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

Freedom to vary and significance tests

Introduction

Statistical tests based on the Binomial distribution (z, χ², log-likelihood and Newcombe-Wilson tests) assume that the item in question is free to vary at each point. This simply means that

  • If we find f items under investigation (what we elsewhere refer to as ‘Type A’ cases) out of N potential instances, the statistical model of inference assumes that it must be possible for f to be any number from 0 to N.
  • Probabilities, p = f / N, are expected to fall in the range [0, 1].

Note: this constraint is a mathematical one. All we are claiming is that the true proportion in the population could conceivably range from 0 to 1. This property is not limited to strict alternation with constant meaning (onomasiological, “envelope of variation” studies). In semasiological studies, where we evaluate alternative meanings of the same word, these tests can also be legitimate.

However, it is common in corpus linguistics to see evaluations carried out against a baseline containing terms that simply cannot plausibly be exchanged with the item under investigation. The most obvious example is statements of the following type: “linguistic Item x increases per million words between category 1 and 2”, with reference to a log-likelihood or χ² significance test to justify this claim. Rarely is this appropriate.

Some terminology: If Type A represents say, the use of modal shall, most words will not alternate with shall. For convenience, we will refer to cases that will alternate with Type A cases as Type B cases (e.g. modal will in certain contexts).

The remainder of cases (other words) are, for the purposes of our study, not evaluated. We will term these invariant cases Type C, because they cannot replace Type A or Type B.

In this post I will explain that not only does introducing such ‘Type C’ cases into an experimental design conflate opportunity and choice, but it also makes the statistical evaluation of variation more conservative. Not only may we mistake a change in opportunity as a change in the preference for the item, but we also weaken the power of statistical tests and tend to reject significant changes (in stats jargon, “Type II errors”).

This problem of experimental design far outweighs differences between methods for computing statistical tests. Continue reading

Testing tests

Introduction

Over the last few months I have been looking at computationally evaluating confidence intervals and significance tests. This process has helped me sharpen up the recommendations I can give to researchers. I have updated some online papers and blog posts as a result.

This analysis has exposed a difference, rarely commented upon, between the optimum test for contingency (“χ²-type”) tests when independent variable samples are drawn from the same population or independent populations.

For 2 × 2 tests it is recommended to use a different test (Newcombe-Wilson) when the IV is sociolinguistic (e.g. genre, time, different subcorpora) or otherwise divides samples by participants, than when the same participant may be sampled in either value (e.g. when the IV is a lexical-grammatical variable).

Meta-comment: In a way this is another benefit of a blog — unlike traditional publication, I can quickly correct any problems or improve papers as a result of my discoveries or those of colleagues. However it also means I need to draw the attention of my readership to any changes.

Confidence intervals and significance tests are closely related, for reasons discussed here. So if we can evaluate a formula for a confidence interval in some way, then we can also potentially evaluate the test. Continue reading

Choosing the right test

Introduction

One of the most common questions a new researcher has to deal with is the following:

what is the right statistical test for my purpose?

To answer this question we must distinguish between

  1. different experimental designs, and
  2. optimum methods for testing significance.

In corpus linguistics, many research questions involve choice. The speaker can say shall or will, choose to add a postmodifying clause to an NP or not, etc. If we want to know what factors influence this choice then these factors are termed independent variables (IVs) and the choice is  the dependent variable (DV). These choices are mutually exclusive alternatives. Framing the research question like this immediately helps us focus in on the appropriate class of tests.  Continue reading

Some bêtes noires

There are a number of common issues in corpus linguistics papers.

  1. an extremely common tendency for authors to primarily cite frequencies normalised per million or thousand words (i.e. a per word baseline or multiple thereof),
  2. data is usually plotted without confidence intervals, so it is not possible to spot visually whether a perceived change might be statistically significant, and
  3. significance tests are often employed without a clear statement of what the test is evaluating.

Experimental design

The first issue may be unique to corpus linguistics, deriving from its particular historical origins.

It concerns the experimenter attempting to identify counterfactual alternates or select baselines. This is an experimental design question.

In the beginning was the Word.

Linguists examining volumes of plain text data (later supported by computing and part-of-speech tagging) invariably concentrated on the idea of the word as the unit of language. Collocation and concordancing sat alongside lexicography as the principal tools of the trade. “Statistics” here primarily concerned probabilistic measures of association between neighbouring words in order to find common patterns. This activity is of course perfectly fine, and allowed researchers to make huge gains in our understanding of language.

But…

Without labouring the point (which I do elsewhere on this blog), the corollary of the statement that language is grammatical is that if, instead of describing the distribution of words, n-grams, etc, we wish to investigate how language is produced, the word cannot be our primary focus. Continue reading

A statistics crib sheet

Confidence intervalsHandout

Confidence intervals on an observed rate p should be computed using the Wilson score interval method. A confidence interval on an observation p represents the range that the true population value, P (which we cannot observe directly) may take, at a given level of confidence (e.g. 95%).

Note: Confidence intervals can be applied to onomasiological change (variation in choice) and semasiological change (variation in meaning), provided that P is free to vary from 0 to 1 (see Wallis 2012). Naturally, the interpretation of significant change in either case is different.

Methods for calculating intervals employ the Gaussian approximation to the Binomial distribution.

Confidence intervals on Expected (Population) values (P)

The Gaussian interval about P uses the mean and standard deviation as follows:

mean xP = F/N,
standard deviation S ≡ √P(1 – P)/N.

The Gaussian interval about P can be written as P ± E, where E = z.S, and z is the critical value of the standard Normal distribution at a given error level (e.g., 0.05). Although this is a bit of a mouthful, critical values of z are constant, so for any given level you can just substitute the constant for z. [z(0.05) = 1.95996 to six decimal places.]

In summary:

Gaussian intervalP ± z√P(1 – P)/N.

Confidence intervals on Observed (Sample) values (p)

We cannot use the same formula for confidence intervals about observations. Many people try to do this!

Most obviously, if p gets close to zero, the error e can exceed p, so the lower bound of the interval can fall below zero, which is clearly impossible! The problem is most apparent on smaller samples (larger intervals) and skewed values of p (close to 0 or 1).

The Gaussian is a reasonable approximation for an as-yet-unknown population probability P, it is incorrect for an interval around an observation p (Wallis 2013a). However the latter case is precisely where the Gaussian interval is used most often!

What is the correct method?

Continue reading

Binomial confidence intervals and contingency tests

Abstract Paper (PDF)

Many statistical methods rely on an underlying mathematical model of probability which is based on a simple approximation, one that is simultaneously well-known and yet frequently poorly understood.

This approximation is the Normal approximation to the Binomial distribution, and it underpins a range of statistical tests and methods, including the calculation of accurate confidence intervals, performing goodness of fit and contingency tests, line-and model-fitting, and computational methods based upon these. What these methods have in common is the assumption that the likely distribution of error about an observation is Normally distributed.

The assumption allows us to construct simpler methods than would otherwise be possible. However this assumption is fundamentally flawed.

This paper is divided into two parts: fundamentals and evaluation. First, we examine the estimation of error using three approaches: the ‘Wald’ (Normal) interval, the Wilson score interval and the ‘exact’ Clopper-Pearson Binomial interval. Whereas the first two can be calculated directly from formulae, the Binomial interval must be approximated towards by computational search, and is computationally expensive. However this interval provides the most precise significance test, and therefore will form the baseline for our later evaluations.

We consider two further refinements: employing log-likelihood in computing intervals (also requiring search) and the effect of adding a correction for the transformation from a discrete distribution to a continuous one.

In the second part of the paper we consider a thorough evaluation of this range of approaches to three distinct test paradigms. These paradigms are the single interval or 2 × 1 goodness of fit test, and two variations on the common 2 × 2 contingency test. We evaluate the performance of each approach by a ‘practitioner strategy’. Since standard advice is to fall back to ‘exact’ Binomial tests in conditions when approximations are expected to fail, we simply count the number of instances where one test obtains a significant result when the equivalent exact test does not, across an exhaustive set of possible values.

We demonstrate that optimal methods are based on continuity-corrected versions of the Wilson interval or Yates’ test, and that commonly-held assumptions about weaknesses of χ² tests are misleading.

Log-likelihood, often proposed as an improvement on χ², performs disappointingly. At this level of precision we note that we may distinguish the two types of 2 × 2 test according to whether the independent variable partitions the data into independent populations, and we make practical recommendations for their use.

Introduction

Estimating the error in an observation is the first, crucial step in inferential statistics. It allows us to make predictions about what would happen were we to repeat our experiment multiple times, and, because each observation represents a sample of the population, predict the true value in the population (Wallis 2013).

Consider an observation that a proportion p of a sample of size n is of a particular type.

For example

  • the proportion p of coin tosses in a set of n throws that are heads,
  • the proportion of light bulbs p in a production run of n bulbs that fail within a year,
  • the proportion of patients p who have a second heart attack within six months after a drug trial has started (n being the number of patients in the trial),
  • the proportion p of interrogative clauses n in a spoken corpus that are finite.

We have one observation of p, as the result of carrying out a single experiment. We now wish to infer about the future. We would like to know how reliable our observation of p is without further sampling. Obviously, we don’t want to repeat a drug trial on cardiac patients if the drug may be adversely affecting their survival.

Continue reading