Logistic regression with Wilson intervals

Introduction

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).
Logistic curve (k = 1) with Wilson score intervals for n = 10, 100.

Logistic curve (k = 1) with Wilson score intervals for n = 10, 100.

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Change and certainty: plotting confidence intervals (2)

Introduction

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).

An example graph plot showing the changing proportions of meanings of the verb think over time in the US TIME Magazine Corpus, with Wilson score intervals, after Levin (2013). Many thanks to Magnus for the data!

In this post I discuss ways in which we can plot intervals on changes (differences) rather than single probabilities.

The clearer our visualisations, the better we can understand our own data, focus our explanations on significant results and communicate our results to others. Continue reading