The confidence of diversity

Introduction

Occasionally it is useful to cite measures in papers other than simple probabilities or differences in probability. When we do, we should estimate confidence intervals on these measures. There are a number of ways of estimating intervals, including bootstrapping and simulation, but these are computationally heavy.

For many measures it is possible to derive intervals from the Wilson score interval by employing a little mathematics. Elsewhere in this blog I discuss how to manipulate the Wilson score interval for simple transformations of p, such as 1/p, 1 – p, etc.

Below I am going to explain how to derive an interval for grammatical diversity, d, which we can define as the probability that two randomly-selected instances have different outcome classes.

Diversity is an effect size measure of a frequency distribution, i.e. a vector of k frequencies. If all frequencies are the same, the data is evenly spread, and the score will tend to a maximum. If all frequencies except one are zero, the chance of picking two different instances will of course be zero. Diversity is well-behaved except where categories have frequencies of 1. Continue reading

Is “grammatical diversity” a useful concept?

Introduction

In a recent paper focusing on distributions of simple NPs (Aarts and Wallis, 2014), we found an interesting correlation across text genres in a corpus between two independent variables. For the purposes of this study, a “simple NP” was an NP consisting of a single-word head. What we found was a strong correlation between

  1. the probability that an NP consists of a single-word head, p(single head), and
  2. the probability that single-word heads were a personal pronoun, p(personal pronoun | single head).

Note that these two variables are independent because they do not compete, unlike, say, the probability that a single-word NP consists of a noun, vs. the probability that it is a pronoun. The scattergraph below illustrates the distribution and correlation clearly.

Scattergraph of text genres in ICE-GB; distributed (horizontally) by the proportion of all noun phrases consisting of a single word and (vertically) by the proportion of those NPs that are personal pronouns; spoken and written, with selected outliers identified.

Scattergraph of text genres in ICE-GB; distributed (horizontally) by the proportion of all noun phrases consisting of a single word and (vertically) by the proportion of those single-word NPs that are personal pronouns; spoken and written, with selected outliers identified.

Continue reading