Shapiro–Wilk test

The Shapiro–Wilk test is a test of normality in frequentist statistics. It was published in 1965 by Samuel Sanford Shapiro and Martin Wilk.^[1]

Theory

The Shapiro–Wilk test utilizes the null hypothesis principle to check whether a sample x₁, ..., x_n came from a normally distributed population. The test statistic is

W={\left(\sum _{i=1}^{n}a_{i}x_{(i)}\right)^{2} \over \sum _{i=1}^{n}(x_{i}-{\overline {x}})^{2}},

where

$x_{{(i)}}$ (with parentheses enclosing the subscript index i) is the ith order statistic, i.e., the ith-smallest number in the sample;
$\overline {x}=\left(x_{1}+\cdots +x_{n}\right)/n$ is the sample mean;
the constants $a_{i}$ are given by^[1]

(a_{1},\dots ,a_{n})={m^{\mathsf {T}}V^{-1} \over (m^{\mathsf {T}}V^{-1}V^{-1}m)^{1/2}},

where

m=(m_{1},\dots ,m_{n})^{{{\mathsf {T}}}}\,

and

m_{1},\ldots ,m_{n}

are the expected values of the order statistics of independent and identically distributed random variables sampled from the standard normal distribution, and

V

is the covariance matrix of those order statistics.

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).^[2] However, since the test is biased by sample size,^[3] 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

Monte Carlo simulation has found 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.^[4]

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.^[5] This technique is used in several software packages including Stata,^[6]^[7] SPSS and SAS.^[8] Rahman and Govidarajulu extended the sample size further up to 5000.^[9]

References

1 2 Shapiro, S. S.; Wilk, M. B. (1965). "An analysis of variance test for normality (complete samples)". Biometrika. 52 (3–4): 591–611. doi:10.1093/biomet/52.3-4.591. JSTOR 2333709. MR 205384. p. 593
↑ "How do I interpret the Shapiro–Wilk test for normality?". JMP. 2004. Retrieved March 24, 2012.
↑ Field, Andy (2009). Discovering statistics using SPSS (3rd ed.). Los Angeles [i.e. Thousand Oaks, Calif.]: SAGE Publications. p. 143. ISBN 978-1-84787-906-6.
↑ Razali, Nornadiah; Wah, Yap Bee (2011). "Power comparisons of Shapiro–Wilk, Kolmogorov–Smirnov, Lilliefors and Anderson–Darling tests" (PDF). Journal of Statistical Modeling and Analytics. 2 (1): 21–33. Retrieved 5 June 2012.
↑ Royston, Patrick (September 1992). "Approximating the Shapiro–Wilk W-test for non-normality". Statistics and Computing. 2 (3): 117–119. doi:10.1007/BF01891203.
↑ Royston, Patrick. "Shapiro–Wilk and Shapiro–Francia Tests". Stata Technical Bulletin, StataCorp LP. 1 (3).
↑ Shapiro–Wilk and Shapiro–Francia tests for normality
↑ Park, Hun Myoung (2002–2008). "Univariate Analysis and Normality Test Using SAS, Stata, and SPSS" (PDF). [working paper]. Retrieved 26 February 2014.
↑ Rahman und Govidarajulu (1997). "A modification of the test of Shapiro and Wilk for normality". Journal of Applied Statistics. 24 (2): 219–236. doi:10.1080/02664769723828.

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