A dynamic run estimator is a run estimator that allows offensive events to interact with each other, such that the value of a given event is not fixed as would be the case in a linear weights formula (e.g. a single is worth .50 runs), but rather is dependent upon all of the other components of the batting line. Dynamic run estimators are great in theory, since the run scoring process for a team is obviously dynamic and not linear. However, there are two issues:

1. They are harder to design than linear estimators. Any idiot with a spreadsheet and a dataset can run a linear regression on runs scored and have a linear estimator when they are done. It may not be a good one, but it will be functional and will probably have a low RMSE when estimating team runs scored. To develop a dynamic model, one must consider the run scoring process and produce a simplified model, but not so simplified as to not produce reasonably accurate estimates.

This is not a series about run estimators, but the most commonly used dynamic run estimator, Bill James’ Runs Created, suffers from flaws that make it unable to handle extreme offenses. A much better model, David Smyth’s Base Runs, is powerful and will be used here.

2. They are not appropriate to apply to individual offensive statistics. Dynamic estimators always involve multiplying base runners by some factor representing advancement of baserunners (in Runs Created that’s the end of the story, Base Runs accounts for the unique nature of home runs). This multiplication is inappropriate when applied to an individual player, as now Frank Thomas’ high OBA is multiplied directly with his high power which advances runners. In reality, there is some interaction, but Thomas’ impact is diluted by being just 1/9th of the lineup. Inputting his statistics into a dynamic run estimator produces an estimate of how many runs a team would score if each batter hit like Thomas.

Due to this issue, I do not advocate applying dynamic run estimators directly to individuals, but this post will still address the rate stat implications of such applications. Later we will discuss theoretical team methods that allow the use of a dynamic run estimator while still accounting for the fact that the player is just one of nine in the lineup.

This series will now discuss what I believe to the be the proper rate stats for a particular framework for evaluating individual offense. One of my objectives is that for each option of a framework for building a rate stat presented, there be at least one variation that is linearly comparable and one that is ratio comparable. I’ve defined those terms as I use them at length before, so here I will be brief:

* A statistic is linearly comparable if the difference between two figures is meaningful. A hitter with a .400 OBA would reach base 100 times more than a hitter with a .300 OBA over 1000 PA.

* A statistic is ratio comparable if the ratio between two figures is meaningful. Our .400 OBA player reached base 33.3% more frequently than the .300 OBA player

Ideally, our metric will facilitate both types of comparison, but if not, I will endeavor to present an alternative formulation that fills the gap. I will not propose any metrics that are neither linearly comparable or ratio comparable because they are the scourge of sabermetrics (hello OPS).

The underlying principle of the discussion that follows for the three frameworks (treating the player as a team, a full linear model, and a theoretical team model) is that the rate stat should be consistent with the run estimator used. If the run estimator treats the player as if he is a team, then the corresponding rate stat should treat the player as if he is a team.

In this case, that makes it very simple. The proper denominator for a team rate stat is outs. If you apply Runs Created, Base Runs, or some other run estimator directly to an individual player, the proper denominator is outs.

At this point in the discussion, this may ring as a somewhat hollow declaration, as I have only indirectly made the case for why we might want to use a denominator other than outs for an individual when it is so clearly the proper choice for a team. Since I’m suggesting that outs are the proper choice for this framework, I’ll defer that case for later.

In this case, I advocate for using outs when applying a dynamic run estimator to a team because it is the only consistent treatment. The only justification for going down this path (other than needing something quick and dirty) is a theoretical exercise – how many runs would a team that hit like Frank Thomas score? While I don’t think this theoretical result is appropriate for attempting to value Thomas’ contribution the 1994 White Sox, it at least does have an interpretation. If you start mixing frameworks, you really have a mess on your hands. There’s no good reason (other than crude estimation) to apply a dynamic run estimator directly to an individual; there’s no sense in deviating from the corresponding rate stat in order to try to make the results more comparable to a better approach to evaluating individual offensive contribution. Just use the better approach, and if you insist on misapplying a dynamic run estimator to individual players, make outs the denominator so that at least you have a theoretically coherent suite of metrics.

I should note that Bill James in the 1980s took this entire process to its logical conclusion. After applying Runs Created to individuals, dividing by outs, and multiplying by a constant that was close to the league outs/game for the definition of outs chosen, he went a step further and used the Pythagorean theorem to estimate the winning percentage that this team would have if it allowed an average number of runs. He then converted it to wins and losses by using the number of outs the player made to define games, which caused all kinds of problems, but at least he was committed.

This will be the first of several times that I’ll run a leaderboard for the 1994 AL using a particular framework. Here we have the top 5 and bottom 5 performers with at least 200 PA in Base Runs/Out. RAA is “Runs Above Average” and is calculated simply as (BsR/O – LgR/O) * Outs. Spoiler alert: No matter how we slice it, Frank Thomas is going to come out as the leading hitter in this league, as he raked .353/.492/.729 on his way to a second consecutive MVP award.

I am showing at least one more decimal place on each metric than I usually would just to allow for a little more precise calculation if you’re following along; it is no way a statement about the significance of the ten-thousandths of runs per out.

Runs per out can of course be scaled; Bill James multiplied it by the league average outs/game appropriate given the categories be considered in the computation of outs. For instance, in this case, since we’re defining outs as AB – H, the average outs/game will be around 25.2 (for the 1994 AL it was 25.19). A more complete accounting of outs, like AB – H + CS + SH + SF + DP, would get close to 27 outs/game. While putting individual contribution on a team games basis is nonsensical on some level, since it is just a scalar multiplier it causes no real distortion and provides a scale that is easily understandable, in the same manner that ERA or K/9 are understood by everyone other than Matt Underwood and Harold Reynolds.