Win Probability Added (or, just WPA to its friends) has received a lot of good and bad publicity lately. After Alan Schwarz covered WPA in his Sunday *New York Times* column, there was a lively debate between Tangotiger and MGL on The Book Blog (the Cliff Notes version: MGL hates it and finds it utterly useless and Tango thinks it’s fascinating). David Pinto also weighed in with a negative opinion of ranking players by WPA.

You may not be surprised to learn that I am in the “fascinated” camp, though I do acknowledge WPA has limits. WPA is a very simple idea (which is part of its appeal to me): calculate the odds of a team winning a game as the game progresses, based on calculations and historical baseball stats, and assign responsibility for changes in WPA to the players involved in each play. I think WPA is a great way to track a ballgame, evaluate in-game tactics and assess the contribution of relievers. I also think it may have some value in identifying “clutch” performers and as a secondary consideration for picking MVP candidates.

Having said that, I’ve never really played with a full season’s worth of WPA stats, or even a half season’s. Thanks to the tremendous effort of David Appelman at Fangraphs.com, we now have a half-season’s worth. So let’s learn a little bit more about WPA. In fact, let’s not even step into the player evaluation brouhaha, let’s just talk about teams. I hope to show you that WPA can help us solve one of the more vexing questions baseball fans often ask.

###### Team Leverage Index

Leverage Index (LI) was invented by Tangotiger. It’s used to measure the criticality of each plate appearance, and it’s perhaps the best tool to come out of WPA. LI is set so that the overall average of a plate appearance is 1.0. Ace relievers have often achieved an LI of 2.0 or more, meaning that their appearances were twice as critical as average. So I wondered, are there differences between teams too?

To find the answer, I used the LI of each player on a team and weighted it by each player’s plate appearances (or batters faced) for a team LI. I calculated two separate LI’s, one for the team’s batters and one for the team’s pitchers, and found that the average LI on both sides ranges roughly from 1.1 to 0.9. Here’s the list of this year’s teams:

Pitching LI Batting LI Team LI Team LI NYN 1.07 PIT 1.08 TB 1.07 ATL 1.07 OAK 1.05 SD 1.07 FLA 1.03 OAK 1.06 HOU 1.03 MIL 1.05 MIL 1.02 WAS 1.03 DET 1.02 PHI 1.03 STL 1.01 NYA 1.02 ATL 1.01 NYN 1.02 PIT 1.01 BAL 1.00 PHI 1.01 SF 1.00 SF 1.00 SEA 1.00 NYA 1.00 KC 1.00 SD 0.99 COL 0.99 ARI 0.99 STL 0.99 WAS 0.99 ARI 0.99 LAN 0.98 BOS 0.98 COL 0.98 LAA 0.98 SEA 0.98 LAN 0.97 CIN 0.98 FLA 0.97 TEX 0.98 CIN 0.97 KC 0.98 TEX 0.97 CHA 0.97 MIN 0.97 BOS 0.96 HOU 0.96 BAL 0.95 TB 0.96 LAA 0.94 CHA 0.96 TOR 0.94 CHN 0.94 MIN 0.92 TOR 0.94 CLE 0.91 DET 0.90 CHN 0.91 CLE 0.89

In general, teams that have played close games will have the highest LI and those that have played in the most runaway games will have the lowest. For instance, the Athletics rank high in both pitching and batting LI, and they have played more close games (50 games won by two runs or less) than any other major league team.

The Mets pitchers are at the top of the pitching list, they have faced more/bigger critical situations than any other team. Meanwhile, the Pirates batters have faced the most critical batting situations. There are also some big differences between batters and pitchers on the same teams. For instance, the Tiger and Devil Ray batters don’t rank highly, but their pitchers have faced a relatively high number of critical situations. I guess the Indians and Cubs have played the most boring games, judging by the low LI’s for both their batters and pitchers.

###### Team WPA

Next, let’s look at each team’s WPA rankings. As you can imagine, batting WPA and pitching WPA closely follow total runs scored and runs allowed. But there are some differences, as the following table shows:

Batting WPA Pitching WPA Team Total RS Team Total RA CHA 8.57 520 DET 12.61 328 BOS 6.26 486 SD 7.70 369 NYA 4.34 479 NYN 6.51 404 TOR 4.10 472 OAK 6.41 394 TEX 3.08 448 CHA 4.43 415 CIN 2.59 448 COL 4.10 399 DET 2.39 455 MIN 3.87 396 CLE 2.14 488 LAA 3.81 416 NYN 1.99 473 BOS 3.74 413 STL 1.98 440 SEA 3.25 421 MIL 1.91 411 STL 2.85 425 LAN 0.57 471 HOU 2.74 435 SF 0.21 419 NYA 2.66 406 MIN 0.13 422 ARI 1.86 450 BAL -0.46 436 LAN 1.43 416 KC -1.30 396 TOR 0.90 432 ATL -1.89 440 SF 0.29 407 PHI -1.97 420 TB 0.18 457 FLA -2.48 409 PHI -1.53 454 ARI -2.96 429 TEX -2.09 427 SD -3.20 393 CIN -2.09 463 COL -3.60 411 CHN -2.48 448 WAS -4.11 407 FLA -2.52 420 HOU -4.24 408 ATL -2.61 449 SEA -4.75 426 WAS -2.89 470 LAA -4.81 407 MIL -2.91 485 OAK -5.41 380 BAL -3.53 501 TB -5.68 383 PIT -5.00 474 CHN -7.52 357 CLE -5.64 443 PIT -10.00 411 KC -11.20 528

Well, that doesn’t really work. I wanted to show you all the data, but the table is kind of overwhelming. Time for a graph; here’s a picture of how each team’s runs scored compares to its batting WPA. Teams above the line have gained relatively more wins with their bats compared to total runs scored, while teams below the line have contributed relatively less of their runs to winning. Check out those Pirates, who have faced more critical situations than any other team’s batters. As the graph shows, they haven’t delivered:

Why do we see variances? Two reasons, I think. One, some teams simply have more opportunities to impact a game than others. Second, some teams actually deliver more in crucial situations than other teams do. LI measures the opportunities. WPA reflects both the opportunities and the actual production. For instance, the Pirates are only batting .220 in “close and late” situations. Combine that with their high LI and you get a really bad WPA.

Here’s the same graph, except for the pitchers. In this graph, I inverted the “runs allowed axis” so that teams above and below the line will have the same impact on their teams’ probability of winning as in the previous graph:

When it comes to pitching WPA, the bullpen and its management have a big impact. For instance, the Indians have had the lowest relative win impact from their runs allowed because their bullpen WPA is the worst in the majors at -5.09. That’s partly because their bullpen LI is also the lowest (0.88) and mostly because they’ve been lousy (4.86 ERA).

###### Pythagorean Breakouts

This isn’t just an academic graphing exercise. In fact, we can get something quite useful out of this stuff. You know the Pythagorean Formula? Invented by Bill James, it projects a team’s won/loss record from its runs scored and allowed and it’s typically very accurate. Baseball analysts like to track teams that vary from their pythagorean formula to see why and how those teams win more or less than predicted. WPA gives us a new way to approach that problem.

Here’s how. First, I used regression analysis to derive formulas that would predict each team’s batting WPA based on its runs scored and pitching WPA based on its runs allowed. As you can imagine, the R squared between WPA and runs is high (between .7 and .8) but not perfect. Next, I ran that formula for each team to see how much the team deviated from its predicted batting and pitching WPA. When I combined the two differences I got a number that is almost exactly each team’s variance from its Pythagorean Formula.

Let me see if I can put that in English. WPA gives us a way to assess how teams are exceeding or falling short of their predicted performance (based on runs allowed and runs scored). Specifically, it allows us to allocate the difference to each team’s offense and defense. The following table is a list of each team’s batting and pitching pythagorean contribution (listed in the second and third columns) based on the WPA analysis. The column labeled “Tot” is the total of the previous two columns, and the column labeled “Pyth” is the actual pythagorean variance for each team.

Bat Pitch Tot Pyth MIL 4.68 1.93 7 6 CHA 0.91 1.85 3 3 BOS 1.85 0.95 3 3 STL 1.97 1.33 3 3 DET 0.95 0.81 2 2 NYN -1.18 2.77 2 2 OAK 0.33 1.60 2 2 TB -0.24 2.06 2 2 CIN 1.81 0.42 2 2 BAL -0.09 3.00 3 2 SD 1.29 0.25 2 1 MIN 1.84 -0.72 1 1 HOU -1.19 2.28 1 1 ARI -1.91 3.00 1 1 TOR 1.03 0.12 1 1 LAA -1.66 1.34 0 0 NYA 0.60 -0.87 0 0 PHI -0.07 0.03 0 0 COL -0.84 -0.17 -1 -1 SF 2.21 -3.14 -1 -1 TEX 2.31 -3.39 -1 -1 CHN 0.42 -1.56 -1 -1 WAS -0.96 0.36 -1 -1 KC 2.90 -1.80 1 -1 SEA -3.42 1.30 -2 -2 LAN -2.41 -1.04 -3 -3 FLA 0.47 -4.56 -4 -4 ATL -1.90 -1.58 -3 -4 CLE -2.46 -5.25 -8 -8 PIT -7.23 -1.32 -9 -9

For example, the Pirates have the worst pythagorean variance in the majors and are on a pace to set the record for most one-run losses ever. As you can see in this table, the issue lies with the team’s batters, who are a whopping 14 wins (or seven games) below what their offense “should” have contributed to the team’s win-loss record.

The Indians have the second worst pythagorean variance, and you can see that their problem is the aforementioned bullpen. Bullpens are often cited as culprits in pythagorean variances but that isn’t always the case. The team with the highest pythagorean variance, the Brewers, has been led by its batters. They’ve added 4.7 games to the variance, while the pitchers have added just 1.9.

This is just one use of WPA that I found while playing with the stats, and I’m sure there are many more. Pretty fascinating, huh?

**References & Resources**

As always, many thanks to Tangotiger for educating us about WPA and making his LI tables available to Fangraphs. Also, thanks to Keith Woolner and Baseball Prospectus for their research in this area. And a huge thanks to David Appelman and Fangraphs.