Now, for discrete variables, there would be different models you could fit.

But let's think about the case where we code y to

be 1 if it's got a characteristic, 0 if it's not.

So this is a binary characteristic.

Yes or no zero one kind of thing.

And then we fit a binary regression.

Could be logistic.

Could be one of those other choices like probe it or complimentary log log.

So we do it with logistic and we fit that model based on complete data on

the logit scale you may remember in a logistic regression.

The logit is the log of the probability of having the characteristic

divided by the probability of not having the characteristic.

So it's log of the odds and that's linear in the x's.

And if we have a missing case,

what we do is we impute first on that logit scale.

I call that z hat k.

So I just use my estimated betas with the hats here and the covariant values for

missing ks, and then I back transform to the probability scale.

So, how do I do that?

This is the back-transform right here.

I take the exponential of the logit valued, divided by one,

plus the exponential of the same thing.

And that gets me back on the p scale, the probability scale.