Shapiro–Wilk test

Not to be confused with the likelihood-ratio test, which is sometimes referred to as Wilks test.

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

Theory

The Shapiro–Wilk test tests the null hypothesis that a sample x₁, ..., x_n came from a normally distributed population. The test statistic is

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

where

• ${\displaystyle x_{(i)}}$ with parentheses enclosing the subscript index i is the ith order statistic, i.e., the ith-smallest number in the sample (not to be confused with ${\displaystyle x_{i}}$).
• ${\displaystyle {\overline {x}}=\left(x_{1}+\cdots +x_{n}\right)/n}$ is the sample mean.

The coefficients ${\displaystyle a_{i}}$ are given by:[1]

${\displaystyle (a_{1},\dots ,a_{n})={m^{\mathsf {T}}V^{-1} \over C},}$

where C is a vector norm:[2]

${\displaystyle C=\|V^{-1}m\|=(m^{\mathsf {T}}V^{-1}V^{-1}m)^{1/2}}$

and the vector m,

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

is made of the expected values of the order statistics of independent and identically distributed random variables sampled from the standard normal distribution; finally, ${\displaystyle V}$ is the covariance matrix of those normal order statistics.[3]

There is no name for the distribution of ${\displaystyle W}$. The cutoff values for the statistics are calculated through Monte Carlo simulations.[2]

Interpretation

The null-hypothesis of this test is that the population is normally distributed. 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 normally distributed.[4]

Like most statistical significance tests, if the sample size is sufficiently large this test may detect even trivial departures from the null hypothesis (i.e., although there may be some statistically significant effect, it may be too small to be of any practical significance); thus, additional investigation of the effect size is typically advisable, e.g., a Q–Q plot in this case.[5]

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, and Lilliefors.[6]^[unreliable source?]

Approximation

Royston proposed an alternative method of calculating the coefficients vector by providing an algorithm for calculating values that extended the sample size from 50 to 2,000.[7] This technique is used in several software packages including GraphPad Prism, Stata,[8][9] SPSS and SAS.[10] Rahman and Govidarajulu extended the sample size further up to 5,000.[11]

See also

• Anderson–Darling test
• Cramér–von Mises criterion
• D'Agostino's K-squared test
• Kolmogorov–Smirnov test
• Lilliefors test
• Normal probability plot
• Shapiro–Francia test

References

1. 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 0205384. p. 593
2. Richard M. Dudley (2015). "The Shapiro-Wilk and related tests for normality" (PDF). Retrieved 2022-06-16.
3. Davis, C. S.; Stephens, M. A. (1978). The covariance matrix of normal order statistics (PDF) (Technical report). Department of Statistics, Stanford University, Stanford, California. Technical Report No. 14. Retrieved 2022-06-17.
4. "How do I interpret the Shapiro–Wilk test for normality?". JMP. 2004. Retrieved March 24, 2012.
5. 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.
6. 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.
7. Royston, Patrick (September 1992). "Approximating the Shapiro–Wilk W-test for non-normality". Statistics and Computing. 2 (3): 117–119. doi:10.1007/BF01891203. S2CID 122446146.
8. Royston, Patrick. "Shapiro–Wilk and Shapiro–Francia Tests". Stata Technical Bulletin, StataCorp LP. 1 (3).
9. Shapiro–Wilk and Shapiro–Francia tests for normality
10. Park, Hun Myoung (2002–2008). "Univariate Analysis and Normality Test Using SAS, Stata, and SPSS". [working paper]. Retrieved 29 July 2023.
11. 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

• Worked example using Excel
• Algorithm AS R94 (Shapiro Wilk) FORTRAN code
• Exploratory analysis using the Shapiro–Wilk normality test in R
• Real Statistics Using Excel: the Shapiro-Wilk Expanded Test