Last time I wrote about the approach of applying BsR to individuals by using static intrinsic linear weights from some entity. This time, we will look at the differential and theoretical team approaches, each of which allow for some interaction between the player and the team, and cause the weights to be different for each individual player.
As I mentioned last time, these too approaches are equivalent if we use the same team totals for each. The TT formulas in use generally assume that the player gets exactly 1/9 of team PAs. This is not an inevitable choice, though.
Let’s first define the team’s A, B, C, and D factors without the player as T_A, T_B, T_C, and T_D. Then, we can estimate the number of runs the team will score with the player as:
(A + T_A)*(B + T_B)/(B + C + T_B + T_C) + D + T_D
The number of runs the team would score without the player is:
T_A*T_B/(T_B + T_C) + T_D
Thus, DBsR (Differential BsR) is the difference between the two, which simplifies a bit to:
DBsR = (A + T_A)*(B + T_B)/(B + C + T_B + T_C) + D - T_A*T_B/(T_B + T_C)
All a standard theoretical team formula does is assume that the player gets 1/9 of team PAs, and thus defines T_A, T_B, and T_C as some entity’s A/PA, B/PA, and C/PA times (eight times individual PA). I like to call A/PA “ROBA” (Runners On Base Average), B/PA “AF” (Advancement Factor), and C/PA “OA” (Out Average). With PA indicating individual PA, the generalized TT formula is):
TT BsR = (A + LgROBA*8*PA)*(B + LgAF*8*PA)/(B + C + (LgAF + LgOA)*8*PA) + D - LgROBA*LgAF/(LgAF + LgOA)*8*PA
Using the basic BsR equation spelled out in Part 5, the 1961-2005 major league averages are: .301 ROBA, .306 AF, .676 OA, producing this TT equation:
TT BsR = (A + 2.41PA)*(B + 2.45PA)/(B + C + 7.86PA) + D - .75PA
TT BsR = (A + 2.46PA)*(B + 2.65PA)/(B + C + 7.97PA) + D - .82PA
Here are each player’s TT BsR figured by each approach; “Long” is the long-term weights, “2007A” is the 2007
Once again, you can see how little difference this makes, and thus why Bill James can get away with using the same TT formula for all of baseball history (incidentally, I did not round off two decimal places in those calculations, so the results may be a little different if you try using the above formulas yourself).
Now let’s apply the differential method, figuring the difference between the player’s team’s BsR with and without them. I am using their actual teams; we could look at each player on any other team, but we would have to do it in TT fashion, or by weighting the player’s PA when figuring the “rest of team” BsR. I realize that is a poorly explained point, but let me try this. Suppose we subtracted ARod’s stats from those of the White Sox, pretending that he was actually a member of that team. Now the team that we’re adding him to is not just the worst offense in the league, they’re the worst offense in the league without the presence of the best hitter in the league--who never actually contributed to that team in the first place.
The point is that if you use the differential method, you need to use it with the entity the player actually belongs to or you need to use a TT approach to scale the “rest of team” stats properly (in the ARod/CHA case, the “rest of team” should hit just as well as they did, just with 600 less PAs (or however many PA ARod actually had)).
So these figures are for the players on each of their actual teams:
Again, you can see that no matter which approach we use, no matter which reasonable team we put the player on, we get very similar final estimates of individual runs created.
Some methods for estimating run contribution (most notably Dick Cramer’s Batter Win Average) have subtracted the player’s stats from those of the league and then found the difference. Cramer did not use an average team, but rather the composite league statistics for all X teams in the league. Here is what our differential figures for each player would look like using that approach:
I do not endorse this approach as a proper method to apply BsR to individuals (recognizing of course that the practical differences between this approach and the ones that I do endorse are minuscule). Runs are not created on a league level; they are created on a team level. If you want to take the player out of the context of a particular or theoretical team, and just get a “global” estimate of the player’s contributions, it is preferable IMO to apply the linear weight values. Pretending that runs are created on the league level and estimating the player’s contribution thusly is wrong in theory, and an inefficient use of time in practice.
So far I’ve shown you the final runs created estimate from each of these approaches. Now I will show the intrinsic weights that led to some of those estimates for Alex Rodriguez. Since many of the differential approaches produce very similar results, there is no need to clutter this up with the weights for each approach. The first column, “BsR”, shows the BsR intrinsic weights for his individual statistics. The second column, “LBsR”, is the linear weights for the 2007
Here we see that while the final estimate stays within a two run range or so, the coefficients that get us there have a little more variation. As expected, the home run and the out are the most stable of the events. You can decide whether these differences are worth it to you to go through a theoretical team procedure, or whether you just want to stick with the linear weights. Personally, while I love the concept of a theoretical team have done as much work with it as anyone other than the aforementioned pioneers of the approach (at least as far as I can tell), I think that it is probably better to stick with the linear weights on a league level for most purposes. In stating that preference, I am not claiming that such an approach is “better” or any such thing. It is just my opinion that the extra effort put into the theoretical team calculation is not justified by the differences in the final estimates.
There is one issue hanging on the periphery of any theoretical team discussion that I would like to acknowledge, although I do not want to go into it here as it really is not so much about run estimation but about moving from run contribution to win contribution. That is the effect that a batter has on his team by creating additional opportunities for his teammates. Traditional runs created estimates, as well as the theoretical team variations presented here, do not account for this. It is pretty easy to tack what David Smyth called PAR (PA ratio) on to TT BsR. Whether you want to or not (and how the PAR approach compares to the old FanHome poster Sibelius’ R+/PA) depends on exactly what you are trying to measure. David Tate’s Marginal Lineup Value was the first method in this vein, and you could also adapt it to use BsR instead of RC.
That’s a topic for another day, but keep in mind that the TT BsR estimate as discussed here for any player measures his direct contribution only.