Plotting the Wilson distribution

Introduction Full Paper (PDF)

We have discussed the Wilson score interval at length elsewhere (Wallis 2013a, b). Given an observed Binomial proportion p = f / n observations, and confidence level 1-α, the interval represents the two-tailed range of values where P, the true proportion in the population, is likely to be found. Note that f and n are integers, so whereas P is a probability, p is a proper fraction (a rational number).

The interval provides a robust method (Newcombe 1998, Wallis 2013a) for directly estimating confidence intervals on these simple observations. It can take a correction for continuity in circumstances where it is desired to perform a more conservative test and err on the side of caution. We have also shown how it can be employed in logistic regression (Wallis 2015).

The point of this paper is to explore methods for computing Wilson distributions, i.e. the analogue of the Normal distribution for this interval. There are at least two good reasons why we might wish to do this.

The first is to shed insight onto the performance of the generating function (formula), interval and distribution itself. Plotting an interval means selecting a single error level α, whereas visualising the distribution allows us to see how the function performs over the range of possible values for α, for different values of p and n.

A second good reason is to counteract the tendency, common in too many presentations of statistics, to present the Gaussian (‘Normal’) distribution as if it were some kind of ‘universal law of data’, a mistaken corollary of the Central Limit Theorem. This is particularly unwise in the case of observations of Binomial proportions, which are strictly bounded at 0 and 1. Continue reading

The variance of Binomial distributions

Introduction

Recently I’ve been working on a problem that besets researchers in corpus linguistics who work with samples which are not drawn randomly from the population but rather are taken from a series of sub-samples. These sub-samples (in our case, texts) may be randomly drawn, but we cannot say the same for any two cases drawn from the same sub-sample. It stands to reason that two cases taken from the same sub-sample are more likely to share a characteristic under study than two cases drawn entirely at random. I introduce the paper elsewhere on my blog.

In this post I want to focus on an interesting and non-trivial result I needed to address along the way. This concerns the concept of variance as it applies to a Binomial distribution.

Most students are familiar with the concept of variance as it applies to a Gaussian (Normal) distribution. A Normal distribution is a continuous symmetric ‘bell-curve’ distribution defined by two variables, the mean and the standard deviation (the square root of the variance). The mean specifies the position of the centre of the distribution and the standard deviation specifies the width of the distribution.

Common statistical methods on Binomial variables, from χ² tests to line fitting, employ a further step. They approximate the Binomial distribution to the Normal distribution. They say, although we know this variable is Binomially distributed, let us assume the distribution is approximately Normal. The variance of the Binomial distribution becomes the variance of the equivalent Normal distribution.

In this methodological tradition, the variance of the Binomial distribution loses its meaning with respect to the Binomial distribution itself. It seems to be only valuable insofar as it allows us to parameterise the equivalent Normal distribution.

What I want to argue is that in fact, the concept of the variance of a Binomial distribution is important in its own right, and we need to understand it with respect to the Binomial distribution, not the Normal distribution. Sometimes it is not necessary to approximate the Binomial to the Normal, and if we can avoid this approximation our results are likely to be stronger as a result.

Continue reading