Appendix: mathematical specifications

These pages state, precisely and in one place, what a test computes: the statistic, its null distribution, the p-value, and, where the test has them, the point estimate and the confidence interval.

They are an appendix rather than part of the manual proper. The manual tells you what a function is called and what its arguments mean; a specification tells you what it computes and the general background behind it, in terms that do not depend on this package or on Julia. The definitions are the published ones, the derivations are given rather than asserted, and the worked values closing each page are conformance vectors that any implementation can be held to.

Sections are numbered so that they can be cited. §6.3 of a specification means what it says, and will go on meaning it.

Contents

Ordered as the manual is: parametric first, then nonparametric.

The t-tests

The one-sample and paired t-tests, and the two-sample tests with and without an assumption of equal variances.

• 1. Preliminaries and notation
• 2. Model and estimand
• 3. One-sample
• 4. Two samples, equal variances (Student)
• 5. Two samples, unequal variances (Welch)
• 6. p-values
• 7. Confidence interval
• 8. Worked values for the t-tests
• 9. The t-tests in this package
• 10. References

Rank-based location inference

The Wilcoxon signed rank and rank sum (Mann-Whitney) procedures, the Hodges-Lehmann estimators, and the distribution-free intervals obtained by inverting either test.

• 1. Preliminaries and notation
• 2. The one-sample procedure (Wilcoxon signed rank)
• 3. The two-sample procedure (Wilcoxon rank sum, Mann-Whitney U)
• 4. Pairwise estimates
• 5. Point estimation
• 6. Interval estimation
• 7. Relation to other implementations
• 8. Worked values for the rank tests
• 9. The rank tests in this package
• 10. References

Conventions

Each page states its estimand before its statistic. A test and the parameter it estimates are separate objects, and conflating them is the most common way to misread one.

Derivations appear where a result would otherwise have to be taken on trust, particularly for a confidence interval obtained by inverting a test.

Where a construction involves a genuine choice rather than a consequence (a continuity correction, a tie-breaking convention, a conservative rather than a nearest-attainable level, a pooled rather than a separate variance estimate), the choice is named as such and the alternative is given.

Differences between implementations are recorded as differences, not as errors, with the reference implementation named. Where this package departs from what the page specifies, the closing section says so.