By removing the effect of defense from the run estimator equation, a DIPS (Defense Independent Pitcher's Index) like equation can be derived.
First, let us review the run estimation formula.
The runs scored per game, R/G, can be decomposed as follows
R/G = R/PA × PA/G R: runs scored, PA: at bats, G: games played
Here, R/PA can be further decomposed as follows
R/PA = R/OB × OB/PA OB: number of runs
OB/PA is, needless to say, the on-base percentage (OBP).
R/OB is the ratio of runs scored to bases, and is also called the score rate.
According to Tom Tango's analysis, this score rate can be roughly approximated by OBP.
Thus, R/PA can be rewritten as follows
R/PA = OPB^2 ・・・ ①
Next, let us break down PA/G.
Since the number of outs in one game is 27,
PA/G = PA / ( OUT/27 ) OUT: Number of outs
= 27 × 1 / ( OUT/PA )
Since the out rate OUT/PA = 1 - OPB
PA/G = 27 × 1 / ( 1 - OPB ) ・・・ ②
Substituting equations (1) and (2) into the equation at the beginning, we finally obtain
R/G = 27 × OBP^2 / ( 1 - OBP )
Thus, with a little approximation and simple arithmetic, R/G can be converted into an expression for the OBP.
By the way, the on-base percentage is calculated by the following formula.
OPB = ( BB+HBP+H ) / ( BB+HBP+SO+BIP+HR )
BB: walks, HBP: hit by pitch, H: hit by pitch, SO: strikeout, BIP: in-play ball, HR: home run
From DIPS theory, the hit rate of in-play batted balls converges to 30% regardless of the pitcher's ability, so the expected on-base percentage xOBP is
xOPB = ( BB+HBP+0.3BIP+HR ) / ( BB+HBP+SO+BIP+HR )
If we apply this xOBP to the scoring structure equation we just used, we get an equation that incorporates DIPS theory. Let us call this xRA for now.
xRA = 27 × xOBP^2 / ( 1 - xOBP )
FIP and tRA are well-known as representative DIPS, but these are equations obtained by analyzing the score value of each event, and the process of deriving them is very complicated. Moreover, the event analysis assumes an average scoring environment, which has the disadvantage that the calculation breaks down if the scoring environment changes significantly. Roki Sasaki's negative FIP is probably well known.
On the other hand, xRA does not go through the event analysis process as FIP does, and thus does not have the above problem of calculation failure. xRA is a more numerically stable indicator than FIP.
However, xRA tends to underestimate homers and overestimate walks and strikeouts (for details, see [ Extra ]), and its accuracy is somewhat limited, so its usefulness will probably never surpass that of FIP. Nevertheless, the clarity of the calculation logic and numerical stability of xRA are remarkable and very attractive.
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[ Extra ] Let's estimate the run value of each event using the Plus One Method.
Here is using Masahiro Tanaka in 2022 as an example. His results are as follows.
BB+HBP (excluding intentional walk): 33
Strikeouts : 126
BIP: 495
HR: 16
Applying these to the xRA formula yields xRA=3.32.
If we calculate the xRA for an increase of one more strikeout and measure the amount of increase, we get 3.35 - 3.32 = 0.03.
Since the number of innings pitched is 163, converted to the number of games, 163/9 = 18.1.
Therefore, the increase in runs is 0.03 × 18.1 = 0.54.
The same calculation is performed for all pitching events,
4 dead balls: 0.54
Strikeouts: -0.22
HR: 0.54
BIP: 0.00
The loss in runs scored for walks, HRs, and strikeouts should be roughly equal to the respective coefficients in the tRA formula, but there is a divergence.
This seems to be the result of the adverse effect of approximating the live-ball rate by the on-base percentage.
When calculating xRA, it is important to keep in mind that such errors are included.
Finally, the results of a similar calculation for Akira Sasaki in 2022 are shown below.
Killed by pitches: 0.45
Strikeouts: -0.14
HR: 0.45
BIP: 0.03
RA: 2.16
xRA: 2.13
It can be seen that the value of runs scored on walks and HRs has decreased. In an environment with fewer baserunners, the impact of walks and homers on score creation should be smaller, and the xRA reflects this effect.
On the other hand, the value of runs scored in BIP has increased slightly, suggesting that in an environment with depleted baserunners, 30% of outs have a greater impact on runs scored than 70% of outs. The results of this study are as follows.
The same idea for strikeouts as for walks and homers should be applied, and the deterrent effect of strikeouts on runs scored should be smaller in an environment with fewer baserunners.
