美式橄欖球與統計學

Omni

感謝眾網友對兄弟關於橄欖球與量子物理的隨筆的捧場。也有網友認為文中關於體育統計學的論斷是盲目拔高此運動項目，他們也許以為橄欖球中只有"計數"而沒有統計學。其實美式橄欖球對於統計學這門嚴格的定量科學的貢獻在所有球類中是首屈一指的，足以令足籃排三大球望塵莫及。在此隨手用兩本在美國大學裡常用的統計學教科書舉一下比較成熟的例子，至於專業文獻中關於橄欖球統計學的文章更是不勝枚舉。

(1) Montgomery, Peck & Vining (2001) "Introduction to Linear Regression Analysis", third edition. 此書第二章和第三章分別講單元和多元線性回歸分析，每章後面的第一道習題2.1和3.1的主要內容如下：

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?

其實以上這些習題只是供“回歸分析”初學者練習的雕蟲小技，回歸分析在實踐應用中最重要的方法是variable selection，這是本書第九章的內容。有趣的是，這個全書最重要的章節後的第一道習題依然是上面用過的美式橄欖球數據，由此可見橄欖球在統計學家心中的地位：

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

在此建議那些質疑橄欖球與統計學關係的網友用SAS或R做一下這3道習題，然後再不妨與兄弟交流一下感想，呵呵。

(2) 某些網友也許會認為Douglas Montgomery在統計學界的“能級”偏低，他的教科書權威性也許不夠，還不足以證明美式橄欖球在統計學中的地位。那麼Wilks Medal得主Donald Rubin夠級別了吧？由他和Gelman等作者參與寫作的"Bayesian Data Analysis"幾乎無人不知。該書將橄欖球置於更加顯赫的地位，頭一章就用了整整一個section (共4頁)舉了橄欖球的例子---1.6 "Example of probability assignment: football point spreads"。在此書之後的兩個重要章節里又用了橄欖球的例子，建議對貝氏統計學有興趣的網友仔細研讀此書。

若有願意花時間讀統計學論文者可以參考以下原始文獻:

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.