The variance of Binomial distributions

Introduction

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.

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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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What might a corpus of parsed spoken data tell us about language?

AbstractPaper (PDF)

This paper summarises a methodological perspective towards corpus linguistics that is both unifying and critical. It emphasises that the processes involved in annotating corpora and carrying out research with corpora are fundamentally cyclic, i.e. involving both bottom-up and top-down processes. Knowledge is necessarily partial and refutable.

This perspective unifies ‘corpus-driven’ and ‘theory-driven’ research as two aspects of a research cycle. We identify three distinct but linked cyclical processes: annotation, abstraction and analysis. These cycles exist at different levels and perform distinct tasks, but are linked together such that the output of one feeds the input of the next.

This subdivision of research activity into integrated cycles is particularly important in the case of working with spoken data. The act of transcription is itself an annotation, and decisions to structurally identify distinct sentences are best understood as integral with parsing. Spoken data should be preferred in linguistic research, but current corpora are dominated by large amounts of written text. We point out that this is not a necessary aspect of corpus linguistics and introduce two parsed corpora containing spoken transcriptions.

We identify three types of evidence that can be obtained from a corpus: factual, frequency and interaction evidence, representing distinct logical statements about data. Each may exist at any level of the 3A hierarchy. Moreover, enriching the annotation of a corpus allows evidence to be drawn based on those richer annotations. We demonstrate this by discussing the parsing of a corpus of spoken language data and two recent pieces of research that illustrate this perspective. Continue reading

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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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.

A methodological progression

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

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Verb Phrase book published

Why this book?

book coverThe grammar of English is often thought to be stable over time. However a new book, edited by Bas Aarts, Joanne Close, Geoffrey Leech and Sean Wallis, The Verb Phrase in English: investigating recent language change with corpora (Cambridge University Press, 2013) presents a body of research from linguists that shows that using natural language corpora one can find changes within a core element of grammar, the Verb Phrase, over a span of decades rather than centuries.

The book draws from papers first presented at a symposium on the verb phrase organised for the Survey of English Usage’s 50th anniversary and on research from the Changing English Verb Phrase project.

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Three kinds of corpus evidence – and two types of constraint

Text corpora permit researchers to find evidence of three distinct kinds.

1. Frequency evidence of known terms (‘performance’)

Suppose you have a plain text corpus which you attempt to annotate automatically. You apply a computer program to the text. This program can be thought of as comprising three elements: a theoretical framework or ‘scheme’, an algorithm, and a knowledge-base (KB). Terms and constituents in this scheme are applied to the corpus according to the algorithm.

Having done so it should be a relatively simple matter to index those terms in the corpus and obtain frequencies for each one (e.g., how many instances of may are classed as a modal verb, noun, etc). The frequency evidence obtained tells you how the program performed against the real-world data in the corpus. However, if you stop at this point you do not know whether this evidence is accurate or complete.

2. Factual evidence of unknown terms (‘discovery’)

The process of annotation presents the opportunity for discovery of novel linguistic events. All NLP algorithms have a particular, and inevitably less-than perfect, performance. The system may misclassify some items, misanalyse constituents, or simply fail. Therefore

  1. first-pass frequency evidence is likely to be inaccurate (and potentially incomplete),
  2. errors may be due to inadequacies in the scheme, algorithm or knowledge-base.

In practice we have two choices: amend the system (scheme, KB or algorithm) and/or correct the corpus manually. A law of diminishing returns applies, and a certain amount of manual editing is inevitably necessary. [As a side comment, part-of-speech annotation is relatively accurate, but full parsing is prone to error. As different systems employ different frameworks accuracy rates vary, but one can anticipate around 95% accuracy for POS-tagging and at best 70% accuracy for parsing. In any case, some errors may be impossible to address without a deeper semantic analysis of the sentence than is feasible.]

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