2022 Blogmas Day 9 – Offensive Wins Above Replacement

Today we move onto one of the gems of advanced baseball statistics¹ that generalizes player value beyond concrete elements like hits and runs, and looks specifically at how many wins a player is worth.

WAR stands for Wins Above Replacement, and attempts to measure how many wins — as in games won by a team — a player has contributed beyond what a replacement-level player would offer. This is a departure from what we’ve seen so far: replacement-level is notably lower than average. When we say replacement-level, we mean a good minor league player who might get called up to replace an injured full-time player.

WAR slowly took over the cultural consciousness of baseball nerds, and while it is still an important career-long statistic that is of significant interest, it doesn’t have the same glitz and glamour that it once held. This is partly due to the need to consider both base-running and defensive metrics when calculating total WAR for position players, and these two factors are very messy. Just offensive WAR is cleaner to handle.²

Perhaps the most interesting element of WAR is that there is not a single way to calculate it. Different organizations (notably Baseball Reference and FanGraphs) make use of slightly different information. You will see bWAR or rWAR both in reference to Baseball Reference’s calculation, while fWAR represents FanGraph’s version. Before we get muddled down with those details, let’s look at the rough way we calculate WAR as it relates to offense. To do so, we’ll explain the entire WAR calculation but ignore the defensive aspect.

There are four factors we sum to get the numerator for WAR:

Runs above average. We’re familiar with this concept.³
Positional Adjustment. Some positions are harder to play than others, so there’s a number of runs per 162 games (1 season) that each position gets to add (or subtract) from their value.
League Adjustment. The American League and National League are still separate⁴ and so their run-scoring environments are a little different. We know wRAA always has a 0 average, but when you add in other factors, the league’s average among them all might be off from 0. This value fixes that up for us.
Replacement Level Runs Adjustment. This is where we differentiate from other statistics that compare to the league average, so we’ll look at it more closely.

Replacement Level

It’s worth realizing that being average is valuable. There are options worse than you, so it’s worth having our statistic about overall value as it relates to a team winning compare players to, in essence, the best bad player we could have: a player on the cusp of breaking into the league, or a long-time bench warmer who was never good enough to be considered a full-time option for any particular position.⁵

With that stage set, we must now gather some numerical minutiae. Let’s establish that every team plays 162 games during the regular season, and each instance of a game is played by 15 pairs of teams for a total of 2430 regular season games. If a team was only made of replacement-level players, the powers that be have determined the team would win 47 or 48 games in a season. If all 30 teams only won that many games, over the season the replacement-level players are worth between 1410 and 1440 wins. Taking the nicest possible choice of 1430 replacement-level wins, that leaves 1000 wins unaccounted for. These are the wins above replacement that we’re calculating.⁶

That means every season there are only 1000 WAR to go around to every player.⁷ Again, based on historical data and general agreement between Baseball Reference and FanGraphs, 57% of WAR is attributed to position players, and 43% to pitchers. That means for the WAR we’re discussing today, at most 570 can be allocated to our position players in a season.

Given all this information, we need to move from our current sum of runs above average, to runs above replacement level. To do so, we add the number of runs above replacement level that the average player is worth. This rescales everything to have it compare to replacement-level. Someone who is truly replacement-level would have such poor runs above average that it would cancel out the average number of runs we give them for free as our replacement adjustment factor, resulting in 0 WAR.

Now, let’s be explicit. We have 570 wins to work with, and we need to convert that into a certain number of runs for a player. To do so, we will have a runs per win statistic (RPW), divide by the number of plate appearances by all players that season, and multiply by the number of plate appearances our particular player whose WAR we’re calculating has had. So, our replacement level runs adjustment (RLR) is:

\text{RLR} = 570\times\text{RPW}\times\frac{\text{PA}}{\text{total PA}}.

Runs per Win is a measure of how many total runs a team needs to have to gain 1 additional win on the season; this is calculated via various models, and each measurement of WAR must choose one. FanGraphs has a slick formula that is extremely accurate when compared to more complicated models.

\text{RPW} = 9\times\frac{\text{total R}}{\text{total Innings}} \times 1.5 + 3.

This is 1.5 times the total runs per nine-inning game, plus 3.⁸

It turns out this value is between 9 and 10 nearly every season, and the rule of thumb is to just use 10 in a pinch.⁹ So, if a player individually gets 10 runs more than a replacement-level player could manage, they are worth about 1 additional win.

The Final Calculation

We have our four factors in hand. They are each in the unit of runs, so what we have are the tools to calculate the number of runs a player is worth above replacement level.¹⁰ But, we already went through the trouble of understanding RPW; if we divide our runs above replacement level by runs per win, we end up with units of wins above replacement level!

\text{WAR} = \frac{\text{RAA} +\text{Pos. Adj.} + \text{Lg. Adj.} + \text{RLR}}{\text{RPW}}

Again, we’re ignoring any defensive matters so this is solely WAR due to hitting and base-running prowess. But, this general formula holds; simply adjust your RAA statistic to include factors that matter to you.

Let’s put it in context. Did your team have 83 wins last year, and you replace your 2.5 WAR shortstop with a 6.5 WAR shortstop? With all other things staying equal, you would expect your team to have 87 wins this year. It’s a rather wild approach to calculating value, but it distills performance down to what really matters, which is why it is such a triumph.

Not everyone is so enamored with WAR, though. It can certainly be overused in analysis, particularly as a single-number statistic devoid of other context.¹¹ It can also overvalue players from bygone eras due to a wider spread in talent, and the lack of good adjustment data compared to what we have now. Yet, when considered alongside other advanced statistics, like wRAA and wRC+, it becomes part of an excellent comparative profile.

This one was a lot of work for you and for me. Tomorrow will be a bit more fun, and we’ll close here with some statistics that emphasize WAR as a career-long counting statistic.

The highest career offensive bWAR in the Live-Ball Era (1920-present) is 143.6, belonging to Barry Bonds. He’s followed by Babe Ruth (136.7), Willie Mays (136.6), and Hank Aaron (132.6).
Jeff Mathis has the lowest offensive bWAR among players in the 21st century, with -6.4. But, don’t count him out before you watch this video.

Continue to Day 10 – Win Probability Added

1
Somehow I’ve gone this long without using the phrase Sabermetrics, which is the catch-all term for many of the derived values we’ve been covering. Its name comes from SABR, the Society for American Baseball Research.
2
It does still include base-running.
3
We’ll use wRAA for batting. Baseball Reference uses something a little different. We also need base-running value, but we won’t discuss that here.
4
Although getting closer together with each passing season.
5
Equivalently, and more pertinent for how the values in the next paragraph would be calculated, replacement-level players are only ever paid the league-minimum salary.
6
Thank you to FanGraphs for having this hard-to-find background information available in their library.
7
Of course, WAR values can be negative.
8
Total innings is total innings pitched, which in a nine-inning game would actually by 17 or 18.
9
RLR ends up being around 17 runs per 600 plate appearances, so the average player is worth about 1.7 WAR.
10
Again, that RLR statistic is our key to moving from runs above average to runs above replacement level.
11
Never use a single number in a statistical analysis, obviously.

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