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How Useful is the Pythagorean Expectation? May 18, 2010

Posted by tomflesher in Baseball.
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The Pythagorean expectation is a method used to approximate how many wins a baseball team “should” have based on its offense (runs scored) and its defense (runs allowed). As the linked article points out, there are some problems with the formula. As far as I’m concerned, the most useful application of an expected win percentage is to compare teams that are otherwise similar. Let’s say, for example, that I have two teams that have identical records and I want to predict which team will win an upcoming series. In that case, an expected win percentage would be useful to indicate which team has more firepower over time.

What’s the perfect way to test this? One-game playoffs. Behind the cut, I have the results of some number-crunching I did to test whether the Pythagorean expectation generates useful results.

There have been 13 tiebreaker games in Major League Baseball. Under the earlier rules, a series format was used. In all cases, though, the team that won the first game of the series went on to win, so for simplicity’s sake I’ll treat all of the playoffs as though they were one-game series.

I created a spreadsheet with data from Wikipedia and Baseball-Reference.com with variables for the winning and losing team’s run for and against, then used that data to generate Pythagorean expectation and run differential. I created binary variables to test whether the winning team was the home team (wHome = 1), the team with the higher Pythagorean expectation (wEx = 1), the team with more runs scored in the season (wRuns = 1), and the team with the higher run differential (wDelta = 1). My expectation is that more often than not, the Home team would win, and that the team with the higher Pythagorean expectation, more runs, and a higher differential would win. The results were quite surprising.

The data for the winners:

Year Winner wRunsF wRunsA wDiff wPyth
1946 Cardinals 700 539 161 62.78%
1948 Indians 840 567 273 68.70%
1951 Giants 781 641 140 59.75%
1959 Dodgers 705 670 35 52.54%
1962 Giants 878 690 188 61.82%
1978 Yankees 735 582 153 61.46%
1980 Astros 637 589 48 53.91%
1995 Mariner 796 708 88 55.83%
1998 Cubs 831 792 39 52.40%
1999 Mets 853 711 142 59.01%
2007 Rockies 860 758 102 56.28%
2008 White Sox 811 729 82 55.31%
2009 Twins 817 765 52 53.28%

And the losers:

Loser lRunsF lRunsA lDiff lPyth
Dodgers 695 561 134 60.55%
Red Sox 907 720 187 61.34%
Dodgers 855 672 183 61.81%
Braves 724 623 101 57.46%
Dodgers 842 697 145 59.34%
Red Sox 796 657 139 59.48%
Dodgers 663 591 72 55.72%
Angels 801 697 104 56.91%
Giants 845 739 106 56.66%
Reds 865 711 154 59.68%
Padres 741 666 75 55.32%
Twins 829 745 84 55.32%
Tigers 743 745 -2 49.87%

And the results:

wHome wEx wRuns wDelta
1 1 1 1
0 1 0 1
0 0 0 0
0 0 0 0
1 1 1 1
0 1 0 1
0 0 0 0
1 0 0 0
1 0 0 0
0 0 0 0
0 1 1 1
1 0 0 0
1 1 1 1
0.46 0.46 0.31 0.46

As you can see, none of the values I expected to be good predictors of winning a one-game playoff actually were. In fewer than half the cases, the expected team wins.

I’m not sure what to think of this. When I played with the Pythagorean expectation a while back, I used it to generate a stat called Wins Above Expectation (which appears to have been invented independently by several other people as well) with the interpretation that a team with a higher WAE value was generating more wins with the same number of runs, so they must be scoring runs when they count and not wasting them on laugher wins. The same interpretation seems to be valid here – a lower expectation given the same record shows some sort of intelligence or ability to play a clutch game, so it makes sense that the team with the lower expecatation would win.

Still, remember that the sample size is only 13 games. We can’t draw conclusions (or do a useful regression analysis) based on such a small n.

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