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.
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 \(\operatorname{rank}(\boldsymbol{C})=q<k\).
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).
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.
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)
#>
#>
#>
#> data:
#> statistic = 0.56536, p-value = 0.2859
#>