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.

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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.

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Taking this analysis in sum, it is probably best to do further analysis numerically, as the derivatives are quite unwieldy.

]]>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…

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