On tests for multivariate normality and associated simulation studies
Bibliographic record
Abstract
We study the empirical size and power of some recently proposed tests for multivariate normality (MVN) and compare them with the existing proposals that performed best in previously published studies. We show that the Royston's [Royston, J.P., 1983b, Some techniques for assessing multivariate normality based on the Shapiro-Wilk W. Applied Statistics, 32, 121–133.] extension to the Shapiro and Wilk [Shapiro, S.S., Wilk, M.B., 1965, An analysis of variance test for normality (complete samples). Biometrika, 52, 591–611.] test is unable to achieve the nominal significance level, and consider a subsequent extension proposed by Royston [Royston, J.P., 1992, Approximating the Shapiro–Wilk W-Test for non-normality. Statistics and Computing, 2, 117–119.] to correct this problem, which earlier studies appear to have ignored. A consistent and invariant test proposed by Henze and Zirkler [Henze, N., Zirkler, B., 1990, A class of invariant consistent tests for multivariate normality. Communications in Statistics—Theory and Methods, 19, 3595–3617.] is found to have good power properties, particularly for sample sizes of 75 or more, while an approach suggested by Royston [Royston, J.P., 1992, Approximating the Shapiro–Wilk W-Test for non-normality. Statistics and Computing, 2, 117–119.] performs effectively at detecting departures from MVN for smaller sample sizes. We also compare our results to those of previous simulation studies, and discuss the challenges associated with generating multivariate data for such investigations.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.187 | 0.655 |
| Meta-epidemiology (narrow) | 0.002 | 0.001 |
| Meta-epidemiology (broad) | 0.003 | 0.003 |
| Bibliometrics | 0.007 | 0.006 |
| Science and technology studies | 0.002 | 0.008 |
| Scholarly communication | 0.004 | 0.009 |
| Open science | 0.004 | 0.006 |
| Research integrity | 0.003 | 0.005 |
| Insufficient payload (model declined to judge) | 0.008 | 0.001 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".