November 26, 2014

Moonshot

A New View of Plate Discipline, Part 3

by Robert Arthur

I’ve recently written about the role and value of plate discipline in hitting. I concluded the last article with the takeaway that plate discipline, while undoubtedly important in hitting, was not fully separable from the other attributes of a hitter. In searching for a complete per-pitch way of evaluating hitters, we have to account for the entire package of skills, because all of the skills interact with each other. So we have to go back to basics.

Despite being athletically impossible, hitting is theoretically simple. Every hitter is confronted on every pitch with a choice: to swing, or to take. A take is valuable when the pitch is likely to be a ball; depending on the count, you can get a walk, or at least advance the count in a favorable direction. If you swing here, you both lose the benefit of the called ball, and also risk whiffing on the pitch or making weak contact. On the other hand, when the pitch is thrown over the middle of the plate and is thus likely to be called a strike, the better choice is to do your best to make contact. If you take, you get a strike, and lose the opportunity of a hittable pitch.

Framed in this way, the focal decision is to swing or not, and so that’s perhaps the information which is most useful in evaluating each hitter. In an economic framework, we can think of the marginal value of a hitter’s decision to swing or take, relative to the alternative (if they swung, what if they decided not to? If they decided not to, what might have happened if they had swung?). We can assign a value to each hitter on each pitch using the straightforward device of linear weights (as calculated by Harry Pavlidis). We then subtract this value according to what might have happened if they made the opposite choice.

I think it helps to walk through how we could score a hitter, which we can divide into two cases, depending on whether the hitter took a cut or held back.

Swing
To figure out what this swing was worth, we first need to know what happened¹. Did the batter whiff, or knock in the game-winning home run? Obviously, the value of his decision to swing will be based in part on what he did with that swing.

I take the value that the hitter actually accrued, and subtract that from the baseline value of what my model believes would have happened if he had decided not to swing. In this case, that’s simply the value of the ball/strike that would have happened, multiplied by the probability that it would have happened (the expected value I introduced last time). So we are left with the actual value of the swing, minus the expected value of the alternative, and that gives us the marginal value of the hitter’s decision to swing².

According to this metric, here are the top 10 players (on a per swing basis):

While the list isn’t perfect, I’m reasonably happy with it. Note well that this metric isn’t park- or league-adjusted (although there’s no reason it couldn’t be, in the future), which is why some Rockies are (forgive the pun) so elevated. Nevertheless, you’ve got your Victor Martinez and Mike Trout, your Michael Brantley and your Paul Goldschmidt, excellent hitters all.

Take
Suppose the hitter has decided to take instead. Good choice, maybe. The logic here follows the same basic formulation as above, but we tweak it to deal with the fact that we are inferring what might have happened had the hitter taken a swing (instead of inferring what would have happened if he decided to take).

First, was it (all things considered) a good choice to take? It was, if the pitch was likely to be called a ball. It was not, if the pitch was likely to be called a strike. If the pitch was more likely to be a ball, it was a better decision to take it the more likely it was to be called a ball. Similarly, if it was a strike, it was a worse decision to take it the more likely it was to be a strike.

As above, we also need to figure out the counterfactual situation, that is, what would have happened if the hitter had decided to swing. This point is where things get kind of tricky, in that we don’t know what would have happened, but we can take a guess. The way that we take a guess could range from the effortlessly simple to the impossibly complex³. As this article represents a preliminary take, I went with something more toward the simple side.

I fit a model to guess the outcome of a swing (in terms of the linear weight value), using the identity of the hitter, the location of the pitch (x- and y-coordinates), and the velocity of the pitch. These variables aren’t a complete list of all of the things which could affect how well a pitch might be struck, but they should touch on the major factors: who the hitter is, where the pitch is, and how difficult the pitch is to hit.

Now that I have the expected value of what would have happened if the batter had chosen to swing, as well as the value of their decision to take, I can calculate the marginal value of their decision by subtraction. This calculation matches our intuition in that the further a pitch is outside, the more value a hitter loses by swinging at it, both in forgoing the called ball they might have had, and probably not being able to get the bat head to the ball very well⁴.

I’m a little happier with the list of best “swing” values than the list of best “take” values. The swing value list accords much more closely with my mental, intuitive list of the best hitters in baseball. The take value list appears a little awry, and is clearly affected in large part by the kinds of pitches different hitters see. Josh Hamilton, Pablo Sandoval, and similar hitters get pitches so far off the plate that to ever consider swinging at them would be folly. So, when they don’t swing, that’s viewed as a significantly positive play by this metric. But on the other hand, they wouldn’t see those pitches to begin with unless they had a prior history of sometimes making completely witless swings at exactly those pitches. In theory, those countering forces should even out, but in practice, it looks like they don’t. Another part of the problem might be that the take values require more modelling and statistical estimation (which is a fancy way of saying “guesswork”). When the models or assumptions which go into the take values are off, biases and distortions can result. With those caveats stated, it’s a start.

If we add together the best swing and best take values, we get (what I consider) the best list yet:

Here’s the full list, with both rate and total statistics for each of the MLB regular hitters, to peruse at your leisure. It’s worth noting that in addition to agreeing well with intuition, it stacks up well with more conventional measures of hitting ability like TAv, with which it has a strong correlation (r = .75). This measurement has the advantage of being calculable for any given pitch, as opposed to TAv or wOBA. It’s more of a process statistic than either of those, because it measures not only the outcomes of a hitter’s plate appearances, but the whole sequence of swings/takes which led to those outcomes.

I want to be clear here that the above measure isn’t a full value-type metric, in the vein of WAR(P) (or even the offensive subset of that metric). For one thing, it doesn’t measure versus a baseline of replacement level. The measurement here is essentially “Runs Above Alternative,” or what actually happened relative to what might have happened if the batter had made the opposite choice. You could call it marginal runs.

This metric can serve instead as a framework to be altered, modified, and mutated by other interested sabermetricians. As a barebones structure, it’s a way to measure the value of hitting at the level of individual pitches, instead of at the level of plate appearances, taking into account the dichotomous nature of hitting.

***

¹Actually, we could get fancier here, and try to figure out what should have happened, instead of simply using the linear weights value of what did happen. More on this below.

²More formally:

Where “LWTS” is shorthand for “linear weight value” (of the subscripted event).

³Effortlessly simple: league-average run value at this count. Impossibly complex: a Bayesian model of the probability of each event (single, double, reached on error, etc.), conditioning on the pitch type, speed and location, which would start with league average effects for each hitter, and then update (in a Bayesian fashion) as data from a hitter was accumulated over the course of his career. If I had unlimited time, I think the impossibly complex approach would produce marginally better results. But I think the margin would be small, and I think doing it properly might take years of work.

⁴Again, but reversed, because the hitter made the opposite choice:

Robert Arthur is an author of Baseball Prospectus. Follow @No_Little_Plans
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