h = swtest (x) performs the Shapiro-Wilk test of the null hypothesis that the sample in the vector x comes from a normal distribution with unknown mean and variance, against the alternative that it does not come from a normal distribution. The result h is 1 if the test rejects the null hypothesis at the 5% significance level, and 0 otherwise. x must be a vector of real values; NaN values are treated as missing and removed.

The test statistic is $$ W = \frac{\left( \sum_{i=1}^n a_i x_{(i)} \right)^2} {\sum_{i=1}^n (x_i - \bar{x})^2}, $$ where the weights a, applied to the sorted sample, are derived from the expected values of the order statistics of a standard normal sample of size n. W lies in (0,1], and small values of W are evidence against normality.

The following Name-Value pairs are supported:

'shapiro-wilk' computes the weights and the p-value with Royston’s algorithm AS R94, which approximates the weights of Shapiro and Wilk and transforms W to a normal deviate. It is the algorithm used by R’s shapiro.test, and it accepts samples of 3 to 5000 values. For n = 3 the p-value is exact.

'shapiro-francia' performs the Shapiro-Francia test instead, whose statistic W' is the squared correlation between the sorted sample and the approximate expected normal order statistics norminv (((1:n) - 3/8) / (n + 1/4)). Its p-value is Royston’s normal approximation for W', as in sf.test of R’s nortest package, and it accepts samples of 5 to 5000 values. The Shapiro-Francia test is known to be more powerful than the Shapiro-Wilk test against leptokurtic alternatives. The test is never selected automatically; the method used is the one requested.

[h, p] = swtest (…) also returns the p-value p of the test, the probability of observing a statistic as small as swstat under the null hypothesis.

[h, p, swstat, critval] = swtest (…) also returns the test statistic swstat, W or W', and the critical value critval at significance level alpha, obtained by inverting the same approximation. The null hypothesis is rejected when swstat < critval, which is the same as p < alpha.

A sample whose values are all equal has no defined statistic and is refused, as is a sample holding an infinite value. Samples of more than 5000 values are refused, since Royston’s approximations are calibrated up to that size; use adtest or jbtest for larger samples.

MATLAB has no Shapiro-Wilk test, so swtest is specific to Octave.

References:

1. S. S. Shapiro and M. B. Wilk. An analysis of variance test for normality (complete samples). Biometrika, 52(3-4):591–611, 1965.
2. S. S. Shapiro and R. S. Francia. An approximate analysis of variance test for normality. Journal of the American Statistical Association, 67(337):215–216, 1972.
3. P. Royston. A pocket-calculator algorithm for the Shapiro-Francia test for non-normality: an application to medicine. Statistics in Medicine, 12(2):181–184, 1993.
4. P. Royston. Remark AS R94: a remark on algorithm AS 181: the W-test for normality. Applied Statistics, 44(4):547–551, 1995.

See also: adtest, jbtest, kstest, lillietest

Source Code: swtest