Tangotiger Blog

A few months ago, a poster in a fantasy baseball forum I participate in came up with a linear model for xERA from xWOBA (he simply plotted 3 years worth of wOBA vs. ERA with some minimum innings threshold). 

I thought something more basic and fundamental was possible.  In particular, I thought xERA should reflect the “expected” out rate, i.e. xBA should figure in the equation for xERA.  My expression is:

The first term in square brackets is runs per plate appearance for the pitcher.

The second term in square brackets is an estimate of plate appearances per 9 innings.  Batting outs per 9 is somewhere between 25.5 and 26.  “bbfrac” is the pitcher’s actual walk rate (i.e. walks per plate appearance… I suppose one could include HB with walks).  And xBA is expected batting average, of course.

Finally, I multiply by the league’s fraction of runs that are earned (0.924 in 2019).

Create an xERA that is proportionate to xwOBA^2.

Then compare that to what you have, and let us know how it looks.

jgf704 might be referring to me.  (I post as Michael@HQ at the BaseballHQ forums.)  I have been using xwERA = xwOBA * 27 - 4.3.  Many of us, not just fantasy players, find it easier to see it in an ERA scale.

Working on it. 😊

Argh.  Dumb typo in my equation in post #1 (wrote the earned run factor backwards).  Should be:

Yep, Detroit Michael, I was referring to your BBHQ work. 😊

For Tom… So digging around the internet, I found a 2017 Reddit thread where you advocated a simple formula for converting wOBA to runs per 9 innings (I call it R9 below):

... which is basically what you are saying in this post without the scaling factors (and ERA would need an earned run factor, of course).

So I calculated R9 by the formula in my post #1 (leaving out the earned run factor) for 2019 pitchers with 50+ IP, along with simple the formula above (in this post).  And, to my surprise (and probably not yours 😊), the simple version works better.  More precisely… there were 341 pitchers with 50+ IP.  The root mean square error in R9 was 0.65 by my formula, and 0.56 by the simple equation.

As you know, my equation in post #1 is based on wRC (along with an estimate of PA9 from xBA and walk rate).  So I view it as more fundamental (in that it is based on something else which we know works).  And this is why I was surprised that the simple equation worked better.

My intuition says that there must be something fundamental in the simple equation as well.  Is there?

Yes, for pitchers, the “interdependence” makes more sense.  As it does for teams.

For batters, you want the linear/additive approach.

By “interdependence”, I assume you mean the fact that RA9 is proportional to wOBA*wOBA.

Is it just serendipitous that the simple formula of yours that I posted in #6 works so well?  Or is there some underlying fundamental principle at work (and that I’m missing)?

There is a fundamental principle.

At a VERY rough level: the % of times you are on base = the % of times that those runners will score.

Obviously at the low end, approaching .000 OBP, it would be natural that 0% of those runners will score.

Similarly at the high end, as your OBP approach 1.000, then 100% of those runners will score.

More to the point, think of the 3 bases as the transition point between getting on base, and scoring.  Every inning you will have 0 to 3 runners left on base.

So, if you are not left on base, you will score. (Setting aside DP,etc)

Let’s assume there’s always 1 runner left on base each inning.  So, you will end up with something like this, starting with 10 runners reaching base for the game:

Batters Put Out 
Runners Reaching Base 
Left on Base 
Runs Scored 
OBP 
% runners scoring

27 10 9 1 0.270 10%
27 11 9 2 0.289 18%
27 12 9 3 0.308 25%
27 13 9 4 0.325 31%
27 14 9 5 0.341 36%
27 15 9 6 0.357 40%
27 16 9 7 0.372 44%
27 17 9 8 0.386 47%
27 18 9 9 0.400 50%

Each of the columns matches to the data in order.

Focus on the last two.  Notice how, roughly, the % of runners that score is close to the % of runners reaching base?

Runs scored, by defintion
= % of runners reaching base
times
% of runners that score

And so, that’s why runs is proportional to OBP squared.

And wOBA is, basically, OBP.

Thanks you.  And, dang, I should have been able to figure that out.  FWIW, I was re-immersing myself in Base Runs after seeing how well the (wOBA)^2 thing worked.  But I was thinking about it an overly-complicated fashion—should have backed off to a simpler model that still captured the salient features as you did here.

Thanks again for coming back to this.

And yeah, you sum up alot of this stuff (i.e. linear vs. non-linear / dynamic run estimators) on your Base Runs wiki page.

I like the end of post #9.  It clearly, succinctly (in my view anyway) explains why it is not a linear relationship.

And yet, look at the graph again.  It’s not linear, but it’s not very far away from linear either within the relevant portion of the graph.  I don’t feel badly at all for using the linear formula from post #3 to estimate xwERA or xERA during the interim before Baseball Savant starting computing it for us.

FWIW, Michael, I agree with you that the linear formula based on least squares minimization likely provides at least as good (and possibly a better) fit to the data.  OTOH, I really like the simplicity of:

Where it saves us is with those great relief seasons that a linear would get you into negative runs.

Especially if you think of it say after 20 or 30 innings into the season, since we are showing the data after every game.