The R Book

Odds

The logistic model for p as a function of x is given by

and there are no prizes for realizing that the model is not linear. But if x = −∞, then p = 0, and if x = +∞ then p = 1, so the model is strictly bounded. If x = 0, then p = exp(a)/[1 + exp(a)]. The trick of linearizing the logistic model actually involves a very simple transformation. You may have come across the way in which bookmakers specify probabilities by quoting the odds against a particular horse winning a race (they might give odds of 2 to 1 on a reasonably good horse or 25 to 1 on an outsider). This is a rather different way of presenting information on probabilities than scientists are used to dealing with. Thus, where the scientist might state a proportion as 0.667 (2 out of 3), the bookmaker would give odds of 2 to 1 (2 successes to 1 failure). In symbols, this is the difference between the scientist stating the probability p, and the bookmaker stating the odds p/q. Now if we take the odds p/q and substitute this into the formula for the logistic, we get

which looks awful. But a little algebra shows that

Now, taking natural logs and recalling that ln(e^x) = x will simplify matters even further, so that

This ...