NAG C Library Function Document

. 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 (nag_double_sort (m01cac) may be used to sort the data) this function calculates the value of Shapiro and Wilk's

W

statistic defined as:

W = \frac{{(\sum_{i = 1}^{n} a_{i} x_{i})}^{2}}{\sum_{i = 1}^{n} {(x_{i} - \bar{x})}^{2}},

where

\bar{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

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. Function nag_normal_scores_exact (g01dac) can be used to obtain the required Normal scores.

4

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

5

Arguments

1: $n$ – IntegerInput: On entry: $n$ , the sample size.

Constraint: $3 \leq n \leq 5000$ .
2: $x [n]$ – const doubleInput: On entry: the ordered sample values, $x_{i}$ , for $i = 1, 2, \dots, n$ .
3: $calc_wts$ – Nag_BooleanInput: On entry: must be set to Nag_TRUE if you wish nag_shapiro_wilk_test (g01ddc) to calculate the elements of a.
calc_wts should be set to Nag_FALSE if you have saved the values in a from a previous call to nag_shapiro_wilk_test (g01ddc).

If in doubt, set calc_wts equal to Nag_TRUE.
4: $a [n]$ – doubleInput/Output: On entry: if calc_wts has been set to Nag_FALSE then before entry a must contain the $n$ weights as calculated in a previous call to nag_shapiro_wilk_test (g01ddc), otherwise a need not be set.

On exit: the $n$ weights required to calculate $w$ .
5: $w$ – double *Output: On exit: the value of the statistic, $w$ .
6: $pw$ – double *Output: On exit: the significance level of $w$ .
7: $fail$ – NagError *Input/Output: The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

NE_ALL_ELEMENTS_EQUAL: On entry, all elements of x are equal.
NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_INT_ARG_GT: On entry, $n = 〈value〉$ .
Constraint: $n \leq 5000$ .
NE_INT_ARG_LT: On entry, $n = 〈value〉$ .
Constraint: $n \geq 3$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
NE_NON_MONOTONIC: On entry, elements of x not in order. $x [〈value〉] = 〈value〉$ , $x [〈value〉] = 〈value〉$ , $x [〈value〉] = 〈value〉$ .

7

Accuracy

There may be a loss of significant figures for large

n

8

Parallelism and Performance

nag_shapiro_wilk_test (g01ddc) is not threaded in any implementation.

9

Further Comments

The time taken by nag_shapiro_wilk_test (g01ddc) depends roughly linearly on the value of

n

For very small samples the power of the test may not be very high.

The contents of the array a should not be modified between calls to nag_shapiro_wilk_test (g01ddc) for a given sample size, unless calc_wts is reset to Nag_TRUE before each call of nag_shapiro_wilk_test (g01ddc).

The Shapiro and Wilk's

W

test is very sensitive to ties. If the data has been rounded the test can be improved by using Sheppard's correction to adjust the sum of squares about the mean. This produces an adjusted value of

w

W A = W \frac{\sum {x_{(i)} - \bar{x}}^{2}}{\{\sum_{i = 1}^{n} {x_{(i)} = \bar{x}}^{2} - \frac{n - 1}{12} ω^{2}\}},

where

ω

is the rounding width.

W A

can be compared with a standard Normal distribution, but a further approximation is given by Royston (1986).

n > 5000

, a value for w and pw is returned, but its accuracy may not be acceptable. See Section 4 for more details.

10

Example

This example tests the following two samples (each of size

20

) for Normality.

Sample Number	Data
1	$0.11$ , $7.87$ , $4.61$ , $10.14$ , $7.95$ , $3.14$ , $0.46$ , $4.43$ , $0.21$ , $4.75$ , $0.71$ , $1.52$ , $3.24$ , $0.93$ , $0.42$ , $4.97$ , $9.53$ , $4.55$ , $0.47$ , $6.66$
2	$1.36$ , $1.14$ , $2.92$ , $2.55$ , $1.46$ , $1.06$ , $5.27$ , $- 1.11$ , $3.48$ , $1.10$ , $0.88$ , $- 0.51$ , $1.46$ , $0.52$ , $6.20$ , $1.69$ , $0.08$ , $3.67$ , $2.81$ , $3.49$

The elements of a are calculated only in the first call of nag_shapiro_wilk_test (g01ddc), and are re-used in the second call.

NAG C Library Function Document

nag_shapiro_wilk_test (g01ddc)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1

Purpose

2

Specification

3

Description