Performs tests of univariate and/or multivariate Normality (M.S. Ridout).

### Option

`PRINT` = string tokens |
Allows the required printed output to be selected: test statistics, tables of critical values and the flagging of significant values with stars (`marginal` , `bivariateangle` , `radius` , `critical` , `stars` ); default `marg` , `biva` , `radi` |
---|

### Parameter

`DATA` = variates or pointers |
Variates whose univariate Normality is to be tested or pointers, each to a set of variates whose Normality and/or multivariate Normality are to be tested |
---|

### Description

This procedure offers three types of test of Normality.

Marginal (univariate) tests – assess the Normality of each variate in turn. The variates are standardized to have mean=0, variance=1 and then transformed with the `NORMAL`

function. The test is based on the idea that, assuming Normality, these transformed values should look like a sample from a uniform distribution on (0,1).

Bivariate angle tests – assess the bivariate Normality of each pair of variates in turn. The variates are standardized so that they are uncorrelated and have mean=0 and variance=1. The test is based on the following idea: if *x* and *y* are the standardized values, then the angle between the *x*-axis and the line joining (0,0) to (*x*,*y*) should, assuming Normality, be uniformly distributed on (0,2π).

Radius test – provides a single overall test of multivariate Normality. The variates are again standardized to have mean=0 and so that their covariance matrix is the identity matrix. The test uses the fact that if *z*_{1}, *z*_{2},…, *z _{n}* are the standardized values then

*z*

_{1}

^{2}+

*z*

_{2}

^{2}+ … +

*z*

_{n}^{2}should, under multivariate Normality, be approximately distributed as chi-square on

*n*degrees of freedom.

For each type of test, the test statistics are empirical distribution function (EDF) statistics – i.e. they compare the empirical distribution function of the sample with the theoretical distribution expected under the null hypothesis. Three EDF statistics are provided for each type of test – the Anderson-Darling statistic, the Cramer-von Mises statistic and the Watson statistic. The idea is to provide good power against a wide range of alternatives. The test statistics are adjusted so that their null distribution is independent of the sample size; critical values can be printed by the procedure (option `PRINT=critical`

).

The `DATA`

parameter is used to indicate the variate(s) whose Normality is to be assessed. If a single variate is supplied, its Normality is tested using the marginal test. Alternatively, `DATA`

can supply a pointer to a set of variates to be tested for multivariate Normality.

The `PRINT`

option can be used to select the type of test using the settings `marginal`

, `bivariateangle`

and `radius`

. The setting `critical`

allows tables of critical values to be printed, and `stars`

requests that significant values of the test statistics be flagged with stars. Settings `bivariateangle`

and `radius`

are relevant only when testing for multivariate Normality. By default `PRINT=marginal,bivariateangle,radius`

Option: `PRINT`

.

Parameter: `DATA`

.

### Method

The calculations are clearly set out in Aitchison (1986; Section 7.3). Bivariate angle and radius tests are described by Andrews, Gnanadesikan & Warner (1973). Stephens (1974) describes the EDF statistics used and gives tables of critical values and information on their comparative power.

### Action with `RESTRICT`

If a variate to which the `DATA`

parameter is set is restricted, the tests will be calculated using only the units included by the restriction. Similarly, the variates in a `DATA`

pointer can be restricted, but then must all be restricted in the same way. The procedure does not work properly with missing values. If missing values are present, `RESTRICT`

should be used (before calling the procedure) to exclude all units for which any of the variates has a missing value.

### References

Aitchison J.A. (1986). *The Statistical Analysis of Compositional Data*. London: Chapman & Hall.

Andrews D.F., Gnanadesikan R. & Warner J.L. (1973). Methods for assessing multivariate normality. In: *Multivariate Analysis III* (ed. P.R. Krishnaiah) 95-116. New York: Academic Press.

Stephens M.A. (1974). EDF statistics for goodness of fit and some comparisons. *Journal of the American Statistical Association*, 69, 730-737.

### See also

Directive: `DISTRIBUTION`

.

Procedures: `EDFTEST`

, `WSTATISTIC`

.

Commands for: Basic and nonparametric statistics.

### Example

CAPTION 'NORMTEST example',\ !t('Data from Aitchison (1986, The Statistical Analysis of',\ 'Compositional Data, pages 354-355), percentages by weight',\ 'of A,B,C,D,E in 25 samples of Hongite.'); STYLE=meta,plain VARIATE [NVALUES=25] V[1...5] READ [SERIAL=yes] V[] 48.8 48.2 37.0 50.9 44.2 52.3 44.6 34.6 41.2 42.6 49.9 45.2 32.7 41.4 46.2 32.3 43.2 49.5 42.3 44.6 45.8 49.9 48.6 45.5 45.9 : 31.7 23.8 9.1 23.8 38.3 26.2 33.0 5.2 11.7 46.6 19.5 37.3 8.5 12.9 17.5 7.3 44.3 32.3 15.8 11.5 16.6 25.0 34.0 16.6 24.9 : 3.8 9.0 34.2 7.2 2.9 4.2 4.6 42.9 26.7 0.7 11.4 2.7 38.9 23.4 15.8 40.9 1.0 3.1 20.4 23.8 16.8 6.8 2.5 17.6 9.7 : 6.4 9.2 9.5 10.1 7.7 12.5 12.2 9.6 9.6 5.6 9.5 5.5 8.0 15.8 8.3 12.9 7.8 8.7 8.3 11.6 12.0 10.9 9.4 9.6 9.8 : 9.3 9.8 10.2 8.0 6.9 4.8 5.6 7.7 10.8 4.5 9.7 9.3 11.9 6.5 12.2 6.6 3.7 6.3 13.2 8.5 8.8 7.4 5.5 10.7 9.7 : " Transform data, as on page 142, and test for multivariate normality." CALCULATE Y[1...4]=LOG(V[1...4]/V[5]) NORMTEST [PRINT=marginal,bivariateangle,radius,critical] Y CAPTION !t(\ 'The results agree with Table 7.4 except those for the Cramer-von Mises and',\ 'Watson forms of the bivariate angle and radius tests. This appears to be',\ 'because in forming the modified test statistics there has been a DIVISION ',\ 'by (1+1/N) or (1+0.8/N) instead of a MULTIPLICATION by these quantities',\ '(see Aitchison, Table 7.3; also Stephens, 1974, J.A.S.A., 69, p.732).')