So I'm just going to give you one example where, let's imagine to motivate this so

we had just a single confounder and let's say it's continuous.

We have a single continuous confounder X.

And let's imagine our propensity score model looks like this,

where it's logit of the probability of treatment is its linear function in X.

And so let's imagine that it looks like this where these tick marks here,

these red marks correspond to actual observations.

And this is sort of what the fitted curve would look like.

So you'll notice on the left side here we have probability of treatment,

so that's all between zero and one.

And our horizontal axis is the actual covariant.

And we'll see that almost all of our data are in the small range,

let's say between roughly negative 0.4 and

positive 0.4, so most of our data are right in here.

So we're estimating those betas from the previous slide from the logistic

regression model.

And it's almost entirely going to be based on how well our data

would sort of correspond to this curve just within this narrow range.

So we really want a particular set of betas that sort of fit well in that

narrow range.

However, we have this one outlier value, so

one person has a very unusual value of X.

So there's going to be varying, that person isn't going to provide very much

information about the fit of this logistic regression model,

about the estimation of betas.

Because almost all of the data are In this narrow range, so

there's this one outlier observation.

And you'll see that for this person, if you believe this logistic regression

curve, their value is here, which is extremely close to 1.

So there's somebody who is very likely to get treated according to our model.

So if they actually didn't get treated, they would have a extremely large weight.

However, we really don't know much about, there's a lot of extrapolation

that is actually taking place outside of the range.

So like, in this direction and this direction, we really don't know what

the true propensity score curve as a function of X should look

like outside of that sort of narrow range defined by those two vertical bars.

Because the data really informed the stuff inside of those vertical bars.

But outside there just really wasn't much data to tell us anything about it.

So that actual shape is determined by an assumption that we had that it's

linear on the logit scale, so that was an assumption we made.

We don't know if it's true or not, and so

much of our sort of inference there's a lot of extrapolation here, essentially.

So I'm drawing a hypothetical sort of alternative curve that you'll notice

that this red curve, this is just a hypothetical kind of curve that I drew in.