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, ..., xn 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

(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.

The user may reject the null hypothesis if W 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.[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 R,[6] Stata,[7][8] SPSS and SAS.[9] Rahman and Govidarajulu extended the sample size further up to 5000.[10]

See also

References

  1. 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
  2. ↑ "How do I interpret the Shapiro–Wilk test for normality?". JMP. 2004. Retrieved March 24, 2012.
  3. ↑ 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.
  4. ↑ 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.
  5. ↑ Royston, Patrick (September 1992). "Approximating the Shapiro–Wilk W-test for non-normality". Statistics and Computing 2 (3): 117–119. doi:10.1007/BF01891203.
  6. ↑ Korkmaz, Selcuk. "Package 'royston'" (PDF). Cran.r-project.org. Retrieved 26 February 2014.
  7. ↑ Royston, Patrick. "Shapiro–Wilk and Shapiro–Francia Tests". Stata Technical Bulletin, StataCorp LP 1 (3).
  8. ↑ Shapiro–Wilk and Shapiro–Francia tests for normality
  9. ↑ Park, Hun Myoung (2002–2008). "Univariate Analysis and Normality Test Using SAS, Stata, and SPSS" (PDF). [working paper]. Retrieved 26 February 2014.
  10. ↑ 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.

External links

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