circular (version 0.3-1)

aov.circular: Analysis of Variance for circular data

Usage

aov.circular(x, group, kappa = NULL,
    method = c("F.test", "LRT"), F.mod = TRUE)
## S3 method for class 'aov.circular':
print(x, digits = max(3, getOption("digits") - 3), ...)

Arguments

x

a vector of class circular.

group

a vector identifying the groups or samples.

kappa

the common value of the concentration parameter. Used when method is "LRT". If left unspecified (by default) the maximum likelihood estimate of kappa is computed and used in the test statistic.

method

the test statistic to use; either a high-concentration F-test or a likelihood ratio test.

F.mod

logical; if TRUE, the AOV F-statistic is modified by a factor of 1+3/8k to improve its approximate F distribution. Default is TRUE.

digits

the number of digits to be printed.

...

additional arguments.

Value

• An object of class aov.circular with the folowing components:
• mu
  mean direction for each sample.
• mu.all
  mean direction of all samples combined.
• kappa
  concentration parameter for each sample.
• kappa.all
  concentration parameter for all samples combined.
• rho
  mean resultant length for each sample.
• rho.all
  mean resultant length for all samples combined.
• method
  the test statistic used.
• df
  degrees of freedom.
• statistic
  the value of the test statistic.
• p.value
  the p.value of the test statistic.
• call
  match.call().
• If the method is "F.test" then the object contains also:
• SSE
  Sum of squares used in F-test.
• MSE
  Mean squares used in F-test.

Details

The samples are assumed to have been drawn from von Mises populations with equal concentration parameter, kappa. The null hypothesis being tested is that all populations also have the same mean direction. If method is "F.test" a high concentration F-test makes use of a decomposition of total sample variation into between groups and within groups variation, analogous to the one-way classification analysis of variance for linear data. Stephens (1972) presented an improved modification to the F-test derived from this decomposition. This is implemented when F.mod is TRUE. A likelihood ratio test is performed when method is "LRT". This function uses the test statistic presented by Cordeiro, Paula, and Botter (1994) which has an approximate chi-squared distribution. If the common concentration parameter is known, it can be specified and used in the computation of the test statistic. Otherwise, the maximum likelihood estimate of the common concentration parameter is used.

References

Cordeiro, G., Paula, G. and Botter, D. (1994). Improved likelihood ratio tests for dispersion models. International Statistical Review, 62, 257-274. Jammalamadaka, S. Rao and SenGupta, A. (2001). Topics in Circular Statistics, Section 5.3, World Scientific Press, Singapore. Mardia, K. and Jupp, P. (1999). Directional Statistics, Section 7.4, John Wiley and Sons, England. Stephens, M. (1972). Multi-sample tests for the von Mises distribution. Technical Report 190, Department of Statistics, Stanford University.

Examples

x <- c(rvonmises(50, 0, 1), rvonmises(100, pi/3, 10))
group <- c(rep(0, 50), rep(1, 100))

aov.circular(x, group)
aov.circular(x, group, method="LRT")