Logistic Regression and Bill James

James reasoned that with \( p_{a,b} \) defined as the probability that Team A will beat team B, \( a \) as Team A's true winning percentage (the probability that they would beat an average team) and, \( b \) as Team B's true winning percentage, the following six statements must be true:

1. \( p_{a,a} = 0.5 \) (note that this rule concerns two teams of equal quality not a team playing itself)

2. \( p_{a,0.5} = a \)

3. If \( a > b \) then \( p_{a,b} > 0.5 \) and if \( a < b \) then \( p_{a,b} < 0.5 \)

4. If \( b < 0.5 \) then \( p_{a,b} > a \) and if \( b > 0.5 \) then \( p_{a,b} < a \)

5. \( 0 \leq p_{a,b} \leq 1 \) and if \( 0 < a < 1 \) then \( p_{a,0}=1 \) and \( p_{a,1}=0. \)

6. \( p_{a,b} + p_{b,a} = 1 \)

log5 <- function(a, b){
  (a - a*b)/(a + b - 2*a*b)
}

log5(0.7, 0.4)

[1] 0.7777778

log5(0.7, 0.5)

[1] 0.7

log5(0.7, 1)

[1] 0

What James had found was that while the probability of Team A beating Team B could not be expressed as a linear combination of team qualities, the log odds of Team A's chance of victory could be.

This suggests that our regression model should predict the log odds of the outcome (a win or a loss) rather than the outcome itself. In the statistics talk, a model that predicts a function of the outcome varible is called a “generalized” linear model and the function that you choose is called the “link function”. The log odds function, \( log(\frac{y}{1-y}) \), is known as the logistic link and a regression model that uses the logistic link is known as logistic regression.