## Wednesday, June 23, 2021

### Rate Stat Series, pt. 6: Rate Stats for Linear Weights I

Last time, I attempted to demonstrate that linear weights producing a result in terms of runs above average properly account for the value of extra plate appearances created by a batter. From here on, I will talk about this contention in the same way as I would any scientifically-demonstrated fact, even though I admit that I have not provided a “proof” in the mathematical sense. The casual use of language is intended to prevent wasting space repeating myself rather than an attempt to claim a more robust result than is appropriate.

Since we have demonstrated that LW_RAA captures the player’s direct and indirect contributions to team offense (at least within the linear framework of metrics), I would contend that it follows that any linear weights-based rate stat we should propose must return the same RAA as we would get from eschewing a rate stat altogether and just applying our linear weights formula with the “-.3 type out value”.  One very simple way to do that is to simply use RAA, rather than a measure of absolute runs created, as the numerator for the rate stat. In this case, the obvious denominator is plate appearances.

Using outs in the denominator would be appropriate for a team metric, but for an individual would overstate the value of his PA generation. For a simple demonstration of this, recall from part three the equation:

PA/G = (O/G)/(1 – OBA)

and the equivalent PA = O*(1 – OBA), given our simple definitions which as a refresher are PA = AB + W, O = AB – H, and OBA = (H + W)/(AB + W)

This direct relationship between plate appearances, outs, and OBA means that if we were to use outs as the denominator rather than PA, we would be inflating the rates for players with higher OBAs, even though we’ve already demonstrated that the PA generation impact of higher OBAs is captured in the numerator (RAA).

Here were the top and bottom 5 performers in terms of RAA/PA, which for ease of use I have restated as RAA/650 PA (This works out to 152.5 full games for a player getting 1/9 of an average 1994 AL team’s PA; sadly, these teams didn’t get anywhere close to playing a 162 game season. There’s no special significance to 650 other than that it is a round number that is reasonably close to the number of PA a full-season batter might accumulate):

You may have noticed that the identity and order of these players did not change from when we used a fully dynamic approach that treated them as if they were their own teams (BsR/O). This recalls a simple truth that I am burying beneath thousands of words of minutia – any reasonable approach will reach similar conclusions for the majority of situations. Even a poorly designed, indefensible metric like OPS will get you 98% of the way there. This series is about the tiny differences that lie beyond that point and the larger differences that arise in values as opposed to rank order.

To wit, while the rank order remains the same, the RAA values are different using linear weights, and with the exception of Griffey, less extreme. All of the other players have moved closer to average, with Frank Thomas losing a whopping 7.7 RAA due to using a linear approach rather than imagining an entire lineup of Big Hurts.

At this point, we could stop, and simply use RAA/PA or RAA/650 PA as our final linear weights-grounded rate stat. However, it lacks the very useful trait of ratio comparability that would be desirable in the ideal rate stat. While linear differences in RAA/PA can be compared (e.g. Thomas contributed an additional .081 RAA/PA beyond what Chili Davis did), the ratios are not particularly useful. Consider two players who each had 600 PA, one contributing +1 RAA and the other -1 RAA. If you can explain the practical baseball interpretation of the resulting ratio of -1, be my guest.

This happens because we have applied the high baseline of average to the metric. However, we have another version of linear weights that is based on absolute runs from which we could build a rate stat. Remember that in order to qualify for consideration, the RAA that results from that rate stat must be equivalent to the RAA from simply applying the linear weights formula and not comparing a rate to the league average.

The obvious first choice is Runs/PA, using our formula for linear weights runs created. In this case, I am not showing the leaders, but rather the same ten players in the same order. RAA in this case is (LW_RC/PA – LgR/PA)*PA:

The results are not even close to what we need, and the reason is simple: we have not accounted in any way for each batter’s PA generation. There is an easy alternative that might correct this: using outs in the denominator. This just takes us to the correct team rate stat, although the numerator is linear rather than dynamic (either in the case of using Base Runs or using actual runs scored on the team level). As such it does implicitly consider PA generation; might it produce satisfactory results for individuals? Here RAA = (LW_RC/O – LgR/O)*O:

A perfect match. Absolute runs per out produces the same RAA as the direct application of LW_RAA. R/O also has the advantages of being meaningfully comparable as both a difference and a ratio and is the same as the correct team rate stat. Everything is great, except we haven’t answered the question: Does it actually work?

Remember, I said that matching LW_RAA was a necessary condition for our proposed alternative linear weight rate stat to meet – it is not a sufficient condition. We’ve already concluded that RAA/PA is a proper rate stat for a linear framework; in order for an alternative to be acceptable, it must produce results that are consistent with RAA/PA. How do we determine this consistency, other than matching the RAA result, when the numerators and the denominators each start from a different basis?

One simple but obvious way to determine if they are consistent is to confirm that they result in the same rank order of players. They did for our most extreme hitters, but does that hold for all hitters?

Apparently not. I didn’t have to go to far to find this little cluster of hitters as they rank 7-10 in RAA/PA. None of them rank in the same spot as Vaughn is first in RAA/650 but second in RC/O; Lofton is second/third; Mack is third/fourth; and Clark is fourth/first.

I slipped OBA onto the chart because it helps to explain what is going on. Will Clark’s .433 OBA ranked fifth among AL hitters with 200 PA; using outs in the denominator helps him as it implicitly assumes that he represents an entire team with a .433 OBA. While all of these hitters had excellent OBAs relative to the league, R/O goes a bit too far in valuing this. Given that R/O still produces the right RAA, any distortions have to be somewhat limited for normal players.

One can come up with extreme thought experiments, like a player with a .999 OBA all from walks versus a player with a .600 OBA, all from home runs. A team made up of the former would score what would certainly feel to the opposing pitching coach like a nearly infinite number of runs; but as a single player in a lineup, his impact would be muted. It’s not necessary to answer the thought experiment as to which would be more productive to see that R/O applied to extreme individual players would break down. Incidentally, it is exactly this type of scenario that I got into trouble trying to “prove” in my last attempt at this series – while I do think that the theoretical team methods I’ll discuss later provide reasonable estimates for this situation, relying too heavily on them for proofs is begging the question.

So, we have a rate stat (LW_RAA/PA) that works, but it lacks ratio comparability. There are (at least) two ways we could go about solving this problem:

1. We could adjust LW_RC/PA in some way to take into account the value of PA generation

2. We could manipulate LW_RAA/PA so that it’s no longer a measure of runs above average, but instead on an absolute runs basis

Next time I’ll explore these two parallel questions...if in fact they are parallel at all.