EDS Resources

This post contains the resources for students taking the UCL English Linguistics MA, all in one place.

Session 15: Introduction to statistics

Sessions 18 and 19: Statistics Workshops

Suggested further reading

Comparing frequencies within a discrete distribution

Note:
This page explains how to compare observed frequencies f₁ and f₂ from the same distributionF = {f₁, f₂,…}. To compare observed frequencies f₁ and f₂ from different distributions, i.e. where F₁ = {f₁,…} and F₂ = {f₂,…}, you need to use a chi-square or Newcombe-Wilson test.

Introduction

In a recent study, my colleague Jill Bowie obtained a discrete frequency distribution by manually classifying cases in a small sample drawn from a large corpus.

Jill converted this distribution into a row of probabilities and calculated Wilson score intervals on each observation, to express the uncertainty associated with a small sample. She had one question, however:

How do we know whether the proportion of one quantity is significantly greater than another?

We might use a Newcombe-Wilson test (see Wallis 2013a), but this test assumes that we want to compare samples from independent sources. Jill’s data are drawn from the same sample, and all probabilities must sum to 1. Instead, the optimum test is a dependent-sample test.

Example

A discrete distribution looks something like this: F = {108, 65, 6, 2}. This is the frequency data for the middle column (circled) in the following chart.

This may be converted into a probability distribution P, representing the proportion of examples in each category, by simply dividing by the total: P = {0.60, 0.36, 0.03, 0.01}, which sums to 1.

We can plot these probabilities, with Wilson score intervals, as shown below.

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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). In this post we discuss the 1960s data (circled). The sum of each column probability is 1. Many thanks to Magnus for the data!

So how do we know if one proportion is significantly greater than another?

  • When comparing values diachronically (horizontally), data is drawn from independent samples. We may use the Newcombe-Wilson test, and employ the handy visual rule that if intervals do not overlap they must be significantly different.
  • However, probabilities drawn from the same sample (vertically) sum to 1 — which is not the case for independent samples! There are k−1 degrees of freedom, where k is the number of classes. It turns out that the relevant significance test we need to use is an extremely basic test, but it is rarely discussed in the literature.

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A methodological progression

(with thanks to Jill Bowie)

Introduction

One of the most controversial arguments in corpus linguistics concerns the relationship between a ‘variationist’ paradigm comparable with lab experiments, and a traditional corpus linguistics paradigm focusing on normalised word frequencies.

Rather than see these two approaches as diametrically opposed, we propose that it is more helpful to view them as representing different points on a methodological progression, and to recognise that we are often forced to compromise our ideal experimental practice according to the data and tools at our disposal.

Viewing these approaches as being represented along a progression allows us to step back from any single perspective and ask ourselves how different results can be reconciled and research may be improved upon. It allows us to consider the potential value in performing more computer-aided manual annotation — always an arduous task — and where such annotation effort would be usefully focused.

The idea is sketched in the figure below.

progression
A methodological progression: from normalised word frequencies to verified alternation.

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