Correcting for continuity

Introduction

Many conventional statistical methods employ the Normal approximation to the Binomial distribution (see Binomial → Normal → Wilson), either explicitly or buried in formulae.

The well-known Gaussian population interval (1) is

Gaussian interval (E⁻, E⁺) ≡ P ± zP(1 – P)/n,(1)

where n represents the size of the sample, and z the two-tailed critical value for the Normal distribution at an error level α, more properly written zα/2. The standard deviation of the population proportion P is S = √P(1 – P)/n, so we could abbreviate the above to (E⁻, E⁺) ≡ P ± zS.

When these methods require us to calculate a confidence interval about an observed proportion, p, we must invert the Normal formula using the Wilson score interval formula (Equation (2)).

Wilson score interval (w⁻, w⁺) ≡ [p + z²/2n ± zp(1 – p)/n + z²/4] / [1 + z²/n].(2)

In a 2013 paper for JQL (Wallis 2013a), I referred to this inversion process as the ‘interval equality principle’. This means that if (1) is calculated for p = E⁻ (the Gaussian lower bound of P), then the upper bound that results, w⁺, will equal P. Similarly, for p = E⁺, the lower bound of pw⁻ will equal P.

We might write this relationship as

p ≡ GaussianLower(WilsonUpper(p, n, α), n, α), or, alternatively
P ≡ WilsonLower(GaussianUpper(P, n, α), n, α), etc. (3)

where E⁻ = GaussianLower(P, n, α), w⁺ = WilsonUpper(p, n, α), etc.

Note. The parameters n and α become useful later on. At this stage the inversion concerns only the first parameter, p or P.

Nonetheless the general principle is that if you want to calculate an interval about an observed proportion p, you can derive it by inverting the function for the interval about the expected population proportion P, and swapping the bounds (so ‘Lower’ becomes ‘Upper’ and vice versa).

In the paper, using this approach I performed a series of computational evaluations of the performance of different interval calculations, following in the footsteps of more notable predecessors. Comparison with the analogous interval calculated directly from the Binomial distribution showed that a continuity-corrected version of the Wilson score interval performed accurately. Continue reading

UCL Summer School in English Corpus Linguistics 2019

I am pleased to announce the seventh annual Summer School in English Corpus Linguistics to be held at University College London from 1-3 July.

The Summer School is a short three-day intensive course aimed at PhD-level students and researchers who wish to get to grips with Corpus Linguistics.

Please note that this course is very popular, and numbers are deliberately limited on a first-come, first-served basis! You will be taught in a small group by a teaching team.

Each day begins with a theory lecture, followed by a guided hands-on workshop with corpora, and a more self-directed and supported practical session in the afternoon.


Continue reading

The other end of the telescope

Introduction

The standard approach to teaching (and thus thinking about) statistics is based on projecting distributions of ranges of expected values. The distribution of an expected value is a set of probabilities that predict what the value will be, according to a mathematical model of what you predict should happen.

For the experimentalist, this distribution is the imaginary distribution of very many repetitions of the same experiment that you may have just undertaken. It is the output of a mathematical model.

  • Note that this idea of a projected distribution is not the same as the term ‘expected distribution’. An expected distribution is a series of values you predict your data should match.
  • Thus in what follows we simply compare a single expected value P with an observed value p. This can be thought of as comparing the expected distribution E = {P, 1 – P} with the observed distribution O = {p, 1 – p}.

Thinking about this projected distribution represents a colossal feat of imagination: it is a projection of what you think would happen if only you had world enough and time to repeat your experiment, again and again. But often you can’t get more data. Perhaps the effort to collect your data was huge, or the data is from a finite set of available data (historical documents, patients with a rare condition, etc.). Actual replication may be impossible for material reasons.

In general, distributions of this kind are extremely hard to imagine, because they are not part of our directly-observed experience. See Why is statistics difficult? for more on this. So we already have an uphill task in getting to grips with this kind of reasoning.

Significant difference (often shortened to ‘significance’) refers to the difference between your observations (the ‘observed distribution’) and what you expect to see (the expected distribution). But to evaluate whether a numerical difference is significant, we have to take into account both the shape and spread of this projected distribution of expected values.

When you select a statistical test you do two things:

  • you choose a mathematical model which projects a distribution of possible values, and
  • you choose a way of calculating significant difference.

The problem is that in many cases it is very difficult to imagine this projected distribution, or — which amounts to the same thing — the implications of the statistical model.

When tests are selected, the main criterion you have to consider concerns the type of data being analysed (an ‘ordinal scale’, a ‘categorical scale’, a ‘ratio scale’, and so on). But the scale of measurement is only one of several parameters that allows us to predict how random selection might affect the resampling of data.

A mathematical model contains what are usually called assumptions, although it might be more accurate to call them ‘preconditions’ or parameters. If these assumptions about your data are incorrect, the test is likely to give an inaccurate result. This principle is not either/or, but can be thought of as a scale of ‘degradation’. The less the data conforms to these assumptions, the more likely your test is to give the wrong answer.

This is particularly problematic in some computational applications. The programmer could not imagine the projected distribution, so they tweaked various parameters until the program ‘worked’. In a ‘black-box’ algorithm this might not matter. If it appears to work, who cares if the algorithm is not very principled? Performance might be less than optimal, but it may still produce valuable and interesting results.

But in science there really should be no such excuse.

The question I have been asking myself for the last ten years or so is simply can we do better? Is there a better way to teach (and think about) statistics than from the perspective of distributions projected by counter-intuitive mathematical models (taken on trust) and significant tests? Continue reading

Plotting the Wilson distribution

Introduction Full Paper (PDF)

We have discussed the Wilson score interval at length elsewhere (Wallis 2013a, b). Given an observed Binomial proportion p = f / n observations, and confidence level 1-α, the interval represents the two-tailed range of values where P, the true proportion in the population, is likely to be found. Note that f and n are integers, so whereas P is a probability, p is a proper fraction (a rational number).

The interval provides a robust method (Newcombe 1998, Wallis 2013a) for directly estimating confidence intervals on these simple observations. It can take a correction for continuity in circumstances where it is desired to perform a more conservative test and err on the side of caution. We have also shown how it can be employed in logistic regression (Wallis 2015).

The point of this paper is to explore methods for computing Wilson distributions, i.e. the analogue of the Normal distribution for this interval. There are at least two good reasons why we might wish to do this.

The first is to shed insight onto the performance of the generating function (formula), interval and distribution itself. Plotting an interval means selecting a single error level α, whereas visualising the distribution allows us to see how the function performs over the range of possible values for α, for different values of p and n.

A second good reason is to counteract the tendency, common in too many presentations of statistics, to present the Gaussian (‘Normal’) distribution as if it were some kind of ‘universal law of data’, a mistaken corollary of the Central Limit Theorem. This is particularly unwise in the case of observations of Binomial proportions, which are strictly bounded at 0 and 1. Continue reading

Mathematical operations with the Normal distribution

This post is a little off-topic, as the exercise I am about to illustrate is not one that most corpus linguists will have to engage in.

However, I think it is a good example of why a mathematical approach to statistics (instead of the usual rote-learning of tests) is extremely valuable.

Case study: The declared ‘deficit’ in the USS pension scheme

At the time of writing (March 2018) nearly two hundred thousand university staff in the UK are active members of a pension scheme called USS. This scheme draws in income from these members and pays out to pensioners. Every three years the pension is valued, which is not a simple process. The valuation consists of two aspects, both uncertain:

  • to value the liabilities of the pension fund, which means the obligations to current pensioners and future pensioners (current active members), and
  • to estimate the future asset value of the pension fund when the scheme is obliged to pay out to pensioners.

What happened in 2017 (and happened in the last two valuations) is that the pension fund has been declared to be in deficit, meaning that the liabilities are greater than the assets. However, in all cases this ‘deficit’ is a projection forwards in time. We do not know how long people will actually live, so we don’t know how much it will cost to pay them a pension. And we don’t know what the future values of assets held by the pension fund will be.

The September valuation

In September 2017, the USS pension fund published a table which included two figures using the method of accounting they employed at the time to value the scheme.

  • They said the best estimate of the outcome was a surplus of £8.3 billion.
  • But they said that the deficit allowing for uncertainty (‘prudence’) was –£5.1 billion.

Now, if a pension fund is in deficit, it matters a great deal! Someone has to pay to address the deficit. Either the rules of the pension fund must change (so cutting the liabilities) or the assets must be increased (so the employers and/or employees, who pay into the pension fund must pay more). The dispute about the deficit engulfed UK universities in March 2018 with strikes by many tens of thousands of staff, lectures cancelled, etc. But is there really a ‘deficit’, and if so, what does this tell us?

The first additional bit of information we need to know is how the ‘uncertainty’ is modelled. In February 2018 I got a useful bit of information. The ‘deficit’ is the lower bound on a 33% confidence interval (α = 2/3). This is an interval that divides the distribution into thirds by area. One third is below the lower bound, one third above the upper bound, and one third is in the middle. This gives us a picture that looks something like this:

Figure 1: Sketch of the probability distribution of the difference between USS assets and liabilities projected on September valuation assumptions (delayed ‘de-risking’).

Of course, experimentalist statisticians will never use such an error-prone confidence interval. We wouldn’t touch anything below 95% (α = 0.05)! To make things a bit more confusing, the actuaries talk about this having a ‘67% level of prudence’ meaning that two-thirds of the distribution is above the lower bound. All of this is fine, but it means we must proceed with care to decode the language and avoid making mistakes.

In any case, the distribution of this interval is approximately Normal. The detailed graphs I have seen of USS’s projections are a bit more shaky (which makes them appear a bit more ‘sciency’), but let’s face it, these are projections with a great deal of uncertainty. It is reasonable to employ a Normal approximation and use a ‘Wald’ interval in this case because the interval is pretty much unbounded – the outcome variable could eventually fall over a large range. (Note that we recommend Wilson intervals on probability ranges precisely because probability p is bounded by 0 and 1.) Continue reading

The confidence of diversity

Introduction

Occasionally it is useful to cite measures in papers other than simple probabilities or differences in probability. When we do, we should estimate confidence intervals on these measures. There are a number of ways of estimating intervals, including bootstrapping and simulation, but these are computationally heavy.

For many measures it is possible to derive intervals from the Wilson score interval by employing a little mathematics. Elsewhere in this blog I discuss how to manipulate the Wilson score interval for simple transformations of p, such as 1/p, 1 – p, etc.

Below I am going to explain how to derive an interval for grammatical diversity, d, which we can define as the probability that two randomly-selected instances have different outcome classes.

Diversity is an effect size measure of a frequency distribution, i.e. a vector of k frequencies. If all frequencies are the same, the data is evenly spread, and the score will tend to a maximum. If all frequencies except one are zero, the chance of picking two different instances will of course be zero. Diversity is well-behaved except where categories have frequencies of 1. Continue reading

Why is statistics difficult?

Imagine you are somewhere on a road that you have never been on before. Picture it. It’s peaceful and calm. A car comes down the road. As it gets to a corner, the driver appears to lose control, and the car crashes into a wall. Fortunately the driver is OK, but they can’t recall what happened.

Let’s think about what you experienced. The car crash might involve a number of variables an investigator would be interested in.

  • How fast was the car going? Where were the brakes applied?
  • Look on the road. Get out a tape measure. How long was the skid before the car finally stopped?
  • How big and heavy was the car? How loud was the bang when the car crashed?

These are all physical variables. We are used to thinking about the world in terms of these kinds of variables: velocity, position, length, volume and mass. They are tangible: we can see and touch them, and we have physical equipment that helps us measure them. Continue reading