Sunday, March 10, 2013

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. 

3 comments:

  1. Very good post and interesting analysis! There is no sport like baseball where there are so many different perspectives and angles with respect to trying to put a valuation on a player. Every event that happens in baseball is an indisputable fact, and you would think that following the information in a logical format could only produce a logical valuation. But baseball players aren't worth their actual results in the real world, only in the theoretical world. The GMs that run these teams aren't basing their evaluations on perfect truth and science, and there are 30 different people with 30 different opinions of how to make an analysis. The true value of a player is not his real and theoretical value, it's only the warped-reality value that these particular GMs come up with. At most only 1 of those 30 can have the best method out of the group, and the other 29 are off the mark in varying degrees.

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  2. It seems that the replacement level player has no value added or subtracted for baserunning, defense, or position yet the actual player does. Does this mean a replacement player is assumed to be an average baserunner and defender? If that's the case wouldn't that be counter to the argument that WAA deflates a players actual value? Shouldn't there be some sort of calculation for what a replacement level baserunner and defender are in any given season, just as this is done for batting? It would seem that applying a modifier for those stats to the player but ignoring it the comparison would skew numbers. Would a replacement level CF for example have the same level of baserunning and defense expectations that a replacement level 1B does? If they are treated as the same for the sake of comparison wouldn't that create a system that artifically inflates numbers for good baserunning/defense while artifically deflating for the opposite? I mean if a replacement level CF is assumed to be the same level in those skills as a 1B it would seem the system is somewhat flawed.

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  3. Good questions. A replacement level player is considered to be an average defender at his position, because there is a large pool of players that are capable of being average defenders. It's not so hard to find a guy like Danny Worth who is an average defender but can't hit. It's a lot harder to find above average hitters. If a player can not defend his position adequately, he is generally moved to another position in the minors. If he hits well enough to play the new position, then he survives. If not, then he usually does not make the majors.

    A first baseman is not assumed to have the same skills as a shortstop. That is why there is a positional adjustment. A shortstop gets runs added to his WAR because the shortstop skill is more difficult to find. Suppose, the SS and 1B both have 100 RC in 600 PA. Both would be 52 Runs Above Replacement after the offensive calculation. However, the shortstop gets 7.5 runs per 162 games added to his WAR and the first baseman gets 12.5 runs subtracted from his WAR.

    As for base running, replacement level is assumed to be average because there are a lot of players capable of average base running. It's the hitters that are more difficult to find.

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