THE BOOK--Playing The Percentages In Baseball

Monday, May 03, 2010

Secondary Average: the sabermetric gateway drug

Patriot gives us an overview of Secondary Average.  To the extent that batting average is in the discussion among your non-saber buddies, then secondary average should be the next thing we talk about before we get to Linear Weights or Runs Created.

It’s a progression, from crawling to walking to running to flying.  Just because we’re used to our BaseRuns 747 doesn’t mean we can ignore the Secondary Average Crocs when people are happily ignorant of being able to stand upright.

Wednesday, April 28, 2010

Player Won-Loss Records, Part 2

Now that we’ve got our feet wet in the conversion, let’s look to see if we can come up with somethig more general.  The first thing to notice is that each player’s individual W/L record can also be expressed as wins above average.  If Greinke shows as a 12-0 pitcher, then we know he’s +6 wins above average.  Therefore, if we figure out his games slices, and we figure out his wins above average, we can express his record as wins and losses.  Therefore, while WPA gave us a logical path to an individual W/L record, we no longer have to rely on WPA.  We can use anything. 

Like WAR.  And that means we can include all our adjustments, like fielding, positions, starter/relief, etc.  We start by figuring out each player’s game slice.  Let’s for the moment presume that 57% of the game slices will go to position players and 43% to pitchers.  We can refine this number in the future, and by era.  Let’s go with it for now.

With 162 games, that means we give 92 game slices to the non-pitchers and 70 to the pitchers.  With about 162*9 innings pitched for each team, this means each pitcher is going to get about 1 game slice for every 20.8 innings pitched.  Relievers with their leverage will complicate this, but, again, let’s set that aside for now.

Baseball Projection.com has Sandy Koufax at 54.5 WAR and 2324 innings.  Ideally, we’d prefer to see it as wins above average (WAA).  Let’s say he’s 28 wins above average.  So, we have all that we need.  The 2324 innings gives us 112 game slices, putting the average pitcher at 56-56.  Koufax is +28 wins above that, or 84 wins and 28 losses.  So, that’s Koufax’s record:
84-28 Koufax

I know there’s a temptation to turn that into a percentage.  Don’t.

Pedro has 2827 innings and 76 WAR (probably +45 WAA).  That converts to 136 game slices, and a 113-23 record.  So, that’s what we have:
113-23 Pedro
84-28 Koufax

For position players, we can follow a similar process.  There’s roughly 6300 PA for each team, and 92 game slices to give out, or one game slice for every 68.5 PA.  Tim Raines had 10244 PA and 65 WAR (or about +33 WAA).  His game slice comes in at 150, and so his W/L record is 108-42
113-23 Pedro
108-42 Raines
84-28 Koufax

And that’s how a player’s individual W/L record looks like.  If we use WPA, we guarantee summable at the team-season level (indeed at the GAME level).  Using WAR or WAA means we lose that, but we gain in terms of being able to add in things like fielding and positions.  And, in any case, we’re more interested to getting something out there for a big number of players, and we can sweat the details later.

Friday, April 09, 2010

Starting Run Expectancy

Two people sent me an email this week asking how I calculated the starting run expectancy in The Book.

Let’s look at Table 3, which shows “runs to end of inning” (REOI) for the regular walk of 0.845 and for the non-intentional walk of 0.849.  What this means is that from the time the batter received a walk, there was an average of 0.845 runs that scored from that point to the end of the inning.  This includes all future batters AND all runners already on base.

Now, you and I both know that IBB are issued in much different circumstances than a regular walk.  What we need to do is quantify that context into a single number: the starting run expectancy.  Table 3 does that for us.  The base/out situation when the IBB was given was .743 while for the regular walk it was .520.  Where did those numbers come from?  Well, you start with the standard run expectancy 24 base/out chart (Table 1), and weight that table based on the frequency that an IBB and a regular walk was handed out.  IBB are disproportionately given out with 1B open and 2 outs.  As an example, with 1b open and 1 out, the RE is either .725 or .983 (or 1.467 to load the bases).  With 2 outs, it’s between .344 and .634.  The weighted average is .743.  For the regular walks, the base/out situation leans heavier toward the bases empty, and the weighted average of Table 1 is .520.

That gives you the starting run expectancy, the situation that the batter faced.  We also know how many runs were scored when the event occurred (including the runners on base = the situation that the batter face = the starting run expectancy).  The difference is the run impact. 

That’s one way to get the Linear Weights values.  The Book details other ways.

Tuesday, March 30, 2010

Secondary Average

Bill James created a great stat 25 years ago called Secondary Average and it was everything that Batting Average was not.  Where Batting Average gave “1” for a single and home run, Secondary Average counted 0 for a single and 3 for a home run.  Basically, the “missing” bases.  It didn’t stop there, as it also included SB and walks and hit batters.  And, as luck would have it, the number of “missing” bases was equal to the number of hits.  And that meant if you divided all these missing bases (these secondary bases), by at bats, you get a league Secondary Average that matched the league Batting Average.  The range was far wider with Secondary Average.

Now, can you just add them?

Using the two metrics in conjunction--either as a linear combination (something I’ve been using pretty frequently as of late and I’ve dubbed “APS"* (Average plus Secondary Average)) or as two separate metrics--gives us a pretty good idea of a player’s overall offensive value. At least a lot better than the metrics that were currently employed.

*It’s my contention that this metric, APS, should have been popularized rather than OPS. Now that we have things like wOBA and EqA, there’s not too much use in crusading for the widespread use of APS. I still prefer to use it, rather than OPS, if I’m trying to get a quick and dirty look at a player’s overall offensive value.

Seeing that we have the same denominator, we should be able to figure this out rather easily.  wOBA already tells us that the weight of a HR relative to a 1B, when the denominator is PA is about 2.2.  And since Batting Average gives a “1” to 1B and HR.  And since Secondary Average gives a HR a “3”, then we simply need to get Secondary Average to count the HR as “1.2”.  And 1.2/3 is 0.4.  And so, using just singles and HR, we’d scale it as:

Batting Average plus 0.4*Secondary Average

However, the walk is 0.8 the value of a single (when the denominator is PA).  Here we have a bit of a problem, since our two metrics have AB, not PA in the denominator.  In any case, we’re going to see that the weight to Secondary Average is going to be close to 0.8 if we’re going to base it on the walk.

If you use the double (should be 1.4 times the single), then we’d have the same formula as the HR version.  And if we use the triple (should be between 1.7 to 1.8 times the single), then we still have the same formula.

On the other hand, the stolen base should be 0.3 times the single.  So, we’d want something like:
Batting Average plus 0.3*Secondary Average

Anyway, you just have to run it through a plus1 method to see what the best-fit weight should be (best-fitting against Linear Weights).  I’m going to guess you’ll get something close to:
Batting Average plus 0.5*Secondary Average

(I’ll do this in the morning, unless someone wants to do it ahead of me.)

And, that pretty much will lead you to the genesis of Equivalent Average.

Friday, March 26, 2010

“OPS, I Did it Again”

The article could have been bare, and I would have still linked it, just for the headline.  Kudos to whoever thought of that.

Tuesday, March 23, 2010

Does Linear Weights out value undervalue Ichiro?

So asks Jeff:

Ichiro hits a lot of grounders, and he hits a lot of grounders with men on and fewer than two outs. A groundout with nobody on is worth the same as a strikeout. A groundout with men on will either not advance the runners, which is worth the same as a strikeout, or it will advance the runners or replace one of them with Ichiro, which is worth a little more than a strikeout. (Rarely will an Ichiro grounder result in a guy getting thrown out at third or home.) Some groundouts will go for double plays, but as Ichiro has demonstrated over his career, this is unusual; his double play rate is roughly a third the league average.

So, I have to wonder - is wOBA undervaluing Ichiro a little bit? Given 450 outs a season, if Ichiro’s average out run value is just 0.013 higher than the league average, then we’re talking about an additional five runs. I could very easily be missing something, of course, and all this may not mean anything at all, but it’s been on my mind.

Sure, he’s right, that each event is not equal for each player.  One man’s out is another man’s, well, not-so-damaging out.  Which is why we have RE24. RE24 counts the run value of each base/out state before and after the batter.  So, this handles the out thing, like productive outs, destructive outs, infield singles, long doubles, etc.

Ichiro’s Linear Weights (wRAA on Fangraphs, for Runs Above Average) is +153 runs since 2002.  That’s a generic set of run values for each generic event.  His RE24 is +223 runs.  So, he’s generated some 70 more runs than his Linear Weights suggests.  While there may be some truth to what Jeff is saying, I would guess the vast majority of his gain is in his situational hitting (i.e., hitting better with men on base).

But, generally-speaking, yeah, Ichiro is cool.

DP skills and chills

JT gives us the list of best and worst at staying out of the DPs.

Sunday, March 14, 2010

More run expectancy articles at BPro

BPro has two articles on the subject:by run environment, and by the 24 base out states.

Thursday, March 11, 2010

Eveyrthing (else) you wanted to know about wOBA… and a bit about EqA

Sit down.  Take the time to understand read and digest.  It’s wordy and mathy… so, there’s no escape.

Monday, March 08, 2010

Splitting up pitching stats

I always say that FIP is one component of pitching, much like OBP is one component of offense.  What you should really do is break up the entire pitching line into its components, and see what you get.  Studes for example has been doing this for a few years with his fantastic batted ball reports. 

Now, Peter takes the same idea, by breaking up RE24 into various components.  He also says this:

The other interesting aspect of the chart for me was the variety of run values for hit-ball runs. Remember these have already been adjusted for the quality of the defense on the pitcher’s team, so what remains should be mostly luck according to DIPS theory. If so, they should regress back toward zero with multi-year sample sizes. You’ll have to wait for Part 2 to find out if they do.

Not to spoil his fun too much, but I split up WPA by batted ball and non-batted ball events on my old blog at Primer:

also present a WAA2 column that assumes that 100% of the BIP goes to the pitcher, just so that you can see what the difference is. Not much is the short answer.

Here’s the top 20, and the bottom 10. I’m sorry, but for the moment, this is all I’m prepared to present.

pitcherid WA LA WAA WAA2
johnr005 83 60 23 24
martp001 55 35 20 21
schic002 67 54 13 14
maddg002 67 55 12 13
...

So, I’m all on-board with what Peter is trying to do, and what the original poster he quoted is saying: instead of discarding data, keep it all, partition it, and massage each partition.  A component-based analysis that treats each component separately, and presents it separately, is the best way to show a pitcher’s performance line. 

This is true of everything, and even to things like Plus/Minus.  You want to show the who and the how that someone can have a +20 while someone else on the same team, possibly a better player, can be a minus 5.  Split things into components, and work things from there. 

Make sure everything adds up.

Wednesday, March 03, 2010

Runs Anything

First off, I love the name Patriot came up with.  Secondly, I’m going to call shotgun on “Runs Something” for my next metric, whatever that’s going to be. 

I think Patriot has a good enough process here.  However he would be better off considering Runs Scored and Runners Driven In (RDI, as opposed to RBI).  I think it’s weak that he says he will only consider R and RBI (and sidestep the HR issue), but then include batting outs.  When I wrote my three-parter Runs Created series, I made the case that runs are split into three categories: getting on, moving over, and inning killer.  Runs scored has a strong relationship to the getting on component.  And outs is obviously the inning killer.  Moving over is best represented by RDI not RBI.  Indeed, MLB should make R and RDI the official categories not R and RBI.  I don’t like the idea that had MLB constrained itself to RDI, then Patriot would have argued on selecting RDI instead and therefore bypassing the HR issue.  Basically, the argument of limiting himself to runs-anything is fine, but that does not mean to limit oneself to what MLB has decided on the accounting of runs.

The next point is that once you do R+RDI (i.e., runs participated in, RPI), then it would likely make more sense to NOT do the R/lgR + RDI/lgRDI, and so you are back to RPI/lgRPI.  I’ve always liked RPI, not the least of which is because at the career level it correlates so highly to Runs Created.

Thursday, February 25, 2010

Wins Above MVP

Continuing in Rally’s Wins above excellence, where excellence can be 3 WAR or 4 WAR or whatever you want it to be, here is WAMVP.

EqA renamed TAv (True Average)

Here’s Jay:

Ilya: Why is TAv better than wOBA?

Jay: 1. The fact that the stat is scaled to batting average makes it easier for the average fan to understand than wOBA being scaled to OBP. “.300 is good” is a notion with t over 100 years of baseball history behind it.

2. EqA is park adjusted, wOBA isn’t, at least as I understand it.

3. The two have virtually identical correlations to runs scored, but TAv produces a smaller RMSE. I’ll leave the defense of that statement and the grisly math to Clay Davenport, who’s got data showing that. He’ll have an article on the topic soon once he gets the PECOTA cards up, but perhaps I can get him to chime in here as well.

Me:
1. This is a feature of wOBA, not a bug.  Some people prefer the BA scale and others prefer the OBP scale.  That doesn’t make one “better” than the other. I can just as well show wBA by dividing by 1.15 instead of multiplying by 1.15.  It’s a choice I make.  It’s different, neither better nor worse.

2. wOBA as presented on Fangraphs is not park-adjusted. wOBA as presented on StatCorner.com IS park-adjusted.  People can use whichever they prefer.

3. If I was interested in having the lowest RMSE at the team-level, I would come up with unacceptable weights.  That’s not the point.  Colin Wyers understands the point, and I hope he’s involved in trying to prove that EqA is “better’ what wOBA.  At least, he’ll prove it at the game or inning level, which is where the real test should take place, not the antiquated team-level.

If you are going to call something better than wOBA, give me a chance to rebut please.

Thursday, February 18, 2010

Using wOBA

Here’s a good piece on being introduced to wOBA and how to use it.  If he’s using the Fangraphs data, it includes SB and CS, so that should be added in there.  The coefficients are around .25 and -.50.

I liked his closing line:

...and the countless other blogs and books that refuse to stop thinking and arguing about baseball.

Great line.

Thursday, February 11, 2010

What does SIERA think of walks?

I started with this baseline data:

PA BB SO BatBall GrnB LinB FlyB
1000 90 180 730 314 146 270

That is, you have 1000 batters, of which 90 are walks, 180 are strikeouts, and 730 are batted balls.  And the batted balls are broken down as 314 GB, 146 LD, 270 FB.  SIERA comes in with an ERA of 4.31.  Estimating 234.7 IP, and .92 ER per R, that means 122.31 runs allowed.

I created a matching line for FIP, using the same PA, BB, SO, IP data, and putting in 27 HR (10% of FB).  Adding in a constant +3.1995 (in place of the “3.2"), and I get an identical ERA and runs allowed of 4.31 and 122.31.

I created a crude BaseRuns equation to give me the same result.

Finally, I put the above in the Markov calculator as:
http://tangotiger.net/markov.html
910 AB, 235 H, 54 2B, 5 3B, 27 HR, 90 BB, 180 SO

This gets me 4.688 runs per game.  Multiplying by .92 and I get 4.31 as an ERA.

All equations are now calibrated to the same baseline.

I then added 1 walk, 1 PA to see what would happen:

Read More

Tuesday, February 09, 2010

Run values of events

This was from Fangraphs, 2006:

Runs Result Description
1.45 Inside the Park HR
1.39 Home Run
1.09 Triple
0.89 Sacrifice With Error
0.81 Sacrifice Fly Error
0.77 Double
0.74 Sacrifice Fielder’s Choice
0.72 Ground Rule Double
0.49 Dropped Third Strike Error
0.47 Single
0.47 Error
0.42 Assist With Error
0.39 Advanced
0.38 Interference
0.35 Dropped Third Strike (PB)
0.35 Hit By Pitch
0.34 Double Steal
0.32 Walked
0.30 Bunt
0.28 Balk
0.28 Passed Ball
0.26 Wild Pitch
0.24 Dropped Third Strike (WP)
0.24 Error
0.23 Caught Stealing With Error
0.18 Intentionally Walked
0.16 Stolen Base
0.12 Defensive Indifference
0.04 Advance On Interference
-0.08 Sacrifice Fly
-0.16 Fielder Choice
-0.18 Additional Base
-0.20 Sacrifice
-0.23 Bunt Out
-0.24 Ground Out
-0.27 Dropped Third Strike
-0.28 Fly Out
-0.28 Strikeout Looking
-0.29 Batter Interference
-0.29 Dropped Third Strike (Taken)
-0.30 Foul Fly Out
-0.30 Strikeout
-0.31 Advance On Throw
-0.31 Fielder’s Choice
-0.33 Line Drive
-0.35 Touched By Own Batted Ball
-0.40 Caught Stealing Double Play
-0.45 Tagged Out
-0.45 Caught Stealing
-0.49 Picked Off
-0.55 Infield Fly
-0.58 Forced Out
-0.72 Non-Force GDP
-0.85 Grounded Into Double Play
-1.06 Double Play
-1.32 Fielder’s Choice GIDP
-1.40 Triple Play

The regular HR and inside the parker should obviously be the same.  But, since this is empirical, this means that there were more runners on base when the inside the parker was hit.  Triple play looks wrong.  This could only happen with, at worst, men on 1b, 2b, 0 outs.  The run value there should be around 1.5 to 1.6 (unless the TP happened in a pitcher’s park or something).  In any case, you get a good feel for how things work.

Monday, February 08, 2010

This week’s primer: WAR

Good job by VEP to get everyone up to speed.

Tuesday, February 02, 2010

Various implementations of WAR; talking WPA/LI

The framework of Wins Above Replacement (WAR) was developed in this blog over a period of months or years. 
Nonpitchers:
- Offense relative to average
- Fielding relative to average
- Positional adjustment
- Common replacement level adjustment
You add those up, and multiply by expected/deserved playing time.

Pitchers
- Pitching relative to average
- Role adjustment (starter / reliever)
- Leverage adjustment (reliever)
- Common replacement Level
You add those up, and multiply by expected/deserved playing time.

That’s the framework of WAR.  There are different implementations of this framework, as Fangraphs has it (fWAR) and Rally’s Baseball Projection has it (rWAR).  They each make some decision as to how to count each component.  The great thing about the framework is how you can slide one thing in or out without affecting anything else.  Prefer UZR to TZ, but like everything else Rally did?  No problem, slide one out, slide the other one in.
Jeremy doesn’t like some of the choices made by some of these implementations.

My main philosophical problem with Fangraphs’ WAR (fWAR) is that relievers are given extra value for having pitched in high-leverage situations. Personally, I don’t understand why we use a pitcher’s actual leverage index and chain from there. Why not just start and end with the deserved leverage index?

In terms of forecasting, I definitely go with deserved LI.  But, in terms of accounting for the past season, I don’t know that I would do that.  After all, imagine if Mariano Rivera was used in mop-up duty all season.  His actual win impact was muted.  Imagine Ozzie Smith at DH or 1B.  Imagine Adam Dunn in CF.  So, in terms of accounting for actual wins and losses, the actual usage is what we care about, and not the optimal / deserved usage.

But, that’s fine.  We can disagree on it.  Jeremy can have his jgWAR if he likes.  As long as we adhere to the basics of the framework, then 90% of the disagreement goes away.  Now the conversation moves to the periphery.  And that’s a good thing.  It’s no longer an RC v BsR debate, but a debate as to the specific component of the “B” variable in BsR.  That’s good.

Now, Jeremy also brings up WPA/LI for pitchers.  In this particular case, there’s a bit of a problem.  WPA/LI is great for hitters because it neutralizes all the PA so that all the PA are equally impactful.  It simply recalibrates each of the components.  Basically, it’s like having a game-state specific version of wOBA.  Sometimes the coefficient for the HR is 1.5 and walk is 0.9, or sometimes the coefficient for the HR is 3.0 and the walk is 0.4.  There’s always some sort of recalibration based on how much impact the walk and HR will have on that particular game state.  But, for pitchers, it’s not so easy.  Prime-Pedro has many fewer men on base specifically because Prime-Pedro is pitching.  So, we don’t want to neutralize each PA so that they count equally and then add them up in a linear fashion, which is what WPA/LI does.  WPA/LI is the first step, but then it needs to be BaseRuns-ized in order to get it into the right scale.

How big a deal is this?  Interestingly not big at all.  When I look at the big 4 of our generation, their career WPA/LI and career WPA has almost no difference.  In any case, to the extent that we have an issue, the bias would work the same for all the same-quality pitchers.  Basically, this is one of those things that is not worth worrying about, other than if you are really someone who enjoys digging into an issue like this.  So, yeah, WPA/LI (for pitchers) gets you at least 95% of the way there, if not 99%.

I love WPA/LI because it balances every PA to be equal to any other PA, while also adjusting for the particular vagaries of the situation.  A runner on 3B and less than 2 outs and down by 1 run is not the same as with 2 outs and up by 5.  WPA/LI handles it properly, and is the only metric to do so, other than its cousin WPA.

Monday, February 01, 2010

Another example of the bias in OPS

This time from PAAPFLY.com:

Continuing my bashing of Bengie Molina, allow me to show you how his terrible OBP can be quite detrimental. Bengie Molina posted a .727 OPS in 2009, which isn’t very good. Ryan Theriot managed to post an even lower OPS of .712 in 2009. He must be the inferior offensive player. Wrong. Molina’s wOBA is actually .308 to Theriot’s .318. Though Theriot slugged 73 points less than Molina, his OBP was 58 points higher, and, wOBA shows us that his 58 OBP points to Molina’s 73 slugging points were actually worth an additional10 points in wOBA. This is just a quick example and a good way to illustrate just how much Molina’s extraordinary out making skills truly do hurt his team, offensively of course.

Bill James turns Win Shares into Wins above replacement (WAR)

He doesn’t come right out and say it, but his explicit rankings shows this implicit acceptance.  Follow me for the proof.

The first thing to note is that he ranks 15 semi-random firstbasemen in… some order.  What order is that?  He didn’t say.  This is the Win Shares and Loss Shares of the 15 guys, in his order:
BJ WS LS
1 319 152
2 289 120
3 269 99
4 257 90
5 265 118
6 291 199
7 181 106
8 218 202
9 143 67
10 150 93
11 146 118
12 162 166
13 144 147
14 109 116
15 79 61

As you can see, it’s not in Win Shares order.  How about if I rank them in Win Shares minus Loss Shares order?
BJ WS-LS
1 167
2 169
3 170
4 167
5 147
6 92
7 75
8 16
9 76
10 57
11 28
12 -4
13 -3
14 -7
15 18

Nope, that’s not it either, though it’s getting closer.  Then I thought “Oh no he di’int.” Yes, he did.  If you do Wins above .333, here’s the list you get:

BJ wsWAR
1 54
2 51
3 49
4 47
5 46
6 43
7 28
8 26
9 24
10 23
11 19
12 18
13 16
14 11
15 11

Fref McGriff, his number one guy in this sample, has 319 win shares and 152 loss shares.  You can just do 2*WS - LS, or WS - LS/2, or WS*.667 - LS*.333, or whatever, as long as you weight the wins twice as much as the losses, and you get the same rankings.

To convert Win Shares to Wins Above Replacement, you do:
WS/3 - .333 * (WS+LS)/3

That “/3” is to convert win shares into wins.  And the “.333” is the win percentage replacement level.  So, the final simple equation is (2*WS-LS)/9.  For McGriff, that’s 319*2 - 152 or 486.  And divide by 9 to get to 54.  This is how you convert Win Shares and Loss Shares into Wins Above Replacement (WAR).  And Bill James has ranked his firstbasemen by (his version) of WAR.

Here is the comparison of Bill James’s WAR and Rally’s WAR, ranked in Bill James’ order:

rWAR bjWAR
51 54 McGriff Fred
61 51 Hernandez Keith
53 49 Cash Norm
52 47 Giambi Jason
44 46 Delgado Carlos
36 43 Garvey Steve
26 28 Vaughn Mo
23 26 May Lee
27 24 Teixeira Mark
22 23 Parker Wes
14 19 Sexson Richie
14 18 Power Vic
9 16 Pepitone Joe
6 11 Stuart Dick
6 11 Aikens Willie

Every player is within 5 wins, except for:
- Keith Hernandez, 10 more wins for Rally than Bill James
- Steve Garvey, 7 more wins for Bill James than Rally
- Joe Pepitone, 7 more wins for Bill than Rally

That’s it. 

It seems to me therefore that Bill has accepted that some sort of wins above baseline is needed.  And he is using a level that pretty much match what we’re using for wins above replacement, since the results are consistent.  I wouldn’t worry about the implication of the “.333 win percentage”, since that number is specific for the way his system is constructed.

This really brings Bill James back to where he always was 25 years ago, when he first wrote about replacment level in the context of Rice v Guidry and Clemens v Mattingly.  And when he ranked players as “chance of being better than a .400 player”.  Bill took a detour along the way in terms of ranking players by Win Shares (without considering Loss Shares).  Him bring Loss Shares into the mix really resets his position back along the Wins above replacement track that the rest of us are on.

Welcome back Bill.

Now, let me ask: if Bill is going to rank players by 2*WS-LS anyway, then why not show that single number as well?  Why show the WS and LS numbers separately (which is fine), but not show the actual single number he’s ranking the players by?  And, to give that number meaning, why not just divide it by 9, so you get the wins above replacement scale that the rest of us use?