More on mERA+by Sal Baxamusa
April 23, 2007
I got a lot of e-mail about mERA+, a pitching metric which makes some marginal improvements to ERA+. Many disagreed, and some said that I simply must have made a mistake someplace.
Well, you were right. In combing through my formulas—this is what I do for fun—I noticed that I had missed a factor of two. Upon correcting this error, I was afraid that the difference between mERA+ and ERA+, which is small to begin with, would get even smaller. But my error was in the opposite direction, which means that the marginal improvement to ERA+ is slightly greater than I first thought. Nothing earth-shattering, but in the interests of full disclosure, I thought that I should share the correct results.
I compared the careers of Roy Halladay and Sandy Koufax and concluded that while Koufax was slightly better by ERA+ (131 to 130), Halladay was better by mERA+ (62.9 to 62.3). I understated the case; Halladay actually has the edge 65.9 to 65.0.
I told you it wasn't a big difference. Here are the career leaders in ERA+
Pitcher ERA lgERA ERA+ mERA+ Pedro Martinez 2.81 4.49 159.8 77.6 Lefty Grove 3.06 4.54 148.4 73.6 Walter Johnson 2.17 3.17 146.1 69.5 Dan Quisenberry 2.76 4.04 146.4 71.8 Hoyt Wilhelm 2.52 3.68 146.0 70.8 Joe Wood 2.03 2.97 146.3 69.0 Ed Walsh 1.82 2.63 144.5 67.4 Roger Clemens 3.10 4.46 143.9 71.7 Johan Santana 3.20 4.59 143.4 71.8 Roy Oswalt 3.05 4.35 142.6 71.0
Because the error was systematic, the trend (and therefore the conclusions) doesn't change, but the absolute number does. How good is Pedro Martinez? In over three-quarters of his career starts, he has delivered a performance better than what you would have expected from an average pitcher.
And as for single-season pitching:
Pitcher ERA lgERA ERA+ mERA+ Pedro Martinez 1.74 4.97 285.6 98.0 Dutch Leonard 0.96 2.68 279.2 91.3 Greg Maddux 1.56 4.26 273.1 96.2 Walter Johnson 1.14 2.96 259.6 91.0 Greg Maddux 1.63 4.23 259.5 95.1 Bob Gibson 1.12 2.90 258.9 90.6 Mordecai Brown 1.04 2.62 251.9 88.4 Pedro Martinez 2.07 5.07 244.9 95.6 Walter Johnson 1.39 3.34 240.3 90.5 Christy Mathewson 1.28 2.93 228.9 87.2
Remember that 100 is a perfect score. Yes, Pedro scored 98.0 in his best season.
Technical points for other researchers
There are a few additional technical points that I wanted to touch upon.
The shape is familiar in both cases, though not perfectly Weibullian. Appier in particular showed a tendency to give up one or two runs less often than expected by the model and four and five more often. Certainly, mERA+ might be improved by using the actual distribution instead of Weibull distribution. But I believe that the assumption that distribution of performance tends to the Weibull is a decent one.
Notice that with ERA+, the plot doesn't quite look symmetrical. This is because a very good pitcher could have an ERA+ approaching infinity. Using the mean and standard deviation to fit it to a normal distribution ("bell-curve") is problematic—you can see that the normal distribution (black line) doesn't do a very good job of matching the actual data. On the other hand, mERA+ fits to a normal distribution much more nicely. It's not perfect, but it is an improvement on ERA+. For reference, the all-time average for mERA+ is 50.6 and the standard deviation is 15.5.
Remember, ERA+ rules!
It is important to remind everybody that ERA+ is still a wonderful metric, suitable for almost all everyday purposes. mERA+ is a slight improvement that is strongest when discussing the very best or very worst pitchers.
References and Resources
The references section from the previous article on mERA+ contains a lot of good reading on Weibull distributions.
Sal Baxamusa is a graduate student in chemical engineering. He can be reached here.