In this new series of Purple Row Academy, I will step on the toes of both RMN and TOGB somewhat--in examining the concept of Wins Above Replacement (WAR)--also called Win Value--in detail, with a historical context slanted toward the best Rockies players as ranked by career WAR in future installments. In this case, with apologies to Edwin Starr, WAR is good for something, and that's quantifying a player's value to his team.

Why is this important? Being able to quantify the value of players that are given a regular role is a great way to evaluate the performance of the front office in getting the best players on the field and their efficiency in doing so. In addition, everybody understands the concept of wins and dollars...and saying Player X was worth 5 wins and $20 million last year is easier to understand than trying to explain to a casual fan (or as I like to call them sometimes, a replacement fan) how Player X's .432 wOBA makes him awesome.

I'm a sucker for valuation stats, and WAR (from Fangraphs) is the one that I've found to be the easiest to both use and understand. In the most recent session of Purple Row Academy, I explained Fangraphs' conversion of win value (WAR) to dollar value as basically a player's WAR value multiplied by that year's market (read: free agent) cost per marginal win. However, I really didn't explain the concept of WAR very well, a fact I'll try to remedy this week accompanying Dave Cameron's articles. This week I'll focus on the WAR of *position players*, with next week covering the pitching calculations.

I'll break down each component and concepts used in WAR after the jump...

By nature, WAR relies on counting stats (HRs, 2Bs, etc.) to be translated into rate stats (OBP, wOBA) and then combines these factors into value stats. The great thing about WAR is that it takes into account a player's value both at the plate and in the field, adjusting it for position and park, and compares it to a replacement player. I'll explain each component individually.

The primary source of the batting inputs into WAR is **wOBA**, a stat that was explained wonderfully by Poseidon's Fist in Counting Rocks and in considerably more detail here and by Cameron. At its heart, wOBA is a linear weight formula converted to a rate statistic that is scaled to OBP (meaning that league-average wOBA = league-average OBP) that emphasizes OBP above slugging percentage. What a linear weight formula does is to properly value the different successful outcomes (HR, 3B, 2B, etc.) relative to each other. For instance, wOBA values a HR as slightly two times more valuable than a single.

Borrowed from PF's article, here's the (simplified) wOBA formula:

PF also provides some context:

Note: .333 is about the league average. A wOBA below .300 is a very poor hitter, between .300 and .333 is below average, between .333 and .370 is above average, between .370 and .400 is a very good All-star caliber player, and above .400 is a superstar.

WAR doesn't use wOBA directly in its calculations, but rather weighted runs above average (**wRAA**) To get the amount of wRAA produced by a player, you simply have to subtract the league-average wOBA number from the player's wOBA and divide that number by 1.15 and multiple the result by the number of a player's plate appearances.

Here's a sample calculation of wRAA for Matt Holliday, using 2008 numbers (MLB-average wOBA = .333):

(.418 - .333) = .087 / 1.15 = .0739 x 623 PA = **46.05 wRAA**

It's a simple calculation that shows that Matt Holliday was worth about 46 runs more to the Rockies than an average player.

However, the problem with wOBA and wRAA is that it fails to adjust for park effects. MLB players do not play in neutral parks, therefore their rate stats must be adjusted accordingly. Therefore, to complete the WAR batting component, a player's wRAA value is adjusted with his home park's environment factor using the Odds Ratio Method, which is really complicated. The basic thing you need to know for that is that numbers (and runs) produced in a friendly hitter's environment are less valuable than those produced in a friendly pitcher's environment.

For instance, Holliday's Coors-inflated wRAA (and yes, with Coors' 2008 park factor of 1.126 it was inflated) was less impressive than if he had produced that line at Petco Park. This adjustment is one of the few limitations of WAR though--in that his value is not similarly calculated for the numbers Holliday produced on the road (with many different park factors involved). Actually, I'm sure that someone has crunched the numbers by individual park, but I don't have access to that data or the time to calculate it myself.

The main takeaway of this component is that a major part of a player's batting value is derived from his OBP, with a smaller emphasis upon his slugging percentage.

WAR's fielding metrics are a little less easy to understand than its hitting component due in large part to their relatively recent development. The stat used by WAR is ultimate zone rating (**UZR**), a concept that I admittedly don't fully understand. However, those who are "in the know" seem to believe it's the best metric currently out there. FanGraphs creator David Appelman has an explanation of some of UZR's components that I'd recommend checking out.

The fielding Win Value calculations is basically **a player’s total UZR at all positions for the given year**. One should note that UZR is relative to the league-average performance at a player's position (in other words, it isn't a universal stat). Therefore, a +15 UZR left-fielder is worth much less than a +15 center-fielder--they're being compared to a different set of players. As Cameron puts it:

This isn’t because CF is any harder to play than LF, but simply because the people he would be compared to are much better defensively than the people he’s compared to as a left fielder.

However, WAR takes the positional adjustment into account separately from UZR (see below). In 2008, Holliday had a UZR of 9.1 in LF.

Takeaway: UZR is king, but the position played is also important...

**Positional Adjustments to Value**:

This part of the WAR calculation was strongly alluded to by RMN when he discussed the Rockies' outfield defense. Basically, the UZR numbers put up by certain positions must be discounted while others require a premium based upon the caliber of defensive players playing that position, the players upon which their UZR rating depends. The most commonly accepted scale of positional adjustment to effectively neutralize defense was developed by Tom Tango and is as follows:

Catcher: +12.5 runs

Shortstop: +7.5 runs

Second Base: +2.5 runs

Third Base: +2.5 runs

Center Field: +2.5 runs

Left Field: -7.5 runs

Right Field: -7.5 runs

First Base: -12.5 runs

Designated Hitter: -17.5 runs

Once again, right fielders are discounted because by and large right-fielders are lousy fielders, not because of any inherent difficulty gap between it and shortstop (though one doubtless exists). Shortstops, meanwhile, are usually their team's best defender, therefore the UZR rating of a shortstop is compared to the league-wide collection of MLB's best fielders. A slick-fielding first baseman would often be considering a butcher with the glove at shortstop, and so forth. These adjustments are multiplied by the proportion of defensive games (DG) played by a player at that position. The resulting number is discounted from or added to a player's UZR and when combined with his park-adjusted wRAA gives us his value (in RAA) relative to a league-average player.

Continuing the Holliday example, Matt Holliday's fielding value in 2008 went from a positive 9.1 UZR rating to a more MLB-average 2.7, due to an adjustment of -6.4 runs. Why -6.4 runs? Holliday played 139/162 games in LF, meaning that his positional adjustment was 86% of LF's -7.5 discount, or -6.4 runs. In all, considering his park-adjusted 39 wRAA, Holliday in 2008 was worth 41.7 runs above the league average.

The problem with calculating value relative to a league-average player is that we don't really know how to quantify league-average. How expensive is it? One route might be to use the mean and median MLB player salaries, but even then finding a league-average value is difficult as these numbers do not have a fixed baseline. This is where the concept of a replacement player comes in.

Takeaway: the positional adjustment scale removes the concept of position from the equation, leveling out all fielding numbers.

**Replacement Players and their Value**

The final component of the WAR calculation is the concept of the **replacement player**, introduced by RMN In his VORP article. Notice that this is NOT a league-average player, but rather a player that is completely replaceable by his team, a AAAA player. Sean Smith at the Hardball Times goes further.

The reason that replacement players are used in these value calculations is that we have a fixed baseline for exactly how much these players are worth--and that is the MLB minimum ($400,000), the lowest possible cost to replace a major league player. From my marginal payroll/marginal wins column, we know that a team full of replacement players (25 active, 3 DL) will cost $11,200,000 and will win 30% (or 48.6) of their games. In other words, a team full of replacement players will be horrible. Meanwhile, a team comprised of league-average players would win 81 games. Replacement players (see Quintanilla, Omar) suck at the major league level and replacing each one with league-average players is worth about two wins to a team.

Basically, the expected value of a replacement level player to his team is 20 runs per 600 PA *below average*--or as described below, a cost of two wins to his team. The more a player plays, the more the value of replacement level (as replacing that player would cost your team comparatively more). What this does is reward durable good players who, by playing often, are keeping a bad replacement level player off the field. To figure out a player's replacement adjustment, divide his PAs by 600 and multiply by 20.

Adding up a player's wRAA, UZR, positional adjustment, and replacement adjustment gives us a player's Value Runs or runs above replacement (**RAR**). This is how many runs above a replacement level player each position player was. The final step to calculating the WAR equation is to convert Value Runs into wins.

In Holliday's case, he had 623 PAs in 2008, meaning that his replacement level cost was a slightly higher 20.8 runs. Therefore, Holliday's total for 2008 was 62.5 runs above a replacement player.

Takeaway: replacement players are NOT league-average...don't play them if you don't have to!

Once you have obtained a player's RAR value, the translation to wins is fairly simple. Using a team's **pythagorean winning percentage **(RS^2/(RS^2 + RA^2)), it can be proved that for every ten runs that are lost or gained a win is similarly lost or gained. For every 10 runs a player is above a replacement level player, he is worth 1 win to his team.

Therefore, Matt Holliday was worth 6.25 wins above replacement in 2008.

Takeaway: **10 RAR = 1 win**

Investigated last session, a player's WAR value multiplied by that year's **market cost per marginal win** allows one to find the dollar value of a player for that year. Last year, according to Cameron, that number was $4.5 million per win as opposed to the average (including non-free agents) of about $2.7 million per MW.

As such, Holliday's performance in 2008 was worth $28 million in 2008.

Takeaway: A player's **$ Value = (Market cost / MW) x player's WAR**

- WAR, due to the lack of quality defensive metrics for catchers, assumes all catchers are average defensively (fair or not)
- WAR does not measure a player's true talent level, merely their past performance
- WAR assumes that talent between the AL and NL is equal, though much evidence suggests that this is not the case and that an AL 2 win player is more valuable than his 2 win NL counterpart
- Fangraphs does not claim decimal point accuracy (a 4.8 win and 4.3 win player are very similar)
- As I discussed in a previous PR Academy session, wins are not linear, meaning that two wins added to a 60 win team are much less valuable than two wins added to a 90 win team
- In that same vein, a two win player for a 60 win team would probably not have that kind of impact on a 90 win team
- Due to the AL having the DH, positional adjustments don't add up to zero

**Sources of Inspiration**: Beyond the Box Score (Sky Kalkman) and Brew Crew Ball (KL Snow)