------------------------------------------------------------------------------- help forcircsummarize-------------------------------------------------------------------------------

Summary statistics for circular data

circsummarizevarlist[weight] [ifexp] [inrange] [,cilevel(#)rayleighkuiperdetail]

circsummarizevarname[weight] [ifexp] [inrange] [,by(byvar)cilevel(#)rayleighkuiperdetail]

fweights andaweights are allowed; see help weights. If weights are specified, they are used in calculating the mean direction and the vector strength (and hence the Rayleigh test and confidence limits if produced, although this may not be justifiable). Weights are not used in calculating the circular range or the Kuiper test.The abbreviation

circsu(only) is allowed.

Description

circsummarizeproduces summary statistics for circular variables with scales between 0 and 360 degrees. These are by defaultthe mean direction (in degrees),

the vector strength or mean resultant length (i.e. length of resultant / number of observations)

and the circular range (length of smallest arc including all observations) (in degrees).

Suppose we have

nmeasurements of a circular variablet. Then calculate in turnC= SUM costS= SUM sintarctan(S/C), the mean directionR= sqrt(C*C+S*S), the length of the resultant vectorR/n, the vector strength or mean resultant length.

Options

by()specifies that results are to be shown for each group defined by values ofbyvar. This option is only available if a singlevarnameis specified.

ciproduces a confidence interval for the vector mean suitable for large samples (at least 25 values, say). This is based on an assumption that sampling variation follows a normal distribution. If we calculate further

m_2 = (1 /n) SUM cos 2(t- vector mean oft),then the circular standard error CSE is estimated by

sqrt[

n* (1 -m_2) / (2 *R*R)and the confidence interval is estimated as

(vector mean - arcsin(

z* CSE), vector mean + arcsin(z* CSE)),where

zis the appropriate percentage point, givenlevel(), from the Normal distribution with mean 0 and standard deviation 1.

level()specifies the confidence level in percent for the confidence interval. See help for ci.

rayleighproduces the Rayleigh test, which tests a null hypothesis of uniformity against an alternative hypothesis of unimodality. The resultingP-value is shown.

kuiperproduces the Kuiper test, which tests a null hypothesis of uniformity against any alternative. The resultingP-value is shown. A numerical approximation due to Stephens (1970, p.118) is used that gives accuracy to 3 decimal places forP< 0.447.

detailis a synonym forci rayleigh kuiper.

Examples

. circsummarize axisasp wallasp

. circsummarize wallasp, by(grade) detail

ReferencesFisher, N.I. 1993.

Statistical analysis of circular data.Cambridge: Cambridge University Press.Stephens, M.A. 1970. Use of the Kolmogorov-Smirnov, Cramér-von Mises and related statistics without extensive tables.

Journal, RoyalStatistical Society Series B32: 115-22.

AuthorNicholas J. Cox, University of Durham, U.K. n.j.cox@durham.ac.uk

Also seeOn-line: help for ci