apval_Bai1996: Asymptotics-Based p-value of the Test Proposed by Bai and...

In highmean: Two-Sample Tests for High-Dimensional Mean Vectors

Description

Calculates p-value of the test for testing equality of two-sample high-dimensional mean vectors proposed by Bai and Saranadasa (1996) based on the asymptotic distribution of the test statistic.

Usage

apval_Bai1996(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}: 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.

Bai and Saranadasa (1996) proposed the following test statistic:

T_{BS} = \frac{(n_1^{-1} + n_2^{-1})^{-1} (\bar{X}_1 - \bar{X}_2)^T (\bar{X}_1 - \bar{X}_2) - tr S}{√{2 n (n + 1) (n - 1)^{-1} (n + 2)^{-1} [tr S^2 - n^{-1} (tr S)^2]}},

and its asymptotic distribution is normal 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 Bai and Saranadasa (1996).

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

Bai ZD and Saranadasa H (1996). "Effect of high dimension: by an example of a two sample problem." Statistica Sinica, 6(2), 311–329.

See Also

epval_Bai1996

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_Bai1996(sam1, sam2)

Example output

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
  Bai1996 
0.5597888 

highmean documentation built on May 2, 2019, 3:45 p.m.