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.

Theory

The Shapiro–Wilk test tests the null hypothesis that a sample x1,..., xn came from a normally distributed population. The test statistic is
where
The coefficients are given by:
where C is a vector norm:
and the vector m,
is made of the expected values of the order statistics of independent and identically distributed random variables sampled from the standard normal distribution; finally, is the covariance matrix of those normal order statistics.
There is no name for the distribution of. The cutoff values for the statistics are calculated through Monte-Carlo simulations.

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 normally distributed. On the other hand, if the p value is greater than the chosen alpha level, then the null hypothesis that the data came from a normally distributed population can not be rejected.
Like most statistical significance tests, if the sample size is sufficiently large this test may detect even trivial departures from the null hypothesis ; thus, additional investigation of the effect size is typically advisable, e.g., a Q–Q plot in this case.

Power analysis

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.

Approximation

Royston proposed an alternative method of calculating the coefficients vector by providing an algorithm for calculating values, which extended the sample size to 2,000. This technique is used in several software packages including Stata, SPSS and SAS. Rahman and Govidarajulu extended the sample size further up to 5,000.