simultest.pe.chisq {PEtests}R Documentation

Two-sample PE simultaneous test using chi-squared approximation

Description

This function implements the two-sample PE simultaneous test on high-dimensional mean vectors and covariance matrices using chi-squared approximation. Suppose {X1,,Xn1}\{\mathbf{X}_1, \ldots, \mathbf{X}_{n_1}\} are i.i.d. copies of X\mathbf{X}, and {Y1,,Yn2}\{\mathbf{Y}_1, \ldots, \mathbf{Y}_{n_2}\} are i.i.d. copies of Y\mathbf{Y}. Let MPEM_{PE} and TPET_{PE} denote the PE mean test statistic and PE covariance test statistic, respectively. (see meantest.pe.comp and covtest.pe.comp for details). The PE simultaneous test statistic via chi-squared approximation is defined as

SPE=MPE2+TPE2.S_{PE} = M_{PE}^2 + T_{PE}^2.

It has been proved that with some regularity conditions, under the null hypothesis H0:μ1=μ2  and  Σ1=Σ2H_0: \boldsymbol{\mu}_1 = \boldsymbol{\mu}_2 \ \text{ and } \ \mathbf{\Sigma}_1 = \mathbf{\Sigma}_2, the two tests are asymptotically independent as n1,n2,pn_1, n_2, p\rightarrow \infty, and therefore SPES_{PE} asymptotically converges in distribution to a χ22\chi_2^2 distribution. The asymptotic pp-value is obtained by

p-value=1Fχ22(SPE),p\text{-value} = 1-F_{\chi_2^2}(S_{PE}),

where Fχ22()F_{\chi_2^2}(\cdot) is the cdf of the χ22\chi_2^2 distribution.

Usage

simultest.pe.chisq(dataX,dataY,delta_mean=NULL,delta_cov=NULL)

Arguments

dataX

an n1n_1 by pp data matrix

dataY

an n2n_2 by pp data matrix

delta_mean

a scalar; the thresholding value used in the construction of the PE component for mean test; see meantest.pe.comp for details.

delta_cov

a scalar; the thresholding value used in the construction of the PE component for covariance test; see covtest.pe.comp for details.

Value

stat the value of test statistic

pval the p-value for the test.

References

Yu, X., Li, D., Xue, L., and Li, R. (2022). Power-enhanced simultaneous test of high-dimensional mean vectors and covariance matrices with application to gene-set testing. Journal of the American Statistical Association, (in press):1–14.

Examples

n1 = 100; n2 = 100; pp = 500
set.seed(1)
X = matrix(rnorm(n1*pp), nrow=n1, ncol=pp)
Y = matrix(rnorm(n2*pp), nrow=n2, ncol=pp)
simultest.pe.chisq(X,Y)

[Package PEtests version 0.1.0 Index]