THE BOOK--Playing The Percentages In Baseball

With experience with the method that Josh is using to measure the zone, I think his methodology for zone measurement itself is perfectly fine.  He essentially does what you say above: uses the contour at which Pr(Strike)=Pr(Ball), given the location.  The methodology is what I use for my zone comparisons and is superior than other methods I’ve seen, IMHO.  This is precisely the technique I use for two academic papers I have under construction.

The advantage of the method is that it doesn’t really matter what the pitcher anticipates the umpire, related to zone size.  We still find that 50% contour that you describe with his method.  This is the case even if 100% of the pitches are thrown down the middle and well outside the zone (though, the imputation would be less accurate...but for most umpires, there isn’t any serious issue with having a full distribution of pitch locations).

If pitchers are throwing to the middle of the plate more due to the smaller strike zone, the hypothesis is that we would pick it up in the kwERA (easier pitches to hit and/or possibly more walks, since fewer are being called strikes if they aren’t changing their behavior).  As you mention, there are important considerations here with non-called pitches as well that may or may not be accounted for.

I agree it would also be interesting to look at the average location overall by umpires...but as for calculation of the strike zone size itself (leaving aside its impact on the game), I find this method to be the most appropriate one out there. 

Of course, the contour chosen is up to the researcher, but I also use the same choice that Josh does here (50% contour...i.e. more likely to be called a strike than a ball means “in the strike zone” and less likely to be called a strike than a ball means “outside the strike zone")

If he did use the countour method (a bunch of concentric circle-ish lines, one inside the other), then that’s fine.  That’s what I meant about the x-width, and 50% of called strikes in that countour region.

But, Josh is NOT using the countour method (as far as I can tell).  This is especially obvious when he talks about intentional balls.  I mean, why even talk about them, if he’s using the countour method, since they’d virtually never be in the sample of the particular 50% contour.

This is how I was reading Josh’s method: if an umpire had 3000 called pitches (including intentional balls), then take the 1500 called strikes closest to the center.  That becomes his “called strike” region.

Did I read him wrong?

Just sounds like a misunderstanding, maybe he didn’t explain well enough.

Technically, calling it a ‘countour’ method is too vague, as you can draw contours for both densities and for probabilities.  But I understand the confusion here.

What you describe that you suspect Josh did is contours for kernel density estimation.  If that is the case, then that is not a good way to do it.  This would be just the % of called pitches that are called strikes.  This would make the strike zone much smaller than it would otherwise be (it would likely be at the 100% ‘contour’ the way you want to see it).  Here, we have:

px*pz contour at 50% == 1500/3000 as you say.

However, Josh uses a generalized additive model, in which it is modeled just like a binomial regression, but with smooth functions for pitch location.  So the 50% contour is the line at which Pr(Strike)=Pr(Ball).  If we say f() is a smooth function, then we have:

g(u) = f(px,pz)

Where u is 0 for a ball call, and 1 for a strike call, and g() is the logit link function.  The only difference between this and a logistic regression is that there aren’t parameter estimates for px and pz since they are non-monotonic due to the spatial features of the zone.  They are estimated non-parametrically and cross-validated using the data to ensure you don’t over/under fit.

This is also what he does for the swing areas, but “u” just equals 1 if the batter swings, and 0 if he does not.  Including intentional balls WOULD affect swing RATE overall, but should not affect the swing area estimation if done as I believe Josh did.

Josh can correct me if I’m wrong here, but I’m nearly 100% certain that he did it the way you are hoping he did.  I generally get rid of intentional balls and pitch outs in my models, but I have found that they don’t have much of an impact on the contour if done the latter way (and they really shouldn’t).  They certainly would in the 1500/3000 that you describe above.

Woops, u above is the mean of the “0” and “1” binomial dependent variable.  Still get the idea though.

Right, exactly on the intentional balls.  Almost all the intentional balls (if not all) are not going to be in the 50% region.  So, even talking about them was pointless in my view.  That’s where I got my confusion.

Given Josh’s other work on the swing areas, where he does use the concentric circles, it seemed odd that he didn’t maintain that.

So, he probably did, and I read him wrong.

I did use the contour method. It was not based on a kernel density estimate, but on a GAM model as Millsy correctly guessed.

I did basically the same thing that I did with swing area with the concentric circles. The circle now is just the 50 percent contour, where as before it was lower. Guess I wasn’t clear about that.

Josh: what percentage of the intentional balls fell at the 50% contour?  If it’s zero, then this means it’s irrelevant how you handled the intentional balls, correct?

Put another way, what is the closest intentional ball to the contour line?  Must be at least 6 inches, no?

I haven’t checked, but I can’t imagine any intentional balls fell within the 50 percent contour. It’s probably 0, and if not, it’s like 1 or 2 intentional balls when the pitchf/x cameras were badly miss-calibrated. The intentional balls are so far away from the contour line that they make no difference. That’s why I included them, because it wouldn’t have made a difference to exclude them.

Same thing with pitches in the dirt, or any other pitch with a very low likelihood of being called a strike. These pitches wouldn’t have changed anything.

Right exactly, hence my initial confusion.

Ok, we’re good.