Package 'PEtests'

Power-Enhanced (PE) Tests for High-Dimensional Data

Description

The package implements several two-sample power-enhanced mean tests, covariance tests, and simultaneous tests on mean vectors and covariance matrices for high-dimensional data.

Details

There are three main functions:
covtest
meantest
simultest

References

Chen, S. X. and Qin, Y. L. (2010). A two-sample test for high-dimensional data with applications to gene-set testing. Annals of Statistics, 38(2):808–835. doi:10.1214/09-AOS716

Cai, T. T., Liu, W., and Xia, Y. (2013). Two-sample covariance matrix testing and support recovery in high-dimensional and sparse settings. Journal of the American Statistical Association, 108(501):265–277. doi:10.1080/01621459.2012.758041

Cai, T. T., Liu, W., and Xia, Y. (2014). Two-sample test of high dimensional means under dependence. Journal of the Royal Statistical Society: Series B: Statistical Methodology, 76(2):349–372. doi:10.1111/rssb.12034

Li, J. and Chen, S. X. (2012). Two sample tests for high-dimensional covariance matrices. The Annals of Statistics, 40(2):908–940. doi:10.1214/12-AOS993

Yu, X., Li, D., and Xue, L. (2022). Fisher’s combined probability test for high-dimensional covariance matrices. Journal of the American Statistical Association, (in press):1–14. doi:10.1080/01621459.2022.2126781

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. doi:10.1080/01621459.2022.2061354

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)
covtest(X, Y)
meantest(X, Y)
simultest(X, Y)

---

Two-sample covariance tests for high-dimensional data

Description

This function implements five two-sample covariance tests on high-dimensional covariance matrices. Let $\mathbf{X} \in \mathbb{R}^p$ and $\mathbf{Y} \in \mathbb{R}^p$ be two $p$-dimensional populations with mean vectors $(\boldsymbol{\mu}_1, \boldsymbol{\mu}_2)$ and covariance matrices $(\mathbf{\Sigma}_1, \mathbf{\Sigma}_2)$, respectively. The problem of interest is to test the equality of the two covariance matrices:

$H_{0c}: \mathbf{\Sigma}_1 = \mathbf{\Sigma}_2.$

Suppose $\{\mathbf{X}_1, \ldots, \mathbf{X}_{n_1}\}$ are i.i.d. copies of $\mathbf{X}$, and $\{\mathbf{Y}_1, \ldots, \mathbf{Y}_{n_2}\}$ are i.i.d. copies of $\mathbf{Y}$. We denote dataX=$(\mathbf{X}_1, \ldots, \mathbf{X}_{n_1})^\top\in\mathbb{R}^{n_1\times p}$ and dataY=$(\mathbf{Y}_1, \ldots, \mathbf{Y}_{n_2})^\top\in\mathbb{R}^{n_2\times p}$.

Usage

covtest(dataX,dataY,method='pe.comp',delta=NULL)

Arguments

dataX	an $n_1$ by $p$ data matrix
dataY	an $n_2$ by $p$ data matrix
method	the method type (default = 'pe.comp'); chosen from 'clx': the $l_\infty$-norm-based covariance test, proposed in Cai et al. (2013); see covtest.clx for details. 'lc': the $l_2$-norm-based covariance test, proposed in Li and Chen (2012); see covtest.lc for details. 'pe.cauchy': the PE covariance test via Cauchy combination; see covtest.pe.cauchy for details. 'pe.comp': the PE covariance test via the construction of PE components; see covtest.pe.comp for details. 'pe.fisher': the PE covariance test via Fisher's combination; see covtest.pe.fisher for details.
delta	This is needed only in method='pe.comp'; see covtest.pe.comp for details. The default is NULL.

Value

method the method type

stat the value of test statistic

pval the p-value for the test.

References

Cai, T. T., Liu, W., and Xia, Y. (2013). Two-sample covariance matrix testing and support recovery in high-dimensional and sparse settings. Journal of the American Statistical Association, 108(501):265–277.

Li, J. and Chen, S. X. (2012). Two sample tests for high-dimensional covariance matrices. The Annals of Statistics, 40(2):908–940.

Yu, X., Li, D., and Xue, L. (2022). Fisher’s combined probability test for high-dimensional covariance matrices. Journal of the American Statistical Association, (in press):1–14.

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)
covtest(X,Y)

---

Two-sample high-dimensional covariance test (Cai, Liu and Xia, 2013)

Description

This function implements the two-sample $l_\infty$-norm-based high-dimensional covariance test proposed in Cai, Liu and Xia (2013). Suppose $\{\mathbf{X}_1, \ldots, \mathbf{X}_{n_1}\}$ are i.i.d. copies of $\mathbf{X}$, and $\{\mathbf{Y}_1, \ldots, \mathbf{Y}_{n_2}\}$ are i.i.d. copies of $\mathbf{Y}$. The test statistic is defined as

$T_{CLX} = \max_{1\leq i,j \leq p} \frac{(\hat\sigma_{ij1}-\hat\sigma_{ij2})^2} {\hat\theta_{ij1}/n_1+\hat\theta_{ij2}/n_2},$

where $\hat\sigma_{ij1}$ and $\hat\sigma_{ij2}$ are the sample covariances, and $\hat\theta_{ij1}/n_1+\hat\theta_{ij2}/n_2$ estimates the variance of $\hat{\sigma}_{ij1}-\hat{\sigma}_{ij2}$. The explicit formulas of $\hat\sigma_{ij1}$, $\hat\sigma_{ij2}$, $\hat\theta_{ij1}$ and $\hat\theta_{ij2}$ can be found in Section 2 of Cai, Liu and Xia (2013). With some regularity conditions, under the null hypothesis $H_{0c}: \mathbf{\Sigma}_1 = \mathbf{\Sigma}_2$, the test statistic $T_{CLX}-4\log p+\log\log p$ converges in distribution to a Gumbel distribution $G_{cov}(x) = \exp(-\frac{1}{\sqrt{8\pi}}\exp(-\frac{x}{2}))$ as $n_1, n_2, p \rightarrow \infty$. The asymptotic $p$-value is obtained by

$p_{CLX} = 1-G_{cov}(T_{CLX}-4\log p+\log\log p).$

Usage

covtest.clx(dataX,dataY)

Arguments

dataX	an $n_1$ by $p$ data matrix
dataY	an $n_2$ by $p$ data matrix

Value

stat the value of test statistic

pval the p-value for the test.

References

Cai, T. T., Liu, W., and Xia, Y. (2013). Two-sample covariance matrix testing and support recovery in high-dimensional and sparse settings. Journal of the American Statistical Association, 108(501):265–277.

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)
covtest.clx(X,Y)

---

Two-sample high-dimensional covariance test (Li and Chen, 2012)

Description

This function implements the two-sample $l_2$-norm-based high-dimensional covariance test proposed by Li and Chen (2012). Suppose $\{\mathbf{X}_1, \ldots, \mathbf{X}_{n_1}\}$ are i.i.d. copies of $\mathbf{X}$, and $\{\mathbf{Y}_1, \ldots, \mathbf{Y}_{n_2}\}$ are i.i.d. copies of $\mathbf{Y}$. The test statistic $T_{LC}$ is defined as

$T_{LC} = A_{n_1}+B_{n_2}-2C_{n_1,n_2},$

where $A_{n_1}$, $B_{n_2}$, and $C_{n_1,n_2}$ are unbiased estimators for $\mathrm{tr}(\mathbf{\Sigma}^2_1)$, $\mathrm{tr}(\mathbf{\Sigma}^2_2)$, and $\mathrm{tr}(\mathbf{\Sigma}_1\mathbf{\Sigma}_2)$, respectively. Under the null hypothesis $H_{0c}: \mathbf{\Sigma}_1 = \mathbf{\Sigma}_2$, the leading variance of $T_{LC}$ is $\sigma^2_{T_{LC}} = 4(\frac{1}{n_1}+\frac{1}{n_2})^2 \rm{tr}^2(\mathbf{\Sigma}^2)$, which can be consistently estimated by $\hat\sigma^2_{LC}$. The explicit formulas of $A_{n_1}$, $B_{n_2}$, $C_{n_1,n_2}$ and $\hat\sigma^2_{T_{LC}}$ can be found in Equations (2.1), (2.2) and Theorem 1 of Li and Chen (2012). With some regularity conditions, under the null hypothesis $H_{0c}: \mathbf{\Sigma}_1 = \mathbf{\Sigma}_2$, the test statistic $T_{LC}$ converges in distribution to a standard normal distribution as $n_1, n_2, p \rightarrow \infty$. The asymptotic $p$-value is obtained by

$p_{LC} = 1-\Phi(T_{LC}/\hat\sigma_{T_{LC}}),$

where $\Phi(\cdot)$ is the cdf of the standard normal distribution.

Usage

covtest.lc(dataX,dataY)

Arguments

dataX	an $n_1$ by $p$ data matrix
dataY	an $n_2$ by $p$ data matrix

Value

stat the value of test statistic

pval the p-value for the test.

References

Li, J. and Chen, S. X. (2012). Two sample tests for high-dimensional covariance matrices. The Annals of Statistics, 40(2):908–940.

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)
covtest.lc(X,Y)

---

Two-sample PE covariance test for high-dimensional data via Cauchy combination

Description

This function implements the two-sample PE covariance test via Cauchy combination. Suppose $\{\mathbf{X}_1, \ldots, \mathbf{X}_{n_1}\}$ are i.i.d. copies of $\mathbf{X}$, and $\{\mathbf{Y}_1, \ldots, \mathbf{Y}_{n_2}\}$ are i.i.d. copies of $\mathbf{Y}$. Let $p_{LC}$ and $p_{CLX}$ denote the $p$-values associated with the $l_2$-norm-based covariance test (see covtest.lc for details) and the $l_\infty$-norm-based covariance test (see covtest.clx for details), respectively. The PE covariance test via Cauchy combination is defined as

$T_{Cauchy} = \frac{1}{2}\tan((0.5-p_{LC})\pi) + \frac{1}{2}\tan((0.5-p_{CLX})\pi).$

It has been proved that with some regularity conditions, under the null hypothesis $H_{0c}: \mathbf{\Sigma}_1 = \mathbf{\Sigma}_2,$ the two tests are asymptotically independent as $n_1, n_2, p\rightarrow \infty$, and therefore $T_{Cauchy}$ asymptotically converges in distribution to a standard Cauchy distribution. The asymptotic $p$-value is obtained by

$p\text{-value} = 1-F_{Cauchy}(T_{Cauchy}),$

where $F_{Cauchy}(\cdot)$ is the cdf of the standard Cauchy distribution.

Usage

covtest.pe.cauchy(dataX,dataY)

Arguments

dataX	an $n_1$ by $p$ data matrix
dataY	an $n_2$ by $p$ data matrix

Value

stat the value of test statistic

pval the p-value for the test.

References

Yu, X., Li, D., and Xue, L. (2022). Fisher’s combined probability test for high-dimensional covariance matrices. 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)
covtest.pe.cauchy(X,Y)

---

Two-sample PE covariance test for high-dimensional data via PE component

Description

This function implements the two-sample PE covariance test via the construction of the PE component. Let $T_{LC}/\hat\sigma_{T_{LC}}$ denote the $l_2$-norm-based covariance test statistic (see covtest.lc for details). The PE component is constructed by

$J_c=\sqrt{p}\sum_{i=1}^p\sum_{j=1}^p T_{ij}\widehat\xi^{-1/2}_{ij} \mathcal{I}\{ \sqrt{2}T_{ij}\widehat\xi^{-1/2}_{ij} +1 > \delta_{cov} \},$

where $\delta_{cov}$ is a threshold for the screening procedure, recommended to take the value of $\delta_{cov}=4\log(\log (n_1+n_2))\log p$. The explicit forms of $T_{ij}$ and $\widehat\xi_{ij}$ can be found in Section 3.2 of Yu et al. (2022). The PE covariance test statistic is defined as

$T_{PE}=T_{LC}/\hat\sigma_{T_{LC}}+J_c.$

With some regularity conditions, under the null hypothesis $H_{0c}: \mathbf{\Sigma}_1 = \mathbf{\Sigma}_2$, the test statistic $T_{PE}$ converges in distribution to a standard normal distribution as $n_1, n_2, p \rightarrow \infty$. The asymptotic $p$-value is obtained by

$p\text{-value}=1-\Phi(T_{PE}),$

where $\Phi(\cdot)$ is the cdf of the standard normal distribution.

Usage

covtest.pe.comp(dataX,dataY,delta=NULL)

Arguments

dataX	an $n_1$ by $p$ data matrix
dataY	an $n_2$ by $p$ data matrix
delta	a scalar; the thresholding value used in the construction of the PE component. If not specified, the function uses a default value $\delta_{cov}=4\log(\log (n_1+n_2))\log p$.

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)
covtest.pe.comp(X,Y)

---

Two-sample PE covariance test for high-dimensional data via Fisher's combination

Description

This function implements the two-sample PE covariance test via Fisher's combination. Suppose $\{\mathbf{X}_1, \ldots, \mathbf{X}_{n_1}\}$ are i.i.d. copies of $\mathbf{X}$, and $\{\mathbf{Y}_1, \ldots, \mathbf{Y}_{n_2}\}$ are i.i.d. copies of $\mathbf{Y}$. Let $p_{LC}$ and $p_{CLX}$ denote the $p$-values associated with the $l_2$-norm-based covariance test (see covtest.lc for details) and the $l_\infty$-norm-based covariance test (see covtest.clx for details), respectively. The PE covariance test via Fisher's combination is defined as

$T_{Fisher} = -2\log(p_{LC})-2\log(p_{CLX}).$

It has been proved that with some regularity conditions, under the null hypothesis $H_{0c}: \mathbf{\Sigma}_1 = \mathbf{\Sigma}_2,$ the two tests are asymptotically independent as $n_1, n_2, p\rightarrow \infty$, and therefore $T_{Fisher}$ asymptotically converges in distribution to a $\chi_4^2$ distribution. The asymptotic $p$-value is obtained by

$p\text{-value} = 1-F_{\chi_4^2}(T_{Fisher}),$

where $F_{\chi_4^2}(\cdot)$ is the cdf of the $\chi_4^2$ distribution.

Usage

covtest.pe.fisher(dataX,dataY)

Arguments

dataX	an $n_1$ by $p$ data matrix
dataY	an $n_2$ by $p$ data matrix

Value

stat the value of test statistic

pval the p-value for the test.

References

Yu, X., Li, D., and Xue, L. (2022). Fisher’s combined probability test for high-dimensional covariance matrices. 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)
covtest.pe.fisher(X,Y)

---

Two-sample mean tests for high-dimensional data

Description

This function implements five two-sample mean tests on high-dimensional mean vectors. Let $\mathbf{X} \in \mathbb{R}^p$ and $\mathbf{Y} \in \mathbb{R}^p$ be two $p$-dimensional populations with mean vectors $(\boldsymbol{\mu}_1, \boldsymbol{\mu}_2)$ and covariance matrices $(\mathbf{\Sigma}_1, \mathbf{\Sigma}_2)$, respectively. The problem of interest is to test the equality of the two mean vectors of the two populations:

$H_{0m}: \boldsymbol{\mu}_1 = \boldsymbol{\mu}_2.$

Suppose $\{\mathbf{X}_1, \ldots, \mathbf{X}_{n_1}\}$ are i.i.d. copies of $\mathbf{X}$, and $\{\mathbf{Y}_1, \ldots, \mathbf{Y}_{n_2}\}$ are i.i.d. copies of $\mathbf{Y}$. We denote dataX=$(\mathbf{X}_1, \ldots, \mathbf{X}_{n_1})^\top\in\mathbb{R}^{n_1\times p}$ and dataY=$(\mathbf{Y}_1, \ldots, \mathbf{Y}_{n_2})^\top\in\mathbb{R}^{n_2\times p}$.

Usage

meantest(dataX,dataY,method='pe.comp',delta=NULL)

Arguments

dataX	an $n_1$ by $p$ data matrix
dataY	an $n_2$ by $p$ data matrix
method	the method type (default = 'pe.comp'); chosen from 'clx': the $l_\infty$-norm-based mean test, proposed in Cai et al. (2014); see meantest.clx for details. 'cq': the $l_2$-norm-based mean test, proposed in Chen and Qin (2010); see meantest.cq for details. 'pe.cauchy': the PE mean test via Cauchy combination; see meantest.pe.cauchy for details. 'pe.comp': the PE mean test via the construction of PE components; see meantest.pe.comp for details. 'pe.fisher': the PE mean test via Fisher's combination; see meantest.pe.fisher for details.
delta	This is needed only in method='pe.comp'; see meantest.pe.comp for details. The default is NULL.

Value

method the method type

stat the value of test statistic

pval the p-value for the test.

References

Chen, S. X. and Qin, Y. L. (2010). A two-sample test for high-dimensional data with applications to gene-set testing. Annals of Statistics, 38(2):808–835.

Cai, T. T., Liu, W., and Xia, Y. (2014). Two-sample test of high dimensional means under dependence. Journal of the Royal Statistical Society: Series B: Statistical Methodology, 76(2):349–372.

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)
meantest(X,Y)

---

Two-sample high-dimensional mean test (Cai, Liu and Xia, 2014)

Description

This function implements the two-sample $l_\infty$-norm-based high-dimensional mean test proposed in Cai, Liu and Xia (2014). Suppose $\{\mathbf{X}_1, \ldots, \mathbf{X}_{n_1}\}$ are i.i.d. copies of $\mathbf{X}$, and $\{\mathbf{Y}_1, \ldots, \mathbf{Y}_{n_2}\}$ are i.i.d. copies of $\mathbf{Y}$. The test statistic is defined as

$M_{CLX}=\frac{n_1n_2}{n_1+n_2}\max_{1\leq j\leq p} \frac{(\bar{X_j}-\bar{Y_j})^2} {\frac{1}{n_1+n_2} [\sum_{u=1}^{n_1} (X_{uj}-\bar{X_j})^2+\sum_{v=1}^{n_2} (Y_{vj}-\bar{Y_j})^2] }$

With some regularity conditions, under the null hypothesis $H_{0c}: \mathbf{\Sigma}_1 = \mathbf{\Sigma}_2$, the test statistic $M_{CLX}-2\log p+\log\log p$ converges in distribution to a Gumbel distribution $G_{mean}(x) = \exp(-\frac{1}{\sqrt{\pi}}\exp(-\frac{x}{2}))$ as $n_1, n_2, p \rightarrow \infty$. The asymptotic $p$-value is obtained by

$p_{CLX} = 1-G_{mean}(M_{CLX}-2\log p+\log\log p).$

Usage

meantest.clx(dataX,dataY)

Arguments

dataX	an $n_1$ by $p$ data matrix
dataY	an $n_2$ by $p$ data matrix

Value

stat the value of test statistic

pval the p-value for the test.

References

Cai, T. T., Liu, W., and Xia, Y. (2014). Two-sample test of high dimensional means under dependence. Journal of the Royal Statistical Society: Series B: Statistical Methodology, 76(2):349–372.

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)
meantest.clx(X,Y)

---

Two-sample high-dimensional mean test (Chen and Qin, 2010)

Description

This function implements the two-sample $l_2$-norm-based high-dimensional mean test proposed by Chen and Qin (2010). Suppose $\{\mathbf{X}_1, \ldots, \mathbf{X}_{n_1}\}$ are i.i.d. copies of $\mathbf{X}$, and $\{\mathbf{Y}_1, \ldots, \mathbf{Y}_{n_2}\}$ are i.i.d. copies of $\mathbf{Y}$. The test statistic $M_{CQ}$ is defined as

$M_{CQ} = \frac{1}{n_1(n_1-1)}\sum_{u\neq v}^{n_1} \mathbf{X}_{u}'\mathbf{X}_{v} +\frac{1}{n_2(n_2-1)}\sum_{u\neq v}^{n_2} \mathbf{Y}_{u}'\mathbf{Y}_{v} -\frac{2}{n_1n_2}\sum_u^{n_1}\sum_v^{n_2} \mathbf{X}_{u}'\mathbf{Y}_{v}.$

Under the null hypothesis $H_{0m}: \boldsymbol{\mu}_1 = \boldsymbol{\mu}_2$, the leading variance of $M_{CQ}$ is $\sigma^2_{M_{CQ}}=\frac{2}{n_1(n_1-1)}\text{tr}(\mathbf{\Sigma}_1^2)+ \frac{2}{n_2(n_2-1)}\text{tr}(\mathbf{\Sigma}_2^2)+ \frac{4}{n_1n_2}\text{tr}(\mathbf{\Sigma}_1\mathbf{\Sigma}_2)$, which can be consistently estimated by $\widehat\sigma^2_{M_{CQ}}= \frac{2}{n_1(n_1-1)}\widehat{\text{tr}(\mathbf{\Sigma}_1^2)}+ \frac{2}{n_2(n_2-1)}\widehat{\text{tr}(\mathbf{\Sigma}_2^2)}+ \frac{4}{n_1n_2}\widehat{\text{tr}(\mathbf{\Sigma}_1\mathbf{\Sigma}_2)}.$ The explicit formulas of $\widehat{\text{tr}(\mathbf{\Sigma}_1^2)}$, $\widehat{\text{tr}(\mathbf{\Sigma}_2^2)}$, and $\widehat{\text{tr}(\mathbf{\Sigma}_1\mathbf{\Sigma}_2)}$ can be found in Section 3 of Chen and Qin (2010). With some regularity conditions, under the null hypothesis $H_{0m}: \boldsymbol{\mu}_1 = \boldsymbol{\mu}_2$, the test statistic $M_{CQ}$ converges in distribution to a standard normal distribution as $n_1, n_2, p \rightarrow \infty$. The asymptotic $p$-value is obtained by

$p_{CQ} = 1-\Phi(M_{CQ}/\hat\sigma_{M_{CQ}}),$

where $\Phi(\cdot)$ is the cdf of the standard normal distribution.

Usage

meantest.cq(dataX,dataY)

Arguments

dataX	an $n_1$ by $p$ data matrix
dataY	an $n_2$ by $p$ data matrix

Value

stat the value of test statistic

pval the p-value for the test.

References

Chen, S. X. and Qin, Y. L. (2010). A two-sample test for high-dimensional data with applications to gene-set testing. Annals of Statistics, 38(2):808–835.

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)
meantest.cq(X,Y)

---

Two-sample PE mean test for high-dimensional data via Cauchy combination

Description

This function implements the two-sample PE covariance test via Cauchy combination. Suppose $\{\mathbf{X}_1, \ldots, \mathbf{X}_{n_1}\}$ are i.i.d. copies of $\mathbf{X}$, and $\{\mathbf{Y}_1, \ldots, \mathbf{Y}_{n_2}\}$ are i.i.d. copies of $\mathbf{Y}$. Let $p_{CQ}$ and $p_{CLX}$ denote the $p$-values associated with the $l_2$-norm-based covariance test (see meantest.cq for details) and the $l_\infty$-norm-based covariance test (see meantest.clx for details), respectively. The PE covariance test via Cauchy combination is defined as

$M_{Cauchy} = \frac{1}{2}\tan((0.5-p_{CQ})\pi) + \frac{1}{2}\tan((0.5-p_{CLX})\pi).$

It has been proved that with some regularity conditions, under the null hypothesis $H_{0m}: \boldsymbol{\mu}_1 = \boldsymbol{\mu}_2,$ the two tests are asymptotically independent as $n_1, n_2, p\rightarrow \infty$, and therefore $M_{Cauchy}$ asymptotically converges in distribution to a standard Cauchy distribution. The asymptotic $p$-value is obtained by

$p\text{-value} = 1-F_{Cauchy}(M_{Cauchy}),$

where $F_{Cauchy}(\cdot)$ is the cdf of the standard Cauchy distribution.

Usage

meantest.pe.cauchy(dataX,dataY)

Arguments

dataX	an $n_1$ by $p$ data matrix
dataY	an $n_2$ by $p$ data matrix

Value

stat the value of test statistic

pval the p-value for the test.

References

Chen, S. X. and Qin, Y. L. (2010). A two-sample test for high-dimensional data with applications to gene-set testing. Annals of Statistics, 38(2):808–835.

Cai, T. T., Liu, W., and Xia, Y. (2014). Two-sample test of high dimensional means under dependence. Journal of the Royal Statistical Society: Series B: Statistical Methodology, 76(2):349–372.

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)
meantest.pe.cauchy(X,Y)

---

Two-sample PE mean test for high-dimensional data via PE component

Description

This function implements the two-sample PE mean via the construction of the PE component. Let $M_{CQ}/\hat\sigma_{M_{CQ}}$ denote the $l_2$-norm-based mean test statistic (see meantest.cq for details). The PE component is constructed by

$J_m = \sqrt{p}\sum_{i=1}^p M_i\widehat\nu^{-1/2}_i \mathcal{I}\{ \sqrt{2}M_i\widehat\nu^{-1/2}_i + 1 > \delta_{mean} \},$

where $\delta_{mean}$ is a threshold for the screening procedure, recommended to take the value of $\delta_{mean}=2\log(\log (n_1+n_2))\log p$. The explicit forms of $M_{i}$ and $\widehat\nu_{j}$ can be found in Section 3.1 of Yu et al. (2022). The PE covariance test statistic is defined as

$M_{PE}=M_{CQ}/\hat\sigma_{M_{CQ}}+J_m.$

With some regularity conditions, under the null hypothesis $H_{0m}: \boldsymbol{\mu}_1 = \boldsymbol{\mu}_2$, the test statistic $M_{PE}$ converges in distribution to a standard normal distribution as $n_1, n_2, p \rightarrow \infty$. The asymptotic $p$-value is obtained by

$p\text{-value}= 1-\Phi(M_{PE}),$

where $\Phi(\cdot)$ is the cdf of the standard normal distribution.

Usage

meantest.pe.comp(dataX,dataY,delta=NULL)

Arguments

dataX	an $n_1$ by $p$ data matrix
dataY	an $n_2$ by $p$ data matrix
delta	a scalar; the thresholding value used in the construction of the PE component. If not specified, the function uses a default value $\delta_{mean}=2\log(\log (n_1+n_2))\log p$.

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)
meantest.pe.comp(X,Y)

---

Two-sample PE mean test for high-dimensional data via Fisher's combination

Description

This function implements the two-sample PE covariance test via Fisher's combination. Suppose $\{\mathbf{X}_1, \ldots, \mathbf{X}_{n_1}\}$ are i.i.d. copies of $\mathbf{X}$, and $\{\mathbf{Y}_1, \ldots, \mathbf{Y}_{n_2}\}$ are i.i.d. copies of $\mathbf{Y}$. Let $p_{CQ}$ and $p_{CLX}$ denote the $p$-values associated with the $l_2$-norm-based covariance test (see meantest.cq for details) and the $l_\infty$-norm-based covariance test (see meantest.clx for details), respectively. The PE covariance test via Fisher's combination is defined as

$M_{Fisher} = -2\log(p_{CQ})-2\log(p_{CLX}).$

It has been proved that with some regularity conditions, under the null hypothesis $H_{0m}: \boldsymbol{\mu}_1 = \boldsymbol{\mu}_2,$ the two tests are asymptotically independent as $n_1, n_2, p\rightarrow \infty$, and therefore $M_{Fisher}$ asymptotically converges in distribution to a $\chi_4^2$ distribution. The asymptotic $p$-value is obtained by

$p\text{-value} = 1-F_{\chi_4^2}(M_{Fisher}),$

where $F_{\chi_4^2}(\cdot)$ is the cdf of the $\chi_4^2$ distribution.

Usage

meantest.pe.fisher(dataX,dataY)

Arguments

dataX	an $n_1$ by $p$ data matrix
dataY	an $n_2$ by $p$ data matrix

Value

stat the value of test statistic

pval the p-value for the test.

References

Chen, S. X. and Qin, Y. L. (2010). A two-sample test for high-dimensional data with applications to gene-set testing. Annals of Statistics, 38(2):808–835.

Cai, T. T., Liu, W., and Xia, Y. (2014). Two-sample test of high dimensional means under dependence. Journal of the Royal Statistical Society: Series B: Statistical Methodology, 76(2):349–372.

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)
meantest.pe.fisher(X,Y)

---

Two-sample simultaneous tests on high-dimensional mean vectors and covariance matrices

Description

This function implements six two-sample simultaneous tests on high-dimensional mean vectors and covariance matrices. Let $\mathbf{X} \in \mathbb{R}^p$ and $\mathbf{Y} \in \mathbb{R}^p$ be two $p$-dimensional populations with mean vectors $(\boldsymbol{\mu}_1, \boldsymbol{\mu}_2)$ and covariance matrices $(\mathbf{\Sigma}_1, \mathbf{\Sigma}_2)$, respectively. The problem of interest is the simultaneous inference on the equality of mean vectors and covariance matrices of the two populations:

$H_0: \boldsymbol{\mu}_1 = \boldsymbol{\mu}_2 \ \text{ and } \ \mathbf{\Sigma}_1 = \mathbf{\Sigma}_2.$

Suppose $\{\mathbf{X}_1, \ldots, \mathbf{X}_{n_1}\}$ are i.i.d. copies of $\mathbf{X}$, and $\{\mathbf{Y}_1, \ldots, \mathbf{Y}_{n_2}\}$ are i.i.d. copies of $\mathbf{Y}$. We denote dataX=$(\mathbf{X}_1, \ldots, \mathbf{X}_{n_1})^\top\in\mathbb{R}^{n_1\times p}$ and dataY=$(\mathbf{Y}_1, \ldots, \mathbf{Y}_{n_2})^\top\in\mathbb{R}^{n_2\times p}$.

Usage

simultest(dataX, dataY, method='pe.fisher', delta_mean=NULL, delta_cov=NULL)

Arguments

dataX	an $n_1$ by $p$ data matrix
dataY	an $n_2$ by $p$ data matrix
method	the method type (default = 'pe.fisher'); chosen from 'cauchy': the simultaneous test via Cauchy combination; see simultest.cauchy for details. 'chisq': the simultaneous test via chi-squared approximation; see simultest.chisq for details. 'fisher': the simultaneous test via Fisher's combination; see simultest.fisher for details. 'pe.cauchy': the PE simultaneous test via Cauchy combination; see simultest.pe.cauchy for details. 'pe.chisq': the PE simultaneous test via chi-squared approximation; see simultest.pe.chisq for details. 'pe.fisher': the PE simultaneous test via Fisher's combination; see simultest.pe.fisher for details.
delta_mean	the thresholding value used in the construction of the PE component for the mean test statistic. It is needed only in PE methods such as method='pe.cauchy', method='pe.chisq', and method='pe.fisher'; see simultest.pe.cauchy, simultest.pe.chisq, and simultest.pe.fisher for details. The default is NULL.
delta_cov	the thresholding value used in the construction of the PE component for the covariance test statistic. It is needed only in PE methods such as method='pe.cauchy', method='pe.chisq', and method='pe.fisher'; see simultest.pe.cauchy, simultest.pe.chisq, and simultest.pe.fisher for details. The default is NULL.

Value

method the method type

stat the value of test statistic

pval the p-value for the test.

References

Chen, S. X. and Qin, Y. L. (2010). A two-sample test for high-dimensional data with applications to gene-set testing. Annals of Statistics, 38(2):808–835.

Li, J. and Chen, S. X. (2012). Two sample tests for high-dimensional covariance matrices. The Annals of Statistics, 40(2):908–940.

Yu, X., Li, D., and Xue, L. (2022). Fisher’s combined probability test for high-dimensional covariance matrices. Journal of the American Statistical Association, (in press):1–14.

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(X,Y)

---

Two-sample simultaneous test using Cauchy combination

Description

This function implements the two-sample simultaneous test on high-dimensional mean vectors and covariance matrices using Cauchy combination. Suppose $\{\mathbf{X}_1, \ldots, \mathbf{X}_{n_1}\}$ are i.i.d. copies of $\mathbf{X}$, and $\{\mathbf{Y}_1, \ldots, \mathbf{Y}_{n_2}\}$ are i.i.d. copies of $\mathbf{Y}$. Let $p_{CQ}$ and $p_{LC}$ denote the $p$-values associated with the $l_2$-norm-based mean test proposed in Chen and Qin (2010) (see meantest.cq for details) and the $l_2$-norm-based covariance test proposed in Li and Chen (2012) (see covtest.lc for details), respectively. The simultaneous test statistic via Cauchy combination is defined as

$C_{n_1, n_2} = \frac{1}{2}\tan((0.5-p_{CQ})\pi) + \frac{1}{2}\tan((0.5-p_{LC})\pi).$

It has been proved that with some regularity conditions, under the null hypothesis $H_0: \boldsymbol{\mu}_1 = \boldsymbol{\mu}_2 \ \text{ and } \ \mathbf{\Sigma}_1 = \mathbf{\Sigma}_2$, the two tests are asymptotically independent as $n_1, n_2, p\rightarrow \infty$, and therefore $C_{n_1,n_2}$ asymptotically converges in distribution to a standard Cauchy distribution. The asymptotic $p$-value is obtained by

$p\text{-value} = 1-F_{Cauchy}(C_{n_1,n_2}),$

where $F_{Cauchy}(\cdot)$ is the cdf of the standard Cauchy distribution.

Usage

simultest.cauchy(dataX,dataY)

Arguments

dataX	an $n_1$ by $p$ data matrix
dataY	an $n_2$ by $p$ data matrix

Value

stat the value of test statistic

pval the p-value for the test.

References

Chen, S. X. and Qin, Y. L. (2010). A two-sample test for high-dimensional data with applications to gene-set testing. Annals of Statistics, 38(2):808–835.

Li, J. and Chen, S. X. (2012). Two sample tests for high-dimensional covariance matrices. The Annals of Statistics, 40(2):908–940.

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.cauchy(X,Y)

---

Two-sample simultaneous test using chi-squared approximation

Description

This function implements the two-sample simultaneous test on high-dimensional mean vectors and covariance matrices using chi-squared approximation. Suppose $\{\mathbf{X}_1, \ldots, \mathbf{X}_{n_1}\}$ are i.i.d. copies of $\mathbf{X}$, and $\{\mathbf{Y}_1, \ldots, \mathbf{Y}_{n_2}\}$ are i.i.d. copies of $\mathbf{Y}$. Let $M_{CQ}/\hat\sigma_{M_{CQ}}$ denote the $l_2$-norm-based mean test statistic proposed in Chen and Qin (2010) (see meantest.cq for details), and let $T_{LC}/\hat\sigma_{T_{LC}}$ denote the $l_2$-norm-based covariance test statistic proposed in Li and Chen (2012) (see covtest.lc for details). The simultaneous test statistic via chi-squared approximation is defined as

$S_{n_1, n_2} = M_{CQ}^2/\hat\sigma^2_{M_{CQ}} + T_{LC}^2/\hat\sigma^2_{T_{LC}}.$

It has been proved that with some regularity conditions, under the null hypothesis $H_0: \boldsymbol{\mu}_1 = \boldsymbol{\mu}_2 \ \text{ and } \ \mathbf{\Sigma}_1 = \mathbf{\Sigma}_2$, the two tests are asymptotically independent as $n_1, n_2, p\rightarrow \infty$, and therefore $S_{n_1,n_2}$ asymptotically converges in distribution to a $\chi_2^2$ distribution. The asymptotic $p$-value is obtained by

$p\text{-value} = 1-F_{\chi_2^2}(S_{n_1,n_2}),$

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

Usage

simultest.chisq(dataX,dataY)

Arguments

dataX	n1 by p data matrix
dataY	n2 by p data matrix

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.chisq(X,Y)

---

Two-sample simultaneous test using Fisher's combination

Description

This function implements the two-sample simultaneous test on high-dimensional mean vectors and covariance matrices using Fisher's combination. Suppose $\{\mathbf{X}_1, \ldots, \mathbf{X}_{n_1}\}$ are i.i.d. copies of $\mathbf{X}$, and $\{\mathbf{Y}_1, \ldots, \mathbf{Y}_{n_2}\}$ are i.i.d. copies of $\mathbf{Y}$. Let $p_{CQ}$ and $p_{LC}$ denote the $p$-values associated with the $l_2$-norm-based mean test proposed in Chen and Qin (2010) (see meantest.cq for details) and the $l_2$-norm-based covariance test proposed in Li and Chen (2012) (see covtest.lc for details), respectively. The simultaneous test statistic via Fisher's combination is defined as

$J_{n_1, n_2} = -2\log(p_{CQ}) -2\log(p_{LC}).$

It has been proved that with some regularity conditions, under the null hypothesis $H_0: \boldsymbol{\mu}_1 = \boldsymbol{\mu}_2 \ \text{ and } \ \mathbf{\Sigma}_1 = \mathbf{\Sigma}_2$, the two tests are asymptotically independent as $n_1, n_2, p\rightarrow \infty$, and therefore $J_{n_1,n_2}$ asymptotically converges in distribution to a $\chi_4^2$ distribution. The asymptotic $p$-value is obtained by

$p\text{-value} = 1-F_{\chi_4^2}(J_{n_1,n_2}),$

where $F_{\chi_4^2}(\cdot)$ is the cdf of the $\chi_4^2$ distribution.

Usage

simultest.fisher(dataX,dataY)

Arguments

dataX	an $n_1$ by $p$ data matrix
dataY	an $n_2$ by $p$ data matrix

Value

stat the value of test statistic

pval the p-value for the test.

References

Chen, S. X. and Qin, Y. L. (2010). A two-sample test for high-dimensional data with applications to gene-set testing. Annals of Statistics, 38(2):808–835.

Li, J. and Chen, S. X. (2012). Two sample tests for high-dimensional covariance matrices. The Annals of Statistics, 40(2):908–940.

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.fisher(X,Y)

---

Two-sample PE simultaneous test using Cauchy combination

Description

This function implements the two-sample PE simultaneous test on high-dimensional mean vectors and covariance matrices using Cauchy combination. Suppose $\{\mathbf{X}_1, \ldots, \mathbf{X}_{n_1}\}$ are i.i.d. copies of $\mathbf{X}$, and $\{\mathbf{Y}_1, \ldots, \mathbf{Y}_{n_2}\}$ are i.i.d. copies of $\mathbf{Y}$. Let $M_{PE}$ and $T_{PE}$ denote the PE mean test statistic and PE covariance test statistic, respectively. (see meantest.pe.comp and covtest.pe.comp for details). Let $p_{m}$ and $p_{c}$ denote their respective $p$-values. The PE simultaneous test statistic via Cauchy combination is defined as

$C_{PE} = \frac{1}{2}\tan((0.5-p_{m})\pi) + \frac{1}{2}\tan((0.5-p_{c})\pi).$

It has been proved that with some regularity conditions, under the null hypothesis $H_0: \boldsymbol{\mu}_1 = \boldsymbol{\mu}_2 \ \text{ and } \ \mathbf{\Sigma}_1 = \mathbf{\Sigma}_2$, the two tests are asymptotically independent as $n_1, n_2, p\rightarrow \infty$, and therefore $C_{PE}$ asymptotically converges in distribution to a standard Cauchy distribution. The asymptotic $p$-value is obtained by

$p\text{-value} = 1-F_{Cauchy}(C_{PE}),$

where $F_{Cauchy}(\cdot)$ is the cdf of the standard Cauchy distribution.

Usage

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

Arguments

dataX	an $n_1$ by $p$ data matrix
dataY	an $n_2$ by $p$ 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.cauchy(X,Y)

---

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 $\{\mathbf{X}_1, \ldots, \mathbf{X}_{n_1}\}$ are i.i.d. copies of $\mathbf{X}$, and $\{\mathbf{Y}_1, \ldots, \mathbf{Y}_{n_2}\}$ are i.i.d. copies of $\mathbf{Y}$. Let $M_{PE}$ and $T_{PE}$ 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

$S_{PE} = M_{PE}^2 + T_{PE}^2.$

It has been proved that with some regularity conditions, under the null hypothesis $H_0: \boldsymbol{\mu}_1 = \boldsymbol{\mu}_2 \ \text{ and } \ \mathbf{\Sigma}_1 = \mathbf{\Sigma}_2$, the two tests are asymptotically independent as $n_1, n_2, p\rightarrow \infty$, and therefore $S_{PE}$ asymptotically converges in distribution to a $\chi_2^2$ distribution. The asymptotic $p$-value is obtained by

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

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

Usage

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

Arguments

dataX	an $n_1$ by $p$ data matrix
dataY	an $n_2$ by $p$ 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)

---