Win Shares Walkthrough, pt. 2

Park Factors in Win Shares
Win Shares requires park adjustments at various points, so I will first go through and calculate the park factors for the 1993 Braves. The Win Shares Park Factor approach is fairly conventional. Five years of data are used when available, but the current year composes 50% of the PF with the other years getting equal shares of the final. We first calculated RPG at home and away for each season of data we are using (this includes Runs and Runs allowed). We then find the PF in two steps:
iPF = RPG(H)/RPG(R)
PF = ((T-1)*iPF+T-iPF)/(2*(T-1))
Where T is the number of teams in the league. In 1993, the Braves scored 366 and allowed 291 in 81 games at home, and scored 401 and allowed 268 in 81 games on the road. So RPG(H) = (366+291)/81 = 8.11 and RPG(R) = (401+268)/81 = 8.26. The iPF = 8.11/8.26 = .982. Then, since there are 14 teams in the league, PF = ((14-1)*.982+14-.982)/(2*(14-1)) = .992. Once we have found the PF for each season, we find the weighted average to get the PF(R), the Runs Park Factor. If more then one season of data is used, then the weight on the focus season is .5 and the weights for the other seasons equally make up the other .5. For instance, if 2 seasons are used, each are weighted at .5. If 5 are used, as in this case, the focus year is .5 and the others are each .125. As best as I can figure, Bill uses only 1993, 1994, and 1995 for ATL in this case, so 1993 is weighted at 1/2 and 1994 and 1995 are each weighted at 1/4. Finding the other year's PFs and applying the formula, we get a PF(R) for the Braves of .998.

PF(HR) is found in the same way, substituting HR data for R data. We get a result of 1.019. Then James figures a third PF, the “Park-S Adjustment”, which is the effect of the park on everything other then Home Runs. The formula for this is:
PF(S) = sqrt((PF(R)-LHR%*PF(HR))/(1-LHR%))
Where LHR% = Lg(HR/R)*1.50
For the Braves, the league hit 1956 HR and scored 10190, so LHR% = 1956/10190*1.5 = .288. Then PF(S) = sqrt((.998-.288*1.019)/(1-.288)) = .995

My Take: The Win Shares Park Factors are very solid. The conversion from the initial factor to the other factor is essentially equivalent to the one I use(which was published by Craig Wright). The one disagreement I have is with the weighting. I think a one-half weighting on the current year is too much, but James cites weather and other factors that may well change from year to year as a reason for this. Just a small difference of opinion.

The Park-S adjustment is actually quite clever, and I wish I would have thought of it myself. The question that I have is why is league data used? The league average is irrelevant to the Braves games--their stats and their opponents’ stats should be considered. Also, the linear weight value of 1.50 for a home run is obviously just a quick and reasonable value, but given the attempt to have precision in other areas of the Win Shares method, it seems very imprecise. So throughout this walkthrough I will just say “that is imprecise”, and that will mean that while it is a reasonable estimate and may work in almost all cases, it could be estimated more precisely as so many things in this method are.

The PF(S) uses a square root because hits and other “run components” have a roughly square relationship with runs. For instance, Pete Palmer uses the square root of the runs PF to adjust OBA and SLG in PRO+. The reason for this is the basic, approximate Runs = OBA*SLG relationship discovered by Dick Cramer in his BRA method, and later used by Bill James to create basic Runs Created.

Dividing Win Shares Between Offense and Defense
The first important step in Win Shares is to divide credit for the team wins between the offense and the defense. This is done by the percentage of marginal runs that each provides.

First, we calculate expected runs and runs allowed
ExpR = Lg(R/I)*PF(R)*IB
ExpRA = Lg(R/I)*PF(R)*IP
In the 1993 NL, the average R/I is 10190/20284 = .5024. The Braves
IB is innings batted, which is estimated as IB = IP - W(H) + L(R), where W(H) is wins at home and L(R) is losses on the road. In the 1993 NL, the average R/I is 10190/20284 = .5024. The Braves pitched 1455 innings and won 51 home games while losing 28 road games, given them an IB of 1455 - 51 + 28 = 1432. So the Braves expected R and RA are:
ExpR = .5024*.998*1432 = 718.16
ExpRA = .5024*.998*1455 = 729.70
Then the marginal runs are found:
MR = R - ExpR*.52
MRA = ExpRA*1.52 – RA
The Braves scored 767 runs and allowed 559, so MR = 767 - 718.16*.52 = 393.56, and MRA = 729.70*1.52 - 559 = 550.14.
Finally, we split the Win Shares(there are 3 win shares for each team win):
OWS = MR/(MR+MRA)*W*3
DWS = MRA/(MR+MRA)*W*3
So the percentage of total marginal runs coming from offense is defined as the percentage of wins coming from offense. The Braves’ offense accounts for 393.56/(393.56+550.14) = 41.7% of the team wins, and the defense the other 58.3%. Since the Braves won 104 games, there are 312 win shares to go around, 130 for the offense and 182 for the defense.

My Take: This is one of the crucial steps in the process, and it is a fairly clever one, certainly not something I would have thought up. I’m not sure it works, but it’s clever either way. What James does is calculate marginal runs against some very low baseline. He never explains exactly what this baseline is supposed to represent, other then that it works. He uses .52 for the offense and 1.52 for the defense. Let’s call the league average runs/game L. James says:
W% = (R-.52L + 1.52L-RA)/(2*L)
In other words, RPW = 2L, since we can rewrite this equation as:
W% = (R-RA)/(2L) + .5
It does not matter if you use .52 and 1.52, or .5 and 1.5, or .6 and 1.6--as long as there is a difference of 1 between the two baselines, the team W% formula will hold.

The team winning percentage formula is correct at .500, or for a team whose RPG (offense and defense) is equal to 2*L (if we assume that Runs per Win = RPG). The league context does not matter in determining the number of games a team will win. The runs/wins converter depends on the scoring context, but it depends on the scoring context of the team, not that of the league. (R-RA)/(R+RA) + .5 is a much better estimate of team W% then James' method. But James must use it so that he can compare to the league. But this will cause increasingly large distortions as team move away from the average situation where the formula holds.

An average team will have R = RA = L, and therefore have MR = L-.52L and MRA = 1.52L-L. Plugging these MR and MRA into the win share splitting formula of MR/(MR+MRA) simplifies to .48. In other words, for a perfectly average team, 48% of wins will be attributed to offense and 52% to defense.

This seems a little off on first blush, as offense and defense are usually assumed to be symmetrical in sabermetrics. There is very little difference in run distribution patterns for offense and defense, which leads to a conclusion of a 50/50 split. James counters by pointing out that the number of runs a poor team could score is limited at zero, while there is no limit to the number of runs a poor team could allow. James also says that the very worst teams, historically, have had relatively worse defenses then offenses. Additionally, he claims that using .52/1.52 gives “better” results then using .5/1.5(i.e. pitchers are not rated as lowly in comparison with batters as they would be otherwise).

I believe that part of the reason that pitchers rate poorly is because pitching is split between pitchers and fielders at the team level, where every pitcher actually relies on their defense to a different extent. I personally don’t feel that his evidence is strong enough to abandon the comfortable .5/1.5 split, but since I don’t have any strong evidence to counter with, I will leave that be. Instead we can look at the properties of this procedure. Suppose we have a league where the average is 5 runs per game and we have a team with an average defense(RA = 5) and a margin-level offense(R = .52*5 = 2.6). This team, according to the W% formula above, will have a (2.6-5+5)/(2*5) = .260 W%. Since there are no marginal runs, all of the teams wins will be credited to the defense(.260).

Suppose we had instead a team which was perfectly average. They would have a .500 W%, and 52% of this would be attributed to the defense, or .260 wins. This illustrates the very useful property of this assumption--an average defense will always get the same number of wins credited to it, regardless of what the offense does.

Of course, this breaks down if either component of your team is sub-marginal, as it will be assigned negative wins. This does not make any sense, but is necessary to keep the other component of your team at its necessary level. The problem is that some individuals on this team will be above the margin, and therefore will be assigned negative win shares for super-marginal performance. Of course, actual teams don’t exist at these extremes, but if a system collapses at .52, what is it doing at .7? There some point at which the system starts to act screwy, and that point may be on the fringe of the real data section. Or it may not.

So the wins that will be assigned to either your offense or defense will be their marginal runs divided by 2 times the league average. In this and all of the above discussion, I am assuming that the team’s actual performance is equivalent to its W% estimation.

Win Shares Walkthrough, pt. 1

This series will be a (hopefully) complete walkthrough of the Win Shares method, similar to that done by Bill James in the Win Shares book. Since James already did this in his book, and since he is the creator of the method, you may ask, why is it necessary for you to do it again? The reason why I am doing it is so that I can comment on what I think are the flaws in Win Shares at the actual stage of the method at which they occur. This way it is easier to see what I mean rather then running through the method, then making a laundry list of complaints at the end.

In the past I have probably been a little too tough on Win Shares. Bill James is a brilliant sabermetrician who has contributed many great things to the field. Win Shares is not, in my opinion, one of those, but it certainly had a lot of work behind it, I’m sure that Bill felt that the assumptions he made were justified, and that he tried to create the best possible method he could. But I simply do not believe that Win Shares taken as a whole is a step forward for sabermetrics. The are of Win Shares which James takes most pride in is the fielding system he developed, which first credits the team fielding as a whole and then distributes the team fielding Win Shares to fielding positions, and then finally to individual fielders. Coincidentally, fielding statistics are the area of sabermetrics that I am least qualified to comment on (if you consider me qualified to comment on any). So the fielding system may well be a conceptual breakthrough, although again I have my doubts.

So this will not be a perfectly balanced assessment. I will admit that I have been a critic of Win Shares since it was first published and you should keep my point of view in mind as you read. I think that my criticisms are valid and based on solid reasoning, but one cannot completely divorce ones prejudices from their analyses.

So I will do this in several installments, and I will calculate the Win Shares for the 1993 Atlanta Braves, who won a classic pennant race against the Giants but were defeated in the NLCS by the Phillies. At various times I will slightly deviate from James’ official method. One example is that James rounds off all Win Shares to even numbers. I will maintain fractions throughout the process. I will attempt to note these departures as I come to them. I have only attempted to use and explain the post-1987 formulas here. I have rounded numbers in the explanation that I have not rounded on my spreadsheet, which will result in some rounding discrepancies.

I have tried to run through the procedure without giving my opinion and then have a “My Take” section which gives my opinion on the formulas I just reviewed.

A very important note before we start: this walkthrough should not be viewed as a substitute for reading James’ explanation of his method. First of all, any discrepancies between my description and his are obviously wrong here, since he designed the method and knows what he’s doing. Secondly, it would be best to read his first and draw your own conclusions and consider his arguments as you go and then read my interpretation. Also, his explanation is much more complete in many areas, while mine does not so much describe how each part is done or why but whether or not I think it is the right thing to do. So this is more of a commentary, where his is an explanation and a commentary.

The first actual piece should appear tomorrow; it will cover park factors and assigning win shares to team offense and defense.

A quick word on the term "Runs Created"

Looking at the title of the last post, I can see that it might cause some confusion. I often use “Runs Created” in a general sense to mean a run estimator, rather then to refer to the Bill James method Runs Created. The James method of RC was once cutting edge, but it has obvious flaws and people have come along and invented new run estimators that are less flawed. I firmly believe that there is no reason to use James’ RC other than for quick estimates.

However, Bill came up with the best name for his method, because it includes an action. Runs are created. Similar terms include runs produced, runs generated, and runs manufactured. I think “created” is the best term, and so I like to use it.

Later run estimator methods have names like Extrapolated Runs and Base Runs and Contextual Runs. Having different names is necessary so that we can tell one run estimator from the other, but these are not good names IMO. I’m not blaming Furtado, Smyth, and Van, because all of the good names were already taken and they had to call it something. But a player does not “extrapolate” or “base” a run. They “create” or “produce” or even “contribute” a run.

So I will continue to use the great term coined by Bill James, even if his method has been surpassed in accuracy.

Writing RC in terms of BA, OBA, and SLG

The primary run estimator that I use here and on my website for evaluating individual hitters is Estimated Runs Produced (ERP). This method was first developed by Paul Johnson and published in the 1985 Baseball Abstract. It is a linear formula, which neither he nor Bill James pointed out in his article, although that is a story for another day.

If I can be considered a good sabermetrician (that is up to you), I believe that it is because I have always experimented. If I see a method published by somebody else, I always try to develop my own version based on the same idea or same general principle, starting from scratch as much as possible. I do not do this out of vanity, thinking that I can come up with a better formula, but because I find that is the best way to find out how and why something works. The point of that digression is that a few years ago I was trying to come up with a linear run estimation skeleton, despite the fact that so many good ones are already in existence. What ended up happening is that I recreated ERP. I got this formula:
(TB + .5H + W - .3(AB - H))*.324
Johnson’s formula (ignoring SB, CS, etc.) was something like this:
(2*(TB + W) + H - .605*(AB - H))*.16
As you can see, my formula is almost exactly the same as Johnson’s, divided by two.

So I have used my version of the formula, and called it ERP, because it is the same formula, so regardless of whether I developed it independently or not, it is still Johnson’s.

Anyway, the only point here is to show a good way to estimate RC/PA or RC/O based only on BA, OBA, and SLG. It is not a “clean” formula that weights each of them, like that you would get from a multiple regression, but it is mathematically equivalent to ERP. We’ll start by writing the formula in terms of per at-bat rates. TB/AB is SLG, H/AB is BA, W/AB is what I call WAB, and (AB-H)/AB is just 1-BA. So we can write the formula as:
RC = (TB + .5H + W - .3(AB - H))*.324
RC/AB = (SLG + .5BA + WAB - .3(1 - BA))*.324
We can write WAB as (OBA - BA)/(1 - OBA), which along with expanding the (1 - BA) term gives:
RC/AB = (SLG + .8BA + (OBA - BA)/(1 - OBA) - .3)*.324
We can convert this to RC/PA:
RC/PA = RC/AB*(1 - OBA)/(1 - BA)
And then RC/PA converts to RC/O:
RC/O = RC/PA*(1/(1 - OBA))
Or, directly, R/O = R/AB*(1 - OBA)/(1 - BA)*(1/(1 - OBA)) = (R/AB)/(1 - BA)

Ball Four and the Boys of Summer

As a fairly well-read baseball fan, it is kind of embarrassing to admit that I had not read two “classics” until very recently: Jim Bouton’s Ball Four and Roger Kahn’s The Boys of Summer. Just to make sure you don’t get the wrong idea, I have read a lot of baseball history type books, not just all of the sabermetric stuff. I have read Seymour’s Baseball series, Voigt’s American Baseball series, Peterson’s Only the Ball Was White, Tygiel’s Baseball’s Great Experiment, Asinof’s Eight Men Out, Helyar’s Lords of the Realm, etc. I am not well-read historically perhaps by the standard of the typical SABR member, but it’s not like I’ve never read anything other than Bill James.

Anyway, with that I can give my brief thoughts on Ball Four and The Boys of Summer. Ball Four caused quite the stir when it came out, but it is difficult to understand why now. It is pretty tame by modern standards: players swear, smoke, and drink, they cheat on their wives and peep at naked women, they take amphetamines. Shocking. Of course, this is someone who was born in the 1980s speaking. I have seen this same point about Ball Four being tame thirty-five years later, but it is even more true when you have always thought about athletes in that way for your entire life. I also realize that this was shocking at the time, but I cannot divorce myself from my cynicism to understand that fully.

With that said, the book is entertaining and is very interesting. I found it a tad bit monotonous, but Bouton’s observations on baseball and other topics are generally interesting, and it is interesting to read the thoughts of a ballplayer throughout the season and the circumstances that it entails. I like Ball Four, and I respect its historical significance and impact, but I cannot consider it one of the great baseball books.

As far as The Boys of Summer goes, I found it tedious and vapid. Dodgers get old and no longer live lives as ballplayers. They have children and they marry and they divorce and they have personal tragedies. In other words, they are like everybody else, except they played baseball. Of course, much of the book is Roger Kahn’s own personal story, which is something he is free to tell of course, but I’m not really interested in sportswriter’s autobiographies.

Most obnoxiously, the author looks at the world, and therefore colors his story, through a view that I do not accept. It is not surprisingly a liberal view, but it also to me at least reflects a condescending attitude toward small towns and rural America. Anyone not living in New York City is somehow living an unfulfilled life. I may exaggerate a little, but that is the vibe that I get. Bill James had similar observations, not about The Boys of Summer, but about various portrayals of his home state of Kansas. While my hometown in suburban/rural Ohio is hardly a Kansas small town of the 1960s, the glorification of the big city still rubs the wrong way. Kahn does not seem to understand that some people could not be paid enough to deal with living in New York City or Los Angeles. The Boys of Summer is overrated, unless you are an old Brooklyn fan.

Rate Stat Series, pt. 3

I had intended this to be the second installment, but as I was writing it I realized I used the concepts of PA generation in it(which I was planning to cover a little later in the series), so I jumbled the order so that what is here makes sense.

Since we know that Runs/Out is the proper rate stat for a team, the obvious solution is to carry that over and apply it to individual batters. We first put it to a logical test. Does it, on first blush, make sense to rate players based on R/O? Yes, it does. Runs and outs are the two crucial factors in an offense. Individuals, as well as teams, want to create runs and avoid outs.

We should also apply a more substantive logical test. What are we trying to measure with a batter’s rate stat? His theoretical individual ability as a batter, or his ability to help his team score runs? The first definition is feasible, but what we really want to know is what this batter will do for a team. An individual is valuable if he helps his team win (or, from an offense only standpoint, help his team score runs). Therefore, our rate stat, along with our quantity stat that we are converting into a rate stat, should reflect his ability to help a team score runs.

The difference between “theoretical individual ability” and “ability to help a team score runs” is very thin; you might even say razor thin. However, it does exist. We see the problem with multiplicative run estimators like RC and BsR. BsR attempts to model the run scoring process; therefore, it represents theoretical offensive ability. But the run scoring function is a team function, and therefore it is inappropriate to apply BsR directly to individual batters. We instead use Linear Weights or a theoretical team method, to model the batter’s impact on the team run scoring process. Of course, the results we get from applying BsR directly to an individual player are not too far off from the result from the more proper approaches. This is because the difference between the team function and the player function are very, very small.

We need to keep this principle in mind when evaluating R/O. Just because R/O is the right rate stat for teams does not mean that it is the right rate stat for individuals (and of course the opposite is true as well; that R/O is right for teams does not disqualify it from being right for players).

R/O passes our rudimentary logical test. Now we should see if it properly reflects the run impact an individual creates within a team context. Suppose we consider two players, admittedly extreme. One has 300 PAs, makes 100 outs, and draws 200 walks. His Runs Created is approximately .32*(200) - .1*(100) = 54, for .540 R/O. Player Two has 500 PA, hits 150 home runs, and makes 350 outs. His RC is about 1.46*(150) - .1*(350) = 184, for .526 R/O. So Player A has a higher R/O and therefore is the “better” hitter.

But suppose that we add these to an otherwise average team, giving them each 1/9 of the team plate appearances. This team has a .330 OBA and scores .12 R/PA and .179 R/O (approximately 4.5 runs/25.2 outs). If we add player A, they will now have an OBA of .330*(8/9) + (200/300)*(1/9) = .3674, which based on the reasoning in the previous installment will result in 25.2/(1 - .3674) = 39.84 PA/G. Their R/PA will be .12*(8/9) + (54/300)*(1/9) = .1267, and their R/G will be 39.84*.1267 = 5.046.

Following the same procedure with Player B, his team will have an OBA of .3267, will generate 37.43 PA/G, score .1476 R/PA, and score 5.523 runs/game. Player B’s team will score .477 R/G more than Player A’s team, and yet Player A had a slight edge in R/O. What is going on here?

What is going on is that we have seen the flaw in applying R/O to individuals. Player A has a tremendous ability to avoid outs with a .667 OBA. A team with an OBA of .667 would score an oodle of runs, because they would make so few outs, and the impact of each PA generated would compound many times because the next batter coming up would also have a stratospheric OBA. Player B is sub-par at reaching base, with a .300 OBA; he reduces the number of PA his team will generate. But he also has 1.200 SLG, and will drive in tons of runs. As a team, he would still be formidable, but his low OBA will be a bigger impediment.

The point to take away from this example is that a player added to a team who avoids an out does not generate subsequent PAs at his OBA--he generates them at his teammates’ OBA(or more precisely, the New OBA of the team with himself included). R/O implicitly assumes that he generates PA at the rate of his own OBA, overstating the importance of avoiding outs.

(A technical note: if you plug Player A’s stats into the BsR version on my site, you will not get an “oodle” of runs--you will get about 3.65 runs/game. There are a couple reasons for this: one, the BsR formula I’m using gives walks a B coefficient of .039, whereas others go as high as .1, which will make a big difference in such an extreme case. Also, the linear weight formula we used to estimate his RC in a team context will be a little off because he is very extreme, and will have significant impact on team LW--of course, Player B will change his team’s values as well. But even if taking these technical issues into account would put Player B’s R/O ahead of Player A’s, it would still be close, and the point here is to illustrate the huge discrepancy between R/O and the impact they actually have on team runs scored).

If R/O does not work, one very obvious alternative is R/PA. It gives the “correct” answer for our players above--.18 for Player A and .368 for Player B. Let’s look at a different set of players and see how R/PA stacks up against what we expect to happen. These two are the 2005 seasons of Chad Tracy and Todd Helton. Tracy had .346 OBA, and created 88 runs in 538 PA and 349 outs, while Helton had a .411 OBA and created 100 runs in 615 PA and 346 outs. Tracy’s RC/PA was .1636 and Helton’s was .1626. But in R/O, Helton was ahead .2890 to .2521.

If add the players to the same team as above (.330 OBA and .12 R/PA), Tracy’s team would score 4.708 runs/game, while Helton’s would score 4.756 runs/game. So now R/PA is not matching our expectations.

What has happened is that with R/PA, we do not account for the player’s avoidance of outs(read: generating additional PA) at all. When Tracy comes to the plate, he creates an equal amount of runs as Helton. But he gets to the plate less often, because he uses up more of his team’s outs.

While we needed an extreme case to show the failure of R/O for an individual, here we took two fairly ordinary players from this season. There are extremes that can be created or more unusual cases in baseball history that show a much larger discrepancy.

In the end, R/PA is useful as a descriptive stat(it is after all the number of direct runs he creates every time he hits), and R/O is useful as a theoretical measure of how good the player’s offense would be as a team, and is actually a pretty decent shortcut rate statistic. But neither R/PA or R/O fulfills everything we want our rate stat to do. Next time, we’ll look at how the flaws of R/O can get really out of hand when you try to stretch it further down the path of batter analysis.

Defining Value

I have been thinking lately about the question of how to define “value” in baseball. I have an article on my website entitled “Ability v. Value” which discusses the differences between what sabermetricians mean when we talk about ability or value.

In the article, I define two classes of value, “Literal Value” and the regular “Value”. Literal value, as I define it, involves only methods that track actual changes in run and win expectancy, like Value-Added Batting Runs or Win Probability Added. Value includes methods which use composite season statistics, but give credit for things like hitting with runners in scoring position or a pitcher who pitches in a lot of high leverage situation.

I also broke down ability into “Ability” and “Literal Ability”. Ability is defined as “theoretical value”, i.e. the value that a player would be expected to accumulate, on average, if he played in a given set of circumstances. Usually this would be our expectation for a player in a neutral park, but it could be “ability to help the team win games in Coors Field” or “in 1915” or “batting fifth in a lineup with A, B, C, and D hitting ahead of him and E, F, G, and H hitting behind him”. There are all sorts of different ways you could define ability, but the mathematical result you get will be specific for the context you choose.

Literal ability goes even further, and attempts to distill the player’s skill in a given area of the game (such as power, or speed, or drawing walks), or his “overall ability”. This is very tricky, because nothing happens in a vacuum, everything happens in some sort of context, and so divorcing a metric from context is pretty much impossible. Therefore literal ability is more of a theoretical concept and not a measurable quantity (although methods like Speed Score are an attempt to measure literal ability in speed, but of course are acknowledged by their creators as approximations).

Anyway, to generalize, value is backwards-looking, and ability is forwards-looking (or at least what might have happened in a different context given the same production in a given timeframe).

The recent signing of BJ Ryan to a large contract by the Blue Jays has put the issue of when to time the value measurement into my head. Literal value methods like Win Probability Added value on a real-time basis. If at a given moment the probability of winning is 60%, and after the next play it increases to 62%, then the player responsible for that play is said to have added .02 wins. So a closer, who pitches as the highest leverage time, will come out with a higher WPA then a starter who had the same performance in the same number of innings.

But if we are ascertaining value after the fact, why do we have to do it in real time? Suppose that Scott Shields is called in to pitch on the road in the bottom of the seventh inning with a one-run lead. According to Tango Tiger’s WE chart (linked below), the win probability is .647. He retires the side and at the end of the inning, the probability is .732, so he is +.085. He starts the eighth inning with a probability of .704, retires the side, and leaves with a probability of .842, so he is +.138 for the inning and +.223 for the game. In the bottom of the ninth, it is still a one-run game and Francisco Rodriguez is summoned with a win probability of .806. He finishes it off and of course the win probability is then 1, so he is +.194 wins. So Shields, for two innings of scoreless work, only gets .029 more wins then Rodriguez did in one inning. Is this fair? Sure, if you define value real-time. Rodriguez pitched in a more critical situation and his performance did more to increase the real-time win probability.

But since we are looking backwards, why can’t we step back and, now, omniscient about what happened in the game, ascertain what value the events actually had? Each out in the game had a win value of 1/27, and since neither allowed any runs or anything else, we don’t have to consider that. So Shields should have added 6/27 wins and Rodriguez 3/27. Viewed from the post-game perspective, Shields performance is much more valuable then Rodriguez’. Now you could also argue that if you took this perspective far enough, any event that didn’t lead to a run in the end(like a hit that does not score) has no value. And that’s a possible outcome.

Now the point is not that real-time value determinations are incorrect or invalid. They are simply a different way of defining literal value. But I would contend that they are not the only way to define literal value. It is one of the easiest to explain and define, and it certainly makes sense. I’m not arguing against it, just arguing that it is not an undeniable choice for what I have called “literal value”. Of course, you can define “value” or “literal value” reasonably, in such a way as to make it an obvious choice.

Tango Tiger's Win Expectancy Chart

Rate Stat Series, pt. 2

Before we proceed with examining options for individual offensive rate stats, we need to establish some theoretical parameters explaining how teams generate Plate Appearances. Every team begins the game with 27 guaranteed Plate Appearances. By avoiding outs, they generate additional PAs throughout the course of the game. We can calculate with RC/PA how many runs a team should score for each PA. If we understand how many PA they will have in a game, we can calculate how many runs they should score per game.

Of course, we already know how many runs a team should score per game, because they have 27 outs and we can just multiply their R/O by the number of outs/game(O/G). But if we know the number of PA the team will have, we can calculate this equivalent number through R/PA.

Our first step is to calculate what I call the Not Out Average(NOA). It is very similar to OBA, except that it subtracts from time on base all outs that are accounted for in the official statistics that come on the bases(i.e. CS and DP). These events eliminate the out avoidance affect of hits, walks, etc. which are credited in the OBA numerator. So, depending on how much detail you are using from the statistics, the formula for NOA is:
NOA = (H + W + HB - CS - DP)/(AB + W + HB + SH + SF)

From NOA, we can easily calculate PA per Game as PA/G = (O/G)/(1-NOA). So a team in a league where the average is 25.5 outs/game(this is about what the average is if you consider just batting outs and CS) and a .330 NOA will get 25.5/(1-.330) = 38.06 PA/G.

Another more intuitive way to think about this is to consider how many extra PAs are generated by each guaranteed PA that the team begins the game with. The probability that any given PA generates another PA is the NOA. A .33 NOA means a 33% of avoiding an out for each guaranteed PA. The probability that 2 PAs are generate equals NOA plus NOA times NOA…ie, the probability that the first batter avoids an out times the probability the second batter avoids an out. The probability of a third is NOA times NOA times NOA. Writing this in a formula:
Extra PA/PA = NOA + NOA*NOA + NOA*NOA^2 + NOA*NOA^3 + … + NOA*NOA^n
Where n is infinity, because theoretically, an infinite number of PA could be generated. We can factor NOA out of the above equation to get:
Extra PA/PA = NOA(1 + NOA + NOA^2 + NOA^3 + … + NOA^n)
The sum of 1 + x + x^2 + … + x^n, where n = infinity and 0 is less then x which is less then 1(as is the case for NOA) is 1/(1-x). So our final equation is:
Extra PA/PA = NOA/(1-NOA)
To convert this to PA/G, we know that O/G number of PA are guaranteed, and that each of those guaranteed PAs will generate (O/G)*(Extra PA/PA) more PA. So:
PA/G = O/G + (O/G)*(Extra PA/PA)
Or:
PA/G = O/G*(1 + Extra PA/PA)
The denominator of Extra PA/PA is 1-NOA. 1 can be rewritten as (1-NOA)/(1-NOA), making the part in parentheses equal to ((1-NOA) + NOA)/(1-NOA), which simplifies to 1/(1-NOA), making the whole thing equal to O/G*1/(1-NOA) = (O/G)/(1-NOA). A lot of math gyrations to show why this formula works, rather then just stating that it does.

Now we want to see how a team’s number of PA will change when we add a player to the team. We will assume that this guy gets a certain fixed percentage of the team PA which we call PA%. Then the new team NOA is:
NewTmNOA = PA%*NOA + (1 - PA%)*TmNOA
So suppose we add a player with a .400 NOA to the .330 team and give him 1/9 of the team PAs. Before adding this guy, the team had 38.06 PA/G(figured above). With him, their NOA will be .338, and their PA/G will be 38.51.

We can also approach this problem through the Extra PA/PA approach. Let me abbreviate Extra PA/PA from here on out as EPA. If we add a player with a given NOA to a team, his NOA will generate the first PA, then the subsequent PAs will be generated by the NewTmNOA rate. So his EPA in this context(which I’ll abbreviate as EPAp) is:
EPAp = NOA + NOA*NewTmNOA + NOA*NewTmNOA^2 + NOA*NewTmNOA^3 + ... + NOA*NewTmNOA^n = NOA(1 + NewTmNOA + NewTmNOA^2 + NewTmNOA^3 + … + NewTmNOA^n) = NOA/(1 - NewTmNOA)
Likewise, the rest of the team(EPAr) will generate the first PA at TmNOA, and the subsequent PAs at NewTmNOA:
EPAr = TmNOA + TmNOA*NewTmNOA + NOA*NewTmNOA^2 + NOA*NewTmNOA^3 + ... + TmNOA*NewTmNOA^n = TmNOA(1 + NewTmNOA + NewTmNOA^2 + NewTmNOA^3 + ... + NewTmNOA^n) = TmNOA/(1 - NewTmNOA)
Then the NewTmEPA = PA%*EPAp + (1 - PA%)*EPAr
This is mathematically equivalent to the approach based directly on NewTmNOA.

We can estimate how many runs a team will score with an added player by calculating the team R/PA and the team’s new PA/G:
NewTmR/PA = PA%*(R/PA) + (1 - PA%)*(TmR/PA)
NewTmR/G = NewTmR/PA*(O/G)/(1 - NewTmNOA)

We get into some problems here, because as covered in other articles on this site and in many other sources, adding a player to a team changes the linear weight values, and so the linear weight RC calculations for both the team and player that we put into that formula will be different then what they produce in the new context. But these differences are often small enough that we can use this procedure to understand some basic concepts, if not have technical precision.

This problem is somewhat similar to another problem that we will encounter in discussing rate stats, which is how the runs created figures for players are calculated. We can divide RC methods into three classes:
1) Team methods(RC, BsR)
2) Linear methods
3) Theoretical team methods based on RC and BsR
Class 3 methods figure a team’s RC with the player minus the team’s RC without the player to give an estimate of runs added by the player. Anyway, class 1 methods treat players as if they are their own team. So using R/O may be justified with them. Class 2 and class 3 methods look at the player as a member of a team, and so R/O is an inappropriate choice, at least in theory. But class 3 methods take into account the effect that the player has on the team LW values, while class 2 methods do not.

This also relates to methods which convert individual run value to win value. In Jim Furtado’s Extrapolated Wins, he uses XR, a linear method, to denote individual run creation. But given that the player in XW is operating within a team, using a class 3 methods might be more appropriate. The moral of the story is that when choosing a rate stat, it is important to consider how the RC estimate is generated as well. The proper rate stat may differ depending on what class of RC method you are using. Mixing contexts may give “good” results, but not have the highest theoretical accuracy.