# NAG Library Routine Document

## 1Purpose

g01ddf calculates Shapiro and Wilk's $W$ statistic and its significance level for testing Normality.

## 2Specification

Fortran Interface
 Subroutine g01ddf ( x, n, a, w, pw,
 Integer, Intent (In) :: n Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: x(n) Real (Kind=nag_wp), Intent (Inout) :: a(n) Real (Kind=nag_wp), Intent (Out) :: w, pw Logical, Intent (In) :: calwts
#include <nagmk26.h>
 void g01ddf_ (const double x[], const Integer *n, const logical *calwts, double a[], double *w, double *pw, Integer *ifail)

## 3Description

g01ddf 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 (m01caf may be used to sort the data) this routine calculates the value of Shapiro and Wilk's $W$ statistic defined as:
 $W= ∑i=1naixi 2 ∑i=1n xi-x- 2 ,$
where $\stackrel{-}{x}=\frac{1}{n}\sum _{1}^{n}{x}_{i}$ 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 $\mathit{i}=1,2,\dots ,n$, are only of interest should you wish to call the routine again to calculate ${\mathbf{w}}$ and its significance level for a different sample of the same size.
It is recommended that the routine is used in conjunction with a Normal $\left(Q-Q\right)$ plot of the data. Routines g01daf and g01dbf can be used to obtain the required Normal scores.

## 4References

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

## 5Arguments

1:     $\mathbf{x}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: the ordered sample values, ${x}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,n$.
2:     $\mathbf{n}$ – IntegerInput
On entry: $n$, the sample size.
Constraint: $3\le {\mathbf{n}}\le 5000$.
3:     $\mathbf{calwts}$ – LogicalInput
On entry: must be set to .TRUE. if you wish g01ddf to calculate the elements of a.
calwts should be set to .FALSE. if you have saved the values in a from a previous call to g01ddf.
If in doubt, set calwts equal to .TRUE..
4:     $\mathbf{a}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput/Output
On entry: if calwts has been set to .FALSE. then before entry a must contain the $n$ weights as calculated in a previous call to g01ddf, otherwise a need not be set.
On exit: the $n$ weights required to calculate ${\mathbf{w}}$.
5:     $\mathbf{w}$ – Real (Kind=nag_wp)Output
On exit: the value of the statistic, ${\mathbf{w}}$.
6:     $\mathbf{pw}$ – Real (Kind=nag_wp)Output
On exit: the significance level of ${\mathbf{w}}$.
7:     $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, . If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value  is recommended. If the output of error messages is undesirable, then the value $1$ is recommended. Otherwise, if you are not familiar with this argument, the recommended value is $0$. When the value  is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 3$.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\le 5000$.
${\mathbf{ifail}}=3$
On entry, all elements of x are equal.
On entry, elements of x not in order. ${\mathbf{x}}\left(〈\mathit{\text{value}}〉\right)=〈\mathit{\text{value}}〉$, ${\mathbf{x}}\left(〈\mathit{\text{value}}〉\right)=〈\mathit{\text{value}}〉$, ${\mathbf{x}}\left(〈\mathit{\text{value}}〉\right)=〈\mathit{\text{value}}〉$.
${\mathbf{ifail}}=-99$
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

## 7Accuracy

There may be a loss of significant figures for large $n$.

## 8Parallelism and Performance

g01ddf is not threaded in any implementation.

The time taken by g01ddf 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 g01ddf for a given sample size, unless calwts is reset to .TRUE. before each call of g01ddf.
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 ${\mathbf{w}}$,
 $WA=W ∑ xi - x- 2 ∑i=1n xi=x- 2 - n-1 12 ω2 ,$
where $\omega$ is the rounding width. $WA$ can be compared with a standard Normal distribution, but a further approximation is given by Royston (1986).
If ${\mathbf{n}}>5000$, a value for w and pw is returned, but its accuracy may not be acceptable. See Section 4 for more details.

## 10Example

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 g01ddf, and are re-used in the second call.

### 10.1Program Text

Program Text (g01ddfe.f90)

### 10.2Program Data

Program Data (g01ddfe.d)

### 10.3Program Results

Program Results (g01ddfe.r)