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).
In a previous post I discussed how to plot confidence intervals on observed probabilities. Using this method we can create graphs like the following. (Data is in the Excel spreadsheet we used previously: for this post I have added a second worksheet.)
The graph depicts both the observed probability of a particular form and the certainty that this observation is accurate. The ‘I’-shaped error bars depict the estimated range of the true value of the observation at a 95% confidence level (see Wallis 2013 for more details).
A note of caution: these probabilities are semasiological proportions (different uses of the same word) rather than onomasiological choices (see Choice vs. use).
In this post I discuss ways in which we can plot intervals on changes (differences) rather than single probabilities.