apval_Sri2008: Asymptotics-Based p-value of the Test Proposed by Srivastava...
In highmean: Two-Sample Tests for High-Dimensional Mean Vectors

Description Usage Arguments Details Value Note References See Also Examples

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.

1	apval_Sri2008(sam1, sam2)

`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.

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}: μ_1 = μ_2 versus H_{A}: μ_1 \neq μ_2. Let \bar{X}_{k} be the sample mean for group k = 1, 2. Also, let S = n^{-1} ∑_{k = 1}^{2} ∑_{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}{√{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.

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).

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.

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.

epval_Sri2008

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)

Loading required package: mvtnorm
$sam.info
 n1  n2   p 
 50  50 200 

$cov.assumption
[1] "the two groups have same covariance"

$method
[1] "asymptotic distribution"

$pval
Sri2008 
0.61321