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Rao spacing test for circular data

circraovarlist[ifexp] [inrange]

Description

circraocarries out a uniformity test for circular variablesvarlistwith scales between 0 and 360 degrees due to Rao (1969, 1976). Sort thenobserved directions and calculate the spacings as differences between successive ordered values: the last spacing is calculated from the last value to the first. Then calculate

U= (1/2) SUM (| spacing - 360 /n|).

Uhas the following interpretation: Placenarcs of fixed length 360 /ndegrees on the circumference, starting with each of the sample points. The circumference would be completely covered by these arcs only if the sample points were uniformly (equally) spaced.Uis the total uncovered portion of the circumference, or equivalently the extent to which the arcs overlap each other. Large values ofUindicate clustering of the sample points or evidence for rejecting the null hypothesis of uniformity. One merit of this test compared with some others is that it works well for data which are not unimodal.Critical values for this statistic are tabulated by Rao (1976, p.333), Batschelet (1981, p.339), Upton and Fingleton (1989, pp.248, 392, for

G= 2U), Mardia and Jupp (2000, p.368) and most extensively by Russell and Levitin (1995). The following are extracted from the last reference, p.885:

n P= 0.05P= 0.01 ----------------------- 4 186.45 221.14 5 183.44 211.93 6 180.65 206.79 7 177.83 202.55 8 175.68 198.46 9 173.68 195.27 10 171.98 192.37 11 170.45 189.88 12 169.09 187.66 13 167.87 185.68 14 166.76 183.90 15 165.75 182.28 16 164.83 180.81 17 163.98 179.46 18 163.20 178.22 19 162.47 177.08 20 161.79 176.01 21 161.16 175.02 22 160.56 174.10 23 160.01 173.23 24 159.48 172.41 25 158.99 171.64 26 158.52 170.92 27 158.07 170.23 28 157.65 169.58 29 157.25 168.96 30 156.87 168.38 35 155.19 165.81 40 153.82 163.73 45 152.68 162.00 50 151.70 160.53 75 148.34 155.49 100 146.29 152.46 150 143.83 148.84 200 142.35 146.67 300 140.57 144.09 400 139.50 142.54 500 138.77 141.48 600 138.23 140.70 700 137.80 140.09 800 137.46 139.60 900 137.18 139.19 1000 136.94 138.84For large

n, it follows from the work of Sherman (1950) that the sampling distribution ofUunder the null hypothesis of uniformity is approximately Normal with mean 360 / exp(1) = 132.4366 and standard deviation 360 * sqrt(2 exp(-1) - 5 exp(-2)) / sqrt(n) = 87.504786 / sqrt(n). The correspondingP-value is shown, irrespective ofn: users should decide whether to trust it.

Examples. circrao axisasp

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

ReferencesBatschelet, E. 1981.

Circular statistics in biology.London: Academic Press.Mardia, K.V. and Jupp, P.E. 2000.

Directional statistics.Chichester: John Wiley.Rao, J.S. 1969. Some contributions to the analysis of circular data. Ph.D. thesis, Indian Statistical Institute, Calcutta.

Rao, J.S. 1976. Some tests based on arc-lengths for the circle.

Sankhya38B, 339-348.Russell, G.S. and Levitin, D.J. 1995. An expanded table of probability values for Rao's spacing test.

Communications in Statistics,Simulation and Computation24, 879-888.Sherman, B. 1950. A random variable related to the spacing of sample values.

Annals of Mathematical Statistics21, 339-361.Upton, G.J.G. and Fingleton, B. 1989.

Spatial data analysis by example.Volume 2: Categorical and directional data.Chichester: John Wiley.

Also seeOn-line: circsummarize