## We Goofed: ERA+ numbers

Posted by Sean Forman on March 25, 2010

Right now on the site you'll see different ERA+ numbers from what you are used to seeing on the site. The old formula was 100*(lgERA/playerERA). The numbers you see now are 100*(2- playerERA/lgERA). This changes the numbers somewhat and bunches the top end a bit more, but doesn't change the ranking of players. The two lists of league leaders are the exact same.

I had not intended to roll out this change just yet as I was mulling a change in name to show that we are presenting a different formula. I was testing the change and ended up rolling it out to the site unintentionally. I apologize, and I am embarassed by the confusion this has caused.

The reasons for making a change are a bit esoteric, but I find them compelling. The old formula is a power equation. The independent variable is in the denominator, so you get a 10% change in the player's ERA showing up as any of a variety of percentage changes in ERA+. It depends where on the curve you are. For example, if you have a league ERA of 4.50 and one pitcher at 3.50 and one at 3.00 and one at 2.50, you get ERA+'s of 129 and 150 and 180. The changes aren't linear.

With the new formula, the equation is linear, so if the league ERA is 4.50 and you have one pitcher at 3.50, one at 3.00 and one at 2.50 you get ERA+'s of 122, 133 and one at 144 (one is 22% better than the league, one is 33%, and one is 44% better). It seems to me the numbers make a little more sense this way.

That said, I've rolled this out in about the worst possible way, so for now, I'm going to take the day to figure out what to do and then implement it tomorrow. I apologize again for the confusion this caused.

**UPDATE:** As I said above, it was not my intent to roll this out at this time. I still believe this to be a good idea, but it needs to be done in a MUCH more organized manner, so I'll be rolling back to the old stat tonight or tomorrow and then taking a more measured approach going forward. I apologize again for the confusion.

March 25th, 2010 at 7:52 am

Here are a couple of discussions on the change.

http://www.baseballthinkfactory.org/files/newsstand/discussion/the_book_blog_tango_thank_you_mr_forman/

http://www.insidethebook.com/ee/index.php/site/comments/thank_you_mr_forman/

March 25th, 2010 at 8:18 am

I was not aware of the existing debate on the stat, but I am nevertheless glad for the change as the numbers now mean something that I can manually calculate.

March 25th, 2010 at 9:22 am

So, people won't need to know how to do harmonic means when they average ERA+'s over multiple seasons anymore? I suppose that's for the best, but the math geek in me is a little sad.

And it used to be relatively simple to convert the old ERA+'s into support-neutral expected WPct's using Pythagoras. It was just (ERA+/100)^2/(1 + (ERA+/100)^2). Now its (100/(200-ERA+))^2/(1 + (100/(200-ERA+))^2). Not going to be able to remember that one off the top of my head.

Closers not having ERA+'s over 400 or 500 will be a good thing. It was hard to wrap my head around that. It will take some adjustment to get used to the changes in the starter numbers. I had gotten used to the high-100's being CYA caliber with 200 meaning historically good. The old 200 scales back to 150 which bunches players up a bit. The ranking of players doesn't change with this calculation switch which is good.

March 25th, 2010 at 2:01 pm

Actually, it's going to be newERA+ divided by 2. That gives you win%. I have the proof on my site if you want to take a look.

March 25th, 2010 at 2:31 pm

[...] They did change ERA+ just yesterday. It produces the same results, in that the players are ranked the same. The formula change just [...]

March 25th, 2010 at 2:33 pm

Sean,

On the list of ERA+ leaders there are no ties listed in the more than 100 seasons. If the values were rounded one would expect a tie or two, so I assume the values are actually fractional. If so, at least in a list of leaders, the fractional values might be useful.

March 25th, 2010 at 3:15 pm

@4

Really? That means ExpWPct for a team level is just 1 - RA/(2*RS)? That seems too simplified, although I'm sure its probably the first couple of term in the series expansion so it likely works pretty well for in most realistic ranges.

No one uses Pythag anymore? I must admit I'm likely a few years out of date with my saber-reading.

March 25th, 2010 at 4:02 pm

I havent had time to read any of the literature and only skimmed Seans post. I intend to read it all later.

That being said, my question is... Since these values are different than the orignal ERA+ values, is it fair to call it "ERA+"? This stat was used as an indicator of how good a pitcher was comapred to his peers. Zack Grienkes old value was 205... meaning he was a little over 2x better than his peers that year (No kidding). Now his number drops to 154.. bringing him down to... 1.5x better. Maybe the new method is the actual answer but I wonder which is the better indicator.

Was Zack Grienke 2x as good or "only" 1.5x times as good?

March 25th, 2010 at 4:13 pm

DavidRF,

I think Tom will agree that the new ERA+/2 being equal to WPct assumes a Pythagorean exponent of 2 (which is not perfect but not bad, either) AND is a linear approximation (which is not so good for extreme situations).

March 25th, 2010 at 4:42 pm

The new ERA+ is in some sense a mirror image of the traditional OPS+. For example, they both allow negative values (which the traditional ERA+ didn't). OPS+ ranges from -100 to a very large number, and the new ERA+ ranges from a very negative number (negative infinity if the ERA is infinite) to +200.

I think the new ERA+ makes sense, in that under the old ERA+, a 0.50 ERA would be twice as good as a 1.00 ERA (and insanely high), but in terms of wins it certainly isn't twice as good.

So the approximate relationship of the new ERA+ to WPct is a plus. It may seem like the scale at the upper end is compressed compared to the old ERA, but it was the old ERA+ scale that was distorted, at least with respect to reality.

We will have to learn how to interpret the new ERA+. But maybe we won't be quite as enamored with pitchers with a very high ERA+ (old version), which might be a good thing if a 250 ERA+ isn't that much more valuable than a 200 ERA+ (these would be 160 and 150 under the new system).

March 25th, 2010 at 5:40 pm

Semi-related to this, I would suggest putting lgERA somewhere on the players stats, maybe in the expanded section like is done with lgAVG/lgOBP/lgSLG/lgOPS for offensive stats. lgERA used to be available but was taken away at some point.

March 25th, 2010 at 9:03 pm

I like this idea, since I've always been annoyed with the "105% better" argument -- Greinke's ERA wasn't 105% lower than average, it was around 50% lower. So NewERA+ (TM of the cap company, of course) of 154 (54% better) seems more in line. This will screw up all my Negative Binomial calculations, however, so back to that drawing board -- and I'd vote for a new name, since it is a new (or at least different) stat.

March 25th, 2010 at 11:09 pm

Not all runs are created equal, so I'm not sure the number makes more sense to me. In fact, the non-linear makes more sense, as the upper tiers of excellence get the big boost they deserve. Still, so long as they're consistent, it doesn't bother me, even if this means I need to re-memorize the leader-boards.

March 25th, 2010 at 11:47 pm

Agree with #11

March 26th, 2010 at 5:26 am

So is OPS+ going to be adjusted in the same manner?

March 26th, 2010 at 11:29 am

[...] update: yesterday, I mentioned that we erroneously changed the definition for ERA+. I am still considering a [...]

March 26th, 2010 at 4:06 pm

Mike,

OPS+ is already linear in OBP and SLG (as opposed to linear in the inverses of those things like the old ERA+ was). However, a certain % increase in OPS+ does not give the same % increase in wins -- an OPS+ of 110 doesn't mean a team wins 10% more, it's more like 5% more).

If you made a new OPS+ as (OBP/lgOBP + SLG/lgSLG)*50, then % changes in OPS+ are the same as % changes in OPS (assuming OBP and SLG scale the same), and dividing OPS+ by 2 would give an approximate WPct, just like the new ERA+. I'm not sure why this wasn't done originally, I think (somebody will correct me if this is not true) the reason was to expand the scale of OPS+, i.e., allow for more granularity in OPS+. But if you want to make a new OPS+ with a similar relationship to wins as the new ERA+, that would be the logical way to do it.

For either the new ERA+ or a potential new OPS+, the relationship to WPct breaks down at extreme values, since it's a linearized approximation.

March 26th, 2010 at 4:31 pm

Oops, a 110 OPS+ means a team wins 2.5% more.

March 26th, 2010 at 9:03 pm

You can get the W% = ERA+/2 trick by assuming R+ = W+ (where R+ is R/lgR and W+ is W/lgW). In other words, assume that runs are proportional to wins, and that W = R/a, where a is equal to league 2 * lgR/G. Of course, in reality, this constant changes as R changes, so the more extreme R is, the worse this assumption is (especially for higher run environments, since 2*lgR/G is a little bit lower than the actual R to W conversion for an average team), but in most cases it works pretty well. Since the league W% is .5, you can express W+ as W%/.5, or 2*W%, and R+ = 2*W%, or W% = R+/2. Like most simple approximations, it isn't as precise as doing the full PythagenPat conversion, but it will give you a good ballpark figure very easily. It does break down at the extremes since it is a linear approximation and the assumption gets further from the truth in more extreme situations.

OPS+ relates to runs in the same way as the new ERA+. Both take the form of:

(lgAVG + RAA) / lgAVG

where lgAVG is the average number of runs (or runs allowed) over the player's PA/IP. OPS+ uses the form it does instead of (OBP/lgOBP + SLG/lgSLG)*50 or OPS/lgOPS because it is designed to model percentage changes in runs, not OPS. If it were instead designed so that going from a 100 OPS+ to 110 represented a 10% increase in OPS, it would no longer represent a 10% increase in runs, because a 10% change in OPS does not translate to a 10% change in runs. OPS+ had to choose to scale to one or the other, and it chose runs since runs are more useful. So the construction of OPS+ gives it the same relationship to runs as the new ERA+ has (and also more or less the same relationship to wins). A 110 OPS+ team will tend to score 10% more runs than average, and, given that it allows an average number of runs, win 10% more games than average. It's the same as with ERA+.

March 26th, 2010 at 9:08 pm

*W = R/a, where a is equal to 2 * lgR/G.

there should be no "league" before the 2 in that sentence.

March 27th, 2010 at 11:13 pm

I grabbed some of the data before it was pulled, and I'm not sure the contention stands up that the weighted average now works. I crunched the numbers for Walter Johnson, and his new career ERA#(sharp) or ERA*(star) or whatever is 100*(2-(1424/2089.5)) = 131.8 => 132. All well and good, and that's the number that was listed as his career value. But the weighted average is 133.0 on the nose -- I used sumproduct in Excel and divided by total IP. Is this suffering from the same thing the "old" ERA+ did -- when a pitcher has lots of innings pitched with a strong ERA+#* in a low-run environment, the "league" run totals are skewed downward and pitcher ERA+#* is pulled down as well?

March 28th, 2010 at 1:10 am

A pitcher's ERA will always influence the league ERA a little bit unless you explicitly remove each pitcher's ERA from the league's to calculate his ERA+, but that isn't what is making the ERA+s not average out properly. It can drop the pitcher's ERA+ by a very small amount, but it will affect both of the calculations you did; it won't cause them to differ from each other.

The weighted average of each season equaling the career value relies on the league ERA being the same in each season that you are averaging into the total. ERA+ is ERA/lgERA (and then flipped to work down from 2 instead of up from 0, but it works out the same as if you treat it as working up from 0). The career value is total ER divided by total league ER, which is the same as taking the sum of the numerators (pitcher ERA) weighted by IP, divided by the sum of the denominators (league ERA), also weighted by IP. As long as the denominator (i.e. the league ERA) in each fraction (i.e. each individual season ERA+) is the same, then the weighted average of each fraction simplifies to the weighted sum of the numerators divided by the weighted sum of the denominators, which is career ERA+. If the denominators (league ERA) are different, then the weighted average no longer simplifies to the weighted sum of the numerators divided by the weighted sum of the denominators, so the weighted average of each season ERA+ is not mathematically equivalent to career ERA+ if the league ERA is not the same in each season.

However, it works out to be fairly close in practice (such as 132 vs. 133 for Walter Johnson). Taking the weighted average is not actually calculating the total ERA+, but it is a good estimate. Also, the same problem existed with averaging multiple ERA+s before because you still had the same issue of the league ERAs being different in different years/ballparks. It's just that now you don't have to take the harmonic mean to get the same issue.

March 28th, 2010 at 12:11 pm

@ Kincaid-

Thank you for expressing very clearly what I wasn't able to sort out in my mind. I guess the question for Sean is, "Which career value is more realistic?" I suspect many of us will be using the weighted average to assess segments of careers (like we used to use the harmonic mean), so I'd argue that the weighted average should also be used for the career value -- and it moves Walter Johnson JUST ahead of Lefty Grove on the career list, which makes for a wonderful debate.

One example in favor of the weighted-average career calculation:

Year W-L IP ERA* ER LgER R/9

1 20-20 360.0 100 80 80 4.00

2 30-10 360.0 150 40 80 4.00

TOT 120 160 125 ERA*

Change Year 2's environment:

Year W-L IP ERA* ER LgER R/9

1 20-20 360.0 100 80 80 4.00

2 30-10 360.0 150 80 160 6.00

TOT 160 240 133 ERA*

Weighted average for both scenarios is 125, which more consistently encapsulates "360 innings of average, 360 innings of 50% better, therefore career total is 25% better."

It'll be interesting to see which way B-R decides to go.