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Rao spacing test for circular data

circrao varlist [if exp] [in range]

Description

circrao carries out a uniformity test for circular variables varlist with
scales between 0 and 360 degrees due to Rao (1969, 1976). Sort the n
observed 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 |).

U has the following interpretation: Place n arcs of fixed length 360 / n
degrees 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. U is the total uncovered
portion of the circumference, or equivalently the extent to which the
arcs overlap each other. Large values of U indicate 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 = 0.05 P = 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   138.84

For large n, it follows from the work of Sherman (1950) that the sampling
distribution of U under 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 corresponding P-value is shown, irrespective of n: users
should decide whether to trust it.

Examples

. circrao axisasp

Author

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

References

Batschelet, 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.  Sankhya
38B, 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 Computation 24, 879-888.

Sherman, B. 1950. A random variable related to the spacing of sample
values.  Annals of Mathematical Statistics 21, 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 see

On-line: circsummarize

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