At its most basic level, our pitching WAR calculation requires only overall Runs Allowed (both earned and unearned) and Innings Pitched. Since we are trying to measure the value of the pitcher's performance to his team, we start with his runs allowed and then adjust that number to put the runs into a more accurate context.

Once we have the pitcher's runs allowed and innings, we set about figuring out what an average pitcher would have done if placed in the same setting as the pitcher we are studying.

Back to 1918, we have gamelogs for every major league pitching appearance. This means that we can, with certainty, determine which team's pitchers we're facing. For each season, we also know the average runs per out for each team and we can adjust this number into a neutral context using park factors. Then, based on this, we can determine what the average number of expected runs would be for this set of teams faced.

This can have a major impact in situations where there is a set of dominant offensive teams and some pitchers face them multiple times while others may never face them. For example, pitchers for the 1927 Yankees never faced Murderers Row.

For seasons with interleague play, we only include the non-interleague games and interleague home games to determine the teams' season average in run scoring. So this would exclude, on average, nine games from a team's average. Our reasoning is that including nine games the Red Sox don't have a DH will skew their offensive averages lower when most pitchers are facing them with a DH.

To account for these out-of-league road games, we then add or subtract 0.2 runs to the team's averages depending on whether they have or do not have a DH in the game. So for the 2011 Red Sox, all AL pitchers or NL pitchers facing them in Boston are expected to give up 5.49 runs per game, but if the Red Sox go to Philly we expect them to score 5.29 runs per game.

When we are in-season, we use the run-scoring averages for offenses for the last 365 days.

The pitcher's expected runs allowed is then the sum of his opposition's run scoring weighted by the innings he faced each team. We call this xRA.

A great deal of work has recently gone into the study of Defense-Independent Pitching Stats (DIPS). We agree with the validity and importance of most all of this work, and some would argue that you shouldn't charge the pitcher for runs allowed in the way we do since it is often not the pitcher's fault, but the defense's. Our view is that, while the pitcher may have been unlucky or lucky in certain ways, we are trying to measure the value of the recorded performance--not its repeatability--and that we can account for defense in different ways.

To account for defense, we find the overall team defensive runs saved, which uses Baseball Info Solutions' Runs Saved from 2003 on and Total Zone before 2003. We then compute the number of balls in play allowed by the team and the number of balls in play allowed by the pitcher, and assign the negative of the proportional team defensive runs to the xRA_ppf values.

`xRA_def = (BIP_pitcher)/(BIP_team)*TeamDefensiveRunsSaved`

In the current MLB environment, relievers have much lower ERAs than starters. Relievers come in, throw gas for an inning or less, and then leave, so for recent years we set this difference league-wide at .1125 runs/game and then from 1960-1973 it is set at .0583 runs/game. This adjustment for starter/reliever ERAs is really only applicable since 1960, and if you look at the difference in starter and reliever ERAs, it is clear that there are two eras of bullpen usage. From 1960-1973 there was a slight starter/reliever effect, and then in 1974 we start to see the current dichotomy.

Before 1960 there is no starter/reliever adjustment.

Since we have gamelogs, we can also customize our park factors to the parks the pitcher actually pitched in. This can have an impact in a division like the NL West where you have three fairly extreme pitcher parks and two fairly extreme hitter parks. Usually, pitchers' custom park factors are less than a point away from their teams' park factor, but on occasion it can be multiple points. For pre-gamelog seasons, we use the team's park factor. All park factors are 3-year average.

In computing the player's wins above average, we use

`xRA_final = PPF_custom * (xRA - xRA_def + xRA_sprp_adj)`

See Runs to Wins. This provides us with WAA (wins above average).

As RA_replacement_adj currently stands, starters would be far, far more valuable than relievers; e.g. most average starting pitchers would be viewed as more valuable than Mariano Rivera in even his best seasons. The flaw in our reasoning above is that closers and many relievers are used in the highest leverage situations. These situations have a much larger ability to impact the outcome of the game than 0-0 top of the 1st, so we adjust the RA_replacement_adj with a leverage multiplier. An average leverage is 1.0, while many closers will approach an average of 2.0 for the season while mopup relievers might be at 0.7. This is applied only for relief innings and the leverage we use in the leverage at the beginning of the pitcher's outing. This way a bad pitcher can't bump up his leverage (and WAA) by walking the bases loaded and striking out the side every time.

`WAA_adj = WAA * (1.00 + leverage_index_pitcher)/2`

The leverage is averaged with 1.00 because of chaining of bullpens. If the closer goes down, the manager is not going to use the AAA callup as the closer. The AAA callup will move to the back of the bullpen while everyone else will move up one slot in the pecking order and the top setup man will become the closer. Modifying the leverage in this way accounts for this difference.

See a full description of leverage and how it is calculated.

One other adjustment occurs here: we re-center WAA for the league at zero, so that the average is exactly zero. This factor is put in this value, which is why you will see some non-zero values in WAA

As with the hitters, we have a replacement level set for each league based on the competitive level of the league. See the replacement level explanation in the WAR for position players page for a full discussion of the multipliers.

See Runs to Wins for an explanation of how we convert the replacement level into wins between the average player and the replacement level player (WAR_rep).

After we make a first pass through the calculations, we determine how the league's current total WAA and WAR differs from the desired overall league WAR. We then add or subtract fractional WAA and replacement runs from each player's WAA or runs_replacement total based on their playing time, and recompute WAR with this adjustment included.

`WAR = WAR_rep + WAA + WAA_adj`

FanGraphs has a long and detailed rundown of their WAR calculation, so we won't fully rehash it here. Our WAR starts with runs allowed by the pitcher and compares it to the league average pitcher (adjusting for quality of opposition), parks pitched in, and quality of defense behind the pitcher.

FanGraphs' WAR begins with FIP, which is a fielding independent pitching stat comparable in scale to ERA that is computed using only pitcher dependent stats.

`FIP = ((13*HR)+(3*(BB+HBP-IBB))-(2*K))/IP + lg_specific_constant(around 3.20 or so)`

In FIP, hits allowed and non-strikeout outs recorded have no role in the calculation other than in the number of total innings pitched. The assumption is that once the ball is put into play (other than a home run) the entire outcome is determined by random chance and team defensive quality. This is definitely true to a greater degree than fans likely believe, but we disagree as to whether this is the best measure of the value of a pitcher's historical performance.

I've crafted some admittedly extreme cases below to illustrate situations where the approaches differ. For most situations, FIP and Runs Allowed Average (RA, essentially what we use) will be very close and are strongly correlated, but there are a number of cases each year where there are large disparities between the two metrics.

Situation #1, Pitcher A throws a perfect game with 20 strikeouts, Pitcher B throws a perfect game with no strikeouts.

FIP: Pitcher A -1.40 FIP, Pitcher B, 3.20 FIP, RA: Pitcher A 0.00 RA, Pitcher B 0.00 RA.

Situation #2, Pitcher A throws one inning w/ sequence, HR, ground out, fly out, BB, BB, BB, fly out, Pitcher B throws one inning w/ sequence BB, BB, BB, HR, SO, SO, SO

FIP: Pitcher A 25.20 FIP, Pitcher B 19.20 FIP, RA: Pitcher A 9.00 RA, Pitcher B 36.00 RA.

As I said, in the average case the two methods will arrive at similar results, but on the edge cases the differences can be quite dramatic.

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