SF2006.GLHT.NABT: Normal-approximation-based test for GLHT problem proposed by...

In HDNRA: High-Dimensional Location Testing with Normal-Reference Approaches

Normal-approximation-based test for GLHT problem proposed by Srivastava and Fujikoshi (2006)

Description

Srivastava and Fujikoshi (2006)'s test for general linear hypothesis testing (GLHT) problem for high-dimensional data with assuming that underlying covariance matrices are the same.

Usage

SF2006.GLHT.NABT(Y,X,C,n,p)

Arguments

Y	A list of k data matrices. The ith element represents the data matrix (n_i \times p) from the ith population with each row representing a p-dimensional observation.
X	A known n\times k full-rank design matrix with \operatorname{rank}(\boldsymbol{X})=k<n.
C	A known matrix of size q\times k with \operatorname{rank}(\boldsymbol{C})=q<k.
n	A vector of k sample sizes. The ith element represents the sample size of group i, n_i.
p	The dimension of data.

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}.

Srivastava and Fujikoshi (2006) proposed the following test statistic:

T_{SF}=\left[2q\hat{a}_2(1+(n-k)^{-1}q)\right]^{-1/2}\left[\frac{\operatorname{tr}(\boldsymbol{B})}{\sqrt{p}}-\frac{q}{\sqrt{n-k}}\frac{\operatorname{tr}(\boldsymbol{W})}{\sqrt{(n-k)p}}\right].

where \boldsymbol{W} and \boldsymbol{B} are the matrix of sum of squares and products due to error and the error, respectively, and \hat{a}_2=[\operatorname{tr}(\boldsymbol{W}^2)-\operatorname{tr}^2(\boldsymbol{W})/(n-k)]/[(n-k-1)(n-k+2)p]. They showed that under the null hypothesis, T_{SF} is asymptotically normally distributed.

Value

A list of class "NRtest" containing the results of the hypothesis test. See the help file for NRtest.object for details.

References

\insertRef

srivastava2006multivariateHDNRA

Examples

library("HDNRA")
data("corneal")
dim(corneal)
group1 <- as.matrix(corneal[1:43, ]) ## normal group
group2 <- as.matrix(corneal[44:57, ]) ## unilateral suspect group
group3 <- as.matrix(corneal[58:78, ]) ## suspect map group
group4 <- as.matrix(corneal[79:150, ]) ## clinical keratoconus group
p <- dim(corneal)[2]
Y <- list()
k <- 4
Y[[1]] <- group1
Y[[2]] <- group2
Y[[3]] <- group3
Y[[4]] <- group4
n <- c(nrow(Y[[1]]),nrow(Y[[2]]),nrow(Y[[3]]),nrow(Y[[4]]))
X <- matrix(c(rep(1,n[1]),rep(0,sum(n)),rep(1,n[2]), rep(0,sum(n)),
            rep(1,n[3]),rep(0,sum(n)),rep(1,n[4])),ncol=k,nrow=sum(n))
q <- k-1
C <- cbind(diag(q),-rep(1,q))
SF2006.GLHT.NABT(Y,X,C,n,p)

HDNRA documentation built on Oct. 30, 2024, 9:28 a.m.