Normal-reference-test with two-cumulant (2-c) matched χ^2-approximation for two-sample BF problem proposed by Zhang et al. (2023)

Zhang et al. (2023)'s test for testing equality of two-sample high-dimensional mean vectors without assuming that two covariance matrices are the same.

Usage

ZZZ2023.TSBF.2cNRT(y1, y2, cutoff)

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.

cutoff

An empirical criterion for applying the adjustment coefficient

Value

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

Details

Suppose we have two independent high-dimensional samples: $$ \boldsymbol{y}_{i1},\ldots,\boldsymbol{y}_{in_i}, \;\operatorname{are \; i.i.d. \; with}\; \operatorname{E}(\boldsymbol{y}_{i1})=\boldsymbol{\mu}_i,\; \operatorname{Cov}(\boldsymbol{y}_{i1})=\boldsymbol{\Sigma}_i,i=1,2. $$ The primary object is to test $$H_{0}: \boldsymbol{\mu}_1 = \boldsymbol{\mu}_2\; \operatorname{versus}\; H_{1}: \boldsymbol{\mu}_1 \neq \boldsymbol{\mu}_2.$$ Zhang et al.(2023) proposed the following test statistic: $$T_{ZZZ}=\frac{n_1 n_2}{np}(\bar{\boldsymbol{y}}_1-\bar{\boldsymbol{y}}_2)^{\top} \hat{\boldsymbol{D}}_n^{-1}(\bar{\boldsymbol{y}}_1-\bar{\boldsymbol{y}}_2),$$ where \(\bar{\boldsymbol{y}}_{i},i=1,2\) are the sample mean vectors, and \(\hat{\boldsymbol{D}}_n=\operatorname{diag}(\hat{\boldsymbol{\Sigma}}_1/n+\hat{\boldsymbol{\Sigma}}_2/n)\) with \(n=n_1+n_2\). They showed that under the null hypothesis, \(T_{ZZZ}\) and a chi-squared-type mixture have the same limiting distribution.

References

Zhang L, Zhu T, Zhang J (2023). “Two-sample Behrens--Fisher problems for high-dimensional data: a normal reference scale-invariant test.” Journal of Applied Statistics, 50(3), 456--476. doi:10.1080/02664763.2020.1834516 .

Examples

library("HDNRA")
data("COVID19")
dim(COVID19)
#> [1]    87 20460
group1 <- as.matrix(COVID19[c(2:19, 82:87), ]) ## healthy group
group2 <- as.matrix(COVID19[-c(1:19, 82:87), ]) ## COVID-19 patients
ZZZ2023.TSBF.2cNRT(group1,group2,cutoff=1.2)
#> 
#> Results of Hypothesis Test
#> --------------------------
#> 
#> Test name:                       Zhang et al. (2023)'s test
#> 
#> Null Hypothesis:                 Difference between two mean vectors is 0
#> 
#> Alternative Hypothesis:          Difference between two mean vectors is not 0
#> 
#> Data:                            group1 and group2
#> 
#> Sample Sizes:                    n1 = 24
#>                                  n2 = 62
#> 
#> Sample Dimension:                20460
#> 
#> Test Statistic:                  T[ZZZ] = 6.511
#> 
#> Approximation method to the      2-c matched chi^2-approximation
#> null distribution of T[ZZZ]: 
#> 
#> Approximation parameter(s):      df  = 10.1280
#>                                  cpn = 17.9488
#> 
#> P-value:                         3.043196e-10
#>