mvhtests (version 1.1)

Log-likelihood ratio test for equality of one covariance matrix: 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)

Value

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

Arguments

x

A matrix containing Euclidean data.

Sigma

The hypothesis covariance matrix.

a

The significance level, set to 0.05 by default.

Author

Michail Tsagris.

R implementation and documentation: Michail Tsagris .

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 \(\frac{1}{2}p\left(p+1\right)\).

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)