This paper reviews the problems that bedevil the selection of an appropriate test for the analysis of a 2 × 2 table. In contradiction to an earlier paper, the author now argues the case for the use of Fisher's exact test. It is noted that all test statistics for the 2 × 2 table have discrete distributions and it is suggested that it is irrational to prescribe an unattainable fixed significance level. The use of mid-P is suggested, if a formula is required for prescribing a variable tail probability. The problems of two-tail tests are discussed

Fisher's Exact Test

PURPOSE: To assess the frequency with which a particular, possibly optimal work-up of noncalcified nodules less than 5.0 mm in diameter identified on initial computed tomographic (CT) images at baseline screening leads to a diagnosis of malignancy prior to first annual repeat screening, compared with a possibly optimal work-up of larger nodules. MATERIALS AND METHODS: Two series of baseline CT screenings in high-risk people were retrospectively reviewed. The first series (n = 1,000) was performed in 1993–1998; the second (n = 1,897), in 1999–2002. In each series, cases in which the largest noncalcified nodule detected was less than 5.0 mm in diameter and those in which it was 5.0–9 mm were reviewed to determine whether diagnostic work-up prior to first annual repeat screening showed or would have shown nodule growth and led or would have led to a diagnosis based on biopsy or surgical specimens. RESULTS: The frequency with which malignancy was or could have been diagnosed when the largest noncalcified nodu...

CT Screening for Lung Cancer: Suspiciousness of Nodules according to Size on Baseline Scans

In discrete distributions, it has often been recommended that an auxiliary random experiment should be carried out so as to make the size of the test equal to the significance level. In certain experimental situations, this procedure would be time-consuming and even embarrassing to the statistician. An alternative criterion of significance is suggested, the mid or median probability. To avoid computations P(χ), the probability of the square root of a chi-square variable, can be used in the binomial, Poisson and hypergeometric distributions as an approximation to the mid probability. In statistical control of counting experiments or where experiments are being repeated P(χ) rather than P(χ c ), the probability of χ corrected for continuity, gives acceptable approximations to size of significance levels. Some computations are given for the multinomial distribution which show that here P(χ2) gives acceptable approximations.

Significance Tests in Discrete Distributions

P-values: Interpretation and Methodology

SUMMARY Significance tests that are based on discrete probabilities are conservative in that the average value of the significance level, when the null hypothesis is true, always exceeds 0.5. An approach suggested by H. 0. Lancaster over 40 years ago overcomes this problem. This is to calculate the mid-P value, where only half of the probability of the observed sample is included in the tail. The average value of the mid-P value is 0.5 and the variance is slightly less than that of a random variable uniformly distributed between 0 and 1. The mid-P concept has usually been advocated in the context of significance testing but it can be extended to the calculation of confidence intervals in an estimation approach by defining, for example, the 95% mid-P confidence limits as the values that have a one-sided mid-P value of 0.025. In this paper we review recent work supporting this approach.

https://www.jstor.org/stable/pdfplus/2348891.pdf

H. O. Lancaster

Papers

Statistical control of counting experiments