Rate Stat Series, pt. 16: Summary

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.

Rate Stat Series, pt. 15: Mixing Frameworks

Thus far I have employed what I’ve described as a “puritanical” approach to matching run estimator, denominator, and win conversion for each of the three frameworks for evaluating an individual’s offensive performance. While I think my logic for this approach is sound, I do not think it is necessarily wrong to mix components in a different manner than I’ve described. This will be a brief discussion of which of these potential hybrids make more sense than others, and a few issues to keep in mind if you choose to do so.

For the player as a team framework, there are many places in the process in which a batter’s value (at least to the extent that we can define and model it) as a member of a real team is distorted. At the very beginning of the process, a dynamic run estimator is applied directly to individual statistics. This creates distortion. Then the rate stat is runs/out; this doesn’t create a tremendous amount of distortion, as runs/out is a defensible if not perfect choice for an individual rate stat even if you don’t use the player as a team framework. Then if we convert to win value, we create a tremendous amount of distortion by essentially multiplying the run distortion by nine – instead of just mis-estimating an individual’s run contribution, we compound the problem by assuming that the entire team hitting like him would shape the run environment to be something radically different from the league average.

This is an opportune juncture to make a point that I should have made earlier in the series – for league average hitters, the distortion will be very small, and the differences across all of the various methodologies we’ve discussed will be lesser. I have focused in this series on a small group of hitters, basically the five most productive and five least productive in each league-season. These are of course the hitters most impacted by different methodological decisions. A league average hitter would by definition have Base Runs = LW_RC = TT_BsR (at least as we’ve linked these formulas in this series), would by definition have 0 RAA or 0 WAA, would by definition have a 100 relative rate, regardless of which approach you take. This series, and more generally my sabermetric interests, tend to focus on the extreme cases, and evaluating methods with an eye to applicability to a wide range of performance levels. Additionally, extreme players are the ones people in general tend to care the most about – you can imagine people debating who had the better offensive season, Frank Robinson in 1966 or Frank Thomas in 1994. I cannot imagine many people doing the same for Earl Battey and Gary Gaetti.

While we could avoid compounding the issues in the player as a team approach, but why bother? The framework is inferior to linear weights or theoretical team in every possible way, except one could argue ease of calculation gives it an advantage over TT. The only value to be had in evaluating a player as a team is a theoretical exercise, and if you lose commitment to that theoretical exercise it ceases to have any value at all.

For the theoretical team framework, you could just calculate TT_BsR, and then treat it in the same way as linear weights, or calculate TT_BsRP and treat it in the same way as R+. However, it would be pretty silly to go through the additional effort needed to calculate the TT run estimates instead of their linear weight analogues only to use them in the same way. Unlike in the case of player as team, there is no argument to be made here that you would be making the results more reasonable by doing so, as the TT estimates can be combined with team-appropriate rate stat denominators and team-appropriate conversions from runs to wins. Here the result of mixing frameworks would be extra work coupled with less pure results.

The only mixture of frameworks that makes sense, then, is to mix the run estimation components of the linear weights framework with the rate stat and win conversion components of the theoretical team framework. A logical path that might defend this approach would be: It is questionable whether the valuation of tertiary offensive contribution claimed by theoretical team approach are accurate or material. Thus our best estimate of a player’s run contribution to a theoretical team remains his linear weights RC or RAA. When it comes to win estimation, we are on much firmer ground in understanding how team runs and runs allowed translate to wins than we are in measuring individual contributions to team runs. We shouldn’t refuse to use this knowledge in the name of methodological consistency, but rather we should use the best possible estimates for each component of the framework. That means using the full Pythagenpat approach coupled with linear weights run estimates.

Convinced? I’m not, but let me walk through an example of how we could apply this hybrid approach to the Franks. We can start with our wRC estimate, which we will now use in place of TT_BsRP as “R_+” going forward for this hybrid linear-theoretical team framework. Then we can use any of our TT rate stats  - I’ll show R+/O+ rather than R+/PA+ here, as I think it’s the former that might serve to make this hybrid framework an attractive option. R+/O+ allows us to express the individual and team rate on the same basis and better yet, doing so while using the most fundamental of rate stats (R/O) as that basis.

O+ remains equal to PA*(1 – LgOBA), and these relative R+/O+ figures are the same as our relative R+/PA, which makes sense – the numerators are the same, and the only difference in the denominators is multiplying PA by (1 – LgOBA). So an alternative way of expressing the relationship is:

R+/O+ = (R+/PA)/(1 – LgOBA)

I will skip some steps here, since they were all covered in the last installment – no need to convert this to a W% and then convert back to a relative adjusted R+/O+.  

TT_R/O = (R+/O+)*(1/9) + LgR/O*(8/9)

TT_x= ((TT_R/O)*LgO/G + LgR/G)^.29

TT_WR = ((TT_R/O)/Lg(R/O))^TT_x

RelAdj R+/O+ = (TT_WR^(1/r) - 1)*9 + 1

It might be helpful to take a step back and look at our results for the relative adjusted metric (whether R+/PA or R+/O+) for each of the four options we’ve considered, which are:

A: linear run estimate and fixed RPG based on league average (final rate based on R+/PA)

B: linear run estimate and dynamic RPG based on player’s impact on team (final rate based on R+/PA)

C: linear run estimate and Pythagenpat theoretical team win estimate (hybrid approach discussed in this installment; final rate based on R+/O+)

D: theoretical team run estimate (full theoretical team approach; final rate based on R+/O+)

I included a third row which shows the percentage by which Thomas’ figure exceeded Robinson’s. The TT (D) approach maximizes each player’s value and also the difference between them. The hybrid approach (C) falls in between the pure linear approach (A) and the TT approach (D). One thing that I did not fully expect to see is that the linear approach that varies RPW based on the estimated RPG of a team with the player added (B) produces a lower estimate than any of the other approaches, and is the outlier of the bunch. I didn’t make enough of this in part 13, but we are actually better off assuming that the individual has no impact on RPW than adjusting RPW based on his own impact on the theoretical team’s RPG.

Note that when we translate to a reference league, we are fixing each theoretical team's RPG to the same level. In reality, the Frank Thomas TT will have a higher RPG than the Matt Walbeck TT for any given reference environment. However, this is not an issue, because the runs value we're reporting is not real. It is intended to be an equivalent run value that reflects the player's win contribution for a common frame of reference. When we use the full Pythagenpat approach, the theoretical team's winning percentage is preserved, so the runs are now an abstract quantity and do not need to tie out to any actual number of runs in this environment. In this sense, when we convert TT to a win-equivalent rate, we're doing something similar to what I said the linear weights framework was doing - we're making the run environments equal after adding the player. The difference is that in this case we are capturing the disparate impact of the players on team wins first, then restating a win ratio as an equivalent run ratio given the assumption of a stable run environment. Thus the runs are an abstraction, but the value they represent is preserved.

When we do the same with a dynamic RPW approach (i.e. when we use the Palmerian approach of adding the batter's RAA/G to the RPG and then calculating RPW), we run into difficulties because while we have fixed a WAA total, and can then translate that WAA to an equivalent for the reference environment, we have not taken into account that the batter would need to contribute more runs to actually produce the same number of wins. This is not a problem for the full Pythagenpat approach because we used a run ratio that modified the batter's abstract runs in concert with the run environment. 

Now there is a way we could address this, but it results in a mess of an equation that I'm not sure has an algebraic solution (whether it does or not, I have no interest in trying to solve it). Basically, we can solve for the RAA that would be needed to preserve the batter's original WAA in the reference environment. To do this, we need to remember to apply the reference PA adjustment. For this application, I think it’s easiest to apply directly to the batter’s original WAA, so WAA*Ref(PA/G)/Lg(PA/G):

We could then say that adjWAA = X/(2*(refRPG + X/G)^.71) where X = the batter's RAA and G could be the batter's actual game or some kind of PA-equivalent games as long as we’re consistent with what was used in the original WAA calculation. I used the Goal Seek function in Excel to solve the following equations for the Franks:

for Robinson: 8.552 = X/(2*(8.83 + X/155)^.71), X = 85.555

for Thomas: 7.217 = X/(2*(8.83 + X/113)^.71), X = 71.176

With these new estimates of RAA, we can get relative adjusted R+/PA by first dividing by each batter’s PA, then adding in the reference R/PA, and then dividing by the reference R/PA:

This approach allows us to more correctly calculate a win-equivalent rate stat for the linear weights framework when allowing RPW to vary based on the batter’s impact on his team’s run environment. However, given that I don't know how to solve for x algebraically, and my pre-existing philosophical issues with using this approach with the linear weights framework to begin with, I think that at this point there are two better choices:

1. If you want to maintain methodological purity, stick with the fixed RPW for the linear weights framework like I advocate

2. Use the hybrid framework, which embraces the "real" run to win conversion and actually makes this math easier

With "B" out of the picture, what we see for A, C, and D make sense in relation to each other. C produces a lower final relativity for great hitters like the Franks because it recognizes that when they inflate their team's run environments, RPW increases. D is higher than both because the initial run estimate is higher, due to using TT_BsR rather than LW, and thus giving each batter credit for their estimated tertiary contributions. And our revised linear weight approach falls somewhere in between.

Of course, it’s possible that I’m wrong about this, and the hybrid or theoretical team approaches that make use of Pythagenpat are overstating the impact of the player on the runs to win conversion. I can’t prove that I am right about this, but I would offer two rejoinders:

1. The full Pythagenpat approach produces the same W% for the theoretical team across different environments by definition, and that is the most important number at the end of the lined if we are trying to build a win-equivalent rate stat.

2. We should expect the results from the linear and the Pythagenpat approach to be similar (which they are after making proper adjustments), as our RPW formula is consistent with Pythagenpat for .500 teams. While the relationship between the RPW formula and Pythagenpat will fray as we insert more extreme teams, the theoretical team approach doesn’t produce any extremes for even great hitters like the Franks. For example, if we use the Palmer approach, Thomas’ 77.8 RAA in 113 games takes an average 1994 AL team that scores 5.23 R/G up to 5.92 R/G, which would rank second in the league, even with the Yankees.  This is not an extreme team performance when it comes to applying the win estimation formulas. If we use Pythagenpat, such a team would be expected to have a .5623 W% using the RPW formula and a .5620 W% using Pythagenpat. So we shouldn’t expect the results to diverge too much when applying the approaches to derive win-equivalent rate stats.

And with that, we have come to the end of everything that I wanted to say about rate stats. I will close out the series with one final installment that attempts to briefly summarize my main points with limited math.

Rate Stat Series, pt. 14: Relativity for the Theoretical Team Framework

Before jumping into win-equivalent rate stats for the theoretical team framework, I think it would be helpful to re-do our theoretical team calculations on a purely rate basis. This is, after all, a rate stat series. In discussing the TT framework in pts. 9-11, I started by using the player’s PA to define the PA of the team, as Bill James chose to do with his TT Runs Created. This allowed our initial estimate of runs created or RAA to remain grounded in the player’s actual season. 

An alternative (and as we will see, equivalent) approach would be to eschew all of the “8*PA” and just express everything in rates to begin with. When originally discussing TT, I didn’t show it that way, but maybe I should have. I found that my own thinking when trying to figure out the win equivalent TT rates was greatly aided by walking through this process first.

Again, everything is equivalent to what we did before – if you just divide a lot of those equations by PA, you will get to the same place a lot quicker than I’m going to. The theoretical team framework we’re working with assumes that the batter gets 1/9 of the PA for the theoretical team. It’s also mathematically true that for Base Runs:

BsR/PA = (A*B/(B+C) + D)/PA = (A/PA)*(B/PA)/(B/PA + C/PA) + D/PA

If for the sake of writing formulas we rename A/PA as ROBA (Runners On Base Average), B/PA as AF (Advancement Factor; I’ve been using this abbreviation long before it came into mainstream usage in other contexts), C/PA as OA (Out Average), and D/PA as HRPA (Home Runs/PA), we can then write:

BsR/PA = ROBA*AF/(AF + OA) + HRPA

Since it is also true that R/O = R/PA/(1 – OBA), in this case it is true that:

BsR/O = (BsR/PA)/OA

We can use these equations to calculate the Base Runs per out for a theoretical team (I’m going to skip over “reference team” notation and just assume that the reference team is a league average team):

TT_ROBA = 1/9*ROBA + 8/9*LgROBA

TT_AF = 1/9*AF + 8/9*LgAF

TT_OA = 1/9*OA + 8/9*LgOA

TT_HRPA = 1/9*HRPA + 8/9*LgHRPA

TT_BsR/PA = TT_ROBA*TT_AF/(TT_AF + TT_OA) + TT_HRPA

TT_BsR/O = (TT_ROBA*TT_AF/(TT_AF + TT_OA) + TT_HRPA)/TT_OA

Here’s a sample calculation for 1994 Frank Thomas:

To calculate a win-equivalent rate stat, we can use the TT_BsR/O figure as a starting point (it suggests that a theoretical team of 1/9 Thomas and 8/9 league average would score .2344 runs/out). We don’t need to go through this additional calculation, though; when we calculated R+/O+ (or R+/PA+, RAA+/O+, or RAA+/PA+), we already had everything we needed for this calculation.

You will see if you do the math that:

TT_BsR/O = (RAA+/O+)*(1/9) + LgR/O

or

TT_BsR/O = (R+/O+)*(1/9) + (LgR/O)*(8/9)

or

TT_BsR/O = (RAA+/PA+)/(1 – LgOBA)*(1/9) + LgR/O

or 

TT_BsR/O = (R+/PA+)/(1 - LgOBA)*(1/9) + (LgR/O)*(8/9)

You could view this as a validation of the R+/O+ approach, as it does what it set out to do, which is to isolate the batter’s contribution to the theoretical team’s runs/out. Once we’ve established the team’s runs/out, it is pretty simple to convert to wins. I will just give formulas as I think they are pretty self-explanatory:

TT_BsR/G = TT_BsR/O*LgO/G

TT_RPG = TT_BsR/G + LgR/G

TT_x = TT_RPG^.29

TT_W% = (TT_BsR/G)^TT_x/((TT_BsR/G)^TT_x + (LgR/G)^TT_x)

Walking through this for the Franks, we have:

One thing to note here is that if we look at the theoretical team’s R/O (or R/G) relative to the league average, subtract one, multiply by nine, and add one back in, we will Thomas and Robinson’s relative R+/O+. This is not a surprising result given what we saw above regarding the relationship between R+/O+ and theoretical team R/O.

We now have a W% for the theoretical team, which we could leave alone as a rate stat, but it’s not very satisfying to me to have an individual rate stat expressed as a team W%. If we subtract .5, we have WAA/Team G; we could interpret this as meaning that Thomas is estimated to add .0609 wins per game and Robinson .0546 to a theoretical team on which they get 1/9 of PA. Another option would be to convert this WAA back to a total, defining “games” as PA/Lg(PA/G), and then we could have WAA+/PA+ or WAA+/O+ as rates. 

In keeping with the general format established in this series, though, my final answer for a win-equivalent rate stat for the TT framework will be to convert the winning percentage (actually, we’ll use win ratio since it  makes the math easier) back to the reference environment, and calculate a relative adjusted R+/O+. Since everything will be on an outs basis (as we’re using O+), we don’t need to worry about league PA/G when calculating our relative adjusted R+/O+.

Instead of calculating TT_W%, we could have left it in the form of team win ratio:

TT_WR = ((TT_BsR/G)/(LgR/G))^(TT_x)

We can convert this back to an equivalent run ratio in the reference environment (which for this series we’ve defined as having Pythagorean exponent r = 1.881) by solving for AdjTT_RR in the equation:

TT_WR = AdjTT_RR^r

so 

AdjTT_RR = TT_WR^(1/r)

We could convert this run ratio back to a team runs/game in the reference environment, and then to a team runs/out, and then use our equation for tying individual R+/O+ to theoretical team R/O to get an equivalent R+/O+ ratio. But why bother with all that, when we will just end up dividing it by the reference environment R/O to get our relative adjusted R+/O+? I noted above that there was a direct relationship between the theoretical team’s run ratio (which is equal to the theoretical team’s R/O divided by league R/O) and the batter’s relative R+/O+:

Rel R+/O+ = (TT_RR – 1)*9 + 1

So our Relative Adjusted R+/O+ can be calculated as:

RelAdj R+/O+ = (AdjTT_RR  - 1)*9 + 1

I brought back our original relative R+/O+ (prior to going through the win-equivalent math) for comparison. Thomas gains slightly and Robinson loses more, because the value of his relative runs is lower in a high scoring environment. This is a similar conclusion to what we saw when comparing relative R+/PA and the relative adjusted R+/PA for Robinson and Thomas. Nominal runs are more valuable when the run scoring environment is lower, because it takes fewer marginal runs to create a marginal win. Relative runs are more valuable when the scoring environment is higher, because the win ratio expected to result from a given run ratio increases due to the higher Pythagenpat exponent.

At this point, we have exhausted my thoughts and ideas concerning the theoretical issues in designing individual batter rate stats. Next time I will discuss mixing up our rate stats and the frameworks within which I assert each should ideally be used.

Rate Stat Series, pt. 13: Relativity for the Linear Weights Framework

Of the three frameworks for evaluating individual offense, linear weights offers the simplest calculation of runs created or RAA, but will be the hardest to convert to a win-equivalent rate – mentally if not computationally. In order to do this, we need to consider what our metrics actually represent and make our choices accordingly. The path that I am going to suggest is not inevitable – it makes sense to me, but there are certainly valid alternative paths.

In attempting to measure the win value of a batter’s performance in the linear weights framework, we could construct a theoretical team and measure his win impact on it. In so doing, one could argue that the batter’s tertiary impact (which would be ignored under such an approach) is immaterial, perhaps even illusory, and that the process of converting runs to wins is independent from the development of the run estimate. Thus we could use a static approach for estimating runs and a dynamic team approach for converting those runs to wins.

I would argue in turn that the most consistent approach is to continue to operate under the assumption that linear weights represents a batter’s impact on a team that is average once he is added to it, and thus not allow any dynamism in the runs to wins conversion. Since under this school of thought all teams are equal, whether we add Frank Thomas or Matt Walbeck, there is no need to account for how those players change the run environment and the run/win conversion – because they both ultimately operate in the same run environment.

One could argue that I am taking a puritanical viewpoint, and that this would become especially clear in a case in which one compared the final result of the linear weights framework to the final result of the theoretical team framework. As we’ve seen, RAA is very similar between the two approaches, but the run/win conversions will diverge more if in one case we ignore the batter’s impact on the run environment. In any event, the methodology we’ll use for the theoretical team framework will be applicable to linear weights as well, if you desire to use it.

Since we will not be modeling any dynamic impact of the batter upon the team’s run environment, it is an easy choice to start with RAA and convert it to wins above average (WAA) by dividing by a runs per win (RPW) value. An example of this is the rule of thumb that 10 runs = 1 win, so 50 RAA would be worth 5 WAA. 

There are any number of methods by which we could calculate RPW, and a couple philosophical paths to doing so. On the latter, I’m assuming that we want our RPW to be represent the best estimate of the number of marginal runs it would take for a .500 team (or more precisely a team with R = RA) to earn a marginal win. Since I’ve presumed that Pythagenpat is the correct run to win conversion, the most consistent is to use the RPW implied by Pythagenpat, which is:

RPW = 2*RPG^(1 – z) where z is the Pythagenpat exponent

so when z = .29, RPW = 2*RPG^.71

For the 1966 AL, this produces 8.588 RPW and for the 1994 AL it is 10.584. So we can calculate LW_WAA = LW_RAA/RPW, and LW_WAA/PA seems like the natural choice for a rate stat:

This tightens the gap between Robinson and Thomas as compared to a RAA/PA comparison, and since we’ve converted to wins, we can look at WAA/PA without having to worry about the underlying contextual differences (note: this is actually not true, but I’m going to pretend like it is for a little bit for the sake of the flow of this discussion). 

There is another step we could take, which is to recognize that the Franks do influence the context in which their wins are earned, driving up their team’s RPGs and thus RPWs and thus their own WAAs. Again, I would contend that a theoretically pure linear weights framework assumes that the team is average after the player is added. Others would contend that by making that assertion I’m elevating individual tertiary offensive contributions to a completely unwarranted level of importance, ignoring a measurable effect of individual contribution because the methodology ignores an immaterial one. This is a perfectly fair critique, and so I will also show how we can adjust for the hitter’s impact on the team RPW in this step. Pete Palmer makes this adjustment as part of converting from Batting Runs (which is what I’m calling LW_RAA) to Batting Wins (what I’m calling LW_WAA), and far be it from me to argue too vociferously against Pete Palmer when it comes to a linear weights framework.

What Palmer would have you do next (conceptually as he uses a different RPW methodology) is take the batter’s RAA, divide by his games played, and add to RPG to get the RPG for an average team with the player in question added. It’s that simple because RPG already represents average runs scored per game by both teams and RAA already captures a batter’s primary and secondary contributions to his team’s offense. One benefit or drawback of this approach, depending on one’s perspective, is that unlike the theoretical team approach it is tethered to the player’s actual plate appearances/games. Using the theoretical team approach from this series, a batter always gets 1/9 of team PA. Under this approach, a batter’s real world team PA, place in the batting order, frequency of being removed from the game, etc. will have a slight impact on our estimate of his impact on an average team. We could also eschew using real games played, and instead use something like “team PA game equivalents”. For example, in the 1994 AL the average team had 38.354 PA/G; Thomas, with 508 PA, had the equivalent of 119.21 games for an average hitter getting 1/9 of an average team’s PA (508/38.354*9). I’ve used real games played, as Palmer did, in the examples that follow.

Applying the Palmerian approach to our RPW equation:

TmRPG = LgRPG + RAA/G

TmRPW = 2*TmRPG^.71

LW_WAA = LW_RAA/TmRPW

For the Franks, we get:

The difference between Thomas and Robinson didn’t change much, but both lose WAA and WAA/PA due to their effect on the team’s run environment as each run is less valuable as more are scored. 

I have used WAA/PA as a win-equivalent rate without providing any justification for doing so. In fact, there is very good theoretical reason for not doing so. One of the key underpinnings of all of our rate stats is that plate appearances are not fixed across contexts – they are a function of team OBA. Wins are fixed across contexts – always exactly one per game. Thus when we compare Robinson and Thomas, it’s not enough to simply look at WAA/PA; we also need to adjust for the league PA/G difference or else the denominator of our win-equivalent rate stat will distort the relativity we have so painstakingly tried to measure.

In the 1966 AL, teams averaged 36.606 PA/G; in 1994, it was 38.354. Imagine that we had two hitters from these leagues with identical WAA and PA. We don’t have to imagine it; in 1966 Norm Siebern had .847 WAA in 399 PA, and in 1994 Felix Jose had .845 WAA in 401 PA (I’m using Palmer-style WAA in this example). It seems that Siebern had a minuscule advantage over Jose. But while wins are fixed across contexts (one per game, regardless of the time and place), plate appearances are not. A batter using 401 PA in 1994 was taking a smaller share of the average PA than one taking 399 in 1966 (you might be yelling about the difference in total games played between the two leagues due to the strike, but remember that WAA already has taken into account the performance of an average player – whether over a 113 or 162 game team-season is irrelevant when comparing their WAA figures). In 1994, 401 PA represented 10.46 team games worth of PA; in 1966, 399 represented 10.90 worth. In fact, Siebern’s WAA rate was not higher than Jose’s; despite having two fewer PA, Siebern took a larger share of his team’s PA to contribute his .85 WAA than Jose did.

If we do not make a correction of this type and just use WAA/PA, we will be suggesting that the hitters of 1966 were more productive on a win-equivalent rate basis than the hitters of 1994 (although this is difficult to prove as by definition the average player’s WAA/PA will be 0, regardless of the environment in which they played). I don’t want to get bogged down in this discussion too much, so I will point you here for a discussion focused just on this aspect of comparing across league-seasons.

There are a number of different ways you could adjust for this; the “team games of PA” approach I used would be one. The approach I will use is to pick a reference PA/G, similar to our reference Pythagenpat exponent from the last installment, and force everyone to this scale. For all seasons 1961-2019, the average PA/G is 37.359 which I will define as refPA/G. The average R/G is 4.415, so the average RPG is 8.83 and the refRPW is 9.390.

If we calculate:

adjWAA/PA = WAA/PA * Lg(PA/G)/ref(PA/G)

Then we will have restated a hitter’s WAA rate in the reference environment. This is an option as our final linear weight win stat:

This increases Thomas’ edge over Robinson, while giving Jose a miniscule lead over Siebern. As a final rate stat, I find it a little unsatisfying for a couple of reasons:

1. while the ultimate objective of an offense is to contribute to wins, runs feels like a more appropriate unit

2. related to #1, wins compress the scale between hitters. There’s nothing wrong with this to the extent that it forces us to recognize that small differences between estimates fall squarely within the margin of error inherent to the exercise, but it makes quoting and understanding the figures more of a challenge.

3. WAA/PA, adjusted for PA context or not, is only differentially comparable; ideally we’d like to have a comparable ratio

My solution to this is to first convert adjusted WAA/PA to an adjusted RAA/PA, which takes care of objections #1 and 2, then to convert it to an adjusted R+/PA, which takes care of objection #3. At each stage we have a perfectly valid rate stat; it’s simply a matter of preference. 

To do this seems simple (let’s not get too attached to this approach, which we’ll revisit in a future post):

adjRAA/PA = adjWAA/PA*refRPW (remember, by adjusting WAA/PA using the refPA/G, we’ve restated everything in the terms of the reference league)

adjR+/PA = adjRAA/PA + ref(R/PA)  (reference R/PA is .1182, which can be obtained by dividing the ref R/G by the ref PA/G)

We can also compute a relative adjusted R+/PA:

reladjR+/PA = (adjR+/PA)/(ref(R/PA))

= ((RAA/PA)/TmRPW * Lg(PA/G)/ref(PA/G) * refRPW + Ref(R/PA))/Ref(R/PA)

= (RAA/PA)/TmRPW * Lg(PA/G)/ref(PA/G) * refRPW/ref(R/PA) + 1

I included raw R+/PA and its ratio the league average (relative R+/PA) to compare to this final relative adjusted R+/PA. For three of the players, the differences are small; it is only Frank Robinson whose standing is significantly diminished. This may seem counterintuitive, but remember that the more ordinary hitters have much smaller impact on RPW than the Franks. Relative to Thomas, Robinson gets more of a boost from his lower RPW (his team RPW was 19% lower than Thomas) than Thomas does from the PA adjustment (the 1994 AL had 4.8% more PA/G than the 1966 AL).

We could also return to the puritan approach (which I actually stubbornly favor for the linear weights framework) and make these adjustments as well. The equations are the same as above except where we use TmRPW, we will instead use LgRPW – reverting to assuming that the batter has no impact on the run/win conversion.

Here the impact on the Franks is similar; both are hurt of course when we consider their impact on TmRPW. Next time, we will quit messing around with half-measures – no more mixing linear run contributions with dynamic run/win converters. We’re going full theoretical team.

Rate Stat Series, pt. 12: Relativity for the Player as a Team Framework

I have now covered all of the ground I wished to cover regarding the construction of rate stats for various frameworks for evaluating individual offense, so I think this would be a good point to take stock of the conclusions I have drawn. From here on out, I will write about these tenets as if they are inviolable, which is not actually the case (“sound” and “well-reasoned to me” are as far as I’m willing to go), but I’m moving on to other topics:

* For team offense, Runs/Out is the only proper rate stat

* If evaluating individuals as teams (i.e. applying a dynamic run estimator like Runs Created or Base Runs directly to an individual’s statistics), Runs/Out is also the proper rate stat. Any other choice breaks consistency with treating the individual as a team, and consistency is the only thing going for such an approach. On a related note, don’t treat individuals as teams.

* If using a linear weights framework, the proper rate stat is RAA/PA, R+/PA, or restatement of those (wOBA is the one commonly used). These metrics all produce consistent rank orders and can be easily restated from one another. Any of them are valid choices at the user’s discretion, with the only objective distinction being whether you want ratio and differential comparability (R+/PA), only care about differential comparability (R+/PA or RAA/PA), or would prefer a different scale altogether at the sacrifice of direct comparisons without futher modifications (wOBA).

* If using a theoretical team framework, R+/O+ and R+/PA+ are equivalent, with the user needing to decide whether they want to think about individual performance in terms of R/O or R/PA.

Throughout this series, I have not worried about context, and originally envisioned a final installment that would briefly discuss some of the issues with comparing player’s rates across league-seasons. In putting it together, I decided that it will take a few installments to do this properly. It will also take another league-season to pair with the 1994 AL in order to make cross-context comparisons.

I have chosen the 1966 AL to fill this role. In the expansion era (1961 – 2019), the AL has averaged 4.52 R/G. 1994 was third-highest at 5.23, 15.6% higher than average. The closest league to being an inverse relative to the average is the 1966 AL (3.89 R/G, 13.8% lower than average). The obvious choice would have been 1968, since it was the most extreme, but as 1994 was not the most extreme I thought a league that was similarly low scoring would be more appropriate.

The other reason I like the 1966 AL for his purpose is that, just as was the case in 1994, the superior offensive player in the league was a future Hall of Fame slugger named Frank. In making comparisons across the twenty-eight year and (more importantly) 1.34 R/G differences between these two league-seasons, the two Franks will serve as our primary reference points.

To look at the 1966 AL we will first need to define our runs created formulas. I will not repeat the explanations of how these are calculated – I used the exact same approach as for the 1994 AL, which are demonstrated primarily in the parts 1, 9, and 10.

Base Runs:

A = H + W – HR = S + D + T + W

B = (2TB - H – 4HR + .05W)*.79920 = .7992S + 2.3976D + 3.996T + 2.3976HR + .0400W

C = AB – H = Outs

D = HR

BsR = (A*B)/(B + C) + D

Linear Weights:

LW_RC = .4560S + .7751D + 1.0942T + 1.4787HR + .3044W - .0841(outs)

LW_RAA = .4560S + .7751D + 1.0942T + 1.4787HR + .3044W - .2369(outs)

wOBA = (.8888S + 1.2981D + 1.7075T + 2.2006HR + .6944W)/PA

Theoretical Team:

TT_BsR = (A + 2.246PA)*(B + 2.346PA)/(B + C + 7.915PA) + D - .6658PA

TT_BsRP = ((A + 2.246PA)*(B + 2.346PA)/(B + C + 7.915PA) + D + .185PA)*PAR – .8519PA

TT_RAA = ((A + 2.246PA)*(B + 2.346PA)/(B + C + 7.915PA) + D + .185PA)*PAR – .9572PA

Using these equations, let’s look at the leaderboards for the key stats for each framework. First, for the player as a team:

Of course, we’ll get slightly different results from the different frameworks, but two things should be obvious: Robinson was the best offensive player in the league (his lead over Mantle in second place is bigger than the gap from Mantle to ninth-place Curt Blefary, and he was also among the leaders in PA), and the difference in league offensive levels carried through to individuals.

Next for the linear weights framework:

And finally for the theoretical team framework:

Now comes the hard part – how do we compare these performances from the 1966 AL against those from the 1994 AL? It should be implied, but in all comparisons that follow, I am only concerned about the value of a given player’s contributions relative to the context in which he played; this is not about whether/to what extent the overall quality of play differed between 1966 and 1994, or about how the existence of pitcher hitting in 1966 but not in 1994 impacted the league average, etc. I also am not going all the way in accounting for context, as I’ve still done nothing to park-adjust. Park adjustments are important, but whether you adjust or not is irrelevant to the question of which rate stat you should use. I’m also not interested in how “impressive” a given rate is due to the variation between players (e.g. z-scores) – I’m simply trying to quantify the win value of the runs a player has contributed.

The obvious first step to compare players across two league-seasons is to compare the difference or ratio of their rate to the league rate. As we’ve discussed throughout this series, some stats can be compared using both differences and ratios, but others can only be compared (at least meaningfully) with one or the others, and others still can be compared but the ratio or difference no longer has any baseball meaning unless some transformation is carried out (wOBA is the most prominent example we’ve touched on). 

The table below shows the rates for our two Franks, the respective league rate, and the difference and ratio (if applicable) for each of the key rate stats we’ve looked at:

One of the other reasons I was drawn to the pairing of these two league-seasons is how close Thomas and Robinson are in the key metrics when compared to the league using a ratio. Thomas has a slight advantage in each metric, except for BsR/O, where the flaw of treating an individual as a team can be seen. Playing in a high offense context, Thomas’ crazy rates (to put it in simple terms related to but not the run unit stats used in this series, Thomas hit .353/.492/.729 to Robinson’s .316/.406/.637) result in a large estimated tertiary contribution when treating him as his own team. Expressing these metrics as ratios hammers home that even for extreme performers (defining “extreme” within the usual confines of a major league season), there isn’t much difference between using the linear weights and theoretical team frameworks, and both are reasonable choices of framework. Treating an individual as a team, not so much.

We’ve expressed each player’s rate relative to the league average, but we haven’t answered the question of which construct is appropriate – the difference or the ratio, or does the answer differ depending on the metric? And how can we make that determination? In lieu of some guiding principle, it is just a matter of personal preference. Most people are drawn to ratios, and the widespread adoption of metrics like ERA+, ERA-, OPS+, and wRC+ speak to that preference. This case illustrates one of the reasons why ratios make sense intuitively – while the Franks are very close when comparing ratios, Thomas has a healthy lead when comparing differences as the league average R/PA and R/O for 1994 are much higher than the same for 1966. It seems intuitive that a batter will be able to exceed the league average by a larger difference when it is .2419 rather than .1528.

Still, in order to really understand how to compare players across contexts, and ensure that our intuition is grounded in reality, we need to think about how runs relate to wins. All of the metrics in that table which I consider the key metrics discussed in this series have been expressed in runs, and this is a natural starting point as the goal of an offense is to score runs. Of course the ultimate reason why an offense wants to score runs is so that they might contribute to wins.  

I will break one of the rules I’ve tried to adhere to by begging the question and asserting that Pythagenpat and associated constructs are the correct way to convert runs to wins. Of course it is just a model, and while it is a good one, it is not perfect, but I think it is the best choice given its accuracy and its seemingly reasonable results for extreme situations. 

Using Pythagorean also lends itself to adoption of a ratio rate stat, as one way to express the Pythagorean Theorem is that the expected ratio of wins to losses is equal to the ratio of runs to runs allowed raised to some power. If we start by thinking about the team/player as team framework for evaluating offense, it is natural to construct a win-unit rate stat by treating the team’s R/O as the numerator and the league average as the denominator. The ratio of these two, raised to some power, is then the equivalent win ratio that results. This is exactly the approach that Bill James took in his early Baseball Abstracts.

We can easily use this relationship to demonstrate that a simple difference of R/O does not capture the differences in win values between players. If one player exceeds the league average R/O by 10% and another 50%, then assuming a Pythagorean exponent of 2, player A will have a “win ratio” of 1.21 and player B will have a win ratio of 2.25. Even if we convert those to winning percentages (.5475 and .6923), the difference between the two player’s R/O does not capture the win value difference.

The ratio doesn’t either, of course, since it would need to be squared, but if we assume a constant Pythagorean exponent like 2, this is simply a matter of scaling, with no impact on the rank order to players. However, if we use a custom Pythagorean exponent, this assumption breaks down, as we can illustrate by comparing the two Franks. Since we’re starting with the assumption that Pythagenpat is correct, this means that we will always need to make some kind of adjustment in order to convert our run rate into an equivalent win rate.

Since we are treating the players as teams, the consistent approach is to first calculate the RPG for each player as a team, then calculate the Pythagenpat exponent for the situation, then convert their relative BsR/O to an equivalent win value. The RPG for the 1966 AL was 3.893 with 25.482 O/G, while in the 1994 AL it was 5.226 with 25.188 O/G.

T_RPG = LgR/G + BsR/O*LgO/G

x = T_RPG^.29 (for a Pythagenpat z value of .29)

Relative BsR/O = (BsR/O)/(LgBsR/O)

Win Ratio = (Relative BsR/O)^x

Offensive Winning Percentage = Win Ratio/(Win Ratio + 1) = (BsR/O)^x/((BsR/O)^x + (LgBsR/O)^x)

Here, we conclude that just looking at Relative BsR/O understates Thomas’ superiority, as Pythagenpat estimates more wins for a given run ratio when the RPG increases (e.g. a run ratio of 8/4 will produce more wins than a run ratio of 7/3.5). 

If we were actually going to use this framework, we could leave our final relative rate stat in the form of a win ratio or OW%, but I would prefer to convert it back to a run rate. Even within the conceit of the player as a team framework, it’s hard to know what to do with an OW% (or the even less familiar win ratio). We can say that Thomas’ OW% of .903 means that a team that hit like Thomas and had league average defense (runs allowed) would be expected to have a .903 W%, but even if you follow the player as team framework, you would probably like to have a result that can be more easily tied back to the player’s performance as the member of a team, not what record the Yuma Mutant Clone Franks would have. One way to return the win ratio to a more familiar scale is to convert it back to a run ratio.

Let’s define “r” to be the Pythagenpat exponent for some reference context, which I will define to be 8.83 RPG – the major league average for the expansion era (1961 – 2019). We can then easily convert our estimated win ratios for the Franks to run ratios that would produce the same estimated win ratio in the reference (8.83 RPG) environment.

The calculation is simply (Win Ratio)^(1/r). Since r = 8.83^.29 = 1.881, this becomes Win Ratio^.5316, which produces 2.614 for Robinson and 3.282 for Thomas. Following the time-honored custom of dropping the decimal place, we wind up with a win-equivalent relative BsR/O of 261 for Robinson and 328 for Thomas, compared to their initial values of 217 and 260 respectively. These both increased because both players would radically alter their run environments, increasing the win value of their relative BsR/O. I would demonstrate the lesser impact on more typical performers if I cared about this framework beyond being thorough.

While I do think that this example demonstrates that just looking at a ratio of R/O does not appropriately capture the win impact between players, the individual OW% approach goes many steps too far, but it is the logical conclusion of the player as a team methodology. If you were on that train until we got to the end of the line, I’d encourage you to consider jumping off at one of the earlier stations (I’d prefer you get on the green or red linear weights or theoretical team lines rather than ever embarking on the player as team blue line). Next time, I’ll explore those paths to win-unit rate stats using those frameworks.

Rate Stat Series, pt. 11: Rate Stats for the Theoretical Team Framework II

Once PAR has been incorporated, it should be clear that a different approach will be needed as our run estimate already includes the batter’s secondary contribution – it is starting out on a similar basis to wRC.  We also can fall back on our necessary test for a rate stat – it must produce the same RAA as the RAA produced by the full implementation of the framework we are looking at.

To argue why this should be the gateway criteria in a slightly different way than I did previously, consider the place of RAA in the theoretical team framework. The theoretical team framework is an attempt to value the batter by estimating the difference between the runs scored for a team on which he accumulates an even share of the plate appearances, and the runs scored for a team on which he does not play. While we have constructed an absolute estimate of runs created using this approach, it inherently is screaming out for a marginal approach – taking the difference between the team with the player and the team without the player. That is exactly what RAA is supposed to represent – and by calculating TT_RAA, we have (to the best of our ability) captured the batter’s primary, secondary, and tertiary contributions. TT_RAA is the marginal impact of the player on a team, and any rate stat that starts by using TT_BsRP should produce the same RAA figure as TT_RAA.

So let’s look at the key figures for our hitters:

Here, “RAA” is what we calculated above, based on TT_BsR/O or TT_BsR+/PA. As you can see, it does not match TT_RAA. If it did, we wouldn’t need to worry about a special set of rate stats for the TT framework – we could just piggyback on the same methodology used for linear weights. It’s clear that we need to apply something different to TT_BsRP when we use the full-blown theoretical team approach including PAR. I would also argue that this is evidence that the theoretical team approach should not be viewed simply as an alternative path to using linear weights, but rather a third unique framework for evaluating a batter’s contribution.

I would now go through a tortured explanation of the logic behind finding a denominator that achieves the desired result, but there is no need – David Smyth developed the answer and explained it clearly and concisely in a June 21, 2001 FanHome post:

On the subject of a rate stat such as R+/PA or R+/O, etc....

The whole idea behind this method is to compute the impact of a player on a theoretical team. On the team level (even a theoretical one), impact is in terms of runs and outs. The R+ generated by the procedure on this thread [NOTE: R+ is theoretically equivalent to what I am calling TT_BsRP] tells us the difference in runs between the theoretical team without the indicated batter (that is, a team of 8 average hitters), and the theoretical team with the indicated batter added. We can also compute the difference in the OUTS between these two teams, and use that total as the rate stat denominator. Call it O+.

And it's easy to calculate. You simply multiply the batter PA by the out percentage (1-OBA) of the reference team...It seems to me that the preferred rate stat for the R+ framework is not R+/PA nor R+/O; it's R+/O+. 

Frank Thomas had 149.9 TT_BsRP in 508 PA. The reference team had a .3433 OBA (same as the league average). Thus, his TT_BsRP/O+ (a specific implementation of Smyth’s R+/O+) would be:

O+ = PA*(1 – LgOBA) = 508*(1 - .3433) = 333.6

TTBsRP/O+ = TT_BsRP/O+ = 149.9/333.6 = .4494

What does this represent? It is not easy to explain it in a sentence, but I will try: Frank Thomas contributed .4494 runs to the theoretical team’s offense for each of its outs distributed to him. When Smyth said that PA*(1 – RefOBA) was equal to the difference in the team’s outs, he was referring to the theoretical team construct, in which the difference in team plate appearances is the batter’s own PA, since in the left hand term (T_BsR) the team has 9*PA, and in the right hand term (R_BsR), the team has 8*PA. The difference in outs is this difference (1*PA) times the team’s out rate.

In this sense, O+ can be thought of in terms of “freezing team outs”. We know that for a team (excluding the list of complicating circumstances like rainouts, foregone batting in the bottom of the ninth, and walkoffs), outs are fixed quality. Regardless of whether we add Frank Thomas or Matt Walbeck to a reference team, the team’s outs are fixed. In the TT/PAR approach, we start by capturing the batter in question’s secondary contribution through the PAR multiplier, but we never directly change the number of PA we start from. Thus, Thomas’ 508 PA and Walbeck’s 355 PA will turn into outs at the same rate for the purpose of the calculation. In reality, Thomas’ team will compile more PA thanks to his greater secondary contribution, but the equation handles this with a multiplier, freezing the original value.

You may not find that explanation convincing – I am struggling to articulate a concept that I may be fooling myself into believing I understand. You might be more convinced by the demonstration of what Thomas’ RAA is under this approach:

TT_RAA = (TT_BSRP/O+ - LgR/O)*O+

which for Thomas = (.4494 - .2075)*333.6 = 80.7 which is the same as his TT_RAA

In fact, I jumped the gun by calling it TT_RAA before I proved it was equal to what we previously called TT_RAA. It is in fact:

This leaves unanswered the question of what the rate stat using TT_RAA should be. Similar to how we had R+/PA and RAA/PA when working with linear weights, we should have an appropriate rate stat for the RAA figure. By manipulating the TT_RAA equation, we can see that TT_RAA/O+ should be consistent with R+/O+:

TT_RAA = (R+/O+ - LgR/O)*O+ 

so divide both sides by O+ to get:

TT_RAA/O+ = R+/O+ - LgR/O

Thus TT_RAA/O+ and R+/O+ are analogous to RAA/PA and R+/PA, except that the difference is LgR/O rather than LgR/PA.

So we’re done, right? Or is there something vaguely unsatisfying about all this, that after an entire series in which I argued that R/O was a team measure and not an individual one, does it bother you that we have left our rate stat for an individual in the form of R/O?

On one hand, it absolutely shouldn’t – R/O is the correct choice of rate stat for a team, and in this case we have modeled the player’s impact on the team and its R/O, and so there’s nothing wrong with expressing a final result in terms of R/O. The problem was not with R/O per se – it was with the inputs that we were putting in for individual batters. Additionally, R+/O+ allows our team and individual rate stats to converge. Team R+/O+ will reduce to team R/O if accept the premise that the proper “reference team” for a team is itself, and that it’s R+ is equal to its actual runs scored since actual runs scored obviously accounts for the team’s primary, secondary, and tertiary offensive actions. Not to mention any quaternary actions we could dream up or the fact that using those terms doesn’t even make sense when talking about teams. 

On the other hand, O+ is not in any way an intuitive metric – just re-read my tortured explanation of what it represents. A batter’s R+/O+ can be contextualized in any number of meaningful ways, just as regular old R/O can, but I’m not sure even those presentations (ala Runs Created/25.2 Outs) are truly relatable to a player’s performance except as a scaling device.

There is a very simple and equivalent alternative, though, and you may have already noticed what it might be from the formulas. We defined O+ as PA*(1 – LgOBA), which is really just plate appearances times a constant. If we just divide by that constant (1 – LgOBA), we can restate everything on the basis of PA, with no loss of ratio comparability. 

Doing this, we now have:

(TT_BsRP/PA – LgR/PA)*PA = TT_RAA

and TT_BsRP/PA = R+/O+ * (1 – LgOBA)

R+/PA+ = TTBsRP/PA

I am going to call TT_BsRP/PA (R+/PA+) even though PA+ is just equal to PA. I’m doing this primarily for ease of discussion, so that R+/PA+ will represent the theoretical team calculation, while R+/PA will represent the linear weights equivalent—and they are equivalent (which also means that R+/O+ could be applied to the linear weights framework – more on this in a later installment). My other justification is that in theoretical team framework, the actual number of plate appearances we plug in is not of importance when dealing with a rate stat, as we are always defining the reference team’s PA to be eight times the batter’s PA. We use the batter’s PA because it allows estimates like TT_BsR to reflect his actual playing time, but if all we care about is the rate at the end, we could use 1 plate appearance or 650 or any number we wanted. Thus I prefer to think about this quantity of plate appearances as the player’s share of the reference team’s PA rather than really being tied to his own, and feel justified in distinguishing it through the abbreviation PA+.

This approach to developing a rate stat for the TT framework can be thought of as “freezing plate appearances” as compared to the “freezing outs” approach of R+/O+. I first became aware of it through a FanHome post in 2007 by David Smyth (surprise). By that time it was over five years since Smyth had published the R+/O+ methodology and I had adopted it myself, so by the time we discussed what I am calling R+/PA+ I was in the interesting position of advocating one Smyth construct against the newer. We quickly verified their equivalence and left it there, which is the position I’m taking now. 

By definition, R+/O+ and R+/PA+ are perfectly correlated since the only difference between the two is multiplying by a constant. One allows us to express a rate stat in terms of R/O, which is consistent with how we would state a team rate stat; the other allow us to express it in terms of R/PA, which is how we would state a rate state from the linear weight framework. Both can be compared using ratios, which will be equivalent; both can be compared using differences, with the question of what the denominator for that difference should be up to the user. Both denominators can be used with RAA as well for RAA+/O+ or RAA+/PA+ (using RAA+ to refer RAA calculated within the full-blown theoretical team framework), and since R+/O+ and R+/PA+ can both be used to calculate RAA, they will also be consistent with RAA+ per O+ or PA+.

I’ll close with a sample calculation. Frank Thomas had 149.9 TT_BsRP in 508 PA, which is .2951 R+/PA+. The league average R/PA was .1363, so Thomas’ TT_RAA was (.2951 - .1363)*508 = 80.7, same as calculated previously using R+/O+ or directly from the original TT_RAA formula. The league OBA was .3433, so Thomas’ R+/O+  is equal to .2951/(1 - .3433) = .4494 as calculated previously. 

Rate Stat Series, pt. 10: Rate Stats for the Theoretical Team Framework I

In calculating TT_BsR for a batter, we have taken into account both his primary and tertiary impact on the offense, but we have neglected to address his secondary impact – that is, the value of the additional plate appearances he generates for his team by avoiding outs. There’s a relatively simple way to apply an adjustment for this using the framework for TT_BsR we’ve already developed. David Smyth called this adjustment PAR for Plate Appearance Ratio, and it is based on the same logic about how PAs are generated that we have relied on many times.

PAR is equal to the ratio of the theoretical team’s plate appearances to the plate appearances a league average team would have had. Remember that:

PA/G = (O/G)/(1 – OBA)

O/G is a constant that we set at the league level – I will call it X in the algebra that follows. We need to know the OBA of the theoretical team; since our player in question gets 1/9 of the PA and the rest of the team is assumed to be league average, this is very simply:

T_OBA = 1/9*OBA + 8/9*LgOBA

Then T_PA/G  = X/(1 - (1/9*OBA + 8/9*LgOBA)) = X/(1 – 1/9*OBA - 8/9*LgOBA), while the league PA/G will be X/(1 – LgOBA). The ratio between the two will be:

(X/(1 – 1/9*OBA – 8/9*LgOBA))/(X/(1 – LgOBA)) = (1 – LgOBA)/(1 – 1/9*OBA – 8/9*LgOBA) = PAR

Since Frank Thomas had a .4921 OBA and the league average was .3433, his PAR is:

(1 - .3433)/(1 – 1/9*.4921 – 8/9*.3433) = 1.0258

This means that a theoretical team on which the Big Hurt had an equal share of the PA would end up generating 2.58% more PA than a league average team. 

In order to take Thomas’ secondary contribution into account, we can return to the definitions from the last installment and calculate:

TT_BsRP = T_BsR*PAR – R_BsR

PAR is only applied to T_BsR (the base runs estimate for the theoretical team with Thomas) because the reference team, filled with league average players, will continue to have the same number of PA as before (which we’ve set to equal eight times Thomas’ PA). Filling in those terms for the 1994, the formula is:

TT_BsRP = ((A + 2.514PA)*(B + 2.722PA)/(B + C + 7.975PA) + D + .232PA)*PAR – 1.090PA

Note that we can no longer combine the D term from T_BsR with the R_BsR term as the former also needs to be inflated by PAR (Thomas’ teammates will hit more homers in those extra 2.58% PA they now enjoy).

Applying PAR increases Thomas’ TT_BsR from 132.2 to 149.9, a significant increase. This figure is more comparable to his wRC (147.1) than to other runs created estimates we’ve examined, as it’s already taken into account the value of his secondary contributions.

You may note that there is the potential of some circularity here, as we are using Thomas’ actual PA as the starting point, but Thomas’ actual PA already inherently include his real secondary contribution to the 1994 White Sox. That is to say that some of the 508 PA that Thomas actually recorded were made possible by his own generation of PA for that team. This is a good argument for using a theoretical number of PA for Thomas rather than his actual PA. Thomas recorded 508 of Chicago’s 4439 PA, or 11.44%. So we could instead use 11.44% of the league average team PA total (4366.9), in which case he would have 499.7 restated PA to plug into the Theoretical Team methodology (this is ignoring that his contribution to the White Sox also had an impact on the league average PA). Of course, in so doing we would also have to proportionally scale back his portion of the T_A, T_B, T_C, and T_D components by 499.7/508. 

On the other hand, the secondary contribution of a batter through generating PA is in the background of the linear framework as well (and any other framework that considers his actual PA), it’s just that the connection leaps to the mind more quickly when modeling the other aspects of a theoretical team. I’m going to ignore this going forward, as this is after all a rate stat series, and also note that we shouldn’t ignore the fact that a batter can benefit from the additional opportunities he helps to create. The fact that the quality of his teammates influences how many opportunities he gets in the real world is at some level unavoidable.

At this point, we should also express Thomas’ contribution in terms of RAA. This is a simple modification; instead of setting R_BsR equal to the league average BsR/PA times 8 times the player’s PA, we would just need to multiply by 9 times the player’s PA so that the lineup isn’t magically shortened and instead we compare T_BsR to what a team would score with an average player in our man’s place. I did not bother running this before introducing PAR, because if there’s one thing we’ve learned from this series is that it doesn’t make a lot of sense to talk about batter RAA without taking out rate into account. So with PAR for the 1994 AL have:

TT_RAA = ((A + 2.514PA)*(B + 2.722PA)/(B + C + 7.975PA) + D + .232PA)*PAR – 1.226PA

We now have three possible theoretical team approaches, and have yet to address the question of this series: what should the rate form be? The guiding principle of this series has been that the properties of the numerator (usually a run estimate) should be logically consistent with the choice of denominator, so we should consider each of the three theoretical team approaches separately.

First is TT_BsR, which is just an estimate of the batter’s impact on the team runs scored, taking into account primary and tertiary (but not secondary) impacts. It is akin to LW_RC, with the key difference being that LW_RC does not attempt to value the batter’s tertiary impact. However, I contend that incorporating tertiary contributions does not alter the considerations when developing a rate stat. The tertiary effect is how the batter’s performance changes the underlying run environment of the team, independently of the change in plate appearances. What we are left which is an estimate of the contribution the batter made in his actual plate appearances – the only difference is that we recognize that those outcomes influenced the value of all of the other offensive events recorded by the team.

So our choices for a rate stat are the same as those for LW_RC. We can first calculate RAA (using R/O), and then take RAA/PA, or we can calculate how many additional PA the batter generated/outs he avoided, add those to his TT_BsR, and divide by PA (the R+/PA) approach. These approaches will be equivalent if we add back in LgR/PA to RAA/PA, we could convert to wOBA, we could calculate wRC along the way...all the same options.

The math will be the same as shown in parts 7 and 8, except we will substitute TT_BsR for LW_RC everywhere it pops up. Here is a leaderboard for some of the key metrics using LW_RC (we’ve seen this all before):

Now the same metrics, except substituting TT_BsR for LW_RC in all calculations:

This was a lot of work to get largely the same results. Maybe applying PAR will make things more interesting? 

Rate Stat Series, pt. 9: Theoretical Teams

We now depart the orderly, neat world of linear weights for the frontiers of offensive evaluation/rate stat development. Allow me to posit that there are three ways in which a batter impacts his team’s run scoring:

1. Through the direct, immediate consequences of his actions (e.g. he draws a walk or flies out). We could call this his primary contribution.

2. Through how those results create or fail to create additional opportunities for his teammates to bat (what I have been calling PA generation). We could call this his secondary contribution (I do so with some reservations because I do like secondary average, which uses “primary” to refer to direct contributions captured batting average and “secondary” to refer to other direct contributions like extra bases on hits, walks, and steals).

3. Through how his impact on the team alters the value of the actions of his teammates. This tertiary effect is hard to define, but we know that the run value of any offensive event is dependent on the context in which it occurs. A walk does no good if no one else in the lineup gets on base; each out is more costly in terms of runs in a higher scoring environment. Dynamic run estimators vary the value of each event based on the frequencies of all offensive events, while linear weights keep them fixed.

I listed and labeled these three elements of offensive production in the order of their magnitude; the third is very small, small enough that it is often ignored. Crucially for this discussion, it is small enough that if we are not careful, in attempting to measure it we could cause more unintended distortion with respect to the evaluation of #1 and #2 so as to make the exercise not just a waste of time, but actively harmful to our understanding.

So far in this series, we have looked at individual offense through two frameworks. Treating the player as a team by plugging his stats directly into a dynamic run estimator, we have captured (but distorted) #1 and #2 and given excessive weight to #3 by pretending as if the 8 teammates all perform at the level of the individual in question. By using linear weights, we have treated the player as if he was part of a semi-static environment where his direct actions and PA generation have an impact on his team, but that no matter how he performs, it has no impact on the offensive environment in which the other eight batters perform.

I believe that a third framework, which captures the impact of all three ways in which a batter affects team runs scored, is theoretically superior to the other approaches. This will involve modeling a team with and without our player – constructing a “theoretical team” in which eight members of the lineup perform at a given level and our player occupies one lineup spot. However, there are cautions, which I alluded to above:

1. The math becomes more complicated. As long as increased complexity corresponds to a more sound approach from a theoretical perspective, this is not objectionable to me, but that’s a minority viewpoint.

2. The impact of #3 is very small relative to #1 and #2, and is arguably negligible, especially when we consider all of the error bars that exist around run estimation, park factors, positional adjustments, and the myriad other variables which will come into play when the estimates are put to full use as part of an overall player evaluation system.

3. If the model which you use to implement this framework is poor, the distortions created when compared to a linear weights framework will swamp the attempt to measure the minuscule impact of #3. Even if your model is good (and I will be using Base Runs and I am quite confident that it is a good model), the linear weights framework is so robust that there is still some risk in abandoning it to chase capturing very small effects.

My original series on rate stats failed on this count, as I begged the question by assuming that a theoretical team approach was correct and using that as one of the testing criteria for other metrics. Again, I believe that the framework is theoretically correct, but the implementation is trickier, and I am not so arrogant today to believe that the model and my implementation are unquestionably superior to using a linear weights framework. To return to a bad and wildly overwrought nautical metaphor, linear weights provide a safe harbor with calm waters in which it is tempting to stay and not venture on to high seas where theoretical team frameworks tempt with the promise of riches but tempests and other dangers lurk.

Before starting, I want to note a handful of people who made significant contributions to the theoretical team concept. One is David Tate, who developed Marginal Lineup Value, which used the framework of basic runs created in conjunction with a theoretical team. Keith Woolner refined and popularized MLV. In 1998, Bill James published the approach that I will use here, although of course he used runs created. Published a year later, Jim Furtado’s Extrapolated Wins methodology used a linear run estimator (his XR) but fleshed out theoretical team concepts with respect to win impact and replacement level. Furtado also, along with G. Jay Walker and Don Malcolm, took apart James' theoretical team RC to understand what was going on behind the scenes. Finally, David Smyth, the developer of Base Runs, was the first to apply a TT construct to BsR and also developed the PAR adjustment which we’ll get to eventually.

Finally, before diving in to the specific implementation of TT I will use in this series, I want to note that by “theoretical team”, I am referring only to constructs that explicitly attempt to place the player on a theoretical/”reference” team, and use a dynamic run estimator to estimate his run impact. It does not refer to other approaches that may be undertaken to apply a dynamic run estimator to an individual hitter. One such example is the technique, so far as I know first used by Dick Cramer with his runs created-like run estimator, of calculating a batter’s runs created as the difference between the league with his stats and he league without them. This is a clever approach for using a dynamic run estimator in evaluating individuals, but not a TT approach. In fact, it more closely resembles the approach we used in this series to develop linear weights from Base Runs. The larger you make the pool to which the player is added, the more you dilute his impact. The differentiation approach takes this to the limit (see what I did there?) by isolating each event and calculating its value if it had no impact at all on the offensive environment.

In contrast, a TT approach uses a realistic scale between the individual and team; a typical approach is to assume that the individual gets 1/9 of team plate appearances. Using a 1/8 ratio between player and reference team does not require us to believe that the player actually had 1/9 of his team’s PA in the real world. One could use a player’s actual percentage of team PA and weight accordingly, but there is a balancing act: one one hand, we want to accurately capture the degree to which the batter impacted the team, but we also don’t want to lose sight of where his impact is actually felt. Consider a batter who plays in just one game in the season, getting four plate appearances. If you use his actual percentage of team PA  (which might be something like 0.05%) to calculate his impact on the team, he will have had essentially no tertiary effect. That is a distortion of reality, though – he really had something closer to 11.1% of the team’s PA, in the game in which he actually played. From the perspective of evaluating his impact on the team, the other 161 games are an accounting fiction, no more relevant to him than to games between other teams played thirty years prior (in fact, we should acknowledge that runs are actually scored at the inning level, which is where we started working out the math on PA generation).

So we will assume that the reference team always has eight times as many plate appearances as the player in question (which of course is equivalent to saying the player gets 1/9 of team PA). We could get cute and recognize that based on a player’s batting order position, his expected share of PA will change, and give different players a different share (while still limiting the scope to games/innings in which the batter actually played), but 1/9 is clean and any alternative approach would leave most batters pretty close to 1/9. The concept is simple; the formula will look a little messy. We start with our Base Runs equation:

A = H + W – HR = S + D + T + W

B = (2TB - H – 4HR + .05W)*.79776 = .7978S + 2.3933D + 3.9888T + 2.3933HR + .0399W

C = AB – H = Outs

D = HR

BsR = (A*B)/(B + C) + D

We will start by calculating the team’s runs with the player. This will take the same form, but now our A, B, C, and D components will start with the player’s stats and add eight reference players. I will assume that the reference player is a league average performer, and thus the reference team is a league average team prior to the addition of our player. One could make the case that with respect to the tertiary effect of a player, linear weights framework sidestep the issue by assuming an inverse relationship between the quality of the player in question and the quality of the reference team. That is, by using static linear weights for all players regardless of their performance, a linear weights framework implicitly assumes that the team is average after the player is added. Thus Frank Thomas is added to a worse team than Matt Walbeck, such that at the end of the day the run values of all events are the same between the Thomas team and the Walbeck team. 

If you are tempted to sweat the details and subtract the player’s stats from the league before determining league average, don’t. It is actually surprising how little impact the choice of reference team has on the outcome (which a cynic might note is a reason for suspecting that the tertiary effect is de minimis, but what’s the fun in that?) This is why James is able to get away with using a single final formula for converting the player’s A, B, and C factors in Runs Created (for which he laid out 24 different versions to cover major league history) to TT RC by using just one equation. It’s not technically correct, of course, but as long as long as the reference team is within a reasonable range of major league offense, it’s not debilitating.

Without our player, the reference team will have a number of plate appearances equal to eight times the individual’s PA, and will perform at the league average, so we can define each factor for the reference team as follows, with the calculation using the 1994 AL averages shown:

R_A = Lg(A/PA)*PA*8 = .3143*PA*8 = 2.514PA

R_B = Lg(B/PA)*PA*8 = .3402*PA*8 = 2.722PA

R_C = Lg(C/PA)*PA*8 = .6567*PA*8 = 5.254PA

R_D = Lg(D/PA)*PA*8 = .0290*PA*8 = .232PA

Then for the team with the player, the team versions of the A, B, C, and D factors are just the player’s factor plus eight times his PA times the league average of the factor/PA:

T_A = A + R_A = A + 2.514PA

T_B = B + R_B  = B + 2.722PA

T_C = C + R_C = C + 5.254PA

T_D = D + R_D = D + .232PA

In order to isolate the individual’s impact, we need to calculate how many runs his new theoretical team would score and subtract the runs that the reference team would have scored with just eight reference players. The team’s BsR will be:

T_BsR = T_A*T_B/(T_B + T_C) + T_D

Some of the PA terms  in the denominator can be combined, so for the 1994 AL we get:

T_BsR = (A + 2.514PA)*(B + 2.722PA)/(B + C + 7.975PA) + D + .232PA

The reference team’s run scored will be equal to the league average BsR/PA times 8 times the player’s PA; to calculate league BsR/PA we can just plug the league average A, B, C, and D factors per PA into the BsR equation to get BsR/PA, then multiply:

R_BsR = (.3143*.3402/(.3402 + .6567) + .0290)*8*PA = 1.090PA

So our estimate of the individual’s run contribution to the theoretical team, which we’ll call Theoretical Team Base Runs (TT_BsR) is just the difference:

TT_BsR = T_BsR – R_BsR

Since we have PA in each term, for the 1994 AL it simplifies to:

TT_BsR = (A + 2.514PA)*(B + 2.722PA)/(B + C + 7.975PA) + D - .8579PA

If we apply Frank Thomas’ statistics directly to Base Runs, we get an estimate of 139.0. If we use Base Runs to estimate linear weight coefficients for the league, we get 131.4 (what we’ve been calling LW_RC in this series). If we use the TT approach, we get 132.2. As you can see, the TT estimate is not that much different than the full linear estimate, which does call into question the need for the TT approach. After all, Thomas is one of the most extreme hitters in the league; if he barely moves the needle, who will?

Regardless of the utility of this approach, I find it useful as an intellectual exercise because I believe the framework is the closest to approximating the real relationship between an individual batter and team performance. For a series ostensibly about rate stats, I’ve spent an entire post just setting up the numerator; don’t rate stats typically have a denominator as well? Seriously, though, if there’s one takeway I would like a reader to glean from this series, it is that if you want to set up an offensive evaluation system, you need to think through all of the pieces as you develop it. Starting with a run estimator, and then slapping on a rate state, and a baseline, and whatever bells and whistles you need, is not a sound approach. The choice of run estimator determines which denominator you should use, and the two should be compatible.