Deconstructing the chi-square

Introduction

Elsewhere in this blog we introduce the concept of statistical significance by considering the reliability of a single sampled observation of a Binomial proportion: an estimate of the probability of selecting an item in the future. This allows us to develop an understanding of the likely distribution of what the true value of that probability in the population might be. In short, were we to make future observations of that item, we could expect that each sampled probability would be found within a particular range – a confidence interval – a fixed proportion of times, such as 1 in 20 or 1 in 100. This ‘fixed proportion’ is termed the ‘error level’ because we predict that the true value will be outside the range 1 in 20 or 1 in 100 times.

This process of inferring about future observations is termed ‘inferential statistics’. Our approach is to build our understanding in a series of stages based on confidence intervals about the single proportion. Here we will approach the same question by deconstructing the chi-square test.

A core idea of statistical inference is this: randomness is a fact of life. If you sample the same phenomenon multiple times, drawing on different data each time, it is unlikely that the observation will be identical, or – to put it in terms of an observed sample – it is unlikely that the mean value of the observation will be the same. But you are more likely than not to find the new mean near the original mean, and the larger the size of your sample, the more reliable your estimate will be. This, in essence, is the Central Limit Theorem.

This principle applies to the central tendency of data, usually the arithmetic mean, but occasionally a median. It does not concern outliers: extreme but rare events (which, by the way, you should include, and not delete, from your data).

We are mainly concerned with Binomial or Multinomial proportions, i.e. the fraction of cases sampled which have a particular property. A Binomial proportion is a statement about the sample, a simple fraction p = f / n. But it is also the sample mean probability of selecting a value. Suppose we selected a random case from the sample. In the absence of any other knowledge about that case, the average chance that X = x₁ is also p.

The same principle applies to the mean of Real or Integer values, for which one might use Welch’s or Student’s t test, and the median rank of Ordinal data, for which a Mann-Whitney U test may be appropriate.

With this in mind, we can form an understanding of significance, or to be precise, significant difference. The ‘difference’ referred to here is the difference between an uncertain observed value and a predicted or known population value, d = pP, or the difference between two uncertain observed values, d = p₂ – p₁. The first of these differences is found in a single-sample z test, the second in a two-sample z test. See Wallis (2013b).

popsamp3
Figure 1. The single-sample population z test. The statistical model assumes that future unobserved samples are Normally distributed, centred on the population mean P. Distance d is compared with a critical threshold, zα/2.S, to carry out the test.

A significance test is created by comparing an observed difference with a second element, a critical threshold extrapolated from the underlying statistical model of variation. Continue reading “Deconstructing the chi-square”

Point tests and multi-point tests for separability of homogeneity

Introduction

I have been recently reviewing and rewriting a paper for publication that I first wrote back in 2011. The paper (Wallis 2019) concerns the problem of how we test whether repeated runs of the same experiment obtain essentially the same results, i.e. results are not significantly different from each other.

These meta-tests can be used to test an experiment for replication: if you repeat an experiment and obtain significantly different results on the first repetition, then, with a 1% error level, you can say there is a 99% chance that the experiment is not replicable.

These tests have other applications. You might be wishing to compare your results with those of others in the literature, compare results with a different operationalisation (definitions of variables), or just compare results obtained with different data – such as comparing a grammatical distribution observed in speech with that found within writing.

The design of tests for this purpose is addressed within the t-testing ANOVA community, where tests are applied to continuously-valued variables. The solution concerns a particular version of an ANOVA, called “the test for interaction in a factorial analysis of variance” (Sheskin 1997: 489).

However, anyone using data expressed as discrete alternatives (A, B, C etc) has a problem: the classical literature does not explain what you should do.

Gradient and point tests

gradient-point-test
Figure 1: Point tests (A) and gradient tests (B), from Wallis (forthcoming).

The rewrite of the paper caused me to distinguish between two types of tests: ‘point tests’, which I describe below, and ‘gradient tests’. Continue reading “Point tests and multi-point tests for separability of homogeneity”

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 “Adapting variance for random-text sampling”