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 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

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)?

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Coping with imperfect data

Introduction

One of the challenges for corpus linguists is that many of the distinctions that we wish to make are either not annotated in a corpus at all or, if they are represented in the annotation, unreliably annotated. This issue frequently arises in corpora to which an algorithm has been applied, but where the results have not been checked by linguists, a situation which is unavoidable with mega-corpora. However, this is a general problem. We would always recommend that cases be reviewed for accuracy of annotation.

A version of this issue also arises when checking for the possibility of alternation, that is, to ensure that items of Type A can be replaced by Type B items, and vice-versa. An example might be epistemic modal shall vs. will. Most corpora, including richly-annotated corpora such as ICE-GB and DCPSE, do not include modal semantics in their annotation scheme. In such cases the issue is not that the annotation is “imperfect”, rather that our experiment relies on a presumption that the speaker has the choice of either type at any observed point (see Aarts et al. 2013), but that choice is conditioned by the semantic content of the utterance.

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Binomial → Normal → Wilson

Introduction

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.
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Choice vs. use

Introduction

Many linguistic researchers are interested in semasiological variation, that is, how the meaning of words and expressions may be observed to vary over time or space. One word might have one dominant meaning or use at one point in time, and other meanings may supplant them. This is of obvious interest to etymology. How do new meanings come about? Why do others decline? Do old meanings die away or retain a specialist use?

Most of the research we have discussed on this blog is, by contrast, concerned with onomasiological variation, or variation in the choice of words or expressions to express the same meaning. In a linguistic choice experiment, the field of meaning is held to be constant, or approximately so, and we are concerned primarily with language production:

  • Given that a speaker (or writer, but we take speech as primary) wishes to express some thought, T, what is the probability that they will use expression E₁ out of the alternate forms {E₁, E₂,…} to express it?

This probability is meaningful in the language production process: it measures the actual use out of the options available to the speaker, at the point of utterance.

Conversely, semasiological researchers are concerned with a different type of probability:

  • Given that a speaker used an expression E, what is the probability that their meaning was T₁ out of the set of {T₁, T₂,…}?

For the hearer, this measure can also be thought of as the exposure rate: what proportion of times should a hearer (reader) interpret E as expressing T₁? This probability is meaningful to a language receiver, but it is not a meaningful statistic at the point of language production.

From the speaker’s point of view we can think of onomasiological variation as variation in choice, and semasiological variation as variation in relative proportion of use.

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Reciprocating the Wilson interval

Introduction

How can we calculate confidence intervals on a property like sentence length (as measured by the number of words per sentence)?

You might want to do this to find out whether or not, say, spoken utterances consist of shorter or longer sentences than those found in writing.

The problem is that the average number of words per sentence is not a probability. If you think about it, this ratio will (obviously) equal or exceed 1. So methods for calculating intervals on probabilities won’t work without recalibration.

Aside: You are most likely to hit this type of problem if you want to plot a graph of some non-probabilistic property, or you wish to cite a property with an upper and lower bound for some reason. Sometimes expressing something as a probability does not seem natural. However, it is a good discipline to think in terms of probabilities, and to convert your hypotheses into hypotheses about probabilities as far as possible. As we shall see, this is exactly what you have to do to apply the Wilson score interval.

Note also that just because you want to calculate confidence intervals on a property, you also have to consider whether the property is freely varying when expressed as a probability.

The Wilson score interval (w⁻, w⁺), is a robust method for computing confidence intervals about probabilistic observations p.

Elsewhere we saw that the Wilson score interval obtained an accurate approximation to the ‘exact’ Binomial interval based on an observed probability p, obtained by search. It is also well-constrained, so that neither upper nor lower bound can exceed the probabilistic range [0, 1].

But the Wilson interval is based on a probability. In this post we discuss how this method can be used for other quantities.

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Binomial algorithm snippets

Introduction

Elsewhere on this blog I summarise an analysis of the performance of a broad range of different confidence interval calculations and 2 × 2 contingency tests against equivalent ‘exact’ Binomial tests calculated from first principles.

For transparency, it is necessary to show how I went about computing these results.

Many of these algorithms are summarised in mathematical terms in this paper. However, for those who wish to recreate the computation, here is the code in the programming language C.

Warning: colleagues have pointed out that this post is not for the faint hearted!

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