Calculates summary statistics and tests of circular data (P.W. Goedhart & R.W. Payne).
|What to print (
||Width of sectors (in degrees) into which to group an
||Defines the centre (in degrees) of the sectors; default 0|
||Direction (in degrees) of the unimodal alternative distribution for the Rayleigh test; default
||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|
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+2*SEGMENT, and so on. The default values for
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
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
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 (
SEGMENT options for a variate or by the factor levels if
ANGLES is set to a factor.
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
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.
ANGLES is restricted, only the unrestricted units are analysed.
Fisher, N.I. (1993). Statistical Analysis of Circular Data. Cambridge University Press, Cambridge.
Commands for: Basic and nonparametric statistics.
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 READ 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