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 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

$(a_1,\dots,a_n) = {m^{\mathsf{T}} V^{-1} \over (m^{\mathsf{T}} V^{-1}V^{-1}m)^{1/2}}$

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. However, since the test is biased by sample size, 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.

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. This technique is used in several software packages including R, Stata, SPSS and SAS.

See also

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

External links

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