Back in 2010 I wrote a short article on the logistic (‘S’) curve in which I described its theoretical justification, mathematical properties and relationship to the Wilson score interval. This observed two key points.
- We can map any set of independent probabilities p ∈ [0, 1] to a flat Cartesian space using the inverse logistic (‘logit’) function, defined as
- logit(p) ≡ log(p / 1 – p) = log(p) – log(1 – p),
- where ‘log’ is the natural logarithm and logit(p) ∈ [-∞, ∞].
- By performing this transformation
- the logistic curve in probability space becomes a straight line in logit space, and
- Wilson score intervals for p ∈ (0, 1) are symmetrical in logit space, i.e. logit(p) – logit(w⁻) = logit(w⁺) – logit(p).