This will borrow heavily from the excellent introduction by Dave Cameron on FanGraphs of their Win Values stat (located below - I'm mostly summarizing, but I highly recommend going through their explanation), as well as Tom Tango's explanations on his blog.

Position Players:

Batting, Fielding, Position, Replacement, Converting Runs To Wins, Dollars, Extra.

Pitchers:

Intro, FIP, Replacement, Run Environments, Runs To Wins, Park Adjustments, Calculations.

Measure of Value:

Wins are the goal for a team, and so the number of wins that a player adds to a team should be the goal we strive for in valuing them. The various metrics are measured in runs, so a runs to wins conversion is required.

The relationship between runs and wins on a team level is seen using the Pythagorean Winning Percentage Formula (Winning % = (Runs Scored ^ 2) / (Runs Scored ^ 2 + Runs Allowed ^ 2), though the PthyagPat formula is slightly more accurate as it adjusts dynamically for the run scoring environment.) A team that scores 800 runs and allows 800 runs will have a .500 winning %, and so go 81-81. Changing either runs scored or runs allowed by 10 would result in the final total changing by about 1 win, so that's what is used. 10 runs ~ 1 win.

Position Players:

Batting: The nice thing about wOBA is that it gives a fairly complete measure of offensive production using (adjusted) linear weights of the run value of it's component stats. That means we can go from wOBA to the value in runs (and thus wins) that a player contributes with the bat pretty quickly.

With an assumed league-average wOBA of .335, we have Batting Wins = (wOBA - .335) / 1.15 * PA / 10. This converts wOBA above-average to runs above average (that's the 1.15) and then to wins above average (the 10). A full season's worth of PA (700) is can be used to start out and then playing time would be adjusted later.

Example: 2009 Nick Markakis (lets just say) has a wOBA of about .395. His value with the bat would be about (.395 - .335) / 1.15 * 700 / 10 = 3.65 wins. He would add about 3.65 wins to the team over what an average hitter would with his bat, given 700 PA. (For less PA just multiply by PA/700).

Fielding: There are a number of ways to evaluate the defensive contributions of a player, but FanGraphs uses UZR with value from outfield throwing and double play turning also available. These are in runs above average for players at the given position, and so can be converted to wins by dividing by 10. For catchers there's even more work to be done (how much is calling a "good game" worth?).

Example: 2009 Nick Markakis has slightly above average range in right, so we say he saves about 2.5 runs by getting to balls that an average RF fielder wouldn't. We also say he has a great arm, and so saves 5 runs by throwing out runners as well as keeping them from advancing due to fear of being thrown out above what an average fielder would. Dividing by 10, we have Nick as being worth 0.75 wins above average with his glove (and arm).

Position: It's a lot easier to find a guy with the skills to play first-base (catch the ball when thrown to you, largely) than it is to find a guy with the skills to play short (range, arm, etc). That means a guy who can handle short is valuable - all else being equal - than a guy who can only handle first-baseman. By looking at players that played multiple position, Tom Tango was available to develop an adjustment to show the relationship between players at all of the positions. For example: when an average shortstop (+0 runs) moves to second-base, he plays +5 run defense. A good left-fielder (+10 runs) could move to center and be about an average center-fielder (+0 runs). And so on. Therefore, the interrelationships between the positions can be adjusted using the following (assuming a full season at the position):

C: + 12.5 runs

1B: - 12.5 runs

2B: + 2.5 runs

3B: + 2.5 runs

SS: + 7.5 runs

LF: - 7.5 runs

CF: + 2.5 runs

RF: - 7.5 runs

DH: - 17.5 runs

Example: 2009 Nick Markakis will play RF, so that's - 7.5 run. If he could play center as well as he plays right then he'd be worth 10 more runs to the team.

We can now say how valuable a player is relative to average. Nick Markakis is (given a full season) 3.65 wins with the bat plus 7.5 wins with the glove minus 7.5 wins for being limited (relatively speaking) to right-field. That's a total of 3.65 wins above average.

Replacement Level: Instead of comparing to average, we would like to compare a player to the guy that would take his spot if he wasn't available - a guy that's just wandering around in the minors looking for a chance - a replacement player. These are guys that are available for the league minimum and any team has a shot at; guys like Justin Christian, Terry Tiffy, and Brandon Fahey. Now in reality, this would depend on the team's bench and what players they have available, but we're going for a general level. A replacement level adjustment is about 22.5 runs. That is, an average player is about 22.5 runs better overall than the Brandon Fahey's of the world. This value is different, however, in the two leagues. The AL has better players, on average, than the NL, and so the adjustment is 25 runs for AL players and 20 runs for NL players. The runs is again converted to wins by diving by 10.

Example: 2009 Nick Markakis is 3.65 wins above average, which makes him (as an AL player) 6.15 wins above replacement. If the O's play Justin Christian in right-field all season instead of Nick Markakis, they'd be worse by close to 6 wins.

Playing Time: There aren't a lot of Cal Ripken Jr.'s around anymore, so one needs to adjust the win values (that are for 700 PA, or a full season in the field) for how much a player actually contributes.

For Example: 2009 Nick Markakis gets 650 PA (and play an equal proportion of innings in right), so he should be worth (650/700) * 6.15 = 5.7 wins. 2009 Nick Markakis is worth 5.7 wins (about 57 runs) more than a replacement level player. That's 5.7 WAR, for short.

Extras: One can (and probably should) add in other things like base-running, but Batting, Fielding, Position, and Playing Time gets the majority of the player's value down.

Pitchers:

Rate of Runs Allowed: We want to know how good a pitcher is, so the proxy for that is how many runs he is responsible for (though not necessarily directly). ERA has some problems with were the responsibility for having given up a run goes, so either FIP or tRA (I prefer the latter converted to ER from R, but they're very similar) do the job as a stand-in. Apply all context adjustments (league, park, etc.) where necessary.

Example: 2009 Jeremy Guthrie has a 4.15 "ERA" (from about a 4.50 tRA).

Replacement Level: Like with the position players, we want to know how much the pitcher actually adds to the team over the guy who'd be there in his place (your Steve Trachsel's of the world). Since it's easier to pitch out of the pen, there's a difference between starters and relievers, in addition to the AL/NL split.

The contribution is distinguished by winning percentage, with a replacement level starter expected to win 38% of his games given league average offense, bullpen, and opponent. With a average starter, offense, and opponent, a replacement level bullpen should win about 47% of their games.

"ERA" to W% Conversion: Since the run scoring environment in which a pitcher plays effects how low of an "ERA" he'll need to maintain a certain W%, an adjustment is needed. Using the league-average ERA and the PythagenPat formula, an individual pitcher's W% can be calculated.

For Example: 2009 Guthrie's 4.15 "ERA" with a league-average ERA of 4.30 results in a W% of 51.7%. His "ERA" is slightly better than average and so he'll win a little more than half his games.

[Step by step: (4.15 + 4.30) / .92 to get total run (the .92 is for earned runs to runs) environment of 9.18 runs. PthagenPat; 9.18 ^ .28 = 1.86. Wins/Losses = (4.30/4.15) ^ 1.86 = 1.068. W% = Wins / Wins+Losses = 1.068 / 1.068+1 = 0.517 = 51.7%.]

W% to Wins: To find how many wins above a replacement level player a pitcher is worth, lets find how many more game she's expected to win. The number of full games he's expected to pitch is IP / 9. Then multiply by the percent of games he's expected to win, above what a replacement level pitcher would.

Example: 2009 Guthrie pitches 20 games (180 IP / 9) with an expected winning percentage above replacement of 14.7 % (.517 - .370 for being an AL starter). That's 2.9 WAR.

Leverage: There's an additional adjustment made for relievers, depending on how important (high leverage, based on score, inning, and base/out situation) their innings are. A closer with a 3.00 ERA is going to be more valuable than a mop-up man with a 3.00 ERA, all else being equal. A relievers base WAR is multiplied by their leverage index (1.0 is average; 1.8 is what closers are likely to see; 0.6 is for mo-up guys) to get their final WAR.

Example: 2009 Jim Johnson was a 3.80 "ERA" in 65 IP. That makes him a 0.7 WAR regular reliever (LI of 1.0) but a 0.9 WAR pitcher as the set-up man (LI of 1.3).

So there we go; we now have a way to say how valuable players are. The next step would be to adjust to dollars, by looking at what teams pay players on the free agent market and dividing how many WAR the players are worth. For 2008 it was about $4.4 M per WAR, so (assuming 10% salary inflation, which is probably high in this economy) it should be about $4.84 M per WAR for 2009.

Example: 2009 Nick Markakis is worth about 5.7 WAR, which on the free agent market would cost about $27.6 M.

## Friday, February 29, 2008

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