This series spans fifteen posts, over thirty tables, and over 25,000 words. I don’t really expect anyone to slog through all that. So here I want to express the key points of the series as succinctly and with as little math as possible. In doing so, it will become apparent that I haven’t broken any new ground in this series, which is even more reason not to slog through the rest.

1. The proper denominator for a rate stat (where “rate stat” is defined as a measure of overall offensive productivity expressed in units of runs or wins, rather than the rate of any given event or subset of events) for a team is outs. This is obviously true if you take a moment to examine it, and is one of the core fundamental insights of sabermetrics. Because when a pitcher is in the game, he functions as his own team, outs are also the proper denominator for any overall pitching rate stat.

2. The number of plate appearances any team gets is a function of their rate of making outs (if we ignore enough statistical categories, this boils down to their On Base Average). On the team level, plate appearances are an inappropriate rate stat denominator as it is illogical to penalize a team for avoiding outs more effectively than another.

3. At the individual batter level, neither outs nor plate appearances are a satisfactory denominator if an estimate of absolute runs created is used as the numerator of the rate stat. Beyond their primary contributions to their team through their direct actions at the plate and on the bases, batters make a secondary contribution by avoiding outs, thus generating additional plate appearances for their teammates. But individual batters don’t operate in a vacuum. An individual contributes to his team’s plate appearance total, but doesn’t individually define it as he only makes up one-ninth of the lineup. Using outs as a denominator treats an individual as if he alone defines his team. Using plate appearances, on the other hand, does not value the secondary contribution that a batter makes by generating additional opportunities for his teammate, absent some adjustment.

4. There are three frameworks through which we can evaluate an individual’s offense. The first, which I do not advocate at all, is to treat the player as a team, plugging the individual’s stats into a dynamic run estimator like Runs Created or Base Runs. The second is to use linear weights to evaluate either absolute runs created (as, for example, Estimated Runs Produced or Extrapolated Runs do) or runs above average (ala Pete Palmer’s Batting Runs). The third is to construct a theoretical team, using a dynamic run estimator to estimate the runs created by a hypothetical team that consists of the batter in question plus eight other (typically league average) players.

5. The selection of approach to run estimation should not be divorced from the choice of rate stat. The assumptions inherent in each of the approaches to run estimation suggest similar, consistently reasoned assumptions that would make sense to use in developing a rate stat. While it is possible and justifiable to mix certain elements across the framework, my point of view is that it makes more sense to keep the “frameworks” pure, and utilize the rate stat that makes the most sense to pair with the chosen run estimator.

6. Using linear weights runs above average (RAA) rather than absolute linear weights runs created as the numerator does enable the use of plate appearances as the denominator, because the RAA estimate already incorporates the batter’s secondary contribution. However, RAA/PA may not be everyone’s ideal choice for a rate stat, because…

7. Some rates can be compared (while maintaining meaningful units) differentially (i.e. subtracting the values for two players makes sense); others are ratio comparable (i.e. dividing the values for two players makes sense); some are neither differentially nor ratio comparable, and some are both. I prefer metrics that can be compared either way, but RAA/PA is only differentially comparable. FanHome poster Sibelius developed an adjustment called R+/PA, that depending on how you look at either adds the league average R/PA to RAA/PA, or makes an adjustment to absolute runs created before dividing by PA, that allows ratio comparisons for the rate stat.

8. wOBA, which is now in wide use thanks to its popularization by Tom Tango and Fangraphs, is a variant of the RAA/PA family as well, although it doesn’t maintain direct differential or ratio comparability.

9. Despite the issues with R/O as a rate stat for an individual, using it to calculate RAA will produce the same result for the RAA total as R+/PA, assuming that the inputs are consistently defined. R/O causes very minor distortion when used to compare normal players, and would cause much distortion with extreme players, but remains a useful shortcut rate stat. There are many worse choices one could make in devising an individual rate stat than using R/O. R/O remains the correct rate stat for a team; the RAA/PA family of metrics is inappropriate for the same reason R/PA is inappropriate for a team, in addition to some issues that would arise if attempting to define terms like “R+” for a team, as their actual runs scored or estimated runs created is already based on the number of plate appearances that they actually generated.

10. One can argue that batters also make tertiary contributions to their team through their impact on the run values of all of their teammate’s actions. The impact is very small for most hitters, dwarfed by their primary and secondary contributions, and if attempting to quantify them one must be careful to ensure that it’s not just measurement error. Attempting to capture these impacts lends itself to use of a theoretical team approach, which uses a dynamic run estimator to model how a batter’s impact on a team.

11. The theoretical team approach gives rise to a rate stat that David Smyth called R+/O+, which is expressed on a R/O scale but produces the same RAA given the same inputs. It can be applied to the linear weights framework as well, and offers an option if one prefers to express results on the R/O scale rather than R/PA, and thus have the same scale for the individual and team rate stat.

12. If you wish to compare rates across run environments, differentials between the individual and the league usually aren’t sufficient as higher run environments make equal differences less valuable in terms of wins. If you assume a fixed Pythagorean exponent for your win conversion, the case can be made that ratios do capture the win difference, but as soon as you introduce a run environment-dependent Pythagorean exponent that better models reality, this assumption fails. It is also necessary to consider that simply comparing the individual to the league average may not properly capture the dynamic of how the individual’s run contribution contributes to his team’s wins. There is also a potential complication from how differences in league PA/G impact rates denominated in PA. All of this is to say that there is no simple solution to converting run rate stats to their win-equivalents, and care should be taken in doing so, especially considering that the impact may be relatively small for many cases.

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