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Monday, December 18, 2006
The Odds Ratio Method
Pure math post on how to calculate the expected matchup rates.
If you have one guy who is a true .600 facing another guy who is a true .400, the resulting win% will be .692, if the league mean is .500.
While the calculations when the mean is .500 is straightfoward (log5), you can use the Odds Ratio method for any mean. For example, assume the league OBP is .333, you have a hitter who is .400 and the pitcher is .250. What’s the resulting OBP?
Odds(H) = .400/.600 = .667
Odds(P) = .250/.750 = .333
Odds(L) = .333/.667 = .500
Odds = Odds(H) * Odds(P) / Odds(L)
= .667*.333/.500=.444
If the Odds are .444 safe to 1 out, the the Rate is .444/(.444+1) = .308
You can further extend this so that the Odds(L) for the hitter and pitcher are different.
The full equation is:
Odds(matchup) Odds(H) * Odds(P)
----------------- = -----------------------
Odds(environment) Odds(envH) * Odds(envP)
So, you have a hitter with an OBP of .400 in a league of .300 facing a pitcher with an OBP of .250 in a league of .350, and they are both playing in a league (or park) where the OBP is expected to be .380 for the league average player. What’s the resulting OBP?
Odds(matchup) (.400/.600) * (.250/.750)
------------- = -------------------------
(.380/.620) (.300/.700) * (.350/.650)
Odds(matchup) = .590
Matchup OBP = .590/1.590 = .371
And, you don’t have to limit yourself to just these variables. You can extend them to infinity.