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).

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. Continue reading

The variance of Binomial distributions


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

Adapting variance for random-text sampling

Introduction Paper (PDF)

Conventional stochastic methods based on the Binomial distribution rely on a standard model of random sampling whereby freely-varying instances of a phenomenon under study can be said to be drawn randomly and independently from an infinite population of instances.

These methods include confidence intervals and contingency tests (including multinomial tests), whether computed by Fisher’s exact method or variants of log-likelihood, χ², or the Wilson score interval (Wallis 2013). These methods are also at the core of others. The Normal approximation to the Binomial allows us to compute a notion of the variance of the distribution, and is to be found in line fitting and other generalisations.

In many empirical disciplines, samples are rarely drawn “randomly” from the population in a literal sense. Medical research tends to sample available volunteers rather than names compulsorily called up from electoral or medical records. However, provided that researchers are aware that their random sample is limited by the sampling method, and draw conclusions accordingly, such limitations are generally considered acceptable. Obtaining consent is occasionally a problematic experimental bias; actually recruiting relevant individuals is a more common problem.

However, in a number of disciplines, including corpus linguistics, samples are not drawn randomly from a population of independent instances, but instead consist of randomly-obtained contiguous subsamples. In corpus linguistics, these subsamples are drawn from coherent passages or transcribed recordings, generically termed ‘texts’. In this sampling regime, whereas any pair of instances in independent subsamples satisfy the independent-sampling requirement, pairs of instances in the same subsample are likely to be co-dependent to some degree.

To take a corpus linguistics example, a pair of grammatical clauses in the same text passage are more likely to share characteristics than a pair of clauses in two entirely independent passages. Similarly, epidemiological research often involves “cluster-based sampling”, whereby each subsample cluster is drawn from a particular location, family nexus, etc. Again, it is more likely that neighbours or family members share a characteristic under study than random individuals.

If the random-sampling assumption is undermined, a number of questions arise.

  • Are statistical methods employing this random-sample assumption simply invalid on data of this type, or do they gracefully degrade?
  • Do we have to employ very different tests, as some researchers have suggested, or can existing tests be modified in some way?
  • Can we measure the degree to which instances drawn from the same subsample are interdependent? This would help us determine both the scale of the problem and arrive at a potential solution to take this interdependence into account.
  • Would revised methods only affect the degree of certainty of an observed score (variance, confidence intervals, etc.), or might they also affect the best estimate of the observation itself (proportions or probability scores)?

Continue reading

Impossible logistic multinomials


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. Continue reading

Binomial → Normal → Wilson


One of the questions that keeps coming up with students is the following.

What does the Wilson score interval represent, and why is it the right way to calculate a confidence interval based around an observation? 

In this blog post I will attempt to explain, in a series of hopefully simple steps, how we get from the Binomial distribution to the Wilson score interval. I have written about this in a more ‘academic’ style elsewhere, but I haven’t spelled it out in a blog post.
Continue reading

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

An unnatural probability?

Not everything that looks like a probability is.

Just because a variable or function ranges from 0 to 1, it does not mean that it behaves like a unitary probability over that range.

Natural probabilities

What we might term a natural probability is a proper fraction of two frequencies, which we might write as p = f/n.

  • Provided that f can be any value from 0 to n, p can range from 0 to 1.
  • In this formula, f and n must also be natural frequencies, that is, n stands for the size of the set of all cases, and f the size of a true subset of these cases.

This natural probability is expected to be a Binomial variable, and the formulae for z tests, χ² tests, Wilson intervals, etc., as well as logistic regression and similar methods, may be legitimately applied to such variables. The Binomial distribution is the expected distribution of such a variable if each observation is drawn independently at random from the population (an assumption that is not strictly true with corpus data).

Another way of putting this is that a Binomial variable expresses the number of individual events of Type A in a situation where an outcome of either A and B are possible. If we observe, say 8 out of 10 cases are of Type A, then we can say we have an observed probability of A being chosen, p(A | {A, B}), of 0.8. In this case, f is the frequency of A (8), and n the frequency of both A and B (10). See Wallis (2013a). Continue reading