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.