It is not the perfect one-to-one ratio as many may think, but it is pretty close.
After every offseason move, baseball analysts (like myself) attempt to evaluate how many “wins” each team gains.
Usually, the exercise is rather simple. One can look at projected WAR for the players currently on the roster, re-estimate the amount of play time for each player and then determine how many wins the team gained from their new addition. For example, with the addition of Andrew Miller, we can estimate that the Cardinals should be about 1.5 wins better. That’s pretty a significant improvement from a relief pitcher alone, and it serves as a testament to Miller’s success.
The problem is that this assumption rests on the belief that the addition of 1 WAR equals a 1 win improvement.
Using 2018 as a case study, I found that, while WAR and wins do have a strong correlation, it isn’t always as simple as a one-to-one ratio. That should not be a surprise. As many critics often point out, if baseball can be distilled down to a spreadsheet, then why do we even play the games? Consider this as Score One for the critics. There is, in fact, a reason why we play the games. What a shocker.
The graph that you see here is the correlation between team wins and WAR wins.
A “WAR win” can be defined as a team’s total WAR + 47.628. We use WAR wins, as opposed to just pure WAR, because a team with 0 WAR does not win zero games. A replacement-level team, rather, is projected to post a .294 winning percentage. Over the course of a 162-game season, this team would win 47.628 games. Thus, for every single WAR above this, a team should be worth WAR + 47.628.
As you can see, however, team wins and WAR wins are strongly correlated (R-squared value of 0.86). The line of best fit is y = 1.06 x – 4.61. So, if your team is projected to produce 100 “WAR wins” (47.628 + 52.372 WAR), they would be expected to win about 101 games. While the correlation does create some variance, the line of best fit demonstrates that the ratio of WAR to wins is pretty close to one-to-one, on average.
The next correlation that I considered was that between WAR and Pythagorean expectation.
For those unfamiliar, Pythagorean expectation effectively assumes that not all wins are created equal. Winning a game 5-4 could be as a result of more luck than winning a game 10-0, for example. Thus, Pythagorean expectation was developed by Bill James to use run differential as a way to calculate an alternative winning percentage. This, then, could be used to show a team’s true strength beyond their win-loss record.
Like with team wins and WAR wins, the correlation between Pythagorean wins and WAR wins was strong in 2018. In fact, with an R-squared value of 0.95. This means that our model (or line of best fit) is able to explain about 95 percent of the variability in our data set; for team wins alone, our model could only explain about 85 percent of the variability.
Thus, for each WAR win that a team gains, it is slightly more likely that they are gaining a Pythagorean win than a true win, at least according to the data from 2018. Of course, it is arguably better to gain Pythagorean wins than it is team wins, especially since the Pythagorean expectation takes luck out of the equation. You’d rather your team be good than lucky.
The line of best fit for this model is y = 1.10 x – 8.26. If your team has 100 WAR wins, they would be projected to post a 102-win Pythagorean expectation, according to the 2018 data. So, again, while it isn’t an exact one-to-one ratio, it’s pretty close. . .which makes sense.
This last chart shows the two graphs that I previously discussed in the top row, while including two new graphs — Second Order Wins vs. fWAR and Third Order Wins vs. fWAR — in the bottom row.
A Second Order Win was developed by Baseball Prospectus. Effectively, it expands upon Pythagorean expectation. Rather than using runs scored and runs allowed in the Pythagorean formula, the Second Order Wins formula uses expected runs scored and expected runs allowed. A Third Order Win is a near relative of the Second Order Win; it does the same thing while adjusting for quality of opponents.
Neither of these metrics correlate stronger with WAR than does Pythagorean win expectation, posting R-squared values of 0.93 and 0.91, respectively. For a team with 100 WAR wins, they would project to produce 103 Second Order Wins and 102 Third Order Wins.
All in all, what do we take away from all of this?
What is most interesting to me is the near one-to-one ratio between real team wins and WAR wins. The line of best fit for this data, while being able to explain the least of the variance, provided the best ratio of wins to WAR wins. A 100-WAR win team would expect to win about 101 games, compared to the 102, 103 and 102 win projections from Pythagorean expectation, Second Order Wins and Third Order Wins, respectively. This may be because the three latter models are all win estimates themselves. We still are still splitting hairs here; even at 100 WAR wins versus 103 Second Order Wins, the two are incredibly close.
Nonetheless, when your favorite team signs the next big free agent, and all of the baseball analysts say that they just got three wins better, just know one thing. . .
We may not be that crazy after all.
Devan Fink is a Featured Writer for Beyond The Box Score. You can follow him on Twitter @DevanFink.