apval_Chen2014 {highmean}

Asymptotics-Based p-value of the Test Proposed by Chen et al (2014)

Description

Calculates p-value of the test for testing equality of two-sample high-dimensional mean vectors proposed by Chen et al (2014) based on the asymptotic distribution of the test statistic.

Usage

apval_Chen2014(sam1, sam2, eq.cov = TRUE)

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.

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.

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 covariances of the two sample populations are \Sigma_1 = (\sigma_{1, ij}) and \Sigma_2 = (\sigma_{2, ij}). 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. For a vector v, we denote v^{(i)} as its ith element.

Chen et al (2014) proposed removing estimated zero components in the mean difference through thresholding; they considered

T_{CLZ}(s) = \sum_{i = 1}^{p} \left\{ \frac{(\bar{X}_1^{(i)} - \bar{X}_2^{(i)})^2}{\sigma_{1,ii}/n_1 + \sigma_{2,ii}/n_2} - 1 \right\} I \left\{ \frac{(\bar{X}_1^{(i)} - \bar{X}_2^{(i)})^2}{\sigma_{1,ii}/n_1 + \sigma_{2,ii}/n_2} > \lambda_{p} (s) \right\},

where the threshold level is \lambda_p(s) := 2 s \log p and I(\cdot) is the indicator function. Since an optimal choice of the threshold is unknown, they proposed trying all possible threshold values, then choosing the most significant one as their final test statistic:

T_{CLZ} = \max_{s \in (0, 1 - \eta)} \{ T_{CLZ}(s) - \hat{\mu}_{T_{CLZ}(s), 0}\}/\hat{\sigma}_{T_{CLZ}(s), 0},

where \hat{\mu}_{T_{CLZ}(s), 0} and \hat{\sigma}_{T_{CLZ}(s), 0} are estimates of the mean and standard deviation of T_{CLZ}(s) under the null hypothesis. They derived its asymptotic null distribution as an extreme value distribution.

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

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 Chen et al (2014).

Note

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

References

Chen SX, Li J, and Zhong PS (2014). "Two-Sample Tests for High Dimensional Means with Thresholding and Data Transformation." arXiv preprint arXiv:1410.2848.

See Also

epval_Chen2014

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_Chen2014(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_Chen2014(sam1, sam2, eq.cov = FALSE)