Talk:Shapiro–Wilk test

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Some questions:

What is the criterion for W to be "too small"? What is the expected value for the order statistics? Is there a multi-variate generalization? PhysPhD 20:55, 16 May 2007 (UTC)[reply]

I think there should be some way of arriving at the a(i)'s... I've seen it like this: ai <- qnorm((i-0.375)/(n+0.25)) where qnorm is the inverse CDF. —Preceding unsigned comment added by 64.122.234.42 (talk) 21:35, 30 October 2007 (UTC)[reply]

I've found some table for critical values of criterion Wcrit in some old Russian book named "Основы математической статистики - Под ред.В.С.Иванова" which is roughly "Fundamentals of Mathematical Statistics - Edited by Ivanov V.S.". The table looks like this:

n	alpha	alpha	n	alpha	alpha	n	alpha	alpha
	0.05	0.01		0.05	0.01		0.05	0.01
4	0.767	0.753	20	0.905	0.868	36	0.935	0.912
5	0.748	0.687	21	0.908	0.873	37	0.936	0.914
6	0.762	0.686	22	0.911	0.878	38	0.938	0.916
7	0.803	0.730	23	0.914	0.881	39	0.939	0.917
8	0.818	0.749	24	0.916	0.884	40	0.940	0.919
9	0.829	0.764	25	0.918	0.888	41	0.941	0.920
10	0.842	0.781	26	0.920	0.891	42	0.942	0.922
11	0.850	0.781	27	0.923	0.894	43	0.943	0.923
12	0.859	0.805	28	0.924	0.896	44	0.944	0.924
13	0.866	0.814	29	0.926	0.898	45	0.945	0.926
14	0.874	0.825	30	0.927	0.900	46	0.945	0.927
15	0.881	0.835	31	0.929	0.902	47	0.946	0.928
16	0.887	0.884	32	0.930	0.904	48	0.947	0.929
17	0.892	0.851	33	0.931	0.906	49	0.947	0.929
18	0.897	0.858	34	0.933	0.908	50	0.947	0.930
19	0.901	0.863	35	0.934	0.910

The null hypothesis is rejected if W < Wcrit. From this table we can deduce that Wcrit depends on so-called statistical significance level alpha (see article http://wiki.riteme.site/wiki/Statistically_significant), and on the actual number of experiments n. This test was specialized for small n (under 40-50), so if you have to test a larger sample, it's better to use other tests like Kolmogorov–Smirnov test (http://wiki.riteme.site/wiki/Kolmogorov%E2%80%93Smirnov_test) —Preceding unsigned comment added by 93.73.35.146 (talk) 07:47, 22 August 2010 (UTC)[reply]

Improve Referencing:

The current reference ^[1] to support highest statistical power claim of the Shapiro-Wilk is dubious, it directly links to a ReasearchGate PDF, and though heavily cited it has no attached DOI on GScholar and the journal with closest matching name, Journal of Statistical Modeling and Analytics (JOSMA) has been created in 2021. Here are a few alternative references ^[2]^[3]^[4]Mystic reveur (talk) 09:17, 23 October 2024 (UTC)[reply]

^ Razali, Nornadiah; Wah, Yap Bee (2011). "Power comparisons of Shapiro–Wilk, Kolmogorov–Smirnov, Lilliefors and Anderson–Darling tests". Journal of Statistical Modeling and Analytics. 2 (1): 21–33. Retrieved 30 March 2017.
^ "Power Comparison of Various Normality Tests". Pakistan Journal of Statistics and Operation Research.
^ "Shapiro–Francia test compared to other normality test using expected p-value". Journal of Statistical Computation and Simulation.
^ "Empirical Power Comparison Of Goodness of Fit Tests for Normality In The Presence of Outliers". Journal of Physics: Conference Series.

[1] Razali, Nornadiah; Wah, Yap Bee (2011). "Power comparisons of Shapiro–Wilk, Kolmogorov–Smirnov, Lilliefors and Anderson–Darling tests". Journal of Statistical Modeling and Analytics. 2 (1): 21–33. Retrieved 30 March 2017.

[2] "Power Comparison of Various Normality Tests". Pakistan Journal of Statistics and Operation Research.

[3] "Shapiro–Francia test compared to other normality test using expected p-value". Journal of Statistical Computation and Simulation.

[4] "Empirical Power Comparison Of Goodness of Fit Tests for Normality In The Presence of Outliers". Journal of Physics: Conference Series.

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