At this point, I want to re-explain how to use the Enby distribution, step-by-step. While I already did this in part 6 of the original series, I now have the new variance estimator as found by Alan Jordan to plug in, and so to avoid any confusion and to make this is easy if anyone ever wants to implement it themselves, I will recount it all in one location. I will also re-introduce a spreadsheet that you can use to estimate the probability of scoring X runs based on the Enby distribution.

Step 1: Estimate the variance of runs scored per game (VAR) as a function of mean runs/game (RG):

VAR = RG^2/9 + (2/c - 1)*RG

where c is the control value from the Tango Distribution. For normal applications, we’ll assume that c = .767.

Step 2: Use the mean and variance to estimate the parameters (r and B) of the negative binomial distribution:

r = RG^2/(VAR - RG)

B = VAR/RG - 1

B will be retained as a parameter for the Enby distribution.

Step 3: Find the probability of zero runs scored as estimated by the negative binomial distribution (we’ll call this value a):

a = (1 + B)^(-r)

Step 4: Use the Tango Distribution to estimate the probability of being shutout. This will become the Enby distribution parameter z:

z =(RI/(RI + c*RI^2))^9

where RI is runs/inning, which we’ll estimate as RG/9.

Step 5: Use trial and error to estimate a new value of r given the modified value at zero. B and z will stay constant, but r must be chosen so as to ensure that the correct mean RG is returned by the Enby distribution. Use the following formula to estimate the probability of k runs scored per game using the non-modified negative binomial distribution:

q(0) = a

q(k) = (r)(r + 1)(r + 2)(r + 3)…(r + k - 1)*B^k/(k!*(1 + B)^(r + k)) for k >=1

Then modify by taking:

p(0) = z

p(k) = (1 - z)*q(k)/(1 - a)for k >=1

The mean is calculated as:

mean = sum (from k = 1 to infinity) of (k*p(k)) = p(1) + 2*p(2) + 3*p(3) + ...

Now you have the parameters r, B, and z and the probability of scoring k runs in a game.

I previously published a spreadsheet that provided the approximate Enby distribution parameters at each .05 increment of RG between 3 and 7. The link below will take you to an updated version of this calculator. It is updated in two ways: first, the Tango Distribution estimate of variance developed by Alan Jordan is used as in the example above. Secondly, I have added lines for RG levels between 0-3 and 7-15 RG (at intervals of .25). Previously, you could enter in any value between 3-7 RG and the calculator would round it to nearest .05; now I’m going to make you enter a legitimate value yourself or accept whatever vlookup() gives you.

P(x) is the probability of scoring x runs in a game, P(<= x) is the probability of scoring that many or fewer, and P(> x) is the probability of scoring more than x runs.

Enby Calculator

## Wednesday, July 12, 2017

### Enby Distribution, pt. 3: Enby Distribution Calculator

## Tuesday, June 20, 2017

### Enby Distribution, pt. 2: Revamping the Variance Estimate

All models are approximations of reality, but some are more useful than others. The notion of being able to estimate the runs per game distribution cleanly in one algorithm (rather than patching together runs per inning distributions or using simulators) is one that can be quite useful in estimating winning percentage or trying to distinguish between the effectiveness of team offense beyond similar noting their runs scored total. I’d argue that a runs per game distribution is a fundamentally useful tool in classical sabermetrics.

However, while such a model would be useful, Enby as currently constructed falls well short of being an ideal tool. There are a few major issues:

1) It is not mathematically feasible to solve directly for the parameters of a zero-modified negative binomial distribution, which forces me to use trial and error to estimate Enby coefficients. In doing so, the distribution is no longer able to exactly match the expected mean and variance--instead, I have chosen to match the mean precisely, and hope that the variance is not too badly distorted.

2) The variance that we should expect for runs per game at any given level of average R/G is itself unknown. I developed a simple formula to estimate variance based on some actual team data, but that formula is far from perfect and there’s no particular reason to expect it to perform well outside of the R/G range represented by the data from which it was developed.

3) An issue with run distribution models found by Tom Tango in the course of his research on runs per inning distribution is that the optimal fit for a single team’s distribution may not return optimal results in situations in which two teams are examined simultaneously (such as using the distribution to model winning percentage). One explanation for this phenomenon is the covariance between runs scored and runs allowed in a given game, due to either environmental or strategic causes.

I have recently attempted to improve the Enby distribution by focusing on these obvious flaws. Unfortunately, my findings were not as useful as I had hoped they would be, but I would argue (hope?) that they represent at least small progress in this endeavor.

During the course of writing the original series on this topic, I was made aware of work being done by Alan Jordan, who was developing a spreadsheet that used the Tango Distribution to estimate scoring distributions and winning percentage. One of the underpinnings was that he found (or found work by Darren Glass and Phillip Lowry that demonstrated) that the variance of runs scored per inning as predicted by the Tango Distribution could be calculated as follows (where RI = runs per inning and c is the Tango Distribution constant):

Variance (inning) = RI*(2/c + RI - 1) = RI^2 + (2/c - 1)*RI

Assuming independence of runs per inning (this is a necessary assumption to use the Tango Distribution to estimate runs per game), the variance of runs per game will simply be nine times the variance of runs per inning (assuming of course that there are precisely nine innings per game, as I did in estimating the z parameter of Enby from the Tango Distribution). If we attempt to simply this further by assuming that RI = RG/9, where RG = runs per game:

Variance (game) = 9*(RI^2 + (2/c - 1)*RI) = 9*((RG/9)^2 + (2/c - 1)*RG/9) = RG^2/9 + (2/c - 1)*RG

The traditional value of c used to estimate runs per inning for one team is .767, so if we substitute that for c, we wind up with:

Variance (game) =1.608*RG + .111*RG^2

When I worked on this problem previously, I did not have any theoretical basis for an estimator of variance as a function of RG, so I experimented with a few possibilities and found what appeared to be a workable correlation between mean RG and the ratio of variance to mean. I used linear regression on a set of actual team data (1981-1996) and wound up with an equation that could be written as:

Variance (game) = 1.43*RG + .1345*RG^2

Note the similarities between this equation and the equation based on the Tango Distribution - they both take the form of a quadratic equation less the constant (I purposefully avoided constants in developing my variance estimator so as to avoid unreasonable results at zero and near-zero RG). The coefficients are somewhat different, but the form of the equation is identical.

On one hand, this is wonderful for me, because it vindicates my intuition that this was a reasonable way to estimate variance. On the other hand, this is very disappointing, because I had hoped that Jordan’s insight would allow me to significantly improve the variance estimate. Instead, any gains to be had here are limited to improving the equation by using a more theoretical basis to estimate its coefficients, but there is no change in the form of this equation.

In fact, any revision to the estimator will reduce accuracy over the 1981-96 sample that I am using, since the linear regression already found optimal coefficients for this particular dataset. This by no means should be taken as a claim on my part that the regression-based equation should be used rather than the more theoretically-grounded Tango Distribution estimate, simply an observation that any improvement will not show up given the confines of the data I have at hand.

What about data from out of that set? I have easy access to the four seasons from 2009-2012. In these seasons, major league teams have averaged 4.401 runs per game and the variance of runs scored per game is 9.373. My equation estimates the variance should be 8.90, while the Tango-based formula estimates 9.23. In this case, we could get a near-precise match by using c = .757.

While we know how accurate each estimator is with respect to variance for this case, what happens when we put Enby to use to estimate the run distribution? The Enby parameters for 4.40 RG using my original equation are (B = 1.0218, r = 4.353, z = .0569). If we instead use the Tango estimated variance of 9.23, the parameters become (B = 1.0970, r = 4.041, z = .0569). With that, we can calculate the estimated frequencies of X runs scored using each estimator and compare to the empirical frequencies from 2009-2012:

Eyeballing this, the Tango-based formula is closer for one run, but exacerbates the recurring issue of over-estimating the likelihood of two or three runs. It makes up for this by providing a better estimate at four and five runs, but a worse estimate at six. After that the two are similar, although the Tango estimate provides for more probability in the tail of the distribution, which in this case is consistent with empirical results.

For now, I will move on to another topic, but I will eventually be coming back to this form of the Tango-based variance estimate, re-estimating the parameters for 3-7 RG, and providing an updated Enby calculator, as I do feel that there are distinct advantages to using the theoretical coefficients of the variance estimator rather than my empirical coefficients.

## Tuesday, May 09, 2017

### Enby Distribution, pt. 1: Pioneers

A few years ago, I attempted to demonstrate that one could do a decent job of estimating the distribution of runs scored per game by using the negative binomial distribution, particularly a zero-modified version given the propensity of an unadulterated negative binomial distribution to underestimate the probability of a shutout. I dubbed this modified distribution Enby.

I’m going to be re-introducing this distribution and adopting a modification to the key formula in this series, but I wanted to start by acknowledging that I am not the first sabermetrician to adopt the negative binomial distribution to the matter of the runs per game distribution. To my knowledge, a zero-modified negative binomial distribution had not been implemented prior to Enby, and while the zero-modification is a significant improvement to the model, it would be disingenuous not to acknowledge and provide an overview of the two previous efforts using the negative binomial distribution of which I am aware.

I acknowledged one of these in the original iteration of this series, but inadvertently overlooked the first. In the early issues of Bill James’ __Baseball Analyst__ newsletter, Dallas Adams published a series of articles on run distributions, ultimately developing an unwieldy formula I discussed in the linked post. What I overlooked was an article in the August 1983 edition in which the author noted that the Poisson distribution worked for hockey, it would not work for baseball because the variance of runs per game is not equal to the mean, but rather is twice the mean. But a "modified Poisson" distribution provided a solution.

The author of the piece? Pete Palmer. Palmer is often overlooked to an undue extent when sabermetric history is recounted. While one could never omit Palmer from such a discussion, his importance is often downplayed. But the sheer volume of methods that he developed or refined is such that I have no qualms about naming him the most important technical sabermetrician by a wide margin. Park factors, run to win converters, linear weights, relative statistics, OPS for better or worse, the construct of an overall metric by adding together runs above average in various discrete components of the game...these were all either pioneered or greatly improved by Palmer. And while it is not nearly as widespread in use as his other innovations, you can add using the negative binomial distribution for the runs per game distribution the list.

Palmer says that he learned about this “modified Poisson” in a book called __Facts From Figures__ by Maroney. The relevant formulas were:

Mean (u) = p/c

Variance (v) = u + u/c

p(0) = (c/(1 + c))^p

p(1) = p(0)*p/(1 + c)

p(2) = p(1)*(p + 1)/(2*(1 + c))

p(3) = p(2)*(p + 2)/(3*(1 + c))

p(n) = p(0)*(p*(p + 1)*(p + 2)*...*(p + n - 1)/(n!*(1 + c)^n)

The text that I used renders the negative binomial distribution as:

p(k) = (1 + B)^(-r) for k = 0

p(k) = (r)(r + 1)(r + 2)(r + 3)…(r + k - 1)*B^k/(k!*(1 + B)^(r + k)) for k >=1

mean (u) = r*B

variance(v) = r*B*(1 + B)

You may be forgiven for not immediately recognizing these two as equivalent; I did not at first glance. But if you recognize that r = p and B = 1/c, then you will find that the mean and variance equations are equivalent and that the formulas for each n or k depending on the nomenclature used are equivalent as well.

So Palmer was positing the negative binomial distribution to model runs scored. He noted that the variance of runs per game is about two times the mean, which is true. In my original Enby implementation, I estimated variance as 1.430*mean + .1345*mean^2, which for the typical mean value of around 4.5 R/G works out to an estimated variance of 9.159, which is 2.04 times the mean. Of course, the model can be made more accurate by allowing the ratio

if variance/mean to vary from two.

The second use of the negative binomial distribution to model runs per game of which I am aware was implemented by Phil Melita. Mr. Melita used it to estimate winning percentage and sent me a copy of his paper (over a decade ago, which is profoundly disturbing in the existential sense). Unfortunately, I am not aware of the paper ever being published so I hesitate to share too much from the copy in my possession.

Melita’s focus was on estimating W%, but he did use negative binomial to look at the run distribution in isolation as well. Unfortunately, I had forgotten his article when I started messing around with various distributions that could be used to model runs per game; when I tried negative binomial and got promising results, I realized that I had seen it before.

So as I begin this update of what I call Enby, I want to be very clear that I am not claiming to have “discovered” the application of the negative binomial distribution in this context. To my knowledge using zero-modification is a new (to sabermetrics) application of the negative binomial, but obviously is a relatively minor twist on the more important task of finding a suitable distribution to use. So if you find that my work in this series has any value at all, remember that Pete Palmer and Phil Melita deserve much of the credit for first applying the negative binomial distribution to runs scored per game.

## Thursday, April 13, 2017

## Saturday, April 01, 2017

### 2017 Predictions

All the usual disclaimers. This is not serious business.

AL EAST

1. Boston

2. Toronto (wildcard)

3. New York

4. Baltimore

5. Tampa Bay

I have noted the last couple years that I always pick the Red Sox--last year was one of the years where that was the right call. Boston has question marks, and they have less talent on hand to fill holes than in past years, but no one else in the division is making a concerted push with the Blue Jays retrenching and the Yankees in transition. While much has been made of the NL featuring more of a clear dichotomy between contenders and rebuilders, the AL features three strong division favorites and a void for wildcard contention that Toronto may well once again fill. New York looks like a .500 team to me, and one with as strong a recent history of overperforming projections/Pythagorean as darlings like Baltimore and Kansas City, but get far less press for it. (I guess the mighty Yankees aren’t a good sell as a team being unfairly dismissed by the statheads). The Orioles offense has to take step back at some point with only Machado and Schoop being young, and if that happens the rotation can’t carry them. It’s not that I think the Rays are bad; this whole division is filled with potential wildcard contenders.

AL CENTRAL

1. Cleveland

2. Detroit

3. Kansas City

4. Minnesota

5. Chicago

I have a general policy of trying to pick against the Indians when reasonable, out of irrational superstition and an attempt to counteract any unconscious fan-infused optimism. Last year I felt they were definitely the best team in this division on paper but picked against them regardless. But the gap is just too big to ignore this season, so I warily pick them in front. There are reasons to be pessimistic--while they didn’t get “every break in the world last season” as Chris Russo says in a commercial that hopefully will be off the air soon, it’s easy to overstate the impact of their pitching injuries since the division was basically wrapped up before the wheels came off the rotation. Consider the volatility of bullpens, the extra workload for the pitchers who were available in October, the fact that the two that weren’t aren’t the best health bets in the world, and you can paint a bleaker picture than the triumphalism that appears to be the consensus. On the other hand, Michael Brantley, the catchers, the fact that the offense didn’t score more runs than RC called for last year. I see them as the fourth-strongest team out of the six consensus division favorites. Detroit is the team best-positioned to challenge them; I used the phrase “dead cat bounce” last year and it remains appropriate. The less said about Kansas City the better, but as much fun as it was to watch the magic dissipate last season, the death throes of this infuriating team could be even better. The Twins have famously gone from worst to first in their franchise history; given the weakness of the division and some young players who may be much better than they’ve shown so far, it’s not that far-fetched, but it’s also more likely that they lose 95 again. The White Sox rebuilding might succeed in helping them compete down the road and finally ridding the world of the disease that is Hawk Harrelson.

AL WEST

1. Houston

2. Seattle (wildcard)

3. Los Angeles

4. Texas

5. Oakland

Houston looks really good to me; if their rotation holds together (or if they patch any holes with the long awaited Jose Quintana acquisition), I see them as an elite team. Maybe the third time is the charm picking Seattle to win the wildcard. Truth be told, I find it hard to distinguish between most AL teams including the middle three in this division. Picking the Angels ahead of the Rangers is more a way to go on record disbelieving that the latter can do it again than an endorsement of the former, but even with a shaky rotation the Angels should be respectable. My Texas pick will probably look terrible when Nomar Mazara breaks out, Yu Darvish returns healthy, and Josh Hamilton rises from the dead or something. Oakland’s outlook for this year looks bleak, but am I crazy to have read their chapter in __Baseball Prospectus__ and thought there were a number of really interesting prospects who could have a sneaky contender season in 2018? Probably.

NL EAST

1. Washington

2. New York (wildcard)

3. Miami

4. Atlanta

5. Philadelphia

It’s very tempting to pick New York over Washington, based on the superficial like the Nationals sad-Giants even year pattern and cashing in most of their trade chits for Adam Eaton, but there remains a significant on-paper gap between the two. Especially since the Mets stood pat from a major league roster perspective. This might be the best division race out there in a season in which there are six fairly obvious favorites. Sadly, Miami is about one 5 WAR player away from being right in the mix…I wonder where on might have found such a player? Atlanta seems like a better bet than Philadelphia in both the present and future tense, but having a great deal of confidence in the ordering of the two seems foolhardy.

NL CENTRAL

1. Chicago

2. Pittsburgh

3. St. Louis

4. Milwaukee

5. Cincinnati

The Cubs’ starting pitching depth is a little shaky? Kyle Schwarber doesn’t have a position and people might be a little too enthusiastic about him? Hector Rondon struggled late in the season and Wade Davis’ health is not a sure thing? These are the straws that one must grasp at to figure out how Chicago might be defeated. You also have to figure out whether Pittsburgh can get enough production from its non-outfielders while also having some good fortune with their pitching. Or whether St. Louis’ offense is good enough. Or whether Milwaukee or Cincinnati might have a time machine that could jump their rebuild forward a few years. You know, the normal questions you ask about a division.

NL WEST

1. Los Angeles

2. San Francisco

3. Arizona

4. Colorado

5. San Diego

Last year I picked the Giants over the Dodgers despite the numbers suggesting otherwise because of injury concerns. I won’t make that mistake again, as it looks as if LA could once again juggle their rotation and use their resources to patch over any holes. The Giants are strong themselves, but while the two appear close in run prevention, the Dodgers have the edge offensively. The Diamondbacks should have a bounce back season, but one that would still probably break Tony LaRussa’s heart if he still cared. The Rockies seem like they should project better than they do, with more promise on the mound than they usually do. The Padres are the consensus worst team in baseball from all of the projection systems, which can be summed up with two words: Jered Weaver.

WORLD SERIES

Los Angeles over Houston

Just about every projection system out there has the Dodgers ever so slightly ahead of the Cubs. That of course does not mean they are all right--perhaps there is some blind spot about these teams that player projection systems and/or collation of said projections into team win estimates share in common. On the other hand, none of these systems *dislike* the Cubs—everyone projects them to win a lot of games. I was leaning towards picking LA even before I saw that it was bordering on a consensus, because the two teams look fairly even to me but the Dodgers have more depth on hand, particularly in the starting pitching department (the natural rebuttal is that the Dodgers are likely to need that depth, while the Cubs have a four pretty reliable starters). The Dodgers bullpen looks better, and their offense is nothing to sneeze at.

AL Rookie of the Year: LF Andrew Benintendi, BOS

AL Cy Young: Chris Sale, BOS

AL MVP: CF George Springer, HOU

NL Rookie of the Year: SS Dansby Swanson, ATL

NL Cy Young: Stephen Strasburg, WAS

NL MVP: 1B Anthony Rizzo, CHN

## Tuesday, March 14, 2017

### Win Value of Pitcher Adjusted Run Averages

The most common class of metrics used in sabermetrics for cross-era comparisons use relative measures of actual or estimated runs per out or sother similar denominator. These include ERA+ for pitchers and OPS+ or wRC+ for batters (OPS+ being an estimate of relative runs per out, wRC+ using plate appearances in the denominator but accounting for the impact of avoiding outs). While these metrics provide an estimate of runs relative to the league average, they implicitly assume that the resulting relative scoring level is equally valuable across all run environments.

This is in fact not the case, as it is well-established that the relationship between run ratio and winning percentage depends on the overall level of run scoring. A team with a run ratio of 1.25 will have a different expected winning percentage if they play in a 9 RPG environment than if they play in a 10 RPG environment. Metrics like ERA+ and OPS+ do not translate relative runs into relative wins, but presumably the users of such metrics are ultimately interested in what they tell us about player contribution to wins.

There are two key points that should be acknowledged upfront. One is that the difference in win value based on scoring level is usually quite small. If it wasn’t, winning percentage estimators that don’t take scoring level into account would not be able to accurately estimate W% across the spectrum of major league teams. While methods that do consider scoring level are more accurate estimators of W% than similar methods that don’t, a method like fixed exponent Pythagorean can still produce useful estimates despite maintaining a fixed relationship between runs and wins.

The second is that players are not teams. The natural temptation (and one I will knowingly succumb to in what follows) is to simply plug the player’s run ratio into the formula and convert to a W%. This approach ignores the fact that an individual player’s run rate does not lead directly to wins, as the performance of his teammates must be included as well. Pitchers are close, because while they are in the game they are the team (more accurately, their runs allowed figures reflect the totality of the defense, which includes contributions from the fielders), but even ignoring fielding, non-complete games include innings pitched by teammates as well.

For the moment I will set that aside and instead pretend (in the tradition of Bill James’ Offensive Winning %) that a player or pitcher’s run ratio can or should be converted directly to wins, without weighting the rest of the team. This makes the figures that follow something of a freak show stat, but the approach could be applied directly to team run ratios as well. Individuals are generally more interesting and obviously more extreme, which means that the impact of considering run environment will be overstated.

I will focus on pitchers for this example and will use Bob Gibson’s 1968 season as an example. Gibson allowed 49 runs in 304.2 innings, which works out to a run average of 1.45 (there will be some rounding discrepancies in the figures). In 1968 the NL average RA was 3.42, so Gibson’s adjusted RA (aRA for the sake of this post) is RA/LgRA = .423 (ideally you would park-adjust as well, but I am ignoring park factors for this post). As an aside, please resist the temptation to instead cite his RA+ of 236 instead. Please.

.423 is a run ratio; Gibson allowed runs at 42.3% of the league average. Since wins are the ultimate unit of measurement, it is tempting to convert this run ratio to a win ratio. We could simply square it, which reflects a Pythagorean relationship. Ideally, though, we should consider the run environment. The 1968 NL was an extremely low scoring league. Pythagenpat suggests that the ideal exponent is around 1.746. Let’s define the Pythagenpat exponent to use as:

x = (2*LgRA)^.29

Note that this simply uses the league scoring level to convert to wins; it does not take into account Gibson’s own performance. That would be an additional enhancement, but it would also strongly increase the distortion that comes from viewing a player as his own team, albeit less so for pitchers and especially those who basically were pitching nine innings/start as in the case of Gibson.

So we could calculate a loss ratio as aRA^x, or .223 for Gibson. This means that a team with Gibson’s aRA in this environment would be expected to have .223 losses for every win (basic ratio transformations apply; the reciprocal would be the win ratio, the loss ratio divided by (1 + itself) would be a losing %, the complement of that W%, etc.)

At this point, many people would like to convert it to a W% and stop there, but I’d like to preserve the scale of a run average while reflecting the win impact. In order to do so, I need to select a Pythagorean exponent corresponding to a reference run environment to convert Gibson’s loss ratio back to an equivalent aRA for that run environment. For 1901-2015, the major league average RA was 4.427, which I’ll use as the reference environment, which corresponds to a 1.882 Pythagenpat exponent (there are actually 8.94 IP/G over this span, so the actual RPG is 8.937 which would be a 1.887 exponent--I'll stick with RA rather than RPG for this example since we are already using it to calculate aRA).

If we call that 1.882 exponent r, then the loss ratio can be converted back to an equivalent aRA by raising it to the (1/r) power. Of course, the loss ratio is just an interim step, and this is equivalent to:

aRA^(x*(1/r)) = aRA^(x/r) = waRA

waRA (excuse the acronyms, which I don’t intend to survive beyond this post) is win-Adjusted Run Average. For Gibson, it works out to .450, which illustrates how small the impact is. Pitching in one of the most extreme run environments in history, Gibsons aRA is only 6.4% higher after adjusting for win impact.

In 1994, Greg Maddux allowed 44 runs in 202 innings for a run average of 1.96. Pitching in a league with a RA of 4.65, his aRA was .421, basically equal to Gibson. But his waRA was better, at .416, since the same run ratio leads to more wins in a higher scoring environment.

It is my guess that consumers of sabermetrics will generally find this result unsatisfactory. There seems to be a commonly-held belief that it is easier to achieve a high ERA+ in a higher run scoring environment, but the result of this approach is the opposite--as RPG increases, the win impact of the same aRA increases as well. Of course, this approach says nothing about how “easy” it is to achieve a given aRA--it converts aRA to an win-value equivalent aRA in a reference run environment. It is possible that it could be simultaneously “easier” to achieve a low aRA in a higher scoring environment and that the value of a low aRA be enhanced in a higher scoring environment. I am making no claim regarding the impressiveness or aesthetic value, etc. of any pitcher’s performance, only attempting to frame it in terms of win value.

Of course, the comparison between Gibson and Maddux need not stop there. I do believe that waRA shows us that Maddux’ rate of allowing runs was more valuable in context than Gibson’s, but there is more to value than the rate of allowing runs. Of course we could calculate a baselined metric like WAR to value the two seasons, but even if we limit ourselves to looking at rates, there is an additional consideration that can be added.

So far, I’ve simply used the league average to represent the run environment, but a pitcher has a large impact on the run environment through his own performance. If we want to take this into account, it would be inappropriate to simply use LgRA + pitcher’s RA as the new RPG to plug into Pythagenpat; we definitely need to consider the extent to which the pitcher’s teammates influence the run environment, since ultimately Gibson’s performance was converted into wins in the context of games played by the Cardinals, not a hypothetical all-Gibson team. So I will calculate a new RPG instead by assuming that the 18 innings in a game (to be more precise for a given context, two times the league average IP/G) is filled in by the pitcher’s RA for his IP/G, and the league’s RA for the remainder.

In the 1968 NL, the average IP/G was 9.03 and Gibson’s 304.2 IP were over 34 appearances (8.96 IP/G), so the new RPG is 8.96*1.45/9 + (2*9.03 - 8.96)* 3.42/9 = 4.90 (rather than 6.84 previously). This converts to a Pythagenpat exponent of 1.59, and an pwaRA (personal win-Adjusted Run Average?) of .485. To spell that all out in a formula:

px = ((IP/G)*RA/9 + (2*Lg(IP/G) - IP/G)*LgRA/9) ^ .29

pwaRA = aRA^(px/r)

Note that adjusting for the pitcher’s impact on the scoring context reduces the win impact of effective pitchers, because as discussed earlier, lowering the RPG lowers the Pythagenpat exponent and makes the same run ratio convert to fewer wins. In fact, considering the pitcher’s effect on the run environment in which he operates actually brings most starting pitchers’ pwaRA closer to league average than their aRA is.

pwaRA is divorced from any real sort of baseball meaning, though, because pitchers aren’t by themselves a team. Suppose we calculated pwaRA for two teammates in a 4.5 RA league. The starter pitches 6 innings and allows 2 runs; the reliever pitches 3 innings and allows 1. Both pitchers have a RA of 3.00, and thus identical aRA (.667) or waRA (.665). Furthermore, their team also has a RA of 3.00 for this game, and whether figured as a whole or as the weighted average of the two individuals, the team also has the same aRA and waRA.

However, if we calculate the starter’s pwaRA, we get .675, while the reliever is at .667. Meanwhile, the team has a pwaRA of .679, which makes this all seem quite counterintuitive. But since all three entities have the same RA, the lower the run environment, the less win value it has on a per inning basis.

I hope this post serves as a demonstration of the difficulty of divorcing a pitcher’s value from the number of innings he pitched. Of course, the effects discussed here are very small, much smaller than the impact of other related differences, like the inherent statistical advantage of pitchers over shorter stints, attempts to model differences in replacement level between starters and relievers, and attempts to detect/value any beneficial side effects of starters working deep into games.

One of my long-standing interests has been the proper rate stat to use to express a batter’s run contribution (I have been promising myself for almost as long as this blog has been existence that I will write a series of posts explaining the various options for such a metric and the rationale for each, yet have failed to do so). I’ve never had the same pull to the question for pitchers, in part because the building block seems obvious: runs/out (which depending on how one defines terms can manifest itself as RA, ERA, component ERA, FIP-type metrics, etc.)

But while there are a few adjustments that can theoretically made between a hitter’s overall performance expressed as a rate and a final value metric (like WAR), the adjustments (such as the hitter’s impact on his team’s run scoring beyond what the metric captures itself, and the secondary effect that follows on the run/win conversion) are quite minor in scale compared to similar adjustments for pitchers. While the pitcher (along with his fielders) can be thought as embodying the entire team while he is the game, that also means that said unit’s impact on the run/win conversion is significant. And while there are certainly cases of batters whose rates may be deceiving because of how they are deployed by their managers (particularly platooning), the additional playing time over which a rate is spread increases value in a WAR-like metric without any special adjustment. Pitchers’ roles and secondary effects thereof (like any potential value generated by “eating” innings) have a more significant (and more difficult to model) impact on value than the comparable effects for position players.

## Monday, February 13, 2017

### Rebuilding a Strip Mall

"Rebuilding", as commonly thrown around in sports discussions, is an interesting term. It inherently implies that something had been built on the same spot previously. It does not, however, give an indication whether what was built there was a blanket fort or the Taj Mahal, a strip mall or the Sears Tower. If one rebuilds on the site of a strip mall, does "re-" imply they are building another strip mall, or might they be building something else?

The baseball program that Greg Beals has presided over for six seasons at The Ohio State University has been much more of a strip mall than a Sears Tower. After his most successful season, which saw OSU tie for third in the Big Ten regular season, win the Big Ten Tournament, and qualify for their first NCAA regional since 2009, Beals is now faced with a rebuilding project in the classic sports sense. Of the nine players with the most PA in 2016, OSU must replace seven, so it would be fair to say that there will be seven new regulars. OSU must also replace two of its three weekend starters; the bullpen is the only area of the roster not decimated by graduation and the draft.

Note: The discussion of potential player roles that follows is my own opinion, informed by my own knowledge of the players and close watching of the program and information released by the SID, particularly the season preview posted here.

Sophomore Jacob Barnwell will almost certainly be the primary catcher; he played sparingly last season (just 29 PA). This is one of the few open positions not due to loss, but rather to a position switch which will be discussed in a moment. Classmate Andrew Fishel (8 PA) will serve as his backup.

First base/DH will be shared by senior Zach Ratcliff, who has flashed power at times during his career but has never earned consistent playing time, and Boo Coolen, a junior Hawaii native who played at Cypress CC in California. Junior Noah McGowan, a transfer from McLennan CC in Texas, would appear to have the inside track at the keystone; his JUCO numbers are impressive but come with obvious caveats. Sophomore Brady Cherry, who got off to a torrid start in 2016 but then cooled precipitously (final line .218/.307/.411 in 143 PA) is likely to play third and bat in the middle of the order. At shortstop, senior captain Jalen Washington moves out from behind the plate to captain the infield; he spent his first two years as a Buckeye as a utility infielder, so it was the move to catcher, not to shortstop that really stands out. Unfortunately, Washington didn’t offer much with the bat as a junior (.249/.331/.343 in 261 PA). Other infield contenders include true freshman shortstop Noah West, redshirt freshman middle infielder Casey Demko, true freshman Conor Pohl at the corners, and redshirt sophomore Nate Romans and redshirt freshman Matt Carpenter in utility roles.

The one thing that appears clear in the outfielder is that junior Tre’ Gantt will take over as center fielder; he struggled offensively last season (.255/.311/.314 in 158 PA). True freshman Dominic Canzone may step in right away in right field, while left field/DH might be split between a pair of transfers. Tyler Cowles, a junior Columbus native who hit well at Sinclair CC in Georgia will attempt to join Coolen and satisfy the Beals’ desperate need for bats with experience and power. Other outfielders include senior former pitcher Shea Murray and little-used redshirt sophomore Ridge Winand.

The pitching staff is slightly more intact, but not much so. Redshirt junior captain Adam Niemeyer will likely be the #1 starter as the only returning weekend starter; his 2016 campaign can be fairly described as average. Sophomore Ryan Feltner was the #4 starter last year and so is a safe bet to pitch on the weekend; his 5.67 eRA was not encouraging but 8 K/3.9 W suggest some raw, harness-able ability. The third spot will apparently go to an erstwhile reliever. Junior Yianni Pavlopoulos was a surprising choice as closer last year, but pitched very well (10.3 K/3.3 W, 3.72 eRA), while senior Jake Post returns from a season wiped out by injury. Neither pitcher has been the picture of health throughout their careers, but Pavlopoulos seems the more likely choice to start. Junior Austin Woodby (7.75 eRA in 39 innings) and sophomore lefty Connor Curlis (six relief innings) will jockey for weekday assignments along with junior JUCO transfer Reece Calvert (a teammate of McGowan) and three true freshmen: lefty Michael McDonough and righties Collin Lollar and Gavin Lyon.

The bullpen will be well-stocked, even assuming Pavlopoulos takes a spot in the rotation. Junior sidearmer Seth Kinker was a workhorse (team-high 38 appearances) and behind departed ace Tanner Tully was arguably Ohio’s most valuable pitcher in 2016. Senior Jake Post will return from a season lost to injury looking to return to a setup role, and junior sidearmer Kyle Michalik pitched well in middle relief last season. These four form a formidable bullpen that will almost certainly be augmented by a lefty specialist, a favorite of Beals. He’ll choose from senior Joe Stoll (twelve unsuccessful appearances), true freshman Andrew Magno, and the favorite in my book is Curlis should be not best Woodby for a starting spot. It appears that sophomore JUCO transfer Thomas Waning (also a sidearmer; one of the few positives about Beals as a coach is his affinity for sidearmers). Other right-handed options for the pen will include junior Dustin Jourdan (a third JUCO transfer from McLennan), sophomore Kent Axcell (making the jump from the club team), and true freshman Jake Vance.

The non-conference schedule is again rather unambitious. The season opens the weekend of February 17 in central Florida with neutral site games against Kansas State (two), Delaware, and Pitt. Two games each against Utah and Oregon State in Arizona will follow as part of the Big Ten/Pac 12 challenge. The Bucks will then play true road series in successive weekends against Campbell and Florida Gulf Coast, then play midweek neutral site games in Port Charlotte, FL against Lehigh and Bucknell. The home schedule opens March 17 with a weekend series against Xavier (the Sunday finale being played in at XU), and the next two weekends see the Buckeyes open Big Ten play by hosting Minnesota and Purdue.

Subsequent weekend series are at Penn State, at Michigan State, home against UNC-Greensboro, home against Nebraska, at the forces of evil, at Iowa, and home against Indiana. Midweek opponents are Youngstown State, OU, Kent State, Cincinnati, Eastern Michigan, Northern Kentucky, Texas Tech (two), Bowling Green, Ball State, and Toledo, all at home, giving OSU 28 scheduled home dates.

Should OSU finish in the top eight in the Big Ten, the Big Ten Tournament is shifting from the recent minor league/MLB/CWS venues (including Huntington Park in Columbus, Target Field, and TD Ameritrade Park in Omaha) to campus sites, although scheduled in advance instead of at the home park of the regular season champ as was the case for many years in the past. This year’s tournament will be in Bloomington, and it speaks to both the volume of players lost and Beals’ uninspiring record that participation in this event should not be taken for granted.

## Thursday, February 09, 2017

### Simple Extra Inning Game Length Probabilities

With the recent news that MLB will be testing starting an extra inning with a runner on second in the low minors, it might be worthwhile to crunch some numbers and estimate the impact on the average length of extra innings game under various base/out situations to start innings. I used empirical data on the probability of scoring X runs in an inning given the base/out situation based on a nifty calculator created by Greg Stoll. Stoll’s description says it is based on MLB games from 1957-2015, including postseason.

Obviously using empirical data doesn’t allow you to vary the run environment…the expected runs for the rest of the inning with no outs, bases empty is .466 so the average R/G here is around 4.2. It also doesn’t account for any behavioral changes due to game situation, as strategy can obviously differ when it is an extra innings situation as opposed to a more mundane point in the game. Plus any quirks in the data are not smoothed over. Still, I think it is a fun exercise to quickly estimate the outcome of various extra inning setups.

These results will be presented in terms of average number of extra innings and probability of Y extra innings assuming that the rule takes effect in the tenth inning (i.e. each extra inning is played under the same rules).

If you know the probability of scoring X runs, assume the two teams are of equal quality, and assume independence between their runs scored (all significant assumptions), then it is very simple to calculate the probabilities of various outcomes in extra innings. If Pa(x) is the probability that team A scores x runs in an inning, and Pb(x) is the probability that team B scores x runs in an inning, then the probability that team A outscores team B in the inning (i.e. wins the game this inning) is:

P(A > B) = Pa(1)*Pb(0) + Pa(2)*[Pb(0) + Pb(1)] + Pa(3)*[Pb(0) + Pb(1) + Pb(2)] + ….

Since we’ve assumed the teams are of equal quality, the probability for team B is the same, just switching the Pas and Pbs. We can calculate the probability of them scoring the same number of runs (i.e. the probability the game extends an additional inning) by taking 1 – P(A > B) – P(B > A) = 1 – 2*P(A >B) since the teams are even, or directly as:

P(A = B) = Pa(0)*Pb(0) + Pa(1)*Pb(1) + Pa(2)*Pb(2) + … = Pa(0)^2 + Pa(1)^2 + Pa(2)^2 + … since the teams are even

I called this P. The probability that game continues past the tenth is equal to P. The probability that the game terminates after the tenth is 1-P. The probability that the game continues past the eleventh is P^2; the probability that the game terminates after the eleventh is P*(1 – P). Continue recursively from here. The average length of the game is 10*P(terminates after 10) + 11*P(terminates after 11) + …

I used Stoll’s data to estimate a few probabilities of game length for a rule that would start each extra innings with the teams in each of the 24 base/out situations. For a given inning-initial base/out situation, P(10) is the probability that the game is over after 10 innings, P(11) the probability it is over after 11 or fewer extra innings, etc. “average” is the average number of innings in an extra inning game played under that rule, and R/I is the average scored in the remainder of the inning from Stoll’s data for teams in that base/out situation.

It will come as no surprise that generally the higher the R/I, the lower the probability of the game continuing is. In a low scoring environment, the teams are more likely to each score zero or one run; as the scoring environment increases, so does the variance (I should have calculated the variance of runs per inning from Stoll’s data to really drive this point home, but I didn’t think of it until after I’d made the tables), and differences in inning run totals between the two teams are what ends extra inning games.

The highlighted roles are bases empty, nobody out (i.e. the status quo); runner at second, nobody out (the proposed MLB rule); runners at first and second, nobody out (the international rule, starting from the eleventh inning; this chart assumes all innings starting with the tenth are played under the same rules, so it doesn’t let you compare these two rules directly); and bases loaded, nobody out, which maximizes the run environment and minimizes the duration of extra innings (making games beyond 12 innings as theoretically rare as games beyond 15 innings are under traditional rules). Of course, these higher scoring innings would take longer to play, so simply looking at the duration of game doesn’t fully address the alleged problems that tinkering with the rules would be intended to solve.

I did separately calculate these probabilities for the international rule--play the tenth inning under standard rules, then start subsequent innings with runners on first and second. It produces longer games than starting with a runner at second in the tenth, which is not surprising.