Baseball fans often debate over how much a given pitcher contributes to their team.

Pitcher A could beat Pitcher B with one arm tied behind their back. Pitcher C has a horrible earned run average and could not pitch his way out a paper bag. Look at Pitcher D’s Win-Loss record…barf.

Sound familiar?

Luckily, the baseball sabermetric community worked together to summarize a player’s total contribution, in one handy statistic, Wins Above Replacement or WAR.

Thank you, sabermetric community!

## Undestanding Wins Above Replacement

WAR is conventionally defined as the number of wins a player adds to his team, when compared to a replacement level player.

Now, a replacement level player is categorized as a minor-league player or AAAA player from the team’s bench. For example, Clayton Kershaw of the Los Angeles Dodgers provided 6.5 wins above replacement level in 2016, if he were to be replaced by a long relief mop up, spot starter pitcher, the team would lose 6.5 wins.

The concept is very simple, right?

Now, as someone with a substantial math background, we know it is necessary to understand the calculations behind Wins Above Replacement. Otherwise we may be utilizing the comparisons improperly.

Oh mathematics, you are so funny! There is no one perfect way here, as with many other things in life.

- FanGraphs calculates WAR and abbreviates it as fWAR.
- Baseball Reference calculates WAR and abbreviates it as rWAR.
- Baseball Prospectus calculates WAR and abbreviates it as WARP.

We will touch on how each statistic is calculated, each slightly different, with a bias towards pitching.

## Wins Above Replacement, fWar

Fangraphs denotes their Wins Above Replacement statistics as fWAR.

The theory behind the calculation, is that a baseball pitcher only has control over so much. If the positional players behind him have poor fielding range or are prone to errors, is it fair to punish the pitcher? Therefore, fWAR looks to strip out this noise and calculates fWAR by utilizing Fielding Independent Pitching (FIP), more on FIP here.

Fangraphs notes that utilizing FIP, instead of expected Fielding Independent Pitching (xFIP), helps better capture past value. Their reasoning is that if a pitcher allowed 20 homeruns (based on regressed xFIP), but actually allowed 30 homeruns, then xFIP may actually make the pitcher seem more valuable.

Now that idea may seem foreign to you, but xFIP applies a league average home-run rate to the statistical measure. So, if you want a true comparison of value, you should utilize the actual rates. To see contextual uses of xFIP, we suggest you read the following article.

Let’s look at the other pieces to the mathematical fWAR puzzle.

Replacement level, run scale, park adjustments, and run environment.

*Replacement Level*.

The replacement level remains relatively the same year-to-year. According to Fangraphs, this is a 0.380 win % for starting pitchers and a 0.470 win % for relief pitchers. Replacement level FIP is recalculated each year based on the performance of the average player.

FanGraphs provides a nice contextual example:

So if league-average for American League starters was 4.40 FIP, then replacement level would be set at the appropriate mark above that for a starting pitcher with a 0.380 win % (in this example: 5.63FIP)

So, FanGraphs calculates the league average FIP and finds where the starting pitcher has a 0.380 win percentage, and utilizes the corresponding value as replacement level FIP.

*Run Scale*. FIP has to be converted into a runs scale, simply divide FIP by 0.92. This ensures that pitching WAR and offensive WAR are operating similarly.

*Park Adjustment*. Fangraphs explains this best, so let’s not rework the wheel here, and give credit where credit is due. “If a 5.63 FIP was the replacement level for an AL starter, and one ballpark depressed offense by 2%, then the replacement level FIP for the park would be 5.52 FIP, 2% lower than the AL replacement level”.

*Run Environment*. The baseball team does not have to score as many runs when their ace is on the mound compared to when an average pitcher is on the mound. The average is considered 10 Runs/Wins, elite pitchers will have a lower Runs/Win rate. FanGraphs conversion formula is ((League RA+Pitcher’s RA)/2)+2*1.5

Hopefully we did not lose you throughout all that fun math!

### Wins Above Replacement, rWar

Baseball Reference calculates Wins Above Replacement (rWAR) slightly different from FanGraphs.

Instead of starting with FIP, as FanGraphs does, Baseball Reference starts with Runs Allowed (both earned and unearned) and Innings Pitched.

Luckily, here there is a nice handy, “easily” to decipher formula.

The final calculation looks something like this:

But, before we get there we need to learn the intermediate steps.

First let’s start with xRA or level of opposition. Baseball Reference determines the average number of expected runs for a given set of teams. This data is generated by taking the average number of runs per out for each team and adjusting a neutral context utilizing park factors.

The level of opposition xRA, is then adjusted for team defense and labeled as xRA_def. To account for defense, Baseball Reference determines the defensive runs saved, computes the number of balls in play allowed by the team and pitcher.

The next step is to adjust the level of opposition xRA, utilizing averages for starters and relievers. Baseball reference applies a .1125 runs/game from 1974 to the present and a 0.0583 runs/game from 1960-1973. These adjustments account for relievers having a lower earned run average (ERA) than starters and is labeled as xRA_sprp.

Lastly, custom park factors (PPfp) are calculated. This factor determines how each ballpark affects the amount of runs scored (due to varying dimensions).

These adjustments are all blended together to form a final level of opposition statistic:

The xRAfinal (or runs allowed final) is then converted into Wins. This is by far the most complicated part.

Baseball Reference notes that if a person calculated the conversion by hand, that the number 10 would be used, since every 10 runs a player adds or subtracts accounts for about 1 win. Simple.

Well fortunately, we have computers to make the calculation more precise.

Let’s utilize Baseball Reference’s example, 2011 Roy Halladay with 59.7 runs in 32 games.

59.7 runs / 32 games = 1.866 runs per game

X = (53.6*0.15447-1.866)^.285 = 1.698

Where 0.15447 represents the 2011 NL average runs/out, and 53.6 and 0.285 are mathematical constants.

Runs Scored (RS) is set to 4.14 and Runs Allowed (RA) is set to 4.14 for league average in 2011.

To calculate RA for Halladay, we subtract 4.14 from 1.866 runs per game = 2.27

Baseball Reference then plugs this into their runs-to-wins estimator formula utilizing the Pythagorean Formula for baseball.

Plugging the numbers into the equation: W-L% = (4.14^2)/(4.14^2+2.27^2) = 0.735

This means that an average team should win 73% of the time when 2011 Roy Halladay was on the mound. We then calculate wins above average= WAA =(0.735-0.500)*32 games started = 7.52 WAA.

The final step is to take the 1.6988 and add it the 7.52, say 1.7 + 7.52 = 9.22 rWAR.

Yeah, we here at Innings Pitched are glad we have excel files and computers to repeat that calculation over and over again.

Kudos to Baseball Reference, who deserves all the credit for the great mathematical breakdown of their Wins Above Replacement statistic.

### Wins Above Replacement, WARP

Baseball Prospectus’ Wins Above Replacement (WARP) statistic is less frequently utilized in our opinion, but still provides an good measure of comparison.

Based on Baseball Prospectus’ website WARP is calculated as.

BWARP=(BRAA+BRR+BRR_ARM+FRAA+POS_ADJ+REP_LEVEL)/RPW

BRAA = Batting Runs Above Average

BRR = Baserunning Runs

BRR_ARM = Fielder Arm Ratings

FRAA = Fielding Runs Above Average

REP_Level= Replacement Level

RPW = Runs per win

Then accounting for the Pitcher:

PWARP=Defensive Runs Allowed, converted into a WARP scale

PWARP and BWARP are relatively proprietary from Baseball Prospectus, so unfortunately we could not provide a further breakdown.

Intuitively though, the BWARP portion of the equation will go to zero for baseball pitchers who never bat or take the field.

### Which Wins Above Replacement Statistic To Use?

So now that you are through the technical portion, let’s discuss the statistic that we here at Innings Pitched prefer.

fWAR, from Fangraphs is our statistic of choice. The reasoning behind this is that FanGraphs generally utilizes FIP for their Wins Above Replacement calculation for baseball pitchers. We love the FIP statistic as it is readily available, simple to understand, and is generally calculated the same across all baseball statisticians.

Therefore, if we know a pitcher has an above average FIP, than we know that he will inherently provide slightly more value to a team. While some regression occurs in the background with FanGraphs, this statistic is much easier to digest than Baseball Reference’s that utilizes many adjustment factors or Baseball Prospectus that generates proprietary coefficients that to us, are more difficult to understand.

However, to get the full picture it is always a good idea to consult each statistic to see if any anomalies have occurred.

So now that we here at Innings Pitch have settled on fWAR from FanGraphs, do not be surprised if we refer to this simply as WAR moving forward in all our articles.

We understand the calculation so let’s move forward on how to use WAR.

### How to Use Wins Above Replacement

The first step is to utilize Wins Above Replacement as a comparison tool, rather than a catch all.

What we mean is, do not look at the statistic in a vacuum.

If Pitcher A has a 5.0 WAR that means the player is superstar caliber, but if Pitcher B has a 4.9 WAR, that does not necessarily mean Pitcher A is superior.

Both baseball pitchers are all-star/superstar caliber.

However, if Pitcher C has a 0.1 WAR we can clearly see that both Pitcher A and Pitcher B are far superior. Also since WAR is context, league and park neutral, the comparison will hold for players between years, leagues, and teams.

Before we leave you, let’s breakdown how FanGraphs categorizes each player with the WAR statistic.

On occasion, some baseball pitchers receive a negative WAR, which means that a minor leaguer could have provided nearly the same result. James Shields (-0.9 WAR) and Jered Weaver (-0.2 WAR) were the only two to complete the feat in 2016, among qualified baseball pitchers.

So there you have it, the complete (almost) breakdown of Wins Above Replacement. Be sure to do your research before jumping to the same conclusion that we here at Innings Pitched have.

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