What if We Took Replacement out of WAR?

Probably the most confusing part of the Wins Above Replacement  (WAR) structure is the replacement level concept.  Skeptical fans are critical of the use of replacement level because we are comparing let's say Miguel Cabrera to a "mythical player".  Some want Cabrera to be compared to the player who would have replaced him if he got injured.  Similarly, they want Prince Fielder, Mike Trout and every other real player to be compared to their actual replacements.

There are two problems with assigning a different replacement player for every player.  First, we don't have any way of knowing what the actual replacement would have done.  Even if we could identify specific players, we don't know whether they would have had good or bad seasons.  Not only that, but it would be quite impractical to do this all for all players throughout the history of the game.

Thus, it is desirable to have a common replacement level for all players.  What if we had no replacement level and just compared Cabrera to a zero player?  We would be assuming that his replacement would have batted .000/.000/.000.  Making that assumption , how many wins would Cabrera have been worth in 2012?

The calculation would start with Weighted Runs Created (RC).  Cabrera had 137 RC in 2012.  The other WAR numbers (for FanGraphs) are:

Ballpark -1.4 runs because Comerica Park slightly favors run scoring.
Base Running: -2.8 Runs below average
Fielding -10 runs according to UZR (yes, fielding metric are iffy but that's for another post)
Position +1.5 because 3B is a relatively difficult position.

Adding the four numbers above yields -12.7 Runs for his non-hitting stuff. Subtracting 12.7 from 137 equals 124.3 Runs.  Every 9.5 runs is worth about one win, so he had 137/9.5 = 13.1 Wins Above Zero (WA0).

Table below 1 shows RC and WA0 for Tigers in 2012.  You can see that everyone had a positive value as even Ryan Raburn was better than nothing.  The problem is that nobody would ever be replaced by a zero player and the result is that players get more credit for their offensive production than they should.

Table 1: Calculating Player Wins for Tigers in 2012 Using Different Baselines

Player	PA	RC	RAA	RAR	WA0	WAA	WAR
Cabrera	697	137	57.3	81.2	13.1	4.7	7.2
Fielder	690	125	46.1	69.8	10.7	2.4	4.9
Jackson	617	98	28	48.6	10.8	3.4	5.5
Hunter	584	86	19.4	39.3	10.2	3.2	5.3
Infante	554	65	-2.4	17.2	8.2	1.1	3.2
Peralta	585	60	-6.7	13.2	7.5	0.5	2.6
Avila	434	54	4.2	19.3	6.2	1.0	2.6
Dirks	344	54	14.7	26.5	4.5	0.4	1.6
Young	608	65	-4.8	16.4	4.4	-2.9	-0.7
Berry	330	35	-2.6	8.6	3.8	-0.2	1
Boesch	503	46	-11	5.8	2.9	-3.1	-1.3
Santiago	259	17	-13	-3.6	1.9	-1.3	-0.3
Raburn	222	8	-17.6	-11.7	0.6	-2.1	-1.5

What if we assume that  Cabrera's replacement would be an average hitter?  Since there were 21,017 runs scored in 184,179 plate appearances in 2012, the average PA resulted in 21,017/184,179 = .114 runs.  Thus, the average hitter would have created .114*697 = 79.5 runs in 697 PA.

Subtract 79.5 from Cabrera's 137 RC and he had 57.5 Runs Above Average (RAA).  Now subtract the -12.7 for non-hitting elements and Cabrera was worth 44.8 runs.  Divide that by 9.5 and he was 4.7 Wins Above Average (WAA).

Just 4.7 wins?  If that seems low for Cabrera, it's because it is in comparison to an average player.  In reality, an average player is a pretty good player and it's not likely that one would be readily available in the case of an injury to Cabrera.

Some analysts believe that WAA is useful in evaluating Hall-of-Fame status, because they don't want to give too much credit to compilers - players who were good (but not great) for a long period of time.  Using a baseline of average makes sense for the Hall of Fame because we are not really interested in how elite players compare to bad players.  We are more interested in how they compare to long-time players, who were pretty good but not Hall-of-Fame quality.

However, WAA usually doesn't work as well for annual analyses where we want to determine how many wins a player added to his team.  This is illustrated by Table 1 which shows Jhonny Peralta (0.5 WAA) and  Andy Dirks (0.4) had almost no value by that measure.  In actuality, It is not likely that either one would haven been easily replaced if they had suffered a season-ending injury in May. 

Since WA0 inflates a player's worth and WAA underestimates it, we need something in between.  Based on research over several year on which past players were added to rosters during a season and able to earn significant playing time, analysts have arrived at the typical replacement player.  The FanGraphs replacement player hits roughly 70% as good as an average player. Other sites have slightly different percents, but I'll go with 70% here.  Does 70% of average sound more real than "mythical player"? 

Remembering that the average PA results in .114 runs, 70% of average is .080.  So, the typical replacement player would create .080 by 697 = 55.8 runs. Subtract that from 137 and Cabrera was 81.2 Runs Above Replacement (RAR).  Subtract the -12.7 for everything beyond hitting and Cabrera was worth 68.5 runs.  Divide by 9.5 and he was 7.2 Wins Above Replacement (WAR).

You don't have to use the 70% of average replacement player if you don't want.  That's a good number to use if you are playing general manager, but a fan could use WAA if he/she just wanted to know how many players on their team are above average. Chances are a team with a lot of those players will win a lot of games.  Choose whatever baseline you want if you are willing to do the calculations.  Just beware that various baselines have different meaning and can produce very different values and ranks. 

OPS+ versus wRC+

Over the last few years, fans have become increasingly familiar with Adjusted OPS (OPS+), a statistic, first introduced by Pete Palmer back in the 1980s.  This OPS+ metric is OPS adjusted for league average and home ballpark.  It is useful because it allows us to compare players who played in different ballparks and/or different eras.  It is calculated using the following items:

OBP = On-Base Percentage
SLG = Slugging Average
MLB OBP = MLB Average OBP (with pitchers removed)
MLB SLG = MLB Average SLG (with pitchers removed)
BFP = Ballpark Factor

The formula is

OPS+ =  (OBP/MLB OBP + SLG/MLB SLG - 1) x 100/BPF

Using Miguel Cabrera's 2012 season as an example yields

OPS+ = ((.393/.324 + .606/.413 - 1) x 100)/ 102 = 165

In general, an OPS+ of 100 is average, an OPS+ of above 100 is above average and an OPS+ of less than 100 is below average.  There is a popular misconception that OPS closely matches the ratio of a player's OPS to league OPS.  However, an OPS+ of 165 does not mean that Cabrera had an OPS 65% better than league average.  We know it's a really high OPS+ because it was the highest in the American League, but it has no concrete meaning.

Another limitation of OPS+ is that it counts OBP and SLG the same when OBP actually contributes about 80% more to run scoring than SLG.  Thus, players who get most of their production from OBP will be short changed by both OPS and OPS+ .  At any rate, the OPS+ figures can be found at Baseball-Reference.com for all players.

The OPS+ metric is OK for many purposes as long as you understand the shortcomings.  If you want a more reliable statistic, you can use Weighted Runs Created Plus (wRC+), a creation of Tom Tango.  Because it is based on Weighted On-Base Average (wOBA), wRC+ more accurately weights batting events (1B, 2B, 3B, HR, BB, HBP, outs) than OPS does.

It is calculated as:

a = MLB Runs per PA ( with pitchers removed)
b = Park Adjusted wRAA/PA (with pitchers removed)
c = b/a + 1
wRC+ =c x 100

Cabrera had the following numbers in 2012:

a = .1164
b = 53.7/697 = .0770
c= .0770/.1164 + 1 = 1.66
wRC = 1.66 x 100 = 166.

Another benefit of wRC+ beyond it's accuracy is that it has a concrete interpretation.  In Cabrera's case, he created 66% more runs than would be expected by an average hitter in 697 PA.  The numbers for all players can be found at FanGraphs.com.

The wRC+ metric is on the same scale as OPS+ and does not generally produce wildly different results.  The biggest difference I found for the Tigers was Austin Jackson (135 wRC+ versus 130 OPS+).  The OPS+ vs. wRC+ comparison for the rest of the Tigers in 2012 is shown in Table 1 below.  The third colum is the percentile among players with 250 or more PAs.

The lesson to be learned here is that wRC+ is a little better than OPS+ and should be used for more serious evaluation of players.  However, if you prefer using the Baseball-Reference site, OPS+ is a reasonably good estimate of a player's relative hitting value in most cases.

Table 1: wRC+ versus OPS+ for Tigers, 2012

Player	OPS+	wRC+	PCTL
Cabrera	165	166	99
Fielder	152	153	97
Jackson	130	135	88
Dirks	130	133	87
Hunter (LAA)	132	130	85
Avila	100	104	52
Infante (DET/MIA)	93	92	34
Berry	86	89	29
Young	89	89	29
Peralta	85	86	24
Boesch	77	77	13
Santiago	52	55	1
Raburn	30	28	0