glht_fhw2004 {HDNRA}

Test proposed by Fujikoshi et al. (2004)

Description

Fujikoshi et al. (2004)'s test for general linear hypothesis testing (GLHT) problem for high-dimensional data with assuming that underlying covariance matrices are the same.

Usage

glht_fhw2004(Y,X,C)

Arguments

Y

An n\times p response matrix obtained by independently observing a p-dimensional response variable for n subjects.

X

A known n\times k full-rank design matrix with \operatorname{rank}(\boldsymbol{G})=k<n.

C

A known matrix of size q\times k with \operatorname{rank}(\boldsymbol{C})=q<k.

Details

A high-dimensional linear regression model can be expressed as

\boldsymbol{Y}=\boldsymbol{X\Theta}+\boldsymbol{\epsilon},

where \Theta is a k\times p unknown parameter matrix and \boldsymbol{\epsilon} is an n\times p error matrix.

It is of interest to test the following GLHT problem

H_0: \boldsymbol{C\Theta}=\boldsymbol{0}, \quad \text { vs. } \quad H_1: \boldsymbol{C\Theta} \neq \boldsymbol{0}.

Fujikoshi et al. (2004) proposed the following test statistic:

T_{FHW}=\sqrt{p}\left[(n-k)\frac{\operatorname{tr}(\boldsymbol{S}_h)}{\operatorname{tr}(\boldsymbol{S}_e)}-q\right],

where \boldsymbol{S}_h and \boldsymbol{S}_e are the matrices of sums of squares and products due to the hypothesis and the error, respecitively.

They showed that under the null hypothesis, T_{FHW} is asymptotically normally distributed.

Value

A (list) object of S3 class htest containing the following elements:

statistic

the test statistic proposed by Fujikoshi et al. (2004).

p.value

the p-value of the test proposed by Fujikoshi et al. (2004).

References

Fujikoshi Y, Himeno T, Wakaki H (2004). “Asymptotic results of a high dimensional MANOVA test and power comparison when the dimension is large compared to the sample size.” Journal of the Japan Statistical Society, 34(1), 19–26. doi:10.14490/jjss.34.19.

Examples

set.seed(1234)
k <- 3
q <- k-1
p <- 50
n <- c(25,30,40)
rho <- 0.01
Theta <- matrix(rep(0,k*p),nrow=k)
X <- matrix(c(rep(1,n[1]),rep(0,sum(n)),rep(1,n[2]),rep(0,sum(n)),rep(1,n[3])),ncol=k,nrow=sum(n))
y <- (-2*sqrt(1-rho)+sqrt(4*(1-rho)+4*p*rho))/(2*p)
x <- y+sqrt((1-rho))
Gamma <- matrix(rep(y,p*p),nrow=p)
diag(Gamma) <- rep(x,p)
U <- matrix(ncol = sum(n),nrow=p)
for(i in 1:sum(n)){
U[,i] <- rnorm(p,0,1)
}
Y <- X%*%Theta+t(U)%*%Gamma
C <- cbind(diag(q),-rep(1,q))
glht_fhw2004(Y,X,C)