Pondering Pythagoras

Some things in baseball are supposed to be inexplicable. We sabermetricians like to try to find explanations for everything, but sometimes, as in the case of clutch hitting, we just give up and toss it up to luck.

Now sometimes, the area we’re investigating is subject completely to random chance, and there really is nothing else to see. But sometimes, as in the case of clutch hitting, we find out that our conclusions were too hasty, that the skill does exist; the effect was just too small to be easily traced.

But at least clutch hitting was always a topic of debate. Some categories, such as a team’s “Pythagorean Differential,” have long been left untouched, at least as far as I can see, thrown into the luck pile by pretty much everyone, when in reality, why should we be so sure that teams outperform or underperform their Pythagorean records just by chance?

Pythagorean record is a method for predicting a team’s winning percentage based on its runs scored and allowed. Bill James originally proposed the idea, and formulated that winning percentage was equal to the runs scored squared divided by the sum of the squares of runs scored and runs allowed. Since then, researchers have found a better exponent than two, one that varies based on the team’s run environment.

“Pythagorean differential” is a statistic that we publish in our “Team” section on The Hardball Times; it simply measures the difference between a team’s actual and expected record. You want, of course, a positive Pythagorean differential—that means that you’re winning more games than your numbers would predict—but most people believe that number is dependent on luck, and nothing else. Today, I want to test that assumption.

I started by assembling four categories that I thought might have some impact on a team’s Pythagorean differential.

  • Balance: The coefficient of variation is the wOBA (a total hitting statistic scaled to look like on-base percentage) of the nine hitters with the most playing time on that team. The coefficient of variation is simply the spread, or standard deviation, divided by the mean to control for the overall level of offense. This measures how balanced a team’s lineup was.
  • Leverage: The number of predicted wins on the team accounting for its distribution of high-leverage outings minus the number of predicted wins without accounting for how its pitchers are leveraged, divided by the number of games the team played.

    Basically, this is a measure of the quality of a team’s bullpen, accounting for the fact that relievers can be leveraged so that the team’s best relievers pitch in high-pressure situations where a good performance is worth more than it is in an average situation. However, a team’s runs allowed do not reflect that fact.

    Because I did not have actual leverage or win probability numbers, I ran a regression to predict Leverage Index based on 2006 data provided to me by David Appleman. The result of the regression was,

    1.11 – .137*GS/G – .86*(GF – Sv)/G + 2.165*SV/G.

    In other words, starters have lower Leverage Indexes as do relievers who mop up games. Closers, and other players who garner saves, have higher Leverage Indexes.

    To measure a team’s “Leverage,” I calculated the Pythagorean record for each pitcher based on his run average and the team’s offense, and then multiplied that by his predicted Leverage Index.

    I summed both terms for each team, and subtracted the former category, which does not account for how each pitcher was used, from the latter, which does, and then divided that by games.

  • Manager: The number of career games managed by the team’s manager, as a stand-in for the quality of the manager. If that team had more than one manager, I used the number of career games managed for the least experienced manager. I also tested to see if the natural logarithm of games managed gave a better fit; it did not.
  • Home Run Reliance: A measure of the team’s reliance on home runs, dividing home runs by Base Runs (which is the team’s predicted number of runs scored based on component statistics).
  • I compiled these statistics for all American League teams from 1976 to 2005, to avoid problems with pitcher hitting, and compiled a large data set. I then ran regressions until I ended with only significant variables (using .10 as my threshold for significance). Every variable except for Home Run Reliance was significant.

    Category	Coefficients	P-Value
    Balance         -.010           .000
    Leverage         .481           .000
    Manager          .00000154      .072
    Correlation = 0.376
    R^2 = 0.141

    The regression indicates that we can predict a team’s Pythagorean differential with relatively accurate results. The R^2 tells us that our formula explains 14% of the variance in a team’s Pythagorean differential, which makes this a more predictable number than, say, how many home runs a pitcher will allow next season.

    Specifically, here is what the results tell us:

    A Hardball Times Update
    Goodbye for now.
  • Teams with balanced lineups outperform their Pythagorean records; one point of balance is worth 1.62 wins above what we would expect. For 95% of all teams, this impact will be limited to +/- 0.05 wins.
  • Each “leveraged win” will add .48 games per season above what is expected. For 95% of all teams, the impact of this category will be within +/- 2.67 wins.
  • For each 1,000 games managed, the manager adds 0.25 wins per season above what is expected of the team. For 95% of all teams, the impact of this category will be within +/- 0.61 wins.
  • In all, using this formula, we would predict 95% of all teams to be within +/- 2.74 wins of their Pythagorean record, with the rest of the variance (95% of all teams land within +/- 7.46 wins of their expected record) explained by luck or factors I have not considered.

    Nonetheless, in a game in which teams pay $5 million per win on the free agent market, this knowledge is a huge deal. It confirms the importance of a good bullpen and suggests the importance of a good manager.

    And perhaps most importantly, it tells us how important it is to never consider a question closed, especially if the best suggested explanation is luck. In the search for the truth, not every question has an easy answer.

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