美式橄榄球与统计学

Omni

感谢众网友对兄弟关于橄榄球与量子物理的随笔的捧场。也有网友认为文中关于体育统计学的论断是盲目拔高此运动项目，他们也许以为橄榄球中只有"计数"而没有统计学。其实美式橄榄球对于统计学这门严格的定量科学的贡献在所有球类中是首屈一指的，足以令足篮排三大球望尘莫及。在此随手用两本在美国大学里常用的统计学教科书举一下比较成熟的例子，至于专业文献中关于橄榄球统计学的文章更是不胜枚举。

(1) Montgomery, Peck & Vining (2001) "Introduction to Linear Regression Analysis", third edition. 此书第二章和第三章分别讲单元和多元线性回归分析，每章后面的第一道习题2.1和3.1的主要内容如下：

Table B.1 gives data concerning the performance of the 26 NFL teams in 1976. It is suspected that the number of yards gained rushing by opponents (x8) has an effect on the number of games won by a team (y).

2.1.a. Fit a simple linear regression model relating games won y to yards gained rushing by opponents x8.

3.1.a. Fit a multiple linear regression model relating the number of games won y to the team's passing yardage (x2), the percentage of rushing plays (x7), and the opponents' yards rushing (x8).

3.1.e. Using the partial F test, determine the contribution of x7 to the model. How is this partial F statistic related to the t test for beta7 calculated in 3.1.c?

其实以上这些习题只是供“回归分析”初学者练习的雕虫小技，回归分析在实践应用中最重要的方法是variable selection，这是本书第九章的内容。有趣的是，这个全书最重要的章节后的第一道习题依然是上面用过的美式橄榄球数据，由此可见橄榄球在统计学家心中的地位：

9.1 Consider the NFL data in Table B.1

a. Use the forward selection algorithm to select a subset regression model
b. Use the backward elimination algorithm to select a subset regression model
c. Use stepwise regression to select a subset regression model
d. Comment on the final model chosen by these three procedures

在此建议那些质疑橄榄球与统计学关系的网友用SAS或R做一下这3道习题，然后再不妨与兄弟交流一下感想，呵呵。

(2) 某些网友也许会认为Douglas Montgomery在统计学界的“能级”偏低，他的教科书权威性也许不够，还不足以证明美式橄榄球在统计学中的地位。那么Wilks Medal得主Donald Rubin够级别了吧？由他和Gelman等作者参与写作的"Bayesian Data Analysis"几乎无人不知。该书将橄榄球置于更加显赫的地位，头一章就用了整整一个section (共4页)举了橄榄球的例子---1.6 "Example of probability assignment: football point spreads"。在此书之后的两个重要章节里又用了橄榄球的例子，建议对贝氏统计学有兴趣的网友仔细研读此书。

若有愿意花时间读统计学论文者可以参考以下原始文献:

Stern, H.S. (1991) On the probability of winning a football game. American Statisticians 45: 179-183.

Harville, D. (1980) Predictions for NFL games with linear-model methodology. JASA 75: 516-524.

Glickman & Stern (1998) A state-space model for NFL scores. JASA 93: 25-35.