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# CDESCRIBE procedure

Calculates summary statistics and tests of circular data (P.W. Goedhart & R.W. Payne).

### Options

`PRINT` = string tokens What to print (`summary`, `fittedvalues`); default `summ` Width of sectors (in degrees) into which to group an `ANGLES` variate for calculation of the test of randomness and the chi-square goodness of fit statistic for the von Mises distribution; default 20 Defines the centre (in degrees) of the sectors; default 0 Direction (in degrees) of the unimodal alternative distribution for the Rayleigh test; default `*` i.e. not known

### Parameters

`ANGLES` = factors or variates Directional observations (in degrees) Saves the summary statistics Saves structures relevant for calculation of the chi-square goodness of fit statistic for the von Mises distribution

### Description

`CDESCRIBE` summarizes data values that consist of directional observations recorded as angles between 0 and 360 degrees. These are supplied using the `ANGLES` parameter, in either a variate or a factor. The procedure mainly uses the methods presented in the book by Fisher (1993). The various statistics are cross-referenced below with the relevant page numbers.

`CDESCRIBE` prints the following summary statistics: number of observations, mean direction (page 31), circular standard deviation (page 32), mean resultant length (page 32), skewness (page 34) and estimate of the parameter Kappa (which provides the concentration parameter of the von Mises distribution for circular data; pages 39 and 88). If the angles are supplied in a factor, a grouping correction is applied to the mean resultant length and to the skewness (page 35).

Two tests of uniformity are presented. The null hypothesis for both of these is that the observations come from a uniform distribution around the circle. The first is a test of randomness against any alternative model. The test is based on counts of the number of observations in a set of angular sectors of equal size (page 67). If `ANGLES` is set to a variate, the width of the sectors is defined by the `SEGMENT` option (in degrees), with centres defined by the `MSEGMENT` option. The sectors are centred at `MSEGMENT`, `MSEGMENT+SEGMENT`, `MSEGMENT+2*SEGMENT`, and so on. The default values for `SEGMENT` and `MSEGMENT` are 20 and 0 respectively. If `ANGLES` is set to a factor with equidistant levels, it is assumed that the levels define the centres of the segments and that the limits of the sectors are at the midpoints between each pair of factor levels. If `ANGLES` is set to factor with non-equidistant levels, the `SEGMENT` and `MSEGMENT` options are used to define the angular sectors.

The second is Rayleigh’s test of uniformity against a unimodel alternative. The test is based on the mean resultant length and has two forms which differ according to whether or not the mean direction of the alternative distribution is known (pages 69 and 70). The direction, if known, is specified using the `DIRECTION` option.

Finally a goodness of fit test is calculated to assess whether the observations follow a von Mises distribution. This is a chi-square test, which compares the observed distribution with the expected distribution from a von Mises distribution with mean direction and concentration parameter (kappa) taking the values estimated from the observations. The observed and expected values are calculated for grouped directional data defined by the (`M`)`SEGMENT` options for a variate or by the factor levels if `ANGLES` is set to a factor.

The `PRINT` options controls whether the summary statistics are printed and whether a table of observed and expected counts for the fit of the von Mises distribution is printed. The summary statistics can be saved by means of the `RESULTS` parameter. The `VONMISESCOUNTS` parameter saves the grouped directional data used for calculation of the chi-square goodness of fit test and tables of observed and expected counts. Note that when `ANGLES` is set to factor, the saved grouped directional data is identical to `ANGLES`.

Options: `PRINT`, `SEGMENT`, `MSEGMENT`, `DIRECTION`.

Parameters: `ANGLES`, `RESULTS`, `VONMISESCOUNTS`.

### Method

`CDESCRIBE` uses methods described by Fisher (1993). A private version (`_SPECIALFUNCTION`) of the Biometris procedure `SPECIALFUNCTION` is used to calculate modified Bessel functions and related functions.

### Action with `RESTRICT`

If `ANGLES` is restricted, only the unrestricted units are analysed.

### Reference

Fisher, N.I. (1993). Statistical Analysis of Circular Data. Cambridge University Press, Cambridge.

Procedures: `CASSOCIATION`, `CCOMPARE`, `DCIRCULAR`, `RCIRCULAR`, `WINDROSE`.

Commands for: Basic and nonparametric statistics.

### Example

```CAPTION 'CDESCRIBE example',!t(\
'Directions chosen by 100 ants in response to an evenly illuminated',\
'black target placed at 180 degrees (see Fisher 1993, page 61).');\
STYLE=meta,plain
VARIATE [NVALUES=100] Direction
330 290 60  200 200 180 280 220 190 180
180 160 280 180 170 190 180 140 150 150
160 200 190 250 180 30  200 180 200 350
200 180 120 200 210 130 30  210 200 230
180 160 210 190 180 230 50  150 210 180
190 210 220 200 60  260 110 180 220 170
10  220 180 210 170 90  160 180 170 200
160 180 120 150 300 190 220 160 70  190
110 270 180 200 180 140 360 150 160 170
140 40  300 80  210 200 170 200 210 190 :
CDESCRIBE Direction
```
Updated on March 8, 2019