naginterfaces.library.stat.test_​shapiro_​wilk

naginterfaces.library.stat.test_shapiro_wilk(x, a=None)

test_shapiro_wilk calculates Shapiro and Wilk’s $W$ statistic and its significance level for testing Normality.

For full information please refer to the NAG Library document for g01dd

https://support.nag.com/numeric/nl/nagdoc_30.3/flhtml/g01/g01ddf.html

Parameters

xfloat, array-like, shape $\left(n\right)$

The ordered sample values, $x_{\text{i}}$, for $\text{i}=1,2,\dots ,n$.

aNone or float, array-like, shape $\left(n\right)$, optional

If $\text{calwts}$ has been set to $False$ then before entry $a$ must contain the $n$ weights as calculated in a previous call to test_shapiro_wilk, otherwise $a$ need not be set.

Returns

afloat, ndarray, shape $\left(n\right)$

The $n$ weights required to calculate $w$.

wfloat

The value of the statistic, $w$.

pwfloat

The significance level of $w$.

Raises

NagValueError

(errno $1$)

On entry, $n=⟨value⟩$.

Constraint: $n\ge 3$.

(errno $2$)

On entry, $n=⟨value⟩$.

Constraint: $n\le 5000$.

(errno $3$)

On entry, elements of $x$ not in order. $x[⟨value⟩]=⟨value⟩$, $x[⟨value⟩]=⟨value⟩$, $x[⟨value⟩]=⟨value⟩$.

(errno $3$)

On entry, all elements of $x$ are equal.

Notes

test_shapiro_wilk calculates Shapiro and Wilk’s $W$ statistic and its significance level for any sample size between $3$ and $5000$. It is an adaptation of the Applied Statistics Algorithm AS R94, see Royston (1995). The full description of the theory behind this algorithm is given in Royston (1992).

Given a set of observations $x_1,x_2,\dots ,x_n$ sorted into either ascending or descending order (sort.realvec_sort may be used to sort the data) this function calculates the value of Shapiro and Wilk’s $W$ statistic defined as:

$$W=\frac{{\left({\sum }_{i=1}^n{a}_i{x}_i\right)}^2}{{\sum }_{i=1}^n{(x_i-\overline{x})}^2}\text{,}$$

where $\overline{x}=\frac{1}{n}{\sum }_1^n{x}_i$ is the sample mean and $a_i$, for $i=1,2,\dots ,n$, are a set of ‘weights’ whose values depend only on the sample size $n$.

On exit, the values of $a_i$, for $\text{i}=1,2,\dots ,n$, are only of interest should you wish to call the function again to calculate $w$ and its significance level for a different sample of the same size.

It is recommended that the function is used in conjunction with a Normal $(Q-Q)$ plot of the data. Functions normal_scores_exact() and normal_scores_approx() can be used to obtain the required Normal scores.

References

Royston, J P, 1982, Algorithm AS 181: the $W$ test for normality, Appl. Statist. (31), 176–180

Royston, J P, 1986, A remark on AS 181: the $W$ test for normality, Appl. Statist. (35), 232–234

Royston, J P, 1992, Approximating the Shapiro–Wilk’s $W$ test for non-normality, Statistics & Computing (2), 117–119

Royston, J P, 1995, A remark on AS R94: A remark on Algorithm AS 181: the $W$ test for normality, Appl. Statist. (44(4)), 547–551