## Adjusting Appearance Data for Base-Out State

So far, we’ve developed some mathematical principles for visualizing appearance data for relief pitchers, and for measuring how apart they are. The goal has been to say something about how pitchers are being used, not only in a vacuum, but in the context of the way in which the team has chosen to divide up its relief innings for the season. We’ve only partially gotten there so far, but today let’s take a slight detour to ask: Is the underlying data conveying the most useful information?

Inning and score differential at the time of entering the game are the critical data elements in answering questions related to usage. The numbers and tables in my previous articles all focused on using these two elements. Here’s an example of the underlying data being used, in the form of three Daniel Hudson appearances which appear identical.

Three (Similar?) Daniel Hudson Appearances
Date Player Season Inning Score
6/28/2016 Daniel Hudson 2016 8 1
8/20/2016 Daniel Hudson 2016 8 1
9/21/2016 Daniel Hudson 2016 8 1

Inning and score differential are critical; however, as data elements are concerned, they are somewhat raw. Fortunately, those aren’t the only data elements we can look at. The next-most impactful data, I would argue, is the base-out state at the time that the pitcher enters the game.

Let’s establish a baseline: It’s the norm for relief pitchers to enter the game in a clean inning (no outs, no runners on base). Among pitchers with 20+ relief appearances in 2016, this was the situation in 68.1% of appearances. That’s a very high percentage, considering that there are 24 base-out states. It’s also very intuitive when we think about the game. Among other reasons, pitchers need time to warm up, and mostly, they do so while their own team is batting. It’s also the only base-out state which is guaranteed to happen every inning.

It would be atypical – and therefore, interesting – for a pitcher to be used frequently in other base-out states. Moreover, we should be giving credit to pitchers who are being used in that way. An appearance where a pitcher enters with a four-run lead but the bases loaded should not be viewed in the same way as an appearance where a pitcher enters with a four-run lead in a clean inning. More than likely, the manager has two different pitchers in mind for each of these scenarios.

Adjusting the inning is easy: Credit partial innings in the event that the pitcher enters with more than zero outs in the inning. This will bump the inning component of every pitcher’s “center of gravity” up a bit, giving credit to players for working slightly later in the game when called upon mid-inning. (Note: we could also define terms in a different way, and say that a pitcher who enters in a “clean” 9th inning is actually entering at inning 8.0, as 8 innings have been recorded prior to his entrance; however, this makes the resulting metric less intuitive.)

Adjusting the score differential doesn’t seem as straightforward at first, but fortunately, we can use the concept of RE24 to accomplish this. Given that entering in a clean inning is the default status, we will make no adjustment to the score differential for a given appearance if the pitcher entered in a clean inning. For any other base-out state, we will add or subtract the difference between expected runs in that base-out state and expected runs in a clean inning state (0 on, 0 out).

Let’s return to the three appearances shown above. As you might have guessed by now, they are not identical. Rather, they illustrate the importance of adjusting for base-out state.

Three Daniel Hudson Appearances (in greater detail)
6/28/2016 Daniel Hudson 8 1 0 ___ 8.00 1.00
8/20/2016 Daniel Hudson 8 1 0 123 8.00 -0.82
9/21/2016 Daniel Hudson 8 1 2 _2_ 8.67 1.16

If you were to ask Daniel Hudson to recall what he could about these three appearances, he’d probably feel very differently about each of them (if he remembers, anyway). In the first case, he’s coming into a clean 8th inning, protecting a one-run lead. It was a situation he found himself in with some regularity in 2016, prior to assuming the closer’s role.

The second situation is an absolute bear. Jake Barrett has allowed a leadoff single to lead off the inning, and poor Steve Hathaway, who shouldn’t be touching this game situation with a 10-foot pole at this point in his career, has subsequently allowed a double and a walk to load the bases. Hudson has been brought in to protect a one-run lead with the bases loaded and nobody out. The opposing team has an expected run value of 2.282. While technically Hudson has been given a lead, it’s one that he would be hard-pressed to keep, even if he does everything right. The reality is that this appearance is associated with an expectation that Arizona will trail by the end of it – as you can see on the play-by-play log, the Padres have a 70.6% win probability at this point. It would be silly to give this appearance the same treatment as the first two. (Hudson, by the way, does a masterful job of escaping this situation without surrendering the lead!)

The third case is the one I want to focus on. Rather than a clean inning, Hudson was asked to get the third out of the 8th inning, with the tying run standing on second base. While the Leverage Index at the time of entry for this appearance is higher (3.50) than in the first instance (2.17), Hudson actually has an easier job: He needs just one out instead of three, and the opposing team is expected to score fewer runs in this situation, all else being equal. In the “clean” 8th inning, he can be expected to give up 0.481 runs, while in the two-out, runner-on-second situation, he can be expected to give up just 0.319 runs. Moreover, the chance of scoring at least one run – presumably the more important question where one-run leads are concerned – is also lower in the “higher leverage” situation. (This doesn’t even account for the batter, Hector Sanchez, who is hardly Wil Myers at the plate, and is probably inferior to the 4-5-6 hitters in the Phillies lineup, as well.)

This brings up an important distinction between leverage and run prevention. Leverage Index, certainly, is an important tool. What it measures, however, is variance in win probability for a single at-bat. Managers rarely have the luxury of giving their pitchers one-batter appearances in the regular season. Even the notoriously fleeting Javier Lopez averaged nearly three batters per appearance in 2016. Managers must therefore determine how to maximize the value of relief appearances as a whole, not just at the time when the reliever is entering the game. Leverage Index shows how much variance can arise from the current plate appearance, but a manager may very well be better served having their best pitcher throw the entirety of the 8th inning, rather than having him get the third out in a situation that commands high leverage but still has relatively low run expectation.

Next time, we’ll look at how base-out state adjustments impacted the raw inning-score matrix data in 2016, to draw conclusions about which relievers were used most often in high-pressure, mid-inning situations, and whether that sort of usage aligns with what we’d expect from an optimal manager.

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I'm an actuary who plays around with baseball data on the side. Brooklyn, NY. Twitter: @malkusm

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Hey Matt, your analysis and writing here is pretty impressive. I like the way you utilize a lot of stats while keeping the prose clean and understandable. If you have any interest in writing for Beyond the Box Score — we’re sort of SB Nation’s FanGraphs — shoot me an email at tripler26 at gmail dot com.