Detecting direction in interaction evidence

IntroductionPaper (PDF)

I have previously argued (Wallis 2014) that interaction evidence is the most fruitful type of corpus linguistics evidence for grammatical research (and doubtless for many other areas of linguistics).

Frequency evidence, which we can write as p(x), the probability of x occurring, concerns itself simply with the overall distribution of a linguistic phenomenon x – such as whether informal written English has a higher proportion of interrogative clauses than formal written English. In order to calculate frequency evidence we must define x, i.e. decide how to identify interrogative clauses. We must also pick an appropriate baseline n for this evaluation, i.e. we need to decide whether to use words, clauses, or any other structure to identify locations where an interrogative clause may occur.

Interaction evidence is different. It is a statistical correlation between a decision that a writer or speaker makes at one part of a text, which we will label point A, and a decision at another part, point B. The idea is shown schematically in Figure 1. A and B are separate ‘decision points’ in a given relationship (e.g. lexical adjacency), which can be also considered as ‘variables’.

Figure 1: Associative inference from lexico-grammatical choice variable A to variable B (sketch).

This class of evidence is used in a wide range of computational algorithms. These include collocation methods, part-of-speech taggers, and probabilistic parsers. Despite the promise of interaction evidence, the majority of corpus studies tend to consist of discussions of frequency differences and distributions.

In this paper I want to look at applications of interaction evidence which are made more-or-less at the same time by the same speaker/writer. In such circumstances we cannot be sure that just because B follows A in the text, the decision relating to B was made after the decision at A.

Impossible logistic multinomials

Introduction

Recently, a number of linguists have begun to question the wisdom of assuming that linguistic change tends to follow an ‘S-curve’ or more properly, logistic, pattern. For example, Nevalianen (2015) offers a series of empirical observations that show that whereas data sometimes follows a continuous ‘S’, frequently this does not happen. In this short article I try to explain why this result should not be surprising.

The fundamental assumption of logistic regression is that a probability representing a true fraction, or share, of a quantity undergoing a continuous process of change by default follows a logistic pattern. This is a reasonable assumption in certain limited circumstances because an ‘S-curve’ is mathematically analogous to a straight line (cf. Newton’s first law of motion).

Regression is a set of computational methods that attempts to find the closest match between an observed set of data and a function, such as a straight line, a polynomial, a power curve or, in this case, an S-curve. We say that the logistic curve is the underlying model we expect data to be matched against (regressed to). In another post, I comment on the feasibility of employing Wilson score intervals in an efficient logistic regression algorithm.

We have already noted that change is assumed to be continuous, which implies that the input variable (x) is real and linear, such as time (and not e.g. probabilistic). In this post we discuss different outcome variable types. What are the ‘limited circumstances’ in which logistic regression is mathematically coherent?

• We assume probabilities are free to vary from 0 to 1.
• The envelope of variation must be constant, i.e. it must always be possible for an observed probability to reach 1.

Taken together this also means that probabilities are Binomial, not multinomial. Let us discuss what this implies.

EDS Resources

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

Session 15: Introduction to statistics

• Binomial – Normal – Wilson
• Binomial spreadsheet

Sessions 18 and 19: Statistics Workshops

• EDS example 1 (Levin 2013)
• EDS example 2 (Bowie et al 2013)
• 2 × 2 or 2 × 1 χ² tests

Suggested further reading

• Plotting confidence intervals (part 1) and part 2
• Robust and sound?
• Choosing the right test
• A statistics crib sheet
• z-squared: the origin and application of χ²
• Plotting the Wilson distribution
• Coping with imperfect data