shapiroWilk#

Purpose#

Computes the Shapiro-Wilk W test for univariate normality.

Format#

out = shapiroWilk(x)#

Parameters:

x (Nx1 vector) – data vector. Missing values are removed automatically.

Returns:

out (struct) –

instance of shapiroWilkOut structure:

out.w	scalar, W test statistic. Range [0, 1], values near 1 indicate normality.
out.p	scalar, p-value. Probability of observing W this small or smaller under normality.
out.n	scalar, effective sample size after removing missing values.

Examples#

Example 1: Normal data#

rndseed 42;

// Generate normal data
x = rndn(100, 1);

// Test normality
struct shapiroWilkOut out;
out = shapiroWilk(x);

print "W statistic:" out.w;
print "p-value:    " out.p;

W statistic:      0.97879310
p-value:          0.10700545

The p-value (0.107) is above 0.05, so we fail to reject normality — consistent with the data being generated from a normal distribution.

Example 2: Non-normal data#

rndseed 42;

// Generate exponential data (non-normal)
x = -ln(rndu(100, 1));

struct shapiroWilkOut out;
out = shapiroWilk(x);

print "W statistic:" out.w;
print "p-value:    " out.p;

W statistic:      0.91599435
p-value:       8.6824924e-06

The very small p-value (< 0.001) strongly rejects normality, correctly detecting that exponential data is not normally distributed.

Example 3: With missing values#

rndseed 42;

// Data with missing values
x = rndn(100, 1);
x[1] = miss(0, 0);
x[50] = miss(0, 0);

struct shapiroWilkOut out;
out = shapiroWilk(x);

print "W statistic:" out.w;
print "p-value:    " out.p;
print "n:          " out.n;

W statistic:      0.97858132
p-value:          0.11011458
n:                 98.000000

Missing values are automatically removed. The effective sample size (out.n = 98) reflects the two removed observations.

Remarks#

The W statistic ranges from 0 to 1. Values close to 1 indicate the data is consistent with a normal distribution.
Sample size must satisfy 3 <= N <= 5000.
Missing values are automatically removed before computation.
The test uses the Blom approximation for expected normal order statistics and applies Royston’s (1992, 1995) transformation for p-value calculation.
For multivariate normality testing, see mvnTest().

References#

Shapiro, S.S. & Wilk, M.B. (1965). “An Analysis of Variance Test for Normality (Complete Samples).” Biometrika, 52(3/4), 591-611.

Royston, J.P. (1995). “Remark AS R94: A Remark on Algorithm AS 181: The W-test for Normality.” Applied Statistics, 44(4), 547-551.