Using the last line in the Google spreadsheet, playerid 1624, we obtain a value of .108 for dwOBA/dPOWH, and to the number of digits displayed this holds numerically.
Put shortly, a raw change of .100 in POWH for this hitter will result in a raw change of .011 in wOBA.
.
We can also pull out how the other three of the Four Factors interact in this case, meaning that an increase in POWH while keeping the other three constant will have a different effect on wOBA depending on what those constant values are.
The least complicated are BB and SO. A higher base rate of BB (or SO) necessarily means that the player will see less impact in wOBA from an increase in POWH – this makes sense in that POWH can only help on a batted ball.
BABIP is significantly more complicated: in the numerator, we have:
(1-BB-SO)*(UB – VB^2)
where U and V are numbers and U > V. Because B can never be greater than 1, a higher B means a higher numerator. In the denominator, we have (B-1), and again because B can never be greater than 1 a higher B means subtracting less of POWH (a positive number), which means a higher denominator. A higher numerator and a higher denominator mean that no general conclusion can be drawn regarding the resulting fraction – it depends on the relative size of X, Y, BB, SO, and POWH.
.
Taking this analysis in sum, it is probably best to do further analysis numerically, as the derivatives are quite unwieldy.
Nick Fleder says:
August 10, 2010 at 12:02 pm
I’m curious… what do you think of George Kottarras’ projections next year?
Seems like the guy is all or nothing, but as you say .195 BABIP among other things prove that luck isn’t on his side…
Craig says:
August 10, 2010 at 12:25 pm
I’m very excited for the afternoon part, and not just because I love algebra. I’ve been hoping someone would take a look at something similar to this.
Eric says:
August 10, 2010 at 2:19 pm
Using the same weightings as the spreadsheet for wOBA, I obtain an expression for wOBA as follows:
wOBA = .72*BB + .9*BABIP*(1-BB-SO) + HR*(1.95+.34*X+.66*Y-.9*BABIP)
where HR is a function of POWH, BABIP, BB, and SO:
HR = [POWH*BABIP*(1-BB-SO)] / [X+2Y+3+(BABIP-1)*POWH]
We can therefore take a derivative of wOBA with respect to any of the four variables, for instance with respect to POWH:
dwOBA/dPOWH = dHR/dPOWH * (1.95+.34*X+.66*Y-.9*BABIP)
and
dHR/dPOWH = [(X+2Y+3+(BABIP-1)*POWH)*BABIP*(1-BB-SO) - POWH*BABIP*(1-BB-SO)*(BABIP-1)] / [X+2Y+3+(BABIP-1)*POWH]^2
= [(X+2Y+3)*BABIP*(1-BB-SO)] / [X+2Y+3+(BABIP-1)*POWH]^2
Using 1.6 for X and .15 for Y, we get:
dHR/dPOWH = [4.9*BABIP*(1-BB-SO)] / [4.9+(BABIP-1)*POWH]^2
and
dwOBA/dPOWH = [(2.53 - .9*BABIP)*4.9*BABIP*(1-BB-SO)] / [4.9+(BABIP-1)*POWH]^2
Using the last line in the Google spreadsheet, playerid 1624, we obtain a value of .108 for dwOBA/dPOWH, and to the number of digits displayed this holds numerically.
Put shortly, a raw change of .100 in POWH for this hitter will result in a raw change of .011 in wOBA.
.
We can also pull out how the other three of the Four Factors interact in this case, meaning that an increase in POWH while keeping the other three constant will have a different effect on wOBA depending on what those constant values are.
The least complicated are BB and SO. A higher base rate of BB (or SO) necessarily means that the player will see less impact in wOBA from an increase in POWH – this makes sense in that POWH can only help on a batted ball.
BABIP is significantly more complicated: in the numerator, we have:
(1-BB-SO)*(UB – VB^2)
where U and V are numbers and U > V. Because B can never be greater than 1, a higher B means a higher numerator. In the denominator, we have (B-1), and again because B can never be greater than 1 a higher B means subtracting less of POWH (a positive number), which means a higher denominator. A higher numerator and a higher denominator mean that no general conclusion can be drawn regarding the resulting fraction – it depends on the relative size of X, Y, BB, SO, and POWH.
.
Taking this analysis in sum, it is probably best to do further analysis numerically, as the derivatives are quite unwieldy.