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# NORMTEST procedure

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 z1, z2,…, zn are the standardized values then z12 + z22 + … + zn2 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.

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.

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]
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)