Wins Above Replacement (WAR)|
WAR is made up of 4 different parts:
In this WAR system, replacement level is defined as a .4136 Winning % , or 67 wins in 162 games. Working backwards with the pythagorean record formula, a 67 win team would score 82.5% as many runs as they would allow.
First, Base Runs are
calculated for each player.
Let's say that "player X" is credited with 104 Base Runs and 400 Outs (104 * 27 / 400 = 7.02 R/27).
Suppose that the park adjusted league average (non-pitchers) for this season
is 4.5 Base Runs per 27 outs.
Next, take the number of games played for each player (since a player can only influence the outcome of the games in which they play). Player X played in 150 Games. Multiply games played by the league average of Base Runs per 27 outs (150 * 4.5 =
675). An "average" team would score 675 runs in 150 games.
Take player X's outs and figure how many Base Runs an "average" player would have with those outs (400 / 27 * 4.5 = 66.67 Base Runs). Calculate the replacement player's rate (4.5 * .825 = 3.725) and do the same for the "replacement" player (400 / 27 * 3.7125 = 55 Base Runs).
We now have Base Runs for player X (104), average player (66.67) and replacement player (55). Now we can figure offensive WAR.
Take 675 (Avg Team's runs) and subtract the average player and add player X (675 - 66.67 + 104 = 712.33 runs). This is the number of runs a team would score if it consisted of player X and all average players.
Do the same for a replacement player (675 - 66.67 + 55 = 663.33 runs).
Then use the pythagorean record formula to figure the winning % of player X's team (712.33 ^ 1.81) / (712.33 ^ 1.81 + 675 ^ 1.81) = .524. Multiply that by 150 games to figure Wins (.524 * 150 = 78.65 Wins).
Do the same for the replacement player's team (663.33 ^ 1.81) / (663.33 ^ 1.81 + 675 ^ 1.81) = .492. Figure Wins (.492 * 150 = 73.82 Wins).
This would give us 78.65 - 73.82 = 4.83 Offensive Wins Above Replacement.
Offensive WAR for Pitchers is basically the same, except that it is assumed that a team consists of 8 position players and 1 pitcher. Also, pitchers are compared to other pitchers and the replacement level hitting pitcher.
The calculation for Fielding WAR is similar to the Fielding Win Shares method, except for that Runs Saved replaces Win Shares.
To explain Runs Saved, let's look at Runs Scored first. Suppose a team scores 750 runs in a season when the park adjusted league average was 700. Using the .5/1.5 Win Shares Run Margin, this team would have 400 offensive marginal runs (750 - 700 * .5 = 400). Working backwards, we can calculate Runs Scored by finding the sum of Marginal Runs and half of the expected runs (700 * .5 + 400 = 750 Runs Scored). Obviously, we do not need to work backwards for Runs Scored, but this is how we calculate Runs Saved.
Let's say this same team allowed 600 runs while the expected runs allowed was still 700. This team's defensive marginal runs would be 450 (700 * 1.5 - 600 = 450). Using the same "backwards" method as above we can calculate Runs Saved (700 * .5 + 450 = 800 Runs Saved). Of course, this was for explanation purposes. An easier method is twice the expected runs allowed minus runs allowed (2 * 700 - 600 = 800 Runs Saved).
It should be noted that Runs Saved used here aren't technically Runs Saved, they are just marginal runs plus sub-marginal runs. Each team's fielders save an infinite number of runs in every game. Without fielders, an opposing team will continue to score until the pitcher strikes out 27 batters.
Once Runs Saved are calculated, the Win Shares method is used to divide Runs Saved amongst pitchers and fielders. Next, they are divided amongst each position and then each player at each position. The only difference is that when dividing Runs Saved amongst positions, .200 is NOT subtracted from each position's claim percentage. This is done (or not done) because sub-marginal runs are included in Offensive and Pitching WAR, so there is no need to take them out of Fielding WAR.
Once each player's Runs Saved is figured, Fielding WAR can be calculated. Instead of going through the entire method of figuring WAR from runs, just refer to the Offensive WAR section, as it is the same method. The rate in which fielders are compared is Runs Saved per Innings at each position.
Just as Offensive WAR is based on Base Runs, Pitching WAR is based on Defensive Independent Pitching Stats (DIPS). More specifically, DIPS version 2.0 is used.
Once defensive independent runs are figured, Pitching WAR can be calculated using the same method as in the Offensive WAR section.
A Positional Adjustment
is needed to make up for the difficulty of each position. Take a look at the Defensive Spectrum to see the order of difficulty of each position.
To quantify the difference in each position, Fielding Win Shares is used. The average WS/Inning for each position is compared to the average for all positions. Playing time is then taken into account and an adjustment is applied.
Win Shares (WS)
Win Shares are calculated using the formula in the book Win Shares by Bill James. If you don't have the book and would like to take a look at the calculations, "Patriot" has a very detailed seven part walk through on his site.
If you are wondering why the Win Shares totals do not exactly match those totals in the Win Shares Book, there are a couple of reasons:
The first, and possibly the biggest reason is that this site uses different park factors than Bill James did. Any difference in park factor can alter the distribution of offensive and defensive Win Shares as well as other parts of the formula.
Possibly, the second biggest reason is that this site keeps decimal points, where as Bill James rounded to the whole number. This is done to keep the values as accurate as possible.
Next, when figuring Catcher's claim points, the book only uses opponent's stolen base data from 1987 until present. This site uses this data from 1954 until present. On the flip side, the book uses sacrifice allowed data from 1931 until present, while this site only uses it from 1956 until present. This difference in data available will alter catcher claim points, which will also slightly alter the other positions as well.
When calculating Runs Created for offensive Win Shares, the same RC formula is used except for that RISP data is not included.
When calculating Pitching Win Shares, the book includes an adjustment for pitchers hitting below a certain level. This site does not make that adjustment since a pitcher's offense (or lack of) should not affect his pitching Win Shares.
Finally, Catcher's ERA is not included in calculating claim points. This is done mainly because of lack of data available, but also because of possible inaccuracies. Some pitchers use a "personal" catcher, which will cause the games caught for each pitcher to be unequal. For example, Eddie Perez was Greg Maddux's personal catcher while Javy Lopez caught the majority of the other games. It is inaccurate to compare their catcher's ERA's since they caught different pitchers.
Win Shares Above Bench (WSAB)
Bench level is set at 75% of league average for all players except Starting Pitchers, whose bench level is 60%
For more information on WSAB, check out the glossary entry at The Hardball Times.
For Offensive WSAB, find the league average of Offensive Win Shares per Plate Appearance. This is done separately for Non-pitchers and pitchers. For explanation purposes, let's say that the league average is .014 WS/PA.
Suppose that player X had 600 Plate Appearances and 12 Offensive Win Shares. Multiply his Plate Appearances by the league average WS/PA. This would give him 8.4 (600 * .014) Expected Offensive Win Shares. To find expected bench, multiply that by the bench level (8.4 * .75 = 6.3 Win Shares). Simply subtract the bench level offensive Win Shares from player X's actual offensive Win Shares to find Offensive WSAB (12 - 6.3 = 5.7).
The same process is done for Fielding WSAB, except that defensive innings is used for the rate instead of Plate Appearances. Bench level is still 75% for fielding.
It is important to note that Fielding Win Shares for each position played are compared to the league at that position. For example, suppose player X played primarily at 2B, but also played some SS and 3B. His total Fielding Win Shares are not just compared to other 2B. Instead, his Fielding Win Shares at 2B are compared to the league at 2B, while his Fielding Win Shares at SS are compared to the league at SS. This is also done for 3B. Once each position's WSAB are calculated, they are added together to find total Fielding WSAB.
For Pitching WSAB, Innings Pitched is used to find the league rate. Starters are compared to other starters while relievers are compared to other relievers. As mentioned earlier, the bench level for starters is 60% while it's still 75% for relievers.
Components Above Replacement
Replacement level for each component is based on the standard deviation of each rate stat. Using the 82.5% from the WAR calculation as a baseline, we can figure that replacement level is approximately .5 standard deviations below league average. Each rate stat's corresponding percentage is calculated based on half of it's standard deviation below (or above in some cases) league average.
The reason that standard deviation is used is because not all rate stats are distributed equally. For example, the replacement level for batting average comes out to 92.8%. If the league average is .260, the replacement level would be a .241 batting average (.260 * .928 = .241). If we just used 82.5%, the replacement level would be .215 which would cause the vast majority of the league to be above replacement and this would not be consistent with our other replacement levels.
Component Stat Calculation
component is calculated using a corresponding rate stat. For example, Hits Above Replacement uses Batting Average (H/AB), Times On Base Above Replacement uses On Base Percentage (H + BB + HBP) / (AB + BB + HBP + SF). The rate stats are needed in order to compare each player to the league.
Let's take Hits Above Replacement for example. Suppose Player X has 150 hits in 500 at bats (.300 Avg). The league's average is .265 and the replacement level is .246 (.265 * .928 = .246). In 500 at bats, a replacement player would collect 123 hits (.246 * 500 = 123). Hits Above Replacement would then be 27 (150 - 123 = 27).
This is done with all Components Above Replacement. Each component just needs a corresponding rate stat so that the denominator can be applied to the replacement level.
To complicate things a little bit more, prior to the calculation, the numbers are also park adjusted for each player. To calculate park adjusted stats, the "Willie Davis Method" is used with an additional adjustment for Home run Factor. The "Willie Davis Method" is explained in The New Bill James Historical Baseball Abstract (pgs 740-743). Essentially, it uses the simple Runs Created formula, a park adjustment is applied and then the quadratic formula is used to find hits. The other stats (2B, BB, etc) are then adjusted at the same rate.