apval_Cai2014: Asymptotics-Based p-value of the Test Proposed by Cai et al...
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 Cai et al (2014) based on the asymptotic distribution of the test statistic.

1	apval_Cai2014(sam1, sam2, eq.cov = TRUE)

`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.
`eq.cov`	a logical value. The default is `TRUE`, indicating that the two sample populations have same covariance; otherwise, the covariances are assumed to be different.

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 covariances of the two sample populations are Σ_1 = (σ_{1, ij}) and Σ_2 = (σ_{2, ij}). 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. For a vector v, we denote v^{(i)} as its ith element.

Cai et al (2014) proposed the following test statistic:

T_{CLX} = \max_{i = 1, …, p} (\bar{X}_1^{(i)} - \bar{X}_2^{(i)})^2/(σ_{1,ii}/n_1 + σ_{2, ii}/n_2),

This test statistic follows an extreme value 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`	the equality assumption on the covariances of the two sample populations; this was specified by the argument `eq.cov`.
`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 Cai et al (2014).

This function does not transform the data with their precision matrix (see Cai et al, 2014). To calculate the p-value of the test statisic with transformation, users can use transformed samples for sam1 and sam2.

Cai TT, Liu W, and Xia Y (2014). "Two-sample test of high dimensional means under dependence." Journal of the Royal Statistical Society: Series B (Statistical Methodology), 76(2), 349–372.

epval_Cai2014

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_Cai2014(sam1, sam2)

# the two sample populations have different covariances
true.cov1 <- 0.2^(abs(outer(1:p, 1:p, "-")))
true.cov2 <- 0.6^(abs(outer(1:p, 1:p, "-")))
sam1 <- mvrnorm(n = n1, mu = mu1, Sigma = true.cov1)
sam2 <- mvrnorm(n = n2, mu = mu2, Sigma = true.cov2)
apval_Cai2014(sam1, sam2, eq.cov = FALSE)