YS2012.GLHT.NABT {HDNRA}R Documentation

Normal-approximation-based test for GLHT problem proposed by Yamada and Srivastava (2012)

Description

Yamada and Srivastava (2012)'test for general linear hypothesis testing (GLHT) problem for high-dimensional data with assuming that underlying covariance matrices are the same.

Usage

YS2012.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. } H_1: \boldsymbol{C\Theta} \neq \boldsymbol{0}.

Yamada and Srivastava (2012) proposed the following test statistic:

T_{YS}=\frac{(n-k)\operatorname{tr}(\boldsymbol{S}_h\boldsymbol{D}_{\boldsymbol{S}_e}^{-1})-(n-k)pq/(n-k-2)}{\sqrt{2q[\operatorname{tr}(\boldsymbol{R}^2)-p^2/(n-k)]c_{p,n}}},

where \boldsymbol{S}_h and \boldsymbol{S}_e are the variation matrices due to the hypothesis and error, respectively, and \boldsymbol{D}_{\boldsymbol{S}_e} and \boldsymbol{R} are diagonal matrix with the diagonal elements of \boldsymbol{S}_e and the sample correlation matrix, respectively. c_{p, n} is the adjustment coefficient proposed by Yamada and Srivastava (2012). They showed that under the null hypothesis, T_{YS} 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

Yamada T, Srivastava MS (2012). “A test for multivariate analysis of variance in high dimension.” Communications in Statistics-Theory and Methods, 41(13-14), 2602–2615. doi:10.1080/03610926.2011.581786.

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))
YS2012.GLHT.NABT(Y,X,C,n,p)



[Package HDNRA version 2.0.1 Index]