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

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.

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

References

Srivastava MS, Du M (2008). “A test for the mean vector with fewer observations than the dimension.” Journal of Multivariate Analysis, 99(3), 386--402. doi:10.1016/j.jmva.2006.11.002 .

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
SD2008.TS.NABT(group1,group2)
#> 
#> Results of Hypothesis Test
#> --------------------------
#> 
#> Test name:                       Srivastava and Du (2008)'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[SD] = 2.5704
#> 
#> Approximation method to the      Normal approximation
#> null distribution of T[SD]: 
#> 
#> Approximation parameter(s):      Adjustment coefficient = 15.2281
#> 
#> P-value:                         0.005078436
#>