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 tests the null hypothesis that a sample x1, ..., xn 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; not to be confused with ${\displaystyle x_{i}}$) is the ith order statistic, i.e., the ith-smallest number in the sample;
• ${\displaystyle {\overline {x}}=\left(x_{1}+\cdots +x_{n}\right)/n}$ is the sample mean;
• the constants ${\displaystyle a_{i}}$ are given by[1]
${\displaystyle (a_{1},\dots ,a_{n})={m^{\mathsf {T}}V^{-1} \over (m^{\mathsf {T}}V^{-1}V^{-1}m)^{1/2}},}$
where
${\displaystyle m=(m_{1},\dots ,m_{n})^{\mathsf {T}}\,}$
and ${\displaystyle m_{1},\ldots ,m_{n}}$ are the expected values of the order statistics of independent and identically distributed random variables sampled from the standard normal distribution, and ${\displaystyle V}$ is the covariance matrix of those order statistics.

Interpretation

The null-hypothesis of this test is that the population is normally distributed. Thus, on the one hand, 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 (e.g., for an alpha level of 0.05, a data set with a p-value of 0.05 rejects the null hypothesis that the data are from a normally distributed population).[2] 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.[3]

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]