On eye analyses

Sir,

The articles by Halberstadt et al,¹ Taner et al,² and Loukovaara et al³ illustrate systemic errors in statistical analysis. They use two-sample t-tests or analysis of variance (ANOVA), but ignore their shortcomings. These compare the means of normal populations assuming unknown homogeneous variances. While the Central Limit Theorem justifies normality for inferences on means, unknown variances need not be equal, making these tests unsuitable for general mean comparisons.

As the joint distribution of sample means of normal populations is a function of the ratio of their unknown variances, tests based on the difference between sample means of normal populations with unknown unequal variances are inexact, and not a t-test.⁴

This problem is not removed by meaninglessly⁵ testing for the equality of variances, or avoiding normality with its nuisance unknown variances with nonparametric rank tests such as the Wilcoxon test. Being a comparison of distributions, these rank tests say nothing specifically about the mean, median, or any moment of the distributions if significant. They are moreover biased⁶ to one side in a two-sided test.

Tsakok⁷ has solved this Behrens–Fisher problem of comparing the means of normal distributions with unknown variances at exact significance levels, showing that the Tsakok solution is more effective in detecting significant mean differences even with unknown equal variances. There is an indication⁸ that the Tsakok technique applies to dependent samples. Its exposition⁹ is available.

The software GSP implements the Tsakok technique. It is now used for mean comparisons at 0.02 significance level (one significant figure) per pair.

Unfortunately, it is not possible to apply GSP to the article by Loukovaara et al³ because, ignoring baseline characteristics, they did not publish the sufficient statistics for ANOVA (sample means and standard deviations), obstructing the minimum requirement of facilitating independent verification.

For Table 3,¹ there are significant mean differences between phakic and pseudophakic patients in their total number of breaks (preoperative and intraoperative), best-corrected visual acuity (BCVA) 6 months after scleral buckling and BCVA 6 months after vitrectomy.

For Table 2,² there is a significant mean difference between basal and after cyclopentolate for the resistive index (RI) of pseudoexfoliation syndrome (PXS).

There is little or no overlap, well below 95% with at least population, between the 99% confidence intervals of the clinical groups concerned.

The care taken with the data means that they deserve correct analysis, which they were denied.

The Tsakok technique is extended to the nonparametric problem of comparing samples using the article on constructing exact unconditional Uniformly Most Powerful Unbiased tests by Tsakok,¹⁰ superseding the χ² test or the Wilcoxon test. The Tsakok articles are reprinted¹¹ with further results.