Writing RC in terms of BA, OBA, and SLG

The primary run estimator that I use here and on my website for evaluating individual hitters is Estimated Runs Produced (ERP). This method was first developed by Paul Johnson and published in the 1985 Baseball Abstract. It is a linear formula, which neither he nor Bill James pointed out in his article, although that is a story for another day.

If I can be considered a good sabermetrician (that is up to you), I believe that it is because I have always experimented. If I see a method published by somebody else, I always try to develop my own version based on the same idea or same general principle, starting from scratch as much as possible. I do not do this out of vanity, thinking that I can come up with a better formula, but because I find that is the best way to find out how and why something works. The point of that digression is that a few years ago I was trying to come up with a linear run estimation skeleton, despite the fact that so many good ones are already in existence. What ended up happening is that I recreated ERP. I got this formula:
(TB + .5H + W - .3(AB - H))*.324
Johnson’s formula (ignoring SB, CS, etc.) was something like this:
(2*(TB + W) + H - .605*(AB - H))*.16
As you can see, my formula is almost exactly the same as Johnson’s, divided by two.

So I have used my version of the formula, and called it ERP, because it is the same formula, so regardless of whether I developed it independently or not, it is still Johnson’s.

Anyway, the only point here is to show a good way to estimate RC/PA or RC/O based only on BA, OBA, and SLG. It is not a “clean” formula that weights each of them, like that you would get from a multiple regression, but it is mathematically equivalent to ERP. We’ll start by writing the formula in terms of per at-bat rates. TB/AB is SLG, H/AB is BA, W/AB is what I call WAB, and (AB-H)/AB is just 1-BA. So we can write the formula as:
RC = (TB + .5H + W - .3(AB - H))*.324
RC/AB = (SLG + .5BA + WAB - .3(1 - BA))*.324
We can write WAB as (OBA - BA)/(1 - OBA), which along with expanding the (1 - BA) term gives:
RC/AB = (SLG + .8BA + (OBA - BA)/(1 - OBA) - .3)*.324
We can convert this to RC/PA:
RC/PA = RC/AB*(1 - OBA)/(1 - BA)
And then RC/PA converts to RC/O:
RC/O = RC/PA*(1/(1 - OBA))
Or, directly, R/O = R/AB*(1 - OBA)/(1 - BA)*(1/(1 - OBA)) = (R/AB)/(1 - BA)