly select anybody from the treated group.

And now, we have done this one-to-one matching.

So now, one person in the control group count the same as

four people from the treatment group.

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So now if we want to think about weights, again,

we would just weight by 1 over the probability of

their specific treatment so on the treatment side.

On the treatment side, we see we would weight by 1 over the propensity score.

So 1 over the probability of treatment, given X = 0.

Well, the propensity score in this case was 0.8.

So, we have 1 over 0.8 for a weight of five-fourths.

In the control group, now these are people who did not get the treatment.

So, we would weight by 1 over the probability of no treatment.

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Well, the probability of no treatment is equal 0.2 which is just 1 minus 0.8.

So, we take 1 over 0.2 for a weight of 5.

So in this case, weighting is accomplishing the same as matching in

the sense that one person in the control group is counting the same as

four people in the treatment group.

So again, if you think of each person in the treatment group as having weight

five-fourths.

If you added up all those weights,

you would get a five which is the same thing as in the control group.