And next, I'm mentioning something that might be helpful and something that

people often do, which is rather than use an untransformed propensity score.

A lot of times, people will first transform it using a logit transformation.

So a logit is just a log-odds.

So before you actually match, you could take the log-odds of the propensity score.

And the reason you would do that is basically to kind of stretch it out

in a sense.

So the propensity score is sort of, it's between 0 and 1.

And a lot of times, let's say you had a rare treatment.

Well, the propensity scores would tend to be very small for everybody.

And it would be very bunch up in a small range.

But if you transformed it by taking a logit transformation,

it will essentially stretch it out.

It's a one-to-one transformation, but it will basically spread it out and

make it easier to find matches.

So the logit of the propensity score is unbounded so

it could take a value anywhere on the real line.

But it still preserves the ranks of the propensity score itself.

So you could match on logit to the propensity score

rather then the propensity score.

So this is, you could do either, but this is something that is often done and

I think there's a lot of situations in which it ends up being helpful.

Another thing that we could do is use a caliper, and

we would do this to make sure that we don't have any bad matches.

So a caliper would basically be our definition of what a bad match is.

So the caliper is just the maximum distance that we are willing to tolerate.

So it's sort of the threshold between an acceptable match and

an unacceptable match.

So in practice,

a lot of times people use a caliper based on standard deviation units.

And in particular, the most common thing I see used in practice is

the following, where your caliper is 0.2 times the standard

deviation of logit of the propensity score.

So that sounds kind of complicated but we can just look at it in steps.

So imagine first, you just estimate the propensity score.

Right, so we use logistic regression for example,

we get a propensity score for each person.

And that's just a value between 0 and 1 for every person.

Then we take a logit transformation of that propensity score.

So we just take the log-odds.

So that's a simple one step calculation, and

now we have logit of the propensity score for every person.

So that's just a variable in our data set, logit of propensity score.

Now, we could just take the standard deviation of that.

So because its a variable in our data set we could just ask our statistical

software for the standard deviation of that variable.