Log-likelihood ratio test for equality of one covariance matrix {mvhtests}

Log-likelihood ratio test for equality of one covariance matrix

Description

Log-likelihood ratio test for equality of one covariance matrix.

Usage

equal.cov(x, Sigma, a = 0.05)

Arguments

x

A matrix containing Euclidean data.

Sigma

The hypothesis covariance matrix.

a

The significance level, set to 0.05 by default.

Details

The hypothesis test is that the the sample covariance is equal to some specified covariance matrix: H_0:\pmb{\Sigma}=\pmb{\Sigma}_0, with \pmb{\mu} unknown. The algorithm for this test is taken from Mardia, Bibby and Kent (1979, pg. 126-127). The test is based upon the log-likelihood ratio test. The form of the test is

-2\log{\lambda}=n \text{tr}\left\lbrace \pmb{\Sigma}_0^{-1}{\bf S}\right\rbrace-n\log{\left|\pmb{\Sigma}_0^{-1}{\bf S} \right|}-np,

where n is the sample size, \pmb{\Sigma}_0 is the specified covariance matrix under the null hypothesis, {\bf S} is the sample covariance matrix and p is the dimensionality of the data (or the number of variables). Let \alpha and g denote the arithmetic mean and the geometric mean respectively of the eigenvalues of \pmb{\Sigma}_0^{-1}{\bf S}, so that tr\left\lbrace \pmb{\Sigma}_0^{-1}{\bf S}\right\rbrace=p\alpha and \left|\pmb{\Sigma}_0^{-1}{\bf S} \right|=g^p, then the test statistic becomes

-2\log{\lambda}=np\left(\alpha-log{(g)}-1 \right).

The degrees of freedom of the \chi^2 distribution are \frac{1}{2}p\left(p+1\right).

Value

A vector with the the test statistic, the p-value, the degrees of freedom and the critical value of the test.

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris [e-mail].

References

Mardia K.V., Kent J.T. and Bibby J.M. (1979). Multivariate Analysis. London: Academic Press.

See Also

likel.cov, Mtest.cov

Examples

x <- as.matrix( iris[, 1:4] )
s <- cov(x) * 1.5
equal.cov(x, s)