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August 28, 2013 Reworking WARPThe Overlooked Uncertainty of OffensePrevious Installments of Reworking WARP
So, we’ll math. But before we math, let’s talk a bit about how sabermetricians measure offense, as opposed to what I like to call “RBI logic.” Traditional accounting of baseball offense works on two basic principles:
Ignoring some pretty silly edge cases, this reconciles with team runs scored. The problem is that it’s such a binary model—either a runner scores or he doesn’t. With baseball, though, there are outcomes that can increase the probability of a runner scoring without driving him in immediately: · You can advance the runner, which makes him more likely to be driven in in a subsequent atbat, and · You can avoid making an out, which—even if you do not advance the runner in doing so—gives additional batters behind you chances to drive him in. So RBI logic does a very good job of reconciling to team runs, by sheer force of will, but it’s a poor reflection of the underlying runscoring process. You end up crediting players for coming up in spots where runners are in scoring position, and ignoring the contributions of players who advance runners over. You also ignore the value of not making outs. The foundation of most modern sabermetric analysis of run scoring is the run expectancy table. Here’s a sample table, derived from 2012 data:
Top to bottom, it goes by the runner on base—a zero indicates no runner on base, one through three indicates a runner on that base. Left to right is the number of outs in an inning. (It’s not explicitly listed on most run expectancy tables, but the threeout state is a special state in which runs expected goes to zero.) The table lists the average number of runs expected to score in the rest of the inning from that state—the lowest is with the bases empty with nobody on and two outs, at 0.101 runs expected, all the way up to the bases loaded with no outs, where 2.262 runs score on average. What’s interesting isn’t so much the run expectancy itself, but the change in run expectancy between events. So let’s run through an example. Say you have runners on first and third, no outs. That’s a run expectancy of 1.677. Now, suppose the next hitter walks. That moves you to a bases loaded, no outs situation. That walk would be worth 0.585 runs—a pretty important walk. What if the hitter strikes out instead? That moves you into a first and third with one out situation, for a value of 0.531. We come up with the value of each event by looking at the average run expectancy change for each event—that’s known as the event’s linear weights value. Here’s a set of linear weights values for official events in 2012:
We’ve separated the intentional walk from other walks. You’ll note that a hitbypitch is worth more runs than a walk—pitchers tend to issue fewer walks with first base occupied, compared to hit batters. Shockingly, a home run is worth more than a triple, a triple is worth more than a double, and so on. Now let’s look at the same table, but with one new piece of information—the standard deviation around that average change in run expectancy:
There is a substantial correlation between the average run value of an event and its standard error, which shouldn’t be surprising. It also tells us that the actual value of a player’s offense is more uncertain the more he relies upon power—the value of a home run is more uncertain that that of a single, after all. We need to get into a bit of gritty math stuff here before getting to the fun stuff. What you have to remember is that the standard deviation is simply the square root of the variance around the average. In order to combine standard deviations, you have to first square them, then combine them, then take the square root again. (In other words, variances add, not standard deviations.) Now, here’s a list of the top 20 players in batting runs above average (derived from linear weights) in 2012, along with the estimated error for each:
So the difference between Mike Trout and Miguel Cabrera in 2012 was 12.5 runs. The combined standard error for the two of them (remember, variances add) is 9.7. How confident are we that Trout was a better hitter (relative to average) than Cabrera in 2012? Divide the difference by the standard error and you get 1.3—that’s what’s known as a zscore. Look up a zscore of 1.3 in a zchart, and you get .9032—in other words, roughly 90 percent. So there’s a 90 percent chance, given our estimates of runs and our estimates of error, that Trout was the better hitter. Now, we should emphasize that a 90 percent chance that he was means there’s a 10 percent chance that he wasn’t. What if we compare Posey to Beltre? That’s a difference of seven runs, which works out to a confidence level of 77 percent that Posey was the better hitter. What about comparing Braun to Votto? That’s a difference of just half a run between them—our confidence is only about 52 percent, essentially a coin flip between them. So what we have is a way to measure our measurement of run production, and then to apply a confidence interval to our estimates. For a fulltime player (one qualified for the batting title, that is) the average standard error is roughly six runs. If you want to compare bad hitters to good hitters, sure, most of the time the difference between them far outstrips the measurement error. But if you want to compare good hitters to good hitters (which is frankly a lot more interesting, and probably a lot more common), then you’ll often find yourself running into cases where the difference between them is close to, if not lower than, the uncertainty of your measurements. So if we can quantify our measurement uncertainty, the next question we can ask is, is there a way to measure offense that’s subject to less measurement uncertainty? I have a handful of ideas on the subject, which we’ll take a look at next week.
Colin Wyers is an author of Baseball Prospectus. Follow @cwyers
35 comments have been left for this article. (Click to hide comments) BP Comment Quick Links jdeich (50647) Fantastic and insightful work. It may be hard to communicate to wider audiences (as uncertainty is in science, medicine, economics, etc.), but it would keep analyticallyminded analysts in check. Too often you'll hear borderline calls stated as absolutes, "Posey was a better hitter than McCutchen in 2012!", when they're within the intuitive measurement error. Aug 28, 2013 06:06 AM I agree that uncertainty may be a hard sell in some spots, but I think there may actually be an audience for it outside of the people currently invested in sabermetrics. I think the stridency and certitude of some people who advocate for sabermetrics can be offputting, and I think an analytical approach that's explicit about measurement error and uncertainty could possibly interest, rather than repel, people who are tired of the current approach. Aug 28, 2013 06:48 AM Sky Kalkman (3454) So the variance for a player's batting runs is based on the population variance for each event's change in run expectancy, yes? Aug 28, 2013 07:27 AM So the question is, why not use a Value Added approach, if your goal is to minimize error in predicting actual changes in RE? Aug 28, 2013 07:32 AM Sky Kalkman (3454) Yes. And if that's not your goal, why worry about the variance in the linear weights numbers? Aug 28, 2013 08:25 AM TangoTiger (57181) Sky is correct. Aug 28, 2013 08:49 AM Mooser (26842) So are you suggesting just use RE24 instead of Linear Weights? Aug 28, 2013 08:54 AM TangoTiger (57181) If the purpose is to track offensive impact by the baseout situation, then yes. That answers that question. Aug 28, 2013 09:16 AM RedsManRick (23592) If we assume that players generally are able to tailor their production to circumstance, than this would suggest that RE24 will be a more accurate representation of the run value produced, but that Linear Weights provides a better estimate of talent/ predictor of future run production. It's analogous to ERA vs. FIP in a way. Aug 28, 2013 19:43 PM eliyahu (11036) I agree with this, and have never understood why the Value Added approach never got more mainstream support. Shifts in run expectancy is an idea that most nonSABR inclined people can get their heads wrapped around. And why use linear weights when you can get the actual impact of a given play given the specific circumstance? Aug 28, 2013 09:09 AM TangoTiger (57181) Well, those on the leading edge need to support RE24 more. Google RE24 and you'll get some good articles. But, we need more people spreading the word. Aug 28, 2013 09:17 AM John Douglass (53235) I don't like the notion that LW 'punishes' a player. What it does is not reward or deduct from their value circumstances that are beyond their control. Aug 28, 2013 09:17 AM ncsaint (67849) Doesn't that undercut one of the main functions of WAR  judging what a player's performance would contribute to an average team? Aug 28, 2013 12:20 PM Sky Kalkman (3454) This is a good point. Maybe do RE24 divided by average RE24_leverage for the season? So if you bat in important situations more often because of your team, we reduce your RE24 total? And vice versa? Aug 28, 2013 12:27 PM Eric M. Van (31218) What is most frustrating about the neglect of RE24 for hitters in favor of a contextneutral approach is that *we do the precise opposite for pitchers.* RA, once adjusted for inherited runners, is essentially a measure of pitchers' net delta Run Expectancy, with context. We should be using linear weights or Base Runs to calculate every pitcher's contextneutral RA, but nobody does that. Bill James' ERC is an attempt at an estimate, but we can do better. Aug 30, 2013 01:37 AM Sean (33374) While I appreciate you highlighting the difference between how we treat hitters and pitchers, there is a fundamental difference: Pitchers pitch to every batter and therefore have a direct hand in the entire baseout situation. Hitters, on the other hand, step into a baseout situation that has been determined for them. So there IS a fundamental difference here, not that I have my mind made up that we should take a contextdependent approach for pitchers and a contextneutral approach for hitters. Aug 30, 2013 08:33 AM Eric M. Van (31218) Yes, that explains why we do it the way we do. It doesn't explain the weird lack of interest that the sabermetric community seems to have for a full contextneutral pitching metric. Yes, we know that some of BABIP is luck, but we know that much of it isn't (for instance, on a startbystart basis, most pitchers have a significant correlation between FIP and BABIP). What I want is that full contextneutral metric (essentially a pitcher's TAv allowed) and the same thing with leagueaverage BABIP substituted, and better yet, with the smartest possible estimate of the pitcher's true BABIP skill. Aug 30, 2013 09:30 AM newsense (5112) I think there is some confusion about standard deviation and standard error here. Aug 28, 2013 08:05 AM ncsaint (67849) So what we're talking about here is all deviation, right? Since you have thousands of doubles and home runs, the standard error would be tiny. Aug 28, 2013 12:15 PM newsense (5112) I think so, but I think you mean NIBBs not IBBs. Aug 28, 2013 12:51 PM ncsaint (67849) Sorry, I wasn't being clear. I was doublechecking that the numbers in the article are all talking about deviation, and then talking about 3Bs and IBBs in terms of error. That is, given the huge sample for everything else on the list, they would have tiny errors, even where the variance is high, but IBBs and 3Bs might have a significant error simply because there are few of them, with 3Bs being obviously much higher because of the higher variance. Aug 28, 2013 13:05 PM ncsaint (67849) Stating that there is only a 90% chance Trout was the better hitter strikes me as framing these interesting results wrong. You are returning halfway to what you call 'RBI logic'. That is, you don't have the binary outcome problem  you get credit for moving runners over and not making outs  but you are going back to rewarding players for coming up in the right situations. Aug 28, 2013 11:56 AM Alan Nathan (53214) I didn't have time to read all the comments, so please excuse me if this has been discussed already. When comparing BRAA for two players, say Trout and Miggy, you have simply added the variances of the two players, then taken the sqrt to get the standard error in their difference. That would be a correct procedure if the variances of the two players are uncorrelated. But are they uncorrelated? I claim no, since they are derived from the same linear weights table. What say you about that? Aug 29, 2013 08:16 AM TangoTiger (57181) Alan: I think this is part of the confusion with what Colin is doing. He's really presuming that the RE24 model is the target model, and he's presuming that he's unaware of Trout and Cabrera's performance in the 24baseout states, and so, that's what the "error" term is about. Aug 29, 2013 11:46 AM Alan Nathan (53214) So let's see if I understand: In Colin's table showing LWTS and STDERR, the STDERR is the standard deviation found from the variation (appropriately weighted perhaps?) among the 24 states contributing to the total. If he had used RE24, then there would be no variance (other than the tiny variance in the actual RE24 numbers, which are based on many contributing events and are therefore small). Am I getting it? Aug 29, 2013 13:24 PM TangoTiger (57181) Perfect! Aug 30, 2013 05:03 AM Nathan Aderhold (68572) I just want to make sure I understand the discussion in the comments about LWTS vs. RE24. I'm still trying to wrap my head around the differences between the two. Sep 02, 2013 11:47 AM Not a subscriber? Sign up today!

I'm really enjoying this series and think it's a real step in the right direction.
I assume you'll get there in the end, but the thing that jumps out to me is that a 6 run uncertainty level on the supposedly easy hitting side of the player value equation has pretty big ramifications for player valuations.
At ~6M/win a 6 run approximation for hitting contributions would suggest a 3.6M level of uncertainty, right?
Seems like if this approach takes hold we may lose a lot of the knee jerk declarations that team X or Y is dumb for every signing (which would be a good thing).
Yeah, 6 runs is roughly 2/3rds of a win, depending on the exact run environment you're in. And that's just for one WARP component. Once we get into stuff like replacement level and positional adjustment, we'll see that margin of error creep up. So yeah, if you start looking at WARP as an estimate rather than a point value, you get into a lot of... well, MAYBE this guy was better, rather than this guy WAS better.