SD2008.TS.NABT: Normal-approximation-based test for two-sample problem...

In HDNRA: High-Dimensional Location Testing with Normal-Reference Approaches

Normal-approximation-based test for two-sample problem proposed by Srivastava and Du (2008)

Description

Srivastava and Du (2008)'s test for testing equality of two-sample high-dimensional mean vectors with assuming that two covariance matrices are the same.

Usage

SD2008.TS.NABT(y1, y2)

Arguments

y1	The data matrix (n_1 \times p) from the first population. Each row represents a p-dimensional observation.
y2	The data matrix (n_2 \times p) from the second population. Each row represents a p-dimensional observation.

Details

Suppose we have two independent high-dimensional samples:

\boldsymbol{y}_{i1},\ldots,\boldsymbol{y}_{in_i}, \;\operatorname{are \; i.i.d. \; with}\; \operatorname{E}(\boldsymbol{y}_{i1})=\boldsymbol{\mu}_i,\; \operatorname{Cov}(\boldsymbol{y}_{i1})=\boldsymbol{\Sigma},i=1,2.

The primary object is to test

H_{0}: \boldsymbol{\mu}_1 = \boldsymbol{\mu}_2\; \operatorname{versus}\; H_{1}: \boldsymbol{\mu}_1 \neq \boldsymbol{\mu}_2.

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

T_{SD} = \frac{n^{-1}n_1n_2(\bar{\boldsymbol{y}}_1 - \bar{\boldsymbol{y}}_2)^\top \boldsymbol{D}_S^{-1}(\bar{\boldsymbol{y}}_1 - \bar{\boldsymbol{y}}_2) - \frac{(n-2)p}{n-4}}{\sqrt{2 \left[\operatorname{tr}(\boldsymbol{R}^2) - \frac{p^2}{n-2}\right] c_{p, n}}},

where \bar{\boldsymbol{y}}_{i},i=1,2 are the sample mean vectors, \boldsymbol{D}_S is the diagonal matrix of sample variance, \boldsymbol{R} is the sample correlation matrix and c_{p, n} is the adjustment coefficient proposed by Srivastava and Du (2008). They showed that under the null hypothesis, T_{SD} is asymptotically normally distributed.

Value

A list of class "NRtest" containing the results of the hypothesis test. See the help file for NRtest.object for details.

References

\insertRef

Srivastava_2008HDNRA

Examples

library("HDNRA")
data("COVID19")
dim(COVID19)
group1 <- as.matrix(COVID19[c(2:19, 82:87), ]) ## healthy group
group2 <- as.matrix(COVID19[-c(1:19, 82:87), ]) ## COVID-19 patients
SD2008.TS.NABT(group1,group2)

HDNRA documentation built on Oct. 30, 2024, 9:28 a.m.