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{title:Rao spacing test for circular data} 

{p 8 17 2}{cmd:circrao} {it:varlist} [{cmd:if} {it:exp}] [{cmd:in} {it:range}] 


{title:Description}

{p 4 4 2}{cmd:circrao} carries out a uniformity test for circular variables 
{it:varlist} with
scales between 0 and 360 degrees due to Rao (1969, 1976). Sort the {it: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 

{p 8 8 2}{it:U} = (1/2) SUM ({c |} spacing - 360 / {it:n} {c |}). 

{p 4 4 2}{it:U} has the following interpretation: Place {it:n} arcs of fixed length
360 / {it: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. {it:U} is the total uncovered
portion of the circumference, or equivalently the extent to which the arcs
overlap each other. Large values of {it: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. 

{p 4 4 2}Critical values for this statistic are tabulated by Rao (1976, p.333),
Batschelet (1981, p.339), Upton and Fingleton (1989, pp.248, 392, for {it:G} =
2{it:U}), Mardia and Jupp (2000, p.368) and most extensively by Russell and
Levitin (1995). The following are extracted from the last reference, p.885: 

        {it:  n  P} = 0.05 {it:P} = 0.01
        {hline 23}
	  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 

{p 4 4 2}For large {it:n}, it follows from the work of Sherman (1950) that the
sampling distribution of {it: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({it:n}) = 87.504786 / sqrt({it:n}).
The corresponding {it:P}-value is shown, irrespective of {it:n}: users 
should decide whether to trust it. 


{title:Examples}

{p 4 8 2}{inp:. circrao axisasp}


{title:Author}

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


{title:References} 

{p 4 8 2}Batschelet, E. 1981. {it:Circular statistics in biology.} London: Academic
Press. 

{p 4 8 2}Mardia, K.V. and Jupp, P.E. 2000. {it:Directional statistics.} Chichester:
John Wiley. 

{p 4 8 2}Rao, J.S. 1969. Some contributions to the analysis of circular data.  Ph.D.
thesis, Indian Statistical Institute, Calcutta. 

{p 4 8 2}Rao, J.S. 1976. Some tests based on arc-lengths for the circle.
{it:Sankhya} 38B, 339-348. 

{p 4 8 2}Russell, G.S. and Levitin, D.J. 1995. An expanded table of probability
values for Rao's spacing test. 
{it:Communications in Statistics, Simulation and Computation} 
24, 879-888. 

{p 4 8 2}Sherman, B. 1950. A random variable related to the spacing of sample values.
{it:Annals of Mathematical Statistics} 21, 339-361. 

{p 4 8 2}Upton, G.J.G. and Fingleton, B. 1989. 
{it:Spatial data analysis by example. Volume 2: Categorical and directional data.} 
Chichester: John Wiley. 


{title:Also see}

{p 4 13 2} 
On-line: {help circsummarize}