## Is scoring different in the AL and the NL? May 31, 2011

Posted by tomflesher in Baseball, Economics.
Tags: , , , , , , , ,

The American League and the National League have one important difference. Specifically, the AL allows the use of a player known as the Designated Hitter, who does not play a position in the field, hits every time the pitcher would bat, and cannot be moved to a defensive position without forfeiting the right to use the DH. As a result, there are a couple of notable differences between the AL and the NL – in theory, there should be slightly more home runs and slightly fewer sacrifice bunts in the AL, since pitchers have to bat in the NL and they tend to be pretty poor hitters. How much can we quantify that difference? To answer that question, I decided to sample a ten-year period (2000 until 2009) from each league and run a linear regression of the form

$\hat{R} = \beta_0 + \beta_1 H + \beta_2 2B + \beta_3 3B + \beta_4 HR + \beta_5 SB + \beta_6 CS + \\ \beta_7 BB + \beta_8 K + \beta_9 HBP + \beta_{10} Bunt + \beta_{11} SF$

Where runs are presumed to be a function of hits, doubles, triples, home runs, stolen bases, times caught stealing, walks, strikeouts, hit batsmen, bunts, and sacrifice flies. My expectations are:

• The sacrifice bunt coefficient should be smaller in the NL than in the AL – in the American League, bunting is used strategically, whereas NL teams are more likely to bunt whenever a pitcher appears, so in any randomly-chosen string of plate appearances, the chance that a bunt is the optimal strategy given an average hitter is much lower. (That is, pitchers bunt a lot, even when a normal hitter would swing away.) A smaller coefficient means each bunt produces fewer runs, on average.
• The strategy from league to league should be different, as measured by different coefficients for different factors from league to league. That is, the designated hitter rule causes different strategies to be used. I’ll use a technique called the Chow test to test that. That means I’ll run the linear model on all of MLB, then separately on the AL and the NL, and look at the size of the errors generated.

The results:

• In the AL, a sac bunt produces about .43 runs, on average, and that number is significant at the 95% level. In the NL, a bunt produces about .02 runs, and the number is not significantly different from saying that a bunt has no effect on run production.
• The Chow Test tells us at about a 90% confidence level that the process of producing runs in the AL is different than the process of producing runs in the NL. That is, in Major League Baseball, the designated hitter has a statistically significant effect on strategy. There’s structural break.

R code is behind the cut.

MLB:

> MLB.lm <- lm(R ~ H + X2B + X3B + HR + SB + CS + BB + SO + HBP + SH + SF)
> summary(MLB.lm)

Call:
lm(formula = R ~ H + X2B + X3B + HR + SB + CS + BB + SO + HBP +
SH + SF)

Residuals:
Min         1Q     Median         3Q        Max
-7.002e+01 -1.174e+01 -9.314e-04  1.498e+01  6.732e+01

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -453.10394   39.32708 -11.521  < 2e-16 ***
H              0.53819    0.02694  19.976  < 2e-16 ***
X2B            0.14640    0.06384   2.293   0.0226 *
X3B            0.69687    0.16290   4.278 2.57e-05 ***
HR             0.93928    0.05036  18.653  < 2e-16 ***
SB             0.05632    0.05459   1.032   0.3031
CS             0.03725    0.14576   0.256   0.7985
BB             0.31244    0.02168  14.410  < 2e-16 ***
SO            -0.02801    0.01513  -1.851   0.0652 .
HBP            0.44309    0.10308   4.298 2.36e-05 ***
SH            -0.14210    0.06689  -2.124   0.0345 *
SF             0.80263    0.18164   4.419 1.41e-05 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 22.46 on 288 degrees of freedom
Multiple R-squared:  0.92,      Adjusted R-squared: 0.9169
F-statistic: 301.1 on 11 and 288 DF,  p-value: < 2.2e-16

> MLB.aov <- aov(R ~ H + X2B + X3B + HR + SB + CS + BB + SO + HBP + SH + SF)
> summary(MLB.aov)
Df  Sum Sq Mean Sq   F value    Pr(>F)
H             1 1111813 1111813 2204.2679 < 2.2e-16 ***
X2B           1   21093   21093   41.8190 4.279e-10 ***
X3B           1    6113    6113   12.1196 0.0005763 ***
HR            1  380838  380838  755.0440 < 2.2e-16 ***
SB            1    2775    2775    5.5022 0.0196709 *
CS            1     776     776    1.5388 0.2158089
BB            1  121628  121628  241.1386 < 2.2e-16 ***
SO            1    1740    1740    3.4504 0.0642589 .
HBP           1   10494   10494   20.8045 7.534e-06 ***
SH            1    3449    3449    6.8387 0.0093899 **
SF            1    9849    9849   19.5263 1.407e-05 ***
Residuals   288  145265     504
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Created by Pretty R at inside-R.org

AL:

> AL.lm <- lm(R ~ H + X2B + X3B + HR + SB + CS + BB + SO + HBP + SH + SF)
> summary(AL.lm)

Call:
lm(formula = R ~ H + X2B + X3B + HR + SB + CS + BB + SO + HBP +
SH + SF)

Residuals:
Min      1Q  Median      3Q     Max
-53.410 -13.107  -1.835  14.574  49.139

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -427.34031   55.72982  -7.668 3.85e-12 ***
H              0.50747    0.03911  12.976  < 2e-16 ***
X2B            0.21243    0.09538   2.227  0.02769 *
X3B            0.85198    0.25274   3.371  0.00099 ***
HR             0.89353    0.07306  12.231  < 2e-16 ***
SB             0.13157    0.08673   1.517  0.13171
CS            -0.41996    0.23826  -1.763  0.08034 .
BB             0.34408    0.03256  10.566  < 2e-16 ***
SO            -0.04241    0.02433  -1.743  0.08368 .
HBP            0.45716    0.15278   2.992  0.00332 **
SH             0.43101    0.19621   2.197  0.02984 *
SF             0.67215    0.26915   2.497  0.01378 *
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 23.14 on 128 degrees of freedom
Multiple R-squared: 0.9209,     Adjusted R-squared: 0.9141
F-statistic: 135.5 on 11 and 128 DF,  p-value: < 2.2e-16

> help()
starting httpd help server ... done
> help(lm)
> AL.aov <- aov(R ~ H + X2B + X3B + HR + SB + CS + BB + SO + HBP + SH + SF)
> summary(AL.aov)
Df Sum Sq Mean Sq  F value    Pr(>F)
H             1 499593  499593 932.8925 < 2.2e-16 ***
X2B           1  17886   17886  33.3988 5.413e-08 ***
X3B           1  15830   15830  29.5595 2.645e-07 ***
HR            1 170063  170063 317.5592 < 2.2e-16 ***
SB            1   1049    1049   1.9592  0.164021
CS            1   4373    4373   8.1658  0.004984 **
BB            1  75600   75600 141.1680 < 2.2e-16 ***
SO            1   2009    2009   3.7505  0.054994 .
HBP           1   5252    5252   9.8080  0.002154 **
SH            1   3234    3234   6.0391  0.015331 *
SF            1   3340    3340   6.2368  0.013781 *
Residuals   128  68548     536
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Created by Pretty R at inside-R.org

NL:

> NL.lm <- lm(R ~ H + X2B + X3B + HR + SB + CS + BB + SO + HBP + SH + SF)
> summary(NL.lm)

Call:
lm(formula = R ~ H + X2B + X3B + HR + SB + CS + BB + SO + HBP +
SH + SF)

Residuals:
Min      1Q  Median      3Q     Max
-71.594  -9.621   1.258  11.946  58.529

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -474.75070   55.58292  -8.541 1.49e-14 ***
H              0.52145    0.03830  13.616  < 2e-16 ***
X2B            0.18340    0.09050   2.027  0.04450 *
X3B            0.65062    0.21683   3.001  0.00316 **
HR             0.99662    0.07093  14.052  < 2e-16 ***
SB             0.02082    0.06957   0.299  0.76515
CS             0.22201    0.18616   1.193  0.23496
BB             0.30956    0.03042  10.177  < 2e-16 ***
SO            -0.01041    0.02050  -0.508  0.61232
HBP            0.38324    0.13761   2.785  0.00605 **
SH             0.01567    0.12655   0.124  0.90164
SF             0.70346    0.24689   2.849  0.00501 **
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 21.14 on 148 degrees of freedom
Multiple R-squared: 0.9171,     Adjusted R-squared: 0.911
F-statistic: 148.9 on 11 and 148 DF,  p-value: < 2.2e-16

> NL.aov <- aov(R ~ H + X2B + X3B + HR + SB + CS + BB + SO + HBP + SH + SF)
> summary(NL.aov)
Df Sum Sq Mean Sq   F value    Pr(>F)
H             1 461597  461597 1032.5156 < 2.2e-16 ***
X2B           1   4837    4837   10.8194  0.001255 **
X3B           1    290     290    0.6481  0.422080
HR            1 204654  204654  457.7763 < 2.2e-16 ***
SB            1   1357    1357    3.0354  0.083546 .
CS            1    490     490    1.0958  0.296899
BB            1  51509   51509  115.2181 < 2.2e-16 ***
SO            1     15      15    0.0327  0.856799
HBP           1   3839    3839    8.5862  0.003925 **
SH            1      9       9    0.0206  0.886062
SF            1   3629    3629    8.1183  0.005007 **
Residuals   148  66165     447
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Created by Pretty R at inside-R.org

Chow’s Test:

$\frac{(S_{MLB} -(S_{AL}+S_{NL}))/(k)}{(S_{NL}+S_{AL})/(N_{NL}+N_{AL}-2k)}$

$\frac{(145265 - (66165 + 68548))/11}{(66165+ 68548)/(140 + 160 - 22)} = 1.9796$

$F_{11, 278, \alpha = .05} = 1.8232$

$F_{11, 278, \alpha = .025} = 2.0394$

Since 1.9796 > 1.8232, the difference is not due to random variation and is significant at the 90% (but not 95%) level.

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