### 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**= {*x*}, which we might write as_{i}*p*(*x*|_{i}**X**), is**bounded**, 0 ≤*p*(*x*|_{i}**X**) ≤ 1, and**exhaustive**, Σ*p*(*x*|_{i}**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:

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

### Excerpt

**S curves and Wilson intervals**

We can sketch the overall behaviour of the system by plotting Wilson intervals for the ‘S’ curve (below). We have plotted a logistic curve for *p* (*k* = 1) and added intervals for *n* = 10 and *n* = 100. Observe that with a small *n* the confidence interval is large and more highly skewed. The difference (*w*⁺ – *w*⁻) is greatest for *p* = 0.5.

### Contents

- The S curve
- Poisson error bars
- S curves and Wilson intervals

### Citation

Wallis, S.A. 2010. *Competition between choices over time*. London: Survey of English Usage, UCL. http://www.ucl.ac.uk/english-usage/statspapers/competition-over-time.pdf