wRAA For Position Player WAR Explained

There has already been a great deal of discussion how to compute Tom Tango's weight on-base average (wOBA), weighted runs above average (wRAA) and related stats. We believe these stats are the best in their class (among linear weights, runs created, and base runs) for assessing batter performance. One major reason for their use is that they rely solely on full-season data sets, which contain H, 2B, 3B, HR, SB, CS, BB, IBB, SO, SH, SF, and HBP. wRAA is commonly converted into wRC+ to appear as a stat centered around 100.

On Baseball-Reference.com we call these stats rOBA (reference-weighted On-Base Average), Rbat and Rbat+.

Here are some rundowns of how to calculate these stats in their basic form:

wOBA Primer from Tom Tango

SQL code from Colin Wyers

wOBA coefficients and linear weights from Beyond the Box Score

For wOBA in its most basic form, we get exactly the results presented by Beyond the Box Score as linked above. A couple of notes on implementing this:

When computing wOBA, we use Plate Appearances = AB + BB - IBB + HBP + SF and we also exclude IBBs from the count of BBs. Our view (and those of the creators of these stats) is that SHs and IBBs are managerial decisions, and in general the best way to handle these is to exclude them from the rate stat (wOBA); then when computing the counting stat (wRAA), we assume the hitter would have done as well as they normally did in the situation if asked to hit.

So for example, in 2010 we get a formula for MLB of: wOBA = (0.70 * uBB + 0.73 * HBP + 0.89 * 1B + 1.27 * 2B + 1.61 * 3B + 2.07 * HR + 0.25 * SB - 0.50 * CS) / (AB+BB-IBB+HBP+SF)

The "wOBAScale" for 2010 is 1.25 and the league wOBA is .320, so for weighted runs above average, use:

wRAA = ( (wOBA - .320) /1.25) * (AB + BB + HBP + SF + SH)

This would be suitable to use as a batting value for WAR after a park factor adjustment, which we would calculate using three-year park factors as follows:

wRAA_pf = wRAA - (BPF/100 - 1) * PA * lgR/PA / (BPF/100) BPF > 100 indicates a hitter's park, and lgR/PA is the league's runs per plate appearance. This is an additive rather than multiplicative park adjustment.

This is very good, but I see a few small places we can improve on this. We'll summarize how rOBA differs from wOBA here and then spell it out in detail later. This was prepared awhile back, so if anything is out of date, please let us know (feedback form at the bottom of the page).

• wOBA includes pitcher hitting in the coefficient calculations, but rOBA excludes pitcher hitting which removes a bias towards pre-1973 hitters.
• rOBA does not group AL and NL batters into one bin. This can have a significant impact. For instance, in 1933 the leagues differed in run scoring by two runs per game.
• rOBA accounts for seasons where caught stealing is missing which is many seasons before 1950.
• rOBA uses ROE calculations back to 1948. This is has been demonstrated to be a repeatable skill.
• rOBA accounts for the batted ball outs vs strikeouts and the varying importance of GIDP's and strikeouts over time.
• rOBA accounts for infield hits vs outfield hits. Some players have a significant number of infield hits and some have none. Not accounting for this will overvalue some and undervalue others.
• At the bottom of this page are a list of players most affected by these changes.

• The standard wOBA includes pitcher hitting. When making a comparison across decades (which WAR is inherently intended to do) this is going to bias things to pre-1973 hitters since there will be twice as many pitchers at bat in their games (actually more than twice as PH appearances have risen as well).
• The leagues for a particular year are grouped together. This can lead to some results where all (maybe not all) of the above average hitters are in the AL and the below average hitters are in the NL. This is true even without the DH effect. In 1933, the AL averaged 5.0 runs per game to the NL's 4.0 runs per game. It seems you really should keep leagues separate, especially since we are already setting different replacement levels for each league as well.
• For many leagues before 1950, we are missing caught stealing data. Not accounting for this will grossly inflate some players' wRAA, because the basic wOBA is going to treat someone like Hugh Nicol as 138 out of 138 stealing bases in 1887. (See our data coverage for leagues in which this is true.)
• With the advent of play-by-play data, we can now calculate reached on error (ROE) back to 1973 exactly, and back to 1948 for most games. This is a repeatable skill that players should be credited for.
• In 19th century baseball (and really through 1915 or so), striking out was much worse than putting the ball in play because fielding percentages were much worse and teams were much worse at turning double plays. Therefore, we should probably account for this difference in cases where we know strikeouts in the league (again, see our data coverage). This effect is true for more modern seasons, but since defenses are so much better, striking out is better than making another kind of out.
• With the advent of batted ball data, we can now track infield singles vs. outfield singles. Infield singles are clearly inferior, as they don't advance the baserunners as far as an outfield hit does, so we are overvaluing Ichiro and Juan Pierre with their 50-60 infield hits and undervaluing Victor Martinez, who one season had two infield hits. So we will account for that as well.

So, starting there, what changes did we make and what does the math look like?

Advanced wOBA?

Taking our six items in order here are the changes we made:

• We filtered out pitcher hitting. This boosts the league averages to a slightly higher level across all of the seasons.
• We computed separate linear weights, wOBA coefficients, wOBA averages and scales for each league season.
• We estimated a player's caught stealing percentage using a regression (on league-seasons we have CS) based on the league SB/CS values (we have these totals for catchers back to 1890), and the rate at which runners stole bases when second was open, which is estimated using (SB/(.8 * 1B + .6 * (BB+HBP))). This regression was divided between live ball and dead ball eras (pre- and post-deadball) and we said that no player can be more than 80% successful (or CS% >= 20%). Based on the regression, our CS% estimators are:
  
  pre 1920, CS% = CS% of Lg - .8535 * SB/(.8 * 1B + .6 * (HBP+BB)) + .1619
  
  1920 on, CS% = CS% of Lg - 1.2834 * SB/(.8 * 1B + .6 * (HBP+BB)) + .1168
  
  Basically, we assume players with high SB totals are better basestealers and the data certainly bears this out; this regression was a far better predictor of CS than just using the league average was.
• From 1953 on, we have play-by-play for well over 98% of the games, so for those games we are including ROE totals in the wOBA calculations for players and their initial coefficient is that of a single plus 0.018.
• Strikeouts and other outs do not appear in the numerator of the wOBA calculation, at least not as it's commonly written. But we could write it as 0.00 * outs = 0.00 * (SO + non-SO outs). So long as the league-wide contribution to wOBA from outs is zero, we could include it. So we now can add two terms to our wOBA calculation: 0.00 * outs = Runs_so * SO + Runs_nonSO * non-SO outs, where SO and non-SO outs are for the league.
  
  The two types of outs have the same value except in a couple of cases: baserunner advancement, reached on error, and double plays. For the other cases, we assume they have the same run value, we'll call that run_so (the capitalization difference is intentional from above). Run_so is the difference between the run value of all outs and a strikeout (or nearly zero), run_so is the absolute value of a runs for a strikeout.
  
  We know ROE exactly back to 1973 and we know back to 1948 that ROE was around .58 * league error totals, so we either have total ROE or can estimate it. And we can set an ROE run value at that of a single plus .018. LgRunsROE = ROE_lg * (runs_1b + .018).
  
  We know the number of double plays in the league, and that the value of a double play above that of a single out is equal to the value of a caught stealing. LgRunsDP = DP_lg * (runs_cs).
  
  For baserunner advances (note, there are more of these when run scoring is higher), we estimate that using a formula suggested by Tom Tango: LgRunsAdvancement = (16*LgR/PA - 1)/40 * (AB - H - SO - ROE).
  
  So we now have run totals for each type of event, and we have the run total for all negative events from the calculation of wOBA tot_runsMinus = (runMinusPerPA*(AB-H)). Then we adjust this by the league runs added by ROE, subtracted by DP and added by baserunner advances.
  
  If you look at this long enough, hopefully you'll see the logic in the following equation (this is for pre-1953, after 1953, we split out the ROE's into their own offensive category and exclude them here):
  tot_runsMinus = run_so * SO_lg + run_nonso * (AB_lg - H_lg - SO_lg) = run_so*SO_lg + run_so * (AB_lg - H_lg - SO_lg - ROE_lg - DP_lg) - LgRunsROE + LgRunsDP - LgRunsAdvancement
  Solving for run_so we get run_so = tot_runsMinus/(AB_lg - H_lg - ROE_lg - DP_lg) and then run_nonso = (tot_runsMinus - (run_so * SO_lg))/(AB_lg - H_lg - SO_lg) and Run_so = run_so - runMinusPerPA, Run_nonSO = run_nonso - runMinusPerPA.
  
  Using these run values, we can now add them into our wOBA calculations. Congratulations if you followed that all of the way through.
• From 2003 on, we know who fielded all balls that were put into play, so we can discern between infield hits and non-infield hits. And we know that the gap between a general single and an infield single is 0.06 runs. So based on that and the number of infield and outfield singles hit, we can compute a run value and wOBA coefficient for infield and outfield hits.

Players most altered by changes to wRAA

Note that the decreases are going to be larger than the increases because we are moving from a system that includes pitchers in the league average to one that doesn't. Also, the numbers presented on the WAR summaries for players will be different because we are separating out baserunning at that point. It is left in here to make an apples to apples comparison with current wRAA.

Note that in the 1930's the AL averaged 5.25 runs per game to 4.62 for the NL. That is a bigger difference than seen in the 1990's DH years.

Below are the wOBA constants and run values we are using for various events.