Over the past couple of months, we have looked at two offensive statistics that evaluate a player's overall offensive contribution, wOBA and wRC+. We will now attempt to evaluate a player's entire contribution with Wins Above Replacement (WAR).

WAR essentially represents the value that a team receives from a player above and beyond the value that a replacement-level player would bring. A replacement-level player is one who can be found on the free agent market at any time, and typically has the ability of a Quad-A player: better than the average Triple-A veteran, but not good enough to be a major-league regular. A position player's value above replacement is determined by four components: batting, fielding, defensive position, and a replacement level adjustment. Sum up the run values of these four components, and you arrive at a player's WAR.

**Batting**

In order to arrive at the number of runs above average that a player is offensively, you convert the player's wOBA to wRAA, which represents offensive runs above average. The conversion formula is as follows:

wRAA = ((wOBA - lgwOBA)/Scale) * PA

The formula essentially subtracts the league average wOBA from a player's actual wOBA in order to calculate the player's contribution above the average contribution. It then divides the difference by a scale factor that is meant to adjust for the specific season's offensive environment. Finally, we multiply the quotient by the number of plate appearances the player accumulated during the season to arrive at his wRAA (Weighted Runs Above Average).

Once you have a player's wRAA, you have to make a slight adjustment to it to control for park effects. A player who calls Petco Park his home will see his wRAA adjusted to reflect the fact that runs are more valuable at Petco than, for example, at Fenway Park; thus, his batting value will be higher than his wRAA. Similarly, a player who calls Fenway Park home will see his wRAA adjusted to reflect that runs are easier to come by than the average park; thus, his batting value will be lower than his wRAA.

That's batting value in a nutshell. Now let's move on to fielding.

**Fielding**

In order to arrive at the number of runs a player is worth in the field, you simply sum up a player's UZR values from each position that he played during the year.

While this part seems very simple, there are some important things to note about the fielding component, but first, let me describe UZR. UZR, or Ultimate Zone Rating, evaluates a player's range and error rate and packages them together to arrive at the number of runs above or below average a fielder is relative to the league average at that player's position. The last few words of the previous statement were important because a shortstop with a +10 UZR is not equivalent to a second baseman with a +10 UZR. Therefore, we need to adjust these UZR values in order to account for these positional differences. However, for the sake of clarity, sabermetricians decided to break up a player's value into as many clearly defined parts as possible. They wanted to strip this positional adjustment factor away from any of the other factors -- batting and fielding -- and thus gave it its own section in the WAR formula.

Let's move on to the defensive position (or positional adjustment) section.

**Defensive Position (or Positional Adjustment)**

The purpose behind valuing a player's defensive position stems from the fact that all positions are not created equal -- some positions are significantly more difficult to play than others. For example, it is much more difficult to find a +5 shortstop than it is to find a +5 first baseman. We need to represent this when we compile WAR.

Here are the positional adjustments that FanGraphs uses:

Catcher: +12.5 runs

First Base: -12.5 runs

Second Base: +2.5 runs

Third Base: +2.5 runs

Shortstop: +7.5 runs

Left Field: -7.5 runs

Center Field: +2.5 runs

Right Field: -7.5 runs

Designated Hitter: -17.5 runs

The adjustments essentially represent (i) the difficulty of defense at the given position, and (ii) the level of scarcity associated with that position at the major league level. For example, (i) catcher is a more difficult position to play than first base, and (ii) it is more difficult to find a major league catcher than it is a major league first baseman. Because of these two facts, a catcher is given an additional 12.5 runs, while a first baseman is docked 12.5 runs.

For more on how exactly these run values were determined, FanGraphs points us to these threads at *The Book* blog.

We're not entirely done with the positional adjustments yet. Since the adjustments are calculated per 162 defensive games, we must multiply a player's positional adjustment by the proportion of games he played in a season. For example, if a shortstop played 140 of his team's 162 games, his positional adjustment would be (+7.5 * (140/162)), which equals 6.48 runs. This correction for playing time also helps provide a more accurate picture of the value of players who play multiple positions. If a player played second base, shortstop, and third base in a given season, we would calculate a positional adjustment for each position and then sum them up to arrive at the player's overall positional adjustment.

Now that we have adjusted for the player's defensive position, we have thoroughly evaluated a player's value above or below average. However, in order to get a complete picture of a player's value, we must define what average means, and add that value to the value that we have thus far calculated. Remember, it is Wins Above Replacement, not Wins Above Average.

**Replacement Level Adjustment**

If we knew how many wins a league average player is worth, then we could just add that value to what we have thus far and we would be done calculating a player's WAR. Unfortunately, by the nature of it, the numbers of a league-average player change every season. Furthermore, there isn't a specific salary associated with league-average players. Thus, we have to look for the next lowest standard. That brings us to the league-minimum player. $490,000 is the least any player in the major leagues can be paid in 2013. We peg this value as replacement level.

Now all we need to complete our WAR system is to find the difference between an average player and a replacement-level player. As I mentioned earlier in this post, a replacement-level player is someone who has more ability than an average Triple-A player, but not enough ability to play in the major leagues regularly. Thanks to some research by Sean Smith, we know that the expected value of a replacement level player is about -20 runs for every 600 PA. Therefore, the formula for calculating the replacement level adjustment is simply (20/600 * PA). Every additional plate appearance that a player collects during a season means that the team will give one less plate appearance to a replacement level player. Therefore, a player is credited this adjustment based on how frequently he plays, and, consequently, how infrequently a replacement level player plays in his place. For example, if a player collected 500 plate appearances during the course of a season, his replacement level adjustment would be (20/600 * 500), which equals 16.67 runs.

With the replacement level adjustment we have now adjusted the scale so that the baseline value is $400,000 at zero wins. We have filled the value gap between replacement level and major league average, and can now appropriately evaluate a player's contribution above replacement level.

Next up, we convert these runs above replacement to wins.

**The Rest**

In order to understand how runs are converted into wins, we will briefly look at the pythagorean formula for expected win-loss records. The Pythagorean Winning Percentage formula states that you can get a good estimate of a team's winning percentage by looking at the number of runs scored and allowed by that team.

Pythag Win % = (Runs Scored^2)/(Runs Scored^2 + Runs Allowed^2)

Now, let's say that a team scored 800 runs and allowed 800 runs. The team's Pythagorean Winning Percentage would be .500, which translates to 81 wins. Now suppose instead that the team scored 10 fewer runs, so it scored 790 runs and allowed 800 runs. The team's Pythagorean Winning Percentage would now be .493, which translates to 79.98 wins. As we can see, a win is effectively worth approximately 10 runs; when the team scored 10 fewer runs, it lost 1.02 wins. This 10 runs per win ratio has become a standard within the sabermetric community.

In order to convert a player's runs above replacement value to Wins Above Replacement, all you have to do is divide the runs above replacement by 10. Voila! We have now calculated a player's value above replacement in the form of wins.

In my next post, I will put this method to the test and walk us through the calculation of a specific player's WAR in all its mathematical glory.

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