LRT03_EqualityCovariances: p-value of the log-transformed LRT statistic and/or its null...

In gajdosandrej/CharFunToolR: Numerical Computation Cumulative Distribution Function and Probability Density Function from Characteristic Function

p-value of the log-transformed LRT statistic and/or its null distribution CF/PDF/CDF

Description

LRT03_EqualityCovariances(W, n, p, q, options) computes p-value of the log-transformed LRT statistic, W = -log(\Lambda), for testing the null hypothesis of equality of covariance matrices (under normality assumptions) of q (q > 1) p-dimensional populations, and/or its null distribution CF/PDF/CDF.

This is based on BALANCED samples of size n for each population!

Usage

LRT03_EqualityCovariances(W, n, p, q, options)

Arguments

W	observed value of the minus log-transformed LRT statistic W = -log(\Lambda). If empty, the algorithm evaluates the CF/PDF/CDF and the quantiles of the null distribution of W.
n	sample size, n > min(p+q-1).
p	common dimension of the vectors X_k, k = 1,...q.
q	number of normal populations, q > 1.
options	option structure, for more details see cf2DistGP. Moreover, x set vector of values where PDF/CDF is evaluated, prob set vector of probabilities for the quantiles, coef set arbitrary multiplicator of the argument t of the characteristic function. If empty, default value is -n/2 (standard value for minus log-transform of LRT). Possible alternative is e.g. coef = -1, leading to W = -(2/n)*log(LRT).

Details

In particular, let X_k ~ N_p(mu_k,\Sigma_k) are p-dimensional random vectors, for k = 1,...,q. We want to test the hypothesis that the covariance matrix \Sigma is common for all X_k, k = 1,...,q. Then, the null hypothesis is given as H0: \Sigma_1 = ... = \Sigma_q, i.e. the covariance matrices are equal in all q populations. Here, the LRT test statistic is given by \Lambda = ( q^{p*q} * prod(det(S_k)) / (det(S))^q )^{n/2}, where S_k are MLEs of \Sigma_k, for k = 1,...,q, and S = S_1 + ... + S_q, based on n samples from each of the the q p-dimensional populations.

Under null hypothesis, distribution of the test statistic \Lambda is \Lambda ~ prod_{k=1}^q prod_{j=1}^{p} (B_{jk})^{n/2}, with B_{jk} ~ Beta((n-j)/2,(j*(q-1)+2*k-1-q)/2), and we set B_{11} = 1 for j=k=1. Here we assume that n > p.

Hence, the exact characteristic function of the null distribution of minus log-transformed LRT statistic \Lambda, say W = -log(\Lambda) is given by cf = function(t) {cf_LogRV_Beta(-(n/2)*t, (n-j)/2, (j*(q-1)+2*k-1-q)/(2*q))}, where k = (1*o,...,q*o) with p-dimensional vector of ones o = (1,...,1) and j = (j_1,...,j_q) with j_k = 1:p.

Value

p-value of the log-transformed LRT statistic, W = -log(\Lambda) and/or its null distribution CF/PDF/CDF.

Note

Ver.: 16-Sep-2018 21:10:08 (consistent with Matlab CharFunTool v1.3.0, 20-Jan-2018 12:43:15).

References

[1] ANDERSON, Theodore Wilbur. An Introduction to Multivariate Statistical Analysis. New York: Wiley, 3rd Ed., 2003.

[2] MARQUES, Filipe J.; COELHO, Carlos A.; ARNOLD, Barry C. A general near-exact distribution theory for the most common likelihood ratio test statistics used in Multivariate Analysis. Test, 2011, 20.1:180-203.

[3] WITKOVSKY, Viktor. Exact distribution of selected multivariate test criteria by numerical inversion of their characteristic functions.
arXiv preprint arXiv:1801.02248, 2018.

See Also

Other Likelihood Ratio Test: LRT01_Independence(), LRT02_EqualityMeans(), LRT04_EqualityPopulations(), LRT05_Sphericity()

Examples

## EXAMPLE
# LRT for testing hypothesis on equality of covariances
# Null distribution of the minus log-transformed LRT statistic
n <- 30              # total sample size
p <- 8               # dimension of X_k, k = 1,...,q where q = 5
q <- 5               # number of populations
# W <- vector()      # observed value of W = -log(Lambda)
options <- list()
# options$coef <- -1
options$prob <- c(0.9, 0.95, 0.99)
output <- LRT03_EqualityCovariances(n = n, p = p, q = q, options = options)
str(output)

gajdosandrej/CharFunToolR documentation built on June 3, 2024, 7:46 p.m.