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.

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]

See also

• Anderson–Darling test
• Cramér–von Mises criterion
• Kolmogorov–Smirnov test
• Normal probability plot
• Ryan–Joiner test
• Watson test
• Lilliefors test

References

1. ^ ^{a b} Shapiro, S. S.;   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.  
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). -test for non-normality"W"Approximating the Shapiro–Wilk . Statistics and Computing 2 (3): 117–119.  
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. 

External links

• Samuel Sanford Shapiro
• Algorithm AS R94 (Shapiro Wilk) FORTRAN code
• Exploratory analysis using the Shapiro–Wilk normality test in R