A test for the significance of the mean direction and the concentration parameter of a circular distribution.
W. S. Rhode
March, 1976
This note describes the development of the Rayleigh test for the significance of the mean direction in a cycle (period) histogram. It is important to know what confidence is to be placed in the mean and concentration parameter (sync coefficient). This is a problem in circular distributions. It is described in the Statistics of Directional Data by K.V. Mardia (AP, 1972).
We shall assume that the underlying population is von Mises,
where U0 is the mean direction and
k
is the concentration parameter
and 
is a modified Bessel function of the first kind.
Tests of Uniformity
Let q 1,...q n be a random sample from a population with p.d.f. f(q We are interested in testing the null hypothesis
against the alternative
where g(q) has the given form but may contain unknown parameters. The parameters U0,k will be assumed unknown in M(U0,k ).
We want to obtain the likelihood ratio test for this situation. Consider a random sample X1,X2,….,Xn from a distribution having p.d.f. f(x,q ),q e W . The joint p.d.f. may be regarded as a function of q . When so regarded, it is called the likelihood function L of the random sample, and we write
We seek a function, say U(x1,….,xn) such that when q is replaced by
U(x1,….,xn) the likelihood function, L, is a maximum.
Therefore 
=
The maximal likelihood statistics are 
and 
.
To determine 
and 
, often the derivative of the log L is used.
(1) 
(2) 
(3) 
Let 
therefore
(4) 
this derives from eqn.(2) 
® eqn.(4).
From eqn.(3)
or, A(k )=R
(4) 
The likelihood ratio statistic is
(5) 
now, 
If q is distributed as M(0,k ) then
since 
and 
therefore 
, using this in eqn. (5) implies that l
is a monotonically
decreasing function of 
.
,therefore is a monotonically increasing function of 
.
The critical region (CR) l
< K reduces to 
and
The pdf of R under H0 is
and the pdf of R under H1 is
Note that CR is that subset of a space which leads us to reject the H0 hypothesis.
The power function of a test of a statistical hypothesis H0 against an alternative
hypothesis H1 is that function which yields the probability that the sample point
falls in the critical region C of the test, i.e., a function which yields the
prob. of rejecting the hypothesis under consideration. The value of the power
function at a parameter point is called the power of the test.
Critical values* of the Rayleigh test of uniformity with the test-statistics
n a 0.10 0.05 0.025 0.01 0.001
5 0.677 0.754 0.816 0.879 0.991
6 .618 .690 .753 .825 .940
7 .572 .642 .702 .771 .891
8 .535 .602 .660 .725 .847
9 .504 .569 .624 .687 .808
10 .478 .540 .594 .655 .775
11 .456 .516 .567 .627 .743
12 .437 .494 .544 .602 .716
13 .420 .475 .524 .580 .692
14 .405 .458 .505 .560 .669
15 .391 .443 .489 .542 .649
16 .379 .429 .474 .525 .630
17 .367 .417 .460 .510 .613
18 .357 .405 .447 .496 .597
19 .348 .394 .436 .484 .583
20 .339 .385 .425 .472 .569
21 .331 .375 .415 .461 .556
22 .323 .367 .405 .451 .544
23 .316 .359 .397 .441 .533
24 .309 .351 .389 .432 .522
25 .303 .344 .381 .423 .512
30 .277 .315 .348 .387 .470
35 .256 .292 .323 .359 .436
40 .240 .273 .302 .336 .409
45 .226 .257 .285 .318 .386
50 .214 .244 .270 .301 .367
100 .15 .17 .19 .21 .26
4.605 5.991 7.378 9.210 13.816
*Based on Table 2 of Stephens (1969d) with the kind permission of the author and the editor of J. Amer. Statist. Ass. and on Batschelet (1971) with the kind permission of the author and the publisher, Amer. Inst. Biol. Sciences and Dr. W.T. Keeton, New York University.