THE BOOK--Playing The Percentages In Baseball

Bill James had this conclusion in one of his early Abstracts - given the same number of runs, consistent offense is better than inconsistent offense, and therefore inconsistent pitching is better than consistent pitching.

Franke/#1

i was just about to ask Tango if he did an article on inconsistent pitching being better.  i know i’ve read some stuff on it before, but i’d like to go back to that topic.

Frank, excellent.

Jamie, yes, it naturally follows.  The fewer shutout innings you have, the more blowout innings you have, if both teams score the same number of runs.  And the more blowout innings you have, the less impact each run has.

So, whatever you say for offense, the opposite is true for defense.

Just a gut intuition here but I suspect there’s an analogy to stock investing. Those who try to “time the market” don’t do as well as those who dollar cost average. In the case of stocks another factor is the transaction cost, but I think the logic still applies. You can’t always predict what your opponent (or the market) will do next inning, and what they did the previous inning may not tell you that much (except on defense: change pitchers!). So do the little things right:  keep investing steadily through thick and thin; don’t rely on perfect timing or the home run.

This, BTW, is why I always feel when the Tigers are playing the Angels somehow the Angels are generating the extra run here or there (often via alert and aggressive baserunning). They accrete runs.

Watching the Red Sox last season shed some light on this topic for me. While their offense scored 872 runs overall (5.38 runs/game), their run distribution was not very even:

2009 Red Sox
Total (162) 95-67 .586 872RS 5.38R/G
Blowouts (50) 34-16 .680 380RS 7.60R/G
Other (112) 61-51 .544 492RS 4.39R/G

For comparison:

2009 Yankees
Total (162) 103-59 .636 915RS 5.65R/G
Blowouts (51) 32-19 .627 394RS 7.73R/G
Other (111) 71-40 .640 521RS 4.69R/G

2009 Angels
Total (162) 97-65 .599 883RS 5.45R/G
Blowouts (46) 30-16 .652 338RS 7.35R/G
Other (116) 67-49 .578 545RS 4.70R/G

The fact that Boston’s offense was inconsistent (ie scored a greater proportion of their runs in blowouts) means that their winning percentage was significantly lower in non-blowout games compared to similar offenses who were more consistent.

I’ve done a number of studies using the Weibull distribution.  I would say that the take home message about run distributions and consistency is that I have never found a persistent skill that allows a team to skew its distribution in any significant way.

salb makes a significant point.  It’s one thing to do what I did on a theoretical level.  It’s quite another to actively create a team that can get anywhere close to an optimal setting.  Ideally, you’d like to have as many Ichiros and Figgins on your team in terms of offensive profile, all other things equal.

I saw this last night and wondered about the practical implications. I’m just asking about this stuff, I can’t follow the more sophisticated mathematics. If the professor is right about SLG, then it seems like on some sort of “win expectancy” (loosely-conceived- level, SLG might be more valuable than run-level lwts say. But if Tango is right that the prof. is wrong about SLG/OBP, than, on a practical level, this isn’t really useful, since you can’t “plan” to be more consistent other than by deliberately trying to score _less_ runs, which, um, I’ll go out on a limb and say is not a good way to win baseball games.

Also, how does this stuff jibe with Sal B.’s long series at THT from a while back?

I basically agree with the professor - it is good to be consistent in run scoring and is good to *in*consistent in run prevention.  That much is true.

However, I do not believe there is a way to make a team more or less inclined to be consistent without changing the total number of runs scored or allowed.  So, it’s all so muc inyourmothersbasementims as far as I’ve concerned.

Tango, at the end of your post, you said, “The [more consistent] first team would have high OBP and low SLG.  The [more streaky] second team would have a low OBP and a high SLG,” but you didn’t explain where that conclusion came from at all.  Can you expand on that a little?  It seems like this issue would be pretty easy to resolve.  Just run a scoring simulation for a first team with 9 .300/.350/.400 guys and a second team 9 .250/.300/.480 guys (ideally, you’d want to set the OBP and SLG of the hypothetical teams such that their average scoring output is equal--I just took a wild guess numbers that might come close), and see which scoring output has the lower standard deviation.

It does make sense that high SLG teams have more consistent scoring than high OBP teams. 

The events which most frequently score runs are extra base hits.  The high slugging team will have more xbh and thus more scoring events, but will score fewer runs on each due to fewer runners on base.  The high OBP team will have fewer xbh but will have more runners on base when they occur.

SLG as a proxy for distribution of run scoring probably breaks down at both extremes of OBP/SLG ratio - once OBP gets so high that xbh stop being a large part of the run scoring process or so low that doubles stop scoring runs.

kcdc/11, your proposal is basically right I think.

I believe the OBP/SLG balance come from the fact that two teams that are equal offensively should have similar wOBAs? You can reach a .340 team wOBA (expected offensive output) by either having more above-average on-base types or more above-average slugging types.

13: Sure, but if you run those two teams through a simulator, you will not find a (large) difference between the two run distributions.  (I tried this a few years back using a simplified simulator and couldn’t find a difference, anyway.)

Moreover, there are practical limits of creating a SLG-heavy team versus an OBP-heavy team because the two are correlated.  (or OBP vs. ISO or however you want to look at it.)

The best/worst spread in OBP in the AL last year was 45 points.  For SLG it was 81 points.  It’s impractical to construct a 350/400 team as opposed to a 300/480, since those OBP and SLG gaps are similar to the gaps between the *best* and *worst* teams (that is, without holding overall output constant).

Finally, we’re dealing with in-season sample sizes of 162 games, and no run-scoring event is going to happen more than about 15% of the time, or 25 games.  So the sample size (and therefore noise) are so large that even if you could skew your run distribution, the actual effect would be almost impossible to tease out.

If there is no optimal strategy from a managerial standpoint, would there nonetheless be an advantage to taking a team’s consistency in run production and run prevention into account as an adjustment factor in Pythaganpat estimates of W/L?

Another idea: How would one test the correlation between SLG and standard deviation of runs scored in MLB data?  I’d guess you’d hit a snag since SLG is positively correlated with runs scored, and higher runs scored would result in a larger standard deviation.  I’m guessing you could determine the relationship between runs scored and standard deviation of the runs scored distribution.  Then you could plot SLG against the standard deviation divided by your result, and see if a correlation shows up.  I’m not a statistician.  Does that make sense?

15: It all depends on what you want pythaganpat to do.  If you want it to be a measure of what happens when you smooth out oddities of run distribution to get an idea of what WPct you would expect from a team that score and allows runs at some level, then Pythaganpat works great.  That is, as a predictive tool, Pythaganpat is fine as it is.

If you want a Pythaganpat to be an analytical tool, then sure you can make some adjustments using run distributions.  Maybe you might help “explain” variations from Pythaganpat based on that, although as I stated before I believe the random error based on sample size will overwhelm any adjustment you might make.  If you really want to take RS/RA distribution into account, then I propose the following (in full snark, natch: break down RS/RA by game, not season, and assign a binary value based on the RS/RA ratio, and sum them over the season.  I call that stat “wins.”

16: If you use a Weibull distribution, then you can determine two parameters, alpha and gamma, then represent (basically) the “average runs scored” and the “shape parameter” of the distribution.  That might be one way to normalize.

What I did a few years ago at THT was to take an aribtrary cutoff (I think I used <= 2 runs) and ask, “how often did a team score <= 2 runs compared to what you would predict for a team of its overall run production?” I then compared that result to a variety of measures (HR/PA, SLG, ISO, etc.) and I don’t think I ever found a strong correlation.

kcdc: perhaps you are correct that I am wrong with my assumption.  I looked at the high scoreless-inning rate and presumed it could only happen with a low OBP team (high out team).

But, it might not be so clear.  It’s a simple enough thing to answer actually.  Bill “wes” expanded my Markov calculator to include the run scoring distribution:
http://tangotiger.net/markov_wes.html

It’s a beautiful thing he did.  Check it out.  Ok, so I dropped HR and increased walks until I got a match in runs scored.  And, doggone it, I was wrong.  The scoreless rate increased when I did that.

I guess it should have been obvious, but it wasn’t.

The professor may be right.

***

There is no real manager benefit with this knowledge.  It would be limited mostly at the GM level.

Tango: Very cool.  Do you think with a bit more robust study, this potential relationship between SLG and scoring distribution could work its way into a stat like wOBA?  I know wOBA is set up to measure run-value, but it sounds like there might be a wrinkle in the relationship between run-value and win-value that hasn’t been fully dealt with.

"Ok, so I dropped HR and increased walks until I got a match in runs scored.  And, doggone it, I was wrong.  The scoreless rate increased when I did that.”

Think of an extreme case, a team that always hits a homer once through the order but never does anything else.  A team of Dave Kingmen.  Go through the order 3 times and that team always scores exactly 3 runs.

@salb#17: I get your point about the interpretation of the Pythag calculation.  But as you know many prognosticators rely on the Pythag projected RS and RA to predict the number of team wins. Further, they take the difference between the Pythag projected wins and the actual wins at any point in the season (including the end) as indicating how lucky or unlucky a team has been.

However, this assumes that the difference between actual and Pythag wins is only due to luck. Suppose the ability to exceed the Pythag win projection were at least partly a “skill” or were predictable in some other systematic way.  Perhaps a team’s style (small ball vs. long ball) or its (unmeasured) baserunning skill (e.g., advancing on a ground out) were PERSISTENT. Suppose a particular team or manager showed an ability over several seasons to win more games than the simple Pythag projection.

In all of those cases, the difference between the Pythag and actual number of wins is not just due to chance.  Rather, it’s a “skill” (perhaps managerial skill or team style).  And if it is persistent and means that, say, an otherwise Pythag 85 win team can be expected to win 89 games, then that’s very meaningful for making projections of team wins.

So the question becomes, is the ability to exceed (or fall short of) a team’s Pythag wins persistent?  If so, then just like “consistency” (IF it is a consistent trait or habit of a team or manager), this ability (or inability) is not just a chance outcome.

(Additional question:  I wonder also whether the return to “consistency” differs depending on the overall run environment.)

22: those are all good points. 

I haven’t found, and I don’t believe anybody has found, a persistent skill to beating Pythag.  I thought the answer was in run distribution, but I haven’t found a skill there either.  Part of the problem is that the random error is something like +/-6 games over the course of a season, so teasing out true differences is very difficult.

Also, I would caution you that (unmeasured) baserunning skill is implicitly accounted for, since that would lead to more runs (sorry Angel fans).  The real question is: can teams *leverage* their run scoring or prevention in a real, persistent, repeatable way.  The answer is that nobody knows for sure, but nobody has found much evidence supporting that notion.

It would be cool if someone *did* find evidence, but I’m not holding my breath.

kcdc: I think that would be one of the last places to worry about.  More important would be baserunning (apart from SB/CS). 

Not to mention that we prefer to think of players as playing in a normal environment, not one that is more conducive to their style of play.

It’s a good thought, but it comes against the reality that the net effect will be pretty small at a player level.

re: size of effect

Assuming the prof’s right that 0.080 points of team slugging is worth 1 win over pythag expectation (and assuming the relationship is roughly linear), a player like Pujols, who slugged about .250 points above league average (and presumably took roughly one tenth of his teams plate appearances) is worth about 0.3 wins more than his WAR suggests.  It’s small, but maybe not insignificant.

It’s not .080 pts of SLG = 1 win; it’s .080 pts of SLG given the same RPG equals 1 win.  That is going to be a minuscule effect when extrapolated to players. 

Even if the professor is right that consistency in run distribution can be modeled (as opposed to Sal, who has also studied the issue and is skeptical), there surely will be a better way to model it than SLG ratio.  I bet even the professor would agree.

Maybe consistency can be modelled with a regression. Some measure of consistency can be created like the standard deviation of runs per game (or STD/runs per game) and that could be the dependent variable. Then OBP & SLG and meaures of speed and maybe some other small ball variables can be the independent variables. That might tell us what leads to consistency.

I tried to estimate the value of consistency once. I ran a regression with team winning pct as the dependent variable and runs per game, runs allowed per game, the standard deviation of runs per game and the standard deviation of runs allowed per game as the independent variables. I looked at all teams from 1996. First the results just using runs and runs allowed

Pct = .502 + .9*RG - .91*RAG

Then I put in the standard deviations

Pct = .072*RG - .099*RAG + .04*STDRG + .016*STDRAG

The standard error fell only .0013. That works out to about .21 per 162 games. So bringing in consistency did not help predict winning pct very much. I have no explanation for why the coefficient on runs per game falls so much. The STDRG is positive, meaning that the less consistent you are, the higher your winning pct. The difference between the highest and lowest is STDRG was about 1.36. That times .04 is about .054. Times 162 games is 8.75 wins. The least consistent team would win alot more games. The t-value was 1.9. But again, the sign is wrong. The t-value on STDRAG was .72. The difference between the highest and lowest was 1.42. That works out to 3.62 wins per season. I also looked at all teams from 1967-68. The regression equations were

Pct = .495 + .113*RG - .111*RAG

and

Pct = .510 + .114*RAG - .11*RAG - .004*STDRG - .007*STDRAG

The standard error actually got higher with the 2nd regression. This time the sign is right for STDRG but not for STDRAG. The difference in STDRG between the highest and lowest was 1.41. For a whole season, that works out to about .84 wins. So the most consistent team in scoring won .84 more games than the least consistent. The t-values were -.21 and -.37 for the STDs. For STDRAG, the difference between the highest and lowest was 1.06 which works out to 1.23 wins. The most consistent team in runs allowed would win 1.23 more games than the least consistent.

The coefficient on runs per game may fall because it’s correlated with the standard deviation of runs per game.

Another way to represent the variance is to use a coefficient of variation (CV), which is the standard deviation divided by the mean. In effect this “norms” the standard deviation—its size is relative to the run environment.

I think you could try substituting CVRG and CVRAG for the two std. deviation variables that you now have. And then I wouldn’t be surprised if the coefficients for RG and RAG became insignificant in such a setup, but that would be perfectly fine, because the CVRG and CVRAG would capture the run-environment-normed effects of variation/consistency in runs scored and allowed.

I’m not even sure that you would end up needing the RG and RAG variables in a final equation of the type I’ve suggested, but they would in principle be interpretable.

I just tried what you suggested for the 1967-8 data. Here is the equation

Pct = .536 + .111*RAG - .111*RAG - .026*CVRG - .01*CVRAG

The t-values for CVRG and CVRAG are -.42 & -.15. So one sign right, one wrong. The difference betweent the highest and lowest CVRG ends up being about 2.18 wins per season.

I wonder if either of the coefficients for the CV’s would become significant if you omited the first two terms.

For 1996

Pct = .265 +.099*RG - .088*RAG + .20*CVRG + .076*CVRAG

The t-values for the last two are 1.88 & .80. Like earlier, being less consistent meant a higher winning pct. The least conistent pitching team would win 3 more games. Of course, I have not looked at that much data. Maybe some day I could do 1996-2000 and 1963-68. The mixed results could be due to small sample sizes.

For 1967-8 without RG and RAG, the equation is

Pct = .583 -.44*CVRG - .34*CVRAG

The t-values were -5.43 and 3.37. So they are significant and both have the same signs.  But the standard error of the regression jumps from about .023 to .045

I meant they both had the right signs

Still, this is looking better in the sense that higher variation (less consistency) in either scoring or being scored upon is associated with a lower win probability.

Now you can try to add back one of the “enviromental variables,” let’s say RS only. That would capture the overall “level” of scoring. Or you could put in total runs in game or total runs per inning to capture the same effect.

Patriot@26--WAR already takes into account the rpg value of slug.  The .080 team slug --> 1 win is on top of what WAR already does.  a player who slugs .08 higher than his team average on 10% of team plate appearances will raise his team SLG by .008 which will be worth 0.1 wins more than is reflected in WAR.  pujols slugs more than .240 higher than league average, so assuming the relationship is correct, he’d be worth 0.3 more wins than his WAR suggests.

#23 - On beating Pythag there is one “trait” of a team that I found that allows teams to beat it, but it is tough to set up.  Basically a relief staff of half studs and half duds.  The studs keep you in games and there is a bunch of one run games, while the duds keep getting shelled when a team is behind.  A team high in one run wins and bunch of blowouts.  Arizona was this way a few years ago.

Also, pythag seems to “break” on high offensive teams.  If the team gets a 3-4 runs lead, teams put in their scrub pitchers and just take the loss.

38: Two things:

1. I looked at this, although not rigorously, and found that the studs and duds relief corp has no correlation to “beating” Pythag.  I have heard this claim often, but the data that are usually presented are of the type: “team X beat Pythag, and look at their studs and duds relief corp - that must be it!” But if you do the exercise backward - that is, look at teams that have a studs and duds relief corp and check the their Pythag differential, I believe you will not find a consistent ability to beat Pythag (but I haven’t checked rigorously).

2. Even if it were true, no GM would be able to use that as a design rule, since it is still better to have as good of a relief corp as possible (allow fewer runs) than it is to try to outgame Pythag.  At best, you can call it a place to skimp, but it’s not like GMs are going out of their way (Ed Wade excluded) to get that fourth righty out of the pen to push them over the top.

the .080 team slug --> 1 win is on top of what WAR already does.  a player who slugs .08 higher than his team average on 10% of team plate appearances will raise his team SLG by .008 which will be worth 0.1 wins more than is reflected in WAR.

That is not what Whisnant’s formula says.  It says that if a team scores the same number of runs as its opponents, but has a SLG .08 higher, then they will win an additional game. 

By raising his team’s SLG, Pujols also raises their R/G.  He only additionally increases their W% according to Whisnant’s formula if his marginal impact on SLG is disproportionate to his marginal impact on R/G.

Suppose you have a player with a wRC+ of 200 and a relative SLG of 167 (equivalent to slugging .700 in a .420 league).  With 10% of the PAs, his team would be estimated to have a run ratio of .1*(2)+.9*(1)= 1.1 and a SLG ratio of .1*(1.67)+.9*(1)= 1.067.  Assuming the averages are 4.5 and .420, they project to win 87.74 games according to his formula.

Now suppose the player had somehow managed to increase his team’s RPG to the same level without changing it’s SLG at all (this is impossible in practice).  They’d project to win 87.44 games.  So there is only a .3 win difference, between players with equal contributions as measured by a run estimator, but wildly divergent SLG.

The types of players for whom the minor effect would come into play at all would be those who have wOBA > OBA, generally.  Pujols does have a higher career wOBA than OBA, but only to a small extent (.436 to .427).  Most of the impact of his SLG has already been captured in his estimated run contribution.

Interestingly, Whisnant’s formula is more conservative than Pythagenpat in many cases.  For example, for the hypothetical team described above, Whisnant projects 87.74 wins.  Pythpat expects 88.38.  So if Whisnant’s actual formula (rather than the principle) is taken at face value, the value estimates of a lot of big hitters will go down regardless of their SLG to other offense relationship.

Back in college, when I was first reading Bill James on the pythagorean theorem, I had thought of this situation where teams might have the same mean runs per game but runs were not evenly distributed.

You win the game by having one more run than your opponent at the end of the game. It’s also the same in elections. Any runs (or votes) greater than n+1 are excess and did not contribute to winning this particular contest.

I thought a better predictor of wins and losses from runs scored would use a gamme distribution, which is used to describe accumulations, such as numbers of people who make x dollars per year.

What percentage of the time did a team score exactly one run? What percentage did they allow zero? Multiply them. Repeat for two runs scored, fewer than two allowed, etc, then add the products to get the team winning percentage.

(I am looking at my notebook now, but have forgotten what 90% of the math I wrote down 30 years ago actually means). Anyway, I have some graphs of the 1979 NL, getting the league values of gamma and beta (describing the mean and the shape of the distribution).

I believe that back then I understood how to take the team’s batting and pitching runs profile and get the estimated winning percentage, and if memory serves me correctly, I came to the same conclusions as here - a consistent offense scores fewer excess runs, so their runs are better leveraged into wins.

Back to politics, it is true that Democrat voters are much more concentrated in urban areas than Republicans, who are more likely to be the majority in rural areas. Democrats voters are packed into fewer districts that they are almost certain to win with 80 or 90%, while Republicans can win more districts with a 60% vote.

Patriot@40

Good point re: wOBA relative to OBP.  A player like Pujols increases his team’s slugging which improves RPG distribution, but he also increases his team’s OBP which, will increasing RPG, also makes for more clustered scoring.  Further study could be conducted to determine how much distribution bonus should be added for a given player’s SLG and OBP.  I’m still guessing it will be at least a few points of wOBA for a few all-power-no-OBP guys.

Phil’s take:

http://sabermetricresearch.blogspot.com/2010/03/improving-on-pythagoras.html