Monday, January 26, 2015

Are umpires biased in favor of star pitchers? Part II

Last post, I talked about the study (.pdf) that found umpires grant more favorable calls to All-Stars because the umps unconsciously defer to their "high status." I suggested alternative explanations that seemed more plausible than "status bias."

Here are a few more possibilities, based on the actual coefficient estimates from the regression itself.

(For this post, I'll mostly be talking about the "balls mistakenly called as strikes" coefficients, the ones in Table 3 of the paper.)


1. The coefficient for "right-handed batter" seems way too high: -0.532. That's so big, I wondered whether it was a typo, but apparently it's not.  

How big? Well, to suffer as few bad calls as his right-handed teammate, a left-handed batter would have to be facing a pitcher with 11 All-Star appearances.

The likely explanation seems to be: umpires don't call strikes by the PITCHf/x (rulebook) standard, and the differences are bigger for lefty batters than righties. Mike Fast wrote, in 2010,

"Many analysts have shown that the average strike zone called by umpires extends a couple of inches outside the rulebook zone to right-handed hitters and several more inches beyond that to left-handed hitters." 

That's consistent with the study's findings in a couple of ways. First, in the other regression, for "strikes mistakenly called as balls", the equivalent coefficient is less than a tenth the size, at -0.047. Which makes sense: if the umpires' strike zone is "too big", it will affect undeserved strikes more than undeserved balls. 

Second: the two coefficients go in the same direction. You wouldn't expect that, right? You'd expect that if lefty batters get more undeserved strikes, they'd also get fewer undeserved balls. But this coefficient is negative both cases. That suggests something external and constant, like the PITCHf/x strike zone overestimating the real one.

And, of course, if the problem is umpires not matching the rulebook, the entire effect could just be that control pitchers are more often hitting the "illicit" part of the zone.  Which is plausible, since that's the part that's closest to the real zone.


2. The "All-Star" coefficient drops when it's interacted with control. Moreover, it drops further for pitchers with poor control than pitchers with good control. 

Perhaps, if there *is* a "status" effect, it's only for the very best pitchers, the ones with the best control. Otherwise, you have to believe that umpires are very sensitive to "status" differences between marginal pitchers' control rates. 

For instance, going into the 2009 season, say, J.C. Romero had a career 12.5% BB/PA rate, while Warner Madrigal's was 9.1%. According to the regression model, you'd expect umpires to credit Madrigal with 37% more undeserved strikes than Warner. Are umpires really that well calibrated?

Suppose I'm right, and all the differences in error rates really accrue to only the very best control pitchers. Since the model assumes the effect is linear all the way down the line, the regression will underestimate the best and worst control pitchers, and overestimate the average ones. (That's what happens when you fit a straight line to a curve; you can see an example in the pictures here.) 

Since the best control pitchers are underestimated, the regression tries to compensate by jiggling one of the other coefficients, something that correlates with only those pitchers with the very best control. The candidate it settles on: All-Star appearances. 

Which would explain why the All-Star coefficient is high, and why it's high mostly for pitchers with good control. 


3. The pitch's location, as you would expect, makes a big difference. The further outside the strike zone, the lower the chance that it will be mistakenly called a strike. 

The "decay rate" is huge. A pitch that's 0.1 feet outside the zone (1.2 inches) has only 43 percent the odds of being called a strike as one that's right on the border (0 feet).  A pitch 0.2 feet outside has only 18 percent the odds (43 percent squared).  And so on.*

(* Actually, the authors used a quadratic to estimate the effect -- which makes sense, since you'd expect the decay rate to increase. If the error rate at 0.1 feet is, say, 10 percent, you wouldn't expect the rate for 1 foot to be 1 percent. It would be much closer to zero. But the quadratic term isn't that big, it turns out, so I'll ignore it for simplicity. That just renders this argument more conservative.) 

The regression coefficient, per foot outside, was 8.292. The coefficient for a single All-Star appearance was 0.047. 

So an All-Star appearance is worth 1/176 of a foot -- which is a bit more than 1/15 of an inch.

That's the main regression. For the one with the lower value for All-Star appearances, it's only an eighteenth of an inch. 

Isn't it more plausible to think that the good pitchers are deceptive enough to fool the umpire by 1/15 inches per pitch, rather than that the umpire is responding to their status? 

Or, isn't it more likely that the good pitchers are hitting the "extra" parts of the umpires' inflated strike zone, at an increased rate of one inch per 15 balls? 


4. The distance from the edge of the strike zone is, I assume, "as the crow flies." So, a high pitch down the middle of the plate is treated as the same distance as a high pitch that's just on the inside edge. 

But, you'd think that the "down the middle" pitch has a better chance of being mistakenly called a strike than the "almost outside" pitch. And isn't it also plausible that control pitchers will have a different ratio of the two types than those with poor control? 

Also, a pitch that's 1 inch high and 1 inch outside registers as the same distance as a pitch over the plate that's 1.4 inches high. Might umpires not be evaluating two-dimensional balls differently than one-dimensional balls?

And, of course: umpires might be calling low balls differently than high balls, and outside pitches differently from inside pitches. If pitchers with poor control throw to the inside part of the plate more than All-Stars (say), and the umpires seldom err on balls inside because of the batter's reaction, that alone could explain the results.


All these explanations may strike you as speculative. But, are they really more speculative than the "status bias" explanation? They're all based on exactly the same data, and the study's authors don't provide any additional evidence other than citations that status bias exists.

I'd say that there are several different possibilities, all consistent with the data:

1.  Good pitchers get the benefit of umpires' "status bias" in their favor.

2.  Good pitchers hit the catcher's glove better, and that's what biases the umpires.

3.  Good pitchers have more deceptive movement, and the umpire gets fooled just as the batter does.

4.  Different umpires have different strike zones, and good pitchers are better able to exploit the differences.

5.  PITCHf/x significantly underestimates umpires in their opinions of what constitutes a strike. Since good pitchers are closer to the strike zone more often, they wind up with more umpire strikes that are PITCHf/x balls. The difference only has to be the equivalent one-fifteenth of an inch per ball.

6.  Umpires are "deliberately" biased. They know that when they're not sure about a pitch, considering the identity of the pitcher gives them a better chance of getting the call right. So that's what they do.

7.  All-Star pitchers have a positive coefficient to compensate for real-life non-linearity in the linear regression model.

8.  Not all pitches the same distance from the strike zone are the same. Better pitchers might err mostly (say) high or outside, and worse pitchers high *and* outside.  If umpires are less likely fooled in two dimensions than one, that would explain the results.


To my gut, #1, unconscious status bias, is the least plausible of the eight. I'd be willing to bet on any of the remaining seven, that they all are contributing to the results to some extent (possibly negatively).  

But I'd bet on #5 being the biggest factor, at least if the differences between umpires and the rulebook really *are* as big as reported.  

As always, your gut may be more accurate than mine.  

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Sunday, January 18, 2015

Are umpires biased in favor of star pitchers?

Are MLB umpires are biased in favor of All-Star pitchers? An academic study, released last spring, says they are. Authored by business professors Braden King and Jerry Kim, it's called "Seeing Stars: Matthew Effects and Status Bias in Major League Baseball Umpiring."

"What Umpires Get Wrong" is the title of an Op-Ed piece in the New York Times where the authors summarize their study. Umps, they write, favor "higher status" pitchers when making ball/strike calls:

"Umpires tend to make errors in ways that favor players who have established themselves at the top of the game's status hierarchy."

But there's nothing special about umpires, the authors say. In deferring to pitchers with high status, umps are just exhibiting an inherent unconscious bias that affects everyone: 

" ... our findings are also suggestive of the way that people in any sort of evaluative role — not just umpires — are unconsciously biased by simple 'status characteristics.' Even constant monitoring and incentives can fail to train such biases out of us."

Well ... as sympathetic as I am to the authors' argument about status bias in regular life, I have to disagree that the study supports their conclusion in any meaningful way.


The authors looked at PITCHf/x data for the 2008 and 2009 seasons, and found all instances where the umpire miscalled a ball or strike, based on the true, measured x/y coordinates of the pitch. After a large multiple regression, they found that umpire errors tend to be more favorable for "high status" pitchers -- defined as those with more All-Star appearances, and those who give up fewer walks per game. 

For instance, in one of their regressions, the odds of a favorable miscall -- the umpire calling a strike on a pitch that was actually out of the strike zone -- increased by 0.047 for every previous All-Star appearance by the pitcher. (It was a logit regression, but for low-probability events like these, the number itself is a close approximation of the geometric difference. So you can think of 0.047 as a 5 percent increase.)

The pitcher's odds also increased 1.4 percent for each year of service, and another 2.5 percent for each percentage point improvement in career BB/PA.

For unfavorable miscalls -- balls called on pitches that should have been strikes -- the effects were smaller, but still in favor of the better pitchers.

I have some issues with the regression, but will get to those in a future post. For now ... well, it seems to me that even if you accept that these results are correct, couldn't there be other, much more plausible explanations than status bias?

1. Maybe umpires significantly base their decisions on how well the pitcher hits the target the catcher sets up. Good pitchers come close to the target, and the umpire thinks, "good control" and calls it a strike. Bad pitchers vary, and the catcher moves the glove, and the umpire thinks, "not what was intended," and calls it a ball.

The authors talk about this, but they consider it an attribute of catcher skill, or "pitch framing," which they adjust for in their regression. I always thought of pitch framing as the catcher's ability to make it appear that he's not moving the glove as much as he actually is. That's separate from the pitcher's ability to hit the target.

2. Every umpire has a different strike zone. If a particular ump is calling a strike on a low pitch that day, a control pitcher is more able to exploit that opportunity by hitting the spot. That shows up as an umpire error in the control pitcher's favor, but it's actually just a change in the definition of the strike zone, applied equally to both pitchers.

3. The study controlled for the pitch's distance from the strike zone, but there's more to pitching than location. Better pitchers probably have better movement on their pitches, making them more deceptive. Those might deceive the umpire as well as the batter. 

Perhaps umpires give deceptive pitches the benefit of the doubt -- when the pitch has unusual movement, and it's close, they tend to call it a strike, either way. That would explain why the good pitchers get favorable miscalls. It's not their status, or anything about their identity -- just the trajectory of the balls they throw. 

4. And what I think is the most important possibility: the umpires are Bayesian, trying to maximize their accuracy. 

Start with this. Suppose that umpires are completely unbiased based on status -- in fact, they don't even know who the pitcher is. In that case, would an All-Star have the same chance of a favorable or unfavorable call as a bad pitcher? Would the data show them as equal?

I don't think so. 

There are times when an umpire isn't really sure about whether a pitch is a ball or a strike, but has to make a quick judgment anyway. It's a given that "high-status" control pitchers throw more strikes overall; that's probably also true in those "umpire not sure" situations. 

Let's suppose a borderline pitch is a strike 60% of the time when it's from an All-Star, but only 40% of the time when it's from a mediocre pitcher.

If the umpire is completely unbiased, what should he do? Maybe call it a strike 50% of the time, since that's the overall rate. 

But then: the good pitcher will get only five strike calls when he deserves six, and the bad pitcher will get five strike calls when he only deserves four. The good pitcher suffers, and the bad pitcher benefits.

So, unbiased umpires benefit mediocre pitchers. Even if umpires were completely free of bias, the authors' methodology would nonetheless conclude that umpires are unfairly favoring low-status pitchers!


Of course, that's not what's happening, since in real life, it's the better pitchers who seem to be benefiting. (But, actually, that does lead to a fifth (perhaps implausible) possibility for what the authors observed: umpires are unbiased, but the *worse* pitchers throw more deceptive pitches for strikes.)

So, there's something else happening. And, it might just be the umpires trying to improve their accuracy.

Our hypothetical unbiased umpire will have miscalled 5 out of 10 pitches for each player. To reduce his miscall rate, he might change his strategy to a Bayesian one. 

Since he understands that the star pitcher has a 60% true strike rate in these difficult cases, he might call *all* strikes in those situations. And, since he knows the bad pitcher's strike rate is only 40%, he might call *all balls* on those pitches. 

That is: the umpire chooses the call most likely to be correct. 60% beats 40%.

With that strategy, the umpire's overall accuracy rate improves to 60%. Even if he has no desire, conscious or unconscious, to favor the ace for the specific reason of "high status", it looks like he does -- but that's just a side-effect of a deliberate attempt to increase overall accuracy.

In other words: it could be that umpires *consciously* take the history of the pitcher into account, because they believe it's more important to minimize the number of wrong calls than to spread them evenly among different skills of pitcher. 

That could just as plausibly be what the authors are observing.

How can the ump improve his accuracy without winding up advantaging or disadvantaging any particular "status" of pitcher? By calling strikes in exactly the proportion he expects from each. For the good pitcher, he calls strikes 60% of the time when he's in doubt. For the bad pitcher, he calls 40% strikes. 

That strategy increases his accuracy rate only marginally -- from 50 percent to 52 percent (60% squared plus 40% squared). But, now, at least, neither pitcher can claim he's being hurt by umpire bias. 

But, even though the result is equitable, it's only because the umpire DOES have a "status bias." He's treating the two pitchers differently, on the basis of their historical performance. But King and Kim's study won't be able to tell there's a bias, because neither pitcher is hurt. The bias is at exactly the right level.

Is that what we should want umpires to do, bias just enough to balance the advantage with the disadvantage? That's a moral question, rather than an empirical one. 

Which are the most ethical instructions to give to the umpires? 


Make what you think is the correct call, on a "more likely than not" basis, *without* taking the pitcher's identity into account.

Advantages: No "status bias."  Every pitcher is treated the same.

Disadvantages: The good pitchers wind up being disadvantaged, and the bad pitchers advantaged. Also, overall accuracy suffers.


Make what you think is the correct call, on a "more likely than not" basis, but *do* take the pitcher's identity into account.

Advantages: Maximizes overall accuracy.

Disadvantages: The bad pitchers wind up being disadvantaged, and the good pitchers advantaged.


Make what you think is the most likely correct call, but adjust only slightly for the pitcher's identity, just enough that, overall, no type of pitcher is either advantaged or disadvantaged.

Advantages: No pitcher has an inherent advantage just because he's better or worse.

Disadvantages: Hard for an umpire to calibrate his brain to get it just right. Also, overall accuracy not as good as it could be. And, how do you explain this strategy to umpires and players and fans?

Which of the three is the right answer, morally? I don't know. Actually, I don't think there necessarily is one -- I think any of the three is fair, if understood by all parties, and applied consistently. Your opinion may vary, and I may be wrong. But, that's a side issue.


Getting back to the study: the fact that umpires make more favorable mistakes for good pitchers than bad pitchers is not, by any means, evidence that they are unconsciously biased against pitchers based on "status." It could just as easily be one of several other, more plausible reasons. 

So that's why I don't accept the study's conclusions. 

There's also another reason -- the regression itself. I'll talk about that next post.

(Hat tip: Charlie Pavitt)

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Wednesday, January 07, 2015

Predicting team SH% from player talent

For NHL teams, shooting percentage (SH%) doesn't seem to carry over all that well from year to year. Here repeated from last post, are the respective correlations: 

-0.19  2014-15 vs. 2013-14
+0.30  2013-14 vs. 2012-13
+0.33  2012-13 vs. 2011-12
+0.03  2011-12 vs. 2010-11
-0.10  2010-11 vs. 2009-10
-0.27  2009-10 vs. 2008-09
+0.04  2008-09 vs. 2007-08

(All data is for 5-on-5 tied situations. Huge thanks to for making the raw data available on their website.)

They're small. Are they real? It's hard to know, because of the small sample sizes. With only 30 teams, even if SH% were totally random, you'd still get coefficients of this size -- the SD of a random 30-team correlation is 0.19.  

That means there's a lot of noise, too much noise in which to discern a small signal. To reduce that noise, I thought I'd look at the individual players on the teams.  (UPDATE: Rob Vollman did this too, see note at bottom of post.)

Start with last season, 2013-14. I found every player who had at least 20 career shots in the other six seasons in the study. Then, I projected his 2013-14 "X-axis" shooting percentage as his actual SH% in those other seasons.  

For every team, I calculated its "X-axis" shooting percentage as the average of the individual player estimates.  

(Notes: I weighted the players by actual shots, except that if a player had more shots in 2013-14 than the other years, I used the "other years" lower shot total instead of the current one. Also, the puckalytics data didn't post splits for players who spent a year with multiple teams -- it listed them only with their last team. To deal with that, when I calculated "actual" for a team, I calculated it for the Puckalytics set of players.  So the team "actual" numbers I used didn't exactly match the official ones.)

If shooting percentage is truly (or mostly) random, the correlation between team expected and team actual should be low.  

It wasn't that low. It was +0.38.  

I don't want to get too excited about that +38, because most other years didn't show that strong an effect. Here are the correlations for those other years:

+0.38  2013-14
+0.45  2012-13
+0.13  2011-12
-0.07  2010-11
-0.34  2009-10
-0.01  2008-09
+0.16  2007-08

They're very similar to the season-by-season correlations at the top of the post ... which, I guess, is to be expected, because they're roughly measuring the same thing.  

If we combine all the years into one dataset, so we have 210 points instead of 30, we get 

+0.13  7 years

That could easily be random luck.  A correlation of +0.13 would be on the edge of statistical significance if the 210 datapoints were independent. But they're not, since every player-year appears up to six different times as part of the "X-axis" variable.

It's "hockey significant," though. The coefficient is +0.30. So, for instance, at the beginning of 2013-14, when the Leafs' players historically had outshot the Panthers' players by 2.96 percentage points ... you'd forecast the actual difference to be 0.89.  (The actual difference came out to be 4.23 points, but never mind.)


The most recent three seasons appear to have higher correlations than the previous four. Again at the risk of cherry-picking ... what happens if we just consider those three?

+0.38  2013-14
+0.45  2012-13
+0.13  2011-12
+0.34  3 years

The +0.34 looks modest, but the coefficient is quite high -- 0.60. That means you have to regress out-of-sample performance only 40% back to the mean.  

Is it OK to use these three years instead of all seven? Not if the difference is just luck; only if there's something that actually makes the 2011-12 to 2013-14 more reliable.  

For instance ... it could be that the older seasons do worse because of selective sampling. If players improve slowly over their careers, then drop off a cliff ... the older seasons will be more likely comparing the player to his post-cliff performance. I have no idea if that's a relevant explanation or not, but that's the kind of argument you'd need to help justify looking at only the three seasons.

Well, at least we can check statistical significance. I created a simulation of seven 30-team seasons, where each identical team had an 8 percent chance of scoring on each of 600 identical shots. Then, I ran a correlation for only three of those seven seasons, like here.

The SD of that correlation coefficient was 0.12. So, the +0.34 in the real-life data was almost three SDs above random.

Still: we did cherry-pick our three seasons, so the raw probability is very misleading.  If it had been 8 SD or something, we would have been pretty sure that we found a real relationship, even after taking the cherry-pick into account. At 3 SD ... not so sure.


Well, suppose we split the difference ... but on the conservative side. The 7-year coefficient is 0.30. The 3-year coefficient is 0.60.  Let's try a coefficient of 0.40, which is only 1/3 of the way between 0.30 and 0.60.

If we do that, we get that the predictive ability of SH% is: one extra goal per X shots in the six surrounding seasons forecasts 0.4 extra goals per X shots this season.

For an average team, 0.4 extra goals is around 5 extra shots, or 9 extra Corsis.

In his study last month, Tango found a goal was only 4 extra Corsis.  Why the difference? Because our studies aren't measuring the same thing.  We were asking the same general question -- "if you combine "goals" and "shots," does that give you a better prediction than "shots" alone? -- but doing so by asking different specific questions.  

Tango asked how you predict half a team's games predict the other half. I was asking how you predict a team's year from its players' six surrounding years. It's possible that the "half-year" method has more luck in it ... or that other differences factor in, also.

My gut says that the answers we found are still fairly consistent.


UPDATE: Rob Vollman, of "Hockey Abstract" fame, did a similar study last summer (which I read, but had forgotten about).  Slightly different methodology, I think, but the results seem consistent.  Sorry, Rob!

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