apval_Sri2008 {highmean}

Asymptotics-Based p-value of the Test Proposed by Srivastava and Du (2008)

Description

Calculates p-value of the test for testing equality of two-sample high-dimensional mean vectors proposed by Srivastava and Du (2008) based on the asymptotic distribution of the test statistic.

Usage

apval_Sri2008(sam1, sam2)

Arguments

sam1

an n1 by p matrix from sample population 1. Each row represents a p-dimensional sample.

sam2

an n2 by p matrix from sample population 2. Each row represents a p-dimensional sample.

Details

Suppose that the two groups of p-dimensional independent and identically distributed samples \{X_{1i}\}_{i=1}^{n_1} and \{X_{2j}\}_{j=1}^{n_2} are observed; we consider high-dimensional data with p \gg n := n_1 + n_2 - 2. Assume that the two groups share a common covariance matrix. The primary object is to test H_{0}: \mu_1 = \mu_2 versus H_{A}: \mu_1 \neq \mu_2. Let \bar{X}_{k} be the sample mean for group k = 1, 2. Also, let S = n^{-1} \sum_{k = 1}^{2} \sum_{i = 1}^{n_{k}} (X_{ki} - \bar{X}_k) (X_{ki} - \bar{X}_k)^T be the pooled sample covariance matrix from the two groups.

Srivastava and Du (2008) proposed the following test statistic:

T_{SD} = \frac{(n_1^{-1} + n_2^{-1})^{-1} (\bar{X}_1 - \bar{X}_2)^T D_S^{-1} (\bar{X}_1 - \bar{X}_2) - (n - 2)^{-1} n p}{\sqrt{2 (tr R^2 - p^2 n^{-1}) c_{p, n}}},

where D_S = diag (s_{11}, s_{22}, ..., s_{pp}), s_{ii}'s are the diagonal elements of S, R = D_S^{-1/2} S D_S^{-1/2} is the sample correlation matrix and c_{p, n} = 1 + tr R^2 p^{-3/2}. This test statistic follows normal distribution under the null hypothesis.

Value

A list including the following elements:

sam.info

the basic information about the two groups of samples, including the samples sizes and dimension.

cov.assumption

this output reminds users that the two sample populations have a common covariance matrix.

method

this output reminds users that the p-values are obtained using the asymptotic distributions of test statistics.

pval

the p-value of the test proposed by Srivastava and Du (2008).

Note

The asymptotic distribution of the test statistic was derived under normality assumption in Bai and Saranadasa (1996). Also, this function assumes that the two sample populations have a common covariance matrix.

References

Srivastava MS and Du M (2008). "A test for the mean vector with fewer observations than the dimension." Journal of Multivariate Analysis, 99(3), 386–402.

See Also

epval_Sri2008

Examples

library(MASS)
set.seed(1234)
n1 <- n2 <- 50
p <- 200
mu1 <- rep(0, p)
mu2 <- mu1
mu2[1:10] <- 0.2
true.cov <- 0.4^(abs(outer(1:p, 1:p, "-"))) # AR1 covariance
sam1 <- mvrnorm(n = n1, mu = mu1, Sigma = true.cov)
sam2 <- mvrnorm(n = n2, mu = mu2, Sigma = true.cov)
apval_Sri2008(sam1, sam2)