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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Plotting confidence intervals on graphs

So: you’ve got some data, you’ve read up on confidence intervals and you’re convinced. Your data is a small sample from a large/infinite population (all of contemporary US English, say), and therefore you need to estimate the error in every observation. You’d like to plot a pretty graph like the one below, but you don’t know where to start.

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!

Of course this graph is not just pretty.

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Competition between choices over time

Introduction Paper (PDF)

Measuring choices over time implies examining competition between alternates.

This is a fairly obvious statement. However, some of the mathematical properties of this system are less well known. These inform the expected behaviour of observations, helping us correctly specify null hypotheses.

  • The proportion of {shall, will} utterances where shall is chosen, p(shall | {shall, will}), is in competition with the alternative probability of will (they are mutually exclusive) and bounded on a probabilistic scale.
  • The probability associated with each member of a set of alternates X = {xi}, which we might write as p(xi | X), is bounded, 0 ≤ p(xi | X) ≤ 1, and exhaustive, Σp(xi | X) = 1.

A bounded system behaves differently from an unbounded one. Every child knows that a ball bouncing in an alley behaves differently than in an open playground. ‘Walls’ direct motion toward the centre.

In this short paper we discuss two properties of competitive choice:

  1. the tendency for change to be S-shaped rather than linear, and
  2. how this has an impact on confidence intervals. Continue reading