glht_zgz2017 {HDNRA}

Test proposed by Zhang et al. (2017)

Description

Zhang et al. (2017)'s test for general linear hypothesis testing (GLHT) problem for high-dimensional data with assuming that underlying covariance matrices are the same.

Usage

glht_zgz2017(Y,G,n,p)

Arguments

Y

A list of k data matrices. The ith element represents the data matrix (p\times n_i) from the ith population with each column representing a p-dimensional observation.

G

A known full-rank coefficient matrix (q\times k) with \operatorname{rank}(\boldsymbol{G})<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

Suppose we have the following k independent high-dimensional samples:

\boldsymbol{y}_{i1},\ldots,\boldsymbol{y}_{in_i}, \;\operatorname{are \; i.i.d. \; with}\; \operatorname{E}(\boldsymbol{y}_{i1})=\boldsymbol{\mu}_i,\; \operatorname{Cov}(\boldsymbol{y}_{i1})=\boldsymbol{\Sigma},\;i=1,\ldots,k.

It is of interest to test the following GLHT problem:

H_0: \boldsymbol{G M}=\boldsymbol{0}, \quad \text { vs. } \quad H_1: \boldsymbol{G M} \neq \boldsymbol{0},

where \boldsymbol{M}=(\boldsymbol{\mu}_1,\ldots,\boldsymbol{\mu}_k)^\top is a k\times p matrix collecting k mean vectors and \boldsymbol{G}:q\times k is a known full-rank coefficient matrix with \operatorname{rank}(\boldsymbol{G})<k.

Zhang et al. (2017) proposed the following test statistic:

T_{ZGZ}=\|\boldsymbol{C \hat{\mu}}\|^2,

where \boldsymbol{C}=[(\boldsymbol{G D G}^\top)^{-1/2}\boldsymbol{G}]\otimes\boldsymbol{I}_p, and \hat{\boldsymbol{\mu}}=(\bar{\boldsymbol{y}}_1^\top,\ldots,\bar{\boldsymbol{y}}_k^\top)^\top, with \bar{\boldsymbol{y}}_{i},i=1,\ldots,k being the sample mean vectors and \boldsymbol{D}=\operatorname{diag}(1/n_1,\ldots,1/n_k).

They showed that under the null hypothesis, T_{ZGZ} and a chi-squared-type mixture have the same normal or non-normal limiting distribution.

Value

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

statistic

the test statistic proposed by Zhang et al. (2017)

p.value

the p-value of the test proposed by Zhang et al. (2017).

beta

the parameters used in Zhang et al. (2017)'s test.

df

estimated approximate degrees of freedom of Zhang et al.(2017)'s test.

References

Zhang J, Guo J, Zhou B (2017). “Linear hypothesis testing in high-dimensional one-way MANOVA.” Journal of Multivariate Analysis, 155, 200–216. doi:10.1016/j.jmva.2017.01.002.

Examples

set.seed(1234)
k <- 3
p <- 50
n <- c(25, 30, 40)
rho <- 0.1
M <- matrix(rep(0, k * p), nrow = k, ncol = p)
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)
Y <- list()
for (g in 1:k) {
  Z <- matrix(rnorm(n[g] * p, mean = 0, sd = 1), p, n[g])
  Y[[g]] <- Gamma %*% Z + t(t(M[g, ])) %*% (rep(1, n[g]))
}
G <- cbind(diag(k - 1), rep(-1, k - 1))
glht_zgz2017(Y, G, n, p)