The Shapiro–Wilk test is a test of normality in frequentist statistics. It was published
in 1965 by Samuel Sanford Shapiro and Martin Wilk.
Theory The Shapiro–Wilk test utilizes the null
hypothesis principle to check whether a sample x1, ..., xn came from a normally distributed
population. The test statistic is:
where is the ith order statistic, i.e., the ith-smallest
number in the sample; is the sample mean;
the constants are given by
where
and are the expected values of the order statistics of independent and identically distributed
random variables sampled from the standard normal distribution, and is the covariance
matrix of those order statistics. The user may reject the null hypothesis if is below
a predetermined threshold. Interpretation
The null-hypothesis of this test is that the population is normally distributed. Thus if
the p-value is less than the chosen alpha level, then the null hypothesis is rejected
and there is evidence that the data tested are not from a normally distributed population.
In other words, the data are not normal. On the contrary, if the p-value is greater than
the chosen alpha level, then the null hypothesis that the data came from a normally distributed
population cannot be rejected. E.g. for an alpha level of 0.05, a data set with a p-value
of 0.02 rejects the null hypothesis that the data are from a normally distributed population.
However, since the test is biased by sample size, the test may be statistically significant
from a normal distribution in any large samples. Thus a Q–Q plot is required for verification
in addition to the test. Power analysis
A research paper concluded that Shapiro–Wilk has the best power for a given significance,
followed closely by Anderson-Darling when comparing the Shapiro–Wilk, Kolmogorov–Smirnov,
Lilliefors, and Anderson-Darling tests. Approximation
Royston proposed an alternative method of calculating the coefficients vector by providing
an algorithm for calculating values, which extended the sample size to 2000. This technique
is used in several software packages including R, Stata, SPSS and SAS.
See also Anderson–Darling test
Cramér–von Mises criterion Kolmogorov–Smirnov test
Normal probability plot Ryan–Joiner test
Watson test Lilliefors test
References ^ Shapiro, S. S.; Wilk, M. B.. "An analysis
of variance test for normality". Biometrika 52: 591–611. doi:10.109352.3-4.591. JSTOR 2333709.
MR 205384. p. 593 ^ "How do I interpret the Shapiro–Wilk test
for normality?". JMP. 2004. Retrieved March 24, 2012.
^ Field, Andy. Discovering statistics using SPSS :. Los Angeles [i.e. Thousand Oaks,
Calif.]: SAGE Publications. p. 143. ISBN 978-1-84787-906-6. ^ Razali, Nornadiah; Wah, Yap Bee. "Power
comparisons of Shapiro–Wilk, Kolmogorov–Smirnov, Lilliefors and Anderson–Darling tests".
Journal of Statistical Modeling and Analytics 2: 21–33. Retrieved 5 June 2012.
^ Royston, Patrick. "Approximating the Shapiro–Wilk W-test for non-normality". Statistics and
Computing 2: 117–119. doi:10.1007/BF01891203. ^ Korkmaz, Selcuk. "Package 'royston'". Cran.r-project.org.
Retrieved 26 February 2014. ^ Royston, Patrick. "Shapiro–Wilk and Shapiro–Francia
Tests". Stata Technical Bulletin, StataCorp LP 1.
^ Park, Hun Myoung. "Univariate Analysis and Normality Test Using SAS, Stata, and SPSS".
[working paper]. Retrieved 26 February 2014. External links
Samuel Sanford Shapiro Algorithm AS R94 FORTRAN code
Exploratory analysis using the Shapiro–Wilk normality test in R