Two-Sample Tests for High-Dimensional Mean Vectors

Description

Provides various tests for comparing high-dimensional mean vectors in two sample populations.

Details

Several two-sample tests for high-dimensional mean vectors have been proposed recently; see, e.g., Bai and Saranadasa (1996), Srivastava and Du (2008), Chen and Qin (2010), Cai et al (2014), and Chen et al (2014). However, these tests are powerful only against certain and limited alternative hypotheses. In practice, since the true alternative hypothesis is unknown, it is unclear how to choose one of these tests to yield high power. Accordingly, Pan et al (2014) and Xu et al (2016) proposed an adaptive test that may maintain high power across a wide range of situations; the asymptotic property of this test was also studied. This package provides functions to calculate p-values of the foregoing tests, using their asymptotic properties and the empirical (permutation or parametric bootstrap resampling) technique.

Author(s)

Lifeng Lin and Wei Pan

Maintainer: Lifeng Lin <linl@umn.edu>

References

Bai ZD and Saranadasa H (1996). "Effect of high dimension: by an example of a two sample problem." Statistica Sinica, 6(2), 311–329.

Cai TT, 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.

Chen SX and Qin YL (2010). "A two-sample test for high-dimensional data with applications to gene-set testing." The Annals of Statistics, 38(2), 808–835.

Chen SX, Li J, and Zhong PS (2014). "Two-Sample Tests for High Dimensional Means with Thresholding and Data Transformation." arXiv preprint arXiv:1410.2848.

Pan W, Kim J, Zhang Y, Shen X, and Wei P (2014). "A powerful and adaptive association test for rare variants." Genetics, 197(4), 1081–1095.

Srivastava MS and Du M (2008). "A test for the mean vector with fewer observations than the dimension." Journal of Multivariate Analysis, 99(3), 386–402.

Xu G, Lin L, Wei P, and Pan W (2016). "An adaptive two-sample test for high-dimensional means." Biometrika, 103(3), 609–624.

Asymptotics-Based p-value of the Test Proposed by Bai and Saranadasa (1996)

Description

Calculates p-value of the test for testing equality of two-sample high-dimensional mean vectors proposed by Bai and Saranadasa (1996) based on the asymptotic distribution of the test statistic.

Usage

apval_Bai1996(sam1, sam2)

Arguments

Details

Suppose that the two groups of p-dimensional independent and identically distributed samples \{X_{1i}\}_{i=1}^{n_1} and \{X_{2j}\}_{j=1}^{n_2} are observed; we consider high-dimensional data with p \gg n := n_1 + n_2 - 2. Assume that the two groups share a common covariance matrix. The primary object is to test H_{0}: \mu_1 = \mu_2 versus H_{A}: \mu_1 \neq \mu_2. Let \bar{X}_{k} be the sample mean for group k = 1, 2. Also, let S = n^{-1} \sum_{k = 1}^{2} \sum_{i = 1}^{n_{k}} (X_{ki} - \bar{X}_k) (X_{ki} - \bar{X}_k)^T be the pooled sample covariance matrix from the two groups.

Bai and Saranadasa (1996) proposed the following test statistic:

T_{BS} = \frac{(n_1^{-1} + n_2^{-1})^{-1} (\bar{X}_1 - \bar{X}_2)^T (\bar{X}_1 - \bar{X}_2) - tr S}{\sqrt{2 n (n + 1) (n - 1)^{-1} (n + 2)^{-1} [tr S^2 - n^{-1} (tr S)^2]}},

and its asymptotic distribution is normal under the null hypothesis.

Value

A list including the following elements:

Note

The asymptotic distribution of the test statistic was derived under normality assumption in Bai and Saranadasa (1996). Also, this function assumes that the two sample populations have a common covariance matrix.

References

Bai ZD and Saranadasa H (1996). "Effect of high dimension: by an example of a two sample problem." Statistica Sinica, 6(2), 311–329.

See Also

epval_Bai1996

Examples

library(MASS)
set.seed(1234)
n1 <- n2 <- 50
p <- 200
mu1 <- rep(0, p)
mu2 <- mu1
mu2[1:10] <- 0.2
true.cov <- 0.4^(abs(outer(1:p, 1:p, "-"))) # AR1 covariance
sam1 <- mvrnorm(n = n1, mu = mu1, Sigma = true.cov)
sam2 <- mvrnorm(n = n2, mu = mu2, Sigma = true.cov)
apval_Bai1996(sam1, sam2)

Asymptotics-Based p-value of the Test Proposed by Cai et al (2014)

Description

Calculates p-value of the test for testing equality of two-sample high-dimensional mean vectors proposed by Cai et al (2014) based on the asymptotic distribution of the test statistic.

Usage

apval_Cai2014(sam1, sam2, eq.cov = TRUE)

Arguments

Details

Suppose that the two groups of p-dimensional independent and identically distributed samples \{X_{1i}\}_{i=1}^{n_1} and \{X_{2j}\}_{j=1}^{n_2} are observed; we consider high-dimensional data with p \gg n := n_1 + n_2 - 2. Assume that the covariances of the two sample populations are \Sigma_1 = (\sigma_{1, ij}) and \Sigma_2 = (\sigma_{2, ij}). The primary object is to test H_{0}: \mu_1 = \mu_2 versus H_{A}: \mu_1 \neq \mu_2. Let \bar{X}_{k} be the sample mean for group k = 1, 2. For a vector v, we denote v^{(i)} as its ith element.

Cai et al (2014) proposed the following test statistic:

T_{CLX} = \max_{i = 1, \ldots, p} (\bar{X}_1^{(i)} - \bar{X}_2^{(i)})^2/(\sigma_{1,ii}/n_1 + \sigma_{2, ii}/n_2),

This test statistic follows an extreme value distribution under the null hypothesis.

Value

A list including the following elements:

Note

This function does not transform the data with their precision matrix (see Cai et al, 2014). To calculate the p-value of the test statisic with transformation, users can use transformed samples for sam1 and sam2.

References

Cai TT, 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.

See Also

epval_Cai2014

Examples

library(MASS)
set.seed(1234)
n1 <- n2 <- 50
p <- 200
mu1 <- rep(0, p)
mu2 <- mu1
mu2[1:10] <- 0.2
true.cov <- 0.4^(abs(outer(1:p, 1:p, "-"))) # AR1 covariance
sam1 <- mvrnorm(n = n1, mu = mu1, Sigma = true.cov)
sam2 <- mvrnorm(n = n2, mu = mu2, Sigma = true.cov)
apval_Cai2014(sam1, sam2)

# the two sample populations have different covariances
true.cov1 <- 0.2^(abs(outer(1:p, 1:p, "-")))
true.cov2 <- 0.6^(abs(outer(1:p, 1:p, "-")))
sam1 <- mvrnorm(n = n1, mu = mu1, Sigma = true.cov1)
sam2 <- mvrnorm(n = n2, mu = mu2, Sigma = true.cov2)
apval_Cai2014(sam1, sam2, eq.cov = FALSE)

Asymptotics-Based p-value of the Test Proposed by Chen and Qin (2010)

Description

Calculates p-value of the test for testing equality of two-sample high-dimensional mean vectors proposed by Chen and Qin (2010) based on the asymptotic distribution of the test statistic.

Usage

apval_Chen2010(sam1, sam2, eq.cov = TRUE)

Arguments

Details

Suppose that the two groups of p-dimensional independent and identically distributed samples \{X_{1i}\}_{i=1}^{n_1} and \{X_{2j}\}_{j=1}^{n_2} are observed; we consider high-dimensional data with p \gg n := n_1 + n_2 - 2. The primary object is to test H_{0}: \mu_1 = \mu_2 versus H_{A}: \mu_1 \neq \mu_2. Let \bar{X}_{k} be the sample mean for group k = 1, 2.

Chen and Qin (2010) proposed the following test statistic:

T_{CQ} = \frac{\sum_{i \neq j}^{n_1} X_{1i}^T X_{1j}}{n_1 (n_1 - 1)} + \frac{\sum_{i \neq j}^{n_2} X_{2i}^T X_{2j}}{n_2 (n_2 - 1)} - 2 \frac{\sum_{i = 1}^{n_1} \sum_{j = 1}^{n_2} X_{1i}^T X_{2j}}{n_1 n_2},

and its asymptotic distribution is normal under the null hypothesis.

Value

A list including the following elements:

References

Chen SX and Qin YL (2010). "A two-sample test for high-dimensional data with applications to gene-set testing." The Annals of Statistics, 38(2), 808–835.

See Also

epval_Chen2010

Examples

library(MASS)
set.seed(1234)
n1 <- n2 <- 50
p <- 200
mu1 <- rep(0, p)
mu2 <- mu1
mu2[1:10] <- 0.2
true.cov <- 0.4^(abs(outer(1:p, 1:p, "-"))) # AR1 covariance
sam1 <- mvrnorm(n = n1, mu = mu1, Sigma = true.cov)
sam2 <- mvrnorm(n = n2, mu = mu2, Sigma = true.cov)
apval_Chen2010(sam1, sam2)

# the two sample populations have different covariances
true.cov1 <- 0.2^(abs(outer(1:p, 1:p, "-")))
true.cov2 <- 0.6^(abs(outer(1:p, 1:p, "-")))
sam1 <- mvrnorm(n = n1, mu = mu1, Sigma = true.cov1)
sam2 <- mvrnorm(n = n2, mu = mu2, Sigma = true.cov2)
apval_Chen2010(sam1, sam2, eq.cov = FALSE)

Asymptotics-Based p-value of the Test Proposed by Chen et al (2014)

Description

Calculates p-value of the test for testing equality of two-sample high-dimensional mean vectors proposed by Chen et al (2014) based on the asymptotic distribution of the test statistic.

Usage

apval_Chen2014(sam1, sam2, eq.cov = TRUE)

Arguments

Details

Suppose that the two groups of p-dimensional independent and identically distributed samples \{X_{1i}\}_{i=1}^{n_1} and \{X_{2j}\}_{j=1}^{n_2} are observed; we consider high-dimensional data with p \gg n := n_1 + n_2 - 2. Assume that the covariances of the two sample populations are \Sigma_1 = (\sigma_{1, ij}) and \Sigma_2 = (\sigma_{2, ij}). The primary object is to test H_{0}: \mu_1 = \mu_2 versus H_{A}: \mu_1 \neq \mu_2. Let \bar{X}_{k} be the sample mean for group k = 1, 2. For a vector v, we denote v^{(i)} as its ith element.

Chen et al (2014) proposed removing estimated zero components in the mean difference through thresholding; they considered

T_{CLZ}(s) = \sum_{i = 1}^{p} \left\{ \frac{(\bar{X}_1^{(i)} - \bar{X}_2^{(i)})^2}{\sigma_{1,ii}/n_1 + \sigma_{2,ii}/n_2} - 1 \right\} I \left\{ \frac{(\bar{X}_1^{(i)} - \bar{X}_2^{(i)})^2}{\sigma_{1,ii}/n_1 + \sigma_{2,ii}/n_2} > \lambda_{p} (s) \right\},

where the threshold level is \lambda_p(s) := 2 s \log p and I(\cdot) is the indicator function. Since an optimal choice of the threshold is unknown, they proposed trying all possible threshold values, then choosing the most significant one as their final test statistic:

T_{CLZ} = \max_{s \in (0, 1 - \eta)} \{ T_{CLZ}(s) - \hat{\mu}_{T_{CLZ}(s), 0}\}/\hat{\sigma}_{T_{CLZ}(s), 0},

where \hat{\mu}_{T_{CLZ}(s), 0} and \hat{\sigma}_{T_{CLZ}(s), 0} are estimates of the mean and standard deviation of T_{CLZ}(s) under the null hypothesis. They derived its asymptotic null distribution as an extreme value distribution.

Value

A list including the following elements:

Note

This function does not transform the data with their precision matrix (see Chen et al, 2014). To calculate the p-value of the test statisic with transformation, users can use transformed samples for sam1 and sam2.

References

Chen SX, Li J, and Zhong PS (2014). "Two-Sample Tests for High Dimensional Means with Thresholding and Data Transformation." arXiv preprint arXiv:1410.2848.

See Also

epval_Chen2014

Examples

library(MASS)
set.seed(1234)
n1 <- n2 <- 50
p <- 200
mu1 <- rep(0, p)
mu2 <- mu1
mu2[1:10] <- 0.2
true.cov <- 0.4^(abs(outer(1:p, 1:p, "-"))) # AR1 covariance
sam1 <- mvrnorm(n = n1, mu = mu1, Sigma = true.cov)
sam2 <- mvrnorm(n = n2, mu = mu2, Sigma = true.cov)
apval_Chen2014(sam1, sam2)

# the two sample populations have different covariances
true.cov1 <- 0.2^(abs(outer(1:p, 1:p, "-")))
true.cov2 <- 0.6^(abs(outer(1:p, 1:p, "-")))
sam1 <- mvrnorm(n = n1, mu = mu1, Sigma = true.cov1)
sam2 <- mvrnorm(n = n2, mu = mu2, Sigma = true.cov2)
apval_Chen2014(sam1, sam2, eq.cov = FALSE)

Asymptotics-Based p-value of the Test Proposed by Srivastava and Du (2008)

Description

Calculates p-value of the test for testing equality of two-sample high-dimensional mean vectors proposed by Srivastava and Du (2008) based on the asymptotic distribution of the test statistic.

Usage

apval_Sri2008(sam1, sam2)

Arguments

Details

Suppose that the two groups of p-dimensional independent and identically distributed samples \{X_{1i}\}_{i=1}^{n_1} and \{X_{2j}\}_{j=1}^{n_2} are observed; we consider high-dimensional data with p \gg n := n_1 + n_2 - 2. Assume that the two groups share a common covariance matrix. The primary object is to test H_{0}: \mu_1 = \mu_2 versus H_{A}: \mu_1 \neq \mu_2. Let \bar{X}_{k} be the sample mean for group k = 1, 2. Also, let S = n^{-1} \sum_{k = 1}^{2} \sum_{i = 1}^{n_{k}} (X_{ki} - \bar{X}_k) (X_{ki} - \bar{X}_k)^T be the pooled sample covariance matrix from the two groups.

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

T_{SD} = \frac{(n_1^{-1} + n_2^{-1})^{-1} (\bar{X}_1 - \bar{X}_2)^T D_S^{-1} (\bar{X}_1 - \bar{X}_2) - (n - 2)^{-1} n p}{\sqrt{2 (tr R^2 - p^2 n^{-1}) c_{p, n}}},

where D_S = diag (s_{11}, s_{22}, ..., s_{pp}), s_{ii}'s are the diagonal elements of S, R = D_S^{-1/2} S D_S^{-1/2} is the sample correlation matrix and c_{p, n} = 1 + tr R^2 p^{-3/2}. This test statistic follows normal distribution under the null hypothesis.

Value

A list including the following elements:

Note

The asymptotic distribution of the test statistic was derived under normality assumption in Bai and Saranadasa (1996). Also, this function assumes that the two sample populations have a common covariance matrix.

References

Srivastava MS and Du M (2008). "A test for the mean vector with fewer observations than the dimension." Journal of Multivariate Analysis, 99(3), 386–402.

See Also

epval_Sri2008

Examples

library(MASS)
set.seed(1234)
n1 <- n2 <- 50
p <- 200
mu1 <- rep(0, p)
mu2 <- mu1
mu2[1:10] <- 0.2
true.cov <- 0.4^(abs(outer(1:p, 1:p, "-"))) # AR1 covariance
sam1 <- mvrnorm(n = n1, mu = mu1, Sigma = true.cov)
sam2 <- mvrnorm(n = n2, mu = mu2, Sigma = true.cov)
apval_Sri2008(sam1, sam2)

Asymptotics-Based p-values of the SPU and aSPU Tests

Description

Calculates p-values of the sum-of-powers (SPU) and adaptive SPU (aSPU) tests based on the asymptotic distributions of the test statistics (Xu et al, 2016).

Usage

apval_aSPU(sam1, sam2, pow = c(1:6, Inf), eq.cov = TRUE, cov.est,
           cov1.est, cov2.est, bandwidth, bandwidth1, bandwidth2,
           cv.fold = 5, norm = "F")

Arguments

Details

Suppose that the two groups of p-dimensional independent and identically distributed samples \{X_{1i}\}_{i=1}^{n_1} and \{X_{2j}\}_{j=1}^{n_2} are observed; we consider high-dimensional data with p \gg n := n_1 + n_2 - 2. Assume that the covariances of the two sample populations are \Sigma_1 = (\sigma_{1, ij}) and \Sigma_2 = (\sigma_{2, ij}). The primary object is to test H_{0}: \mu_1 = \mu_2 versus H_{A}: \mu_1 \neq \mu_2. Let \bar{X}_{k} be the sample mean for group k = 1, 2. For a vector v, we denote v^{(i)} as its ith element.

For any 1 \le \gamma < \infty, the sum-of-powers (SPU) test statistic is defined as:

L(\gamma) = \sum_{i = 1}^{p} (\bar{X}_1^{(i)} - \bar{X}_2^{(i)})^\gamma.

For \gamma = \infty,

L (\infty) = \max_{i = 1, \ldots, p} (\bar{X}_1^{(i)} - \bar{X}_2^{(i)})^2/(\sigma_{1,ii}/n_1 + \sigma_{2,ii}/n_2).

The adaptive SPU (aSPU) test combines the SPU tests and improve the test power:

T_{aSPU} = \min_{\gamma \in \Gamma} P_{SPU(\gamma)},

where P_{SPU(\gamma)} is the p-value of SPU(\gamma) test, and \Gamma is a candidate set of \gamma's. Note that T_{aSPU} is no longer a genuine p-value. The asymptotic properties of the SPU and aSPU tests are studied in Xu et al (2016).

Value

A list including the following elements:

References

Bickel PJ and Levina E (2008). "Regularized estimation of large covariance matrices." The Annals of Statistics, 36(1), 199–227.

Pan W, Kim J, Zhang Y, Shen X, and Wei P (2014). "A powerful and adaptive association test for rare variants." Genetics, 197(4), 1081–1095.

Pourahmadi M (2013). High-Dimensional Covariance Estimation. John Wiley & Sons, Hoboken, NJ.

Xu G, Lin L, Wei P, and Pan W (2016). "An adaptive two-sample test for high-dimensional means." Biometrika, 103(3), 609–624.

See Also

cpval_aSPU, epval_aSPU

Examples

library(MASS)
set.seed(1234)
n1 <- n2 <- 50
p <- 200
mu1 <- rep(0, p)
mu2 <- mu1
mu2[1:10] <- 0.2
true.cov <- 0.4^(abs(outer(1:p, 1:p, "-"))) # AR1 covariance
sam1 <- mvrnorm(n = n1, mu = mu1, Sigma = true.cov)
sam2 <- mvrnorm(n = n2, mu = mu2, Sigma = true.cov)
# use true covariance matrix
apval_aSPU(sam1, sam2, cov.est = true.cov)
# fix bandwidth as 10
apval_aSPU(sam1, sam2, bandwidth = 10)
# use the optimal bandwidth from a candidate set
#apval_aSPU(sam1, sam2, bandwidth = 0:20)

# the two sample populations have different covariances
#true.cov1 <- 0.2^(abs(outer(1:p, 1:p, "-")))
#true.cov2 <- 0.6^(abs(outer(1:p, 1:p, "-")))
#sam1 <- mvrnorm(n = n1, mu = mu1, Sigma = true.cov1)
#sam2 <- mvrnorm(n = n2, mu = mu2, Sigma = true.cov2)
#apval_aSPU(sam1, sam2, eq.cov = FALSE,
#	bandwidth1 = 10, bandwidth2 = 10)

Permutation-And-Asymptotics-Based p-values of the SPU and aSPU Tests

Description

Calculates p-values of the sum-of-powers (SPU) and adaptive SPU (aSPU) tests based on the combination of permutation method and asymptotic distributions of the test statistics (Xu et al, 2016).

Usage

cpval_aSPU(sam1, sam2, pow = c(1:6, Inf), n.iter = 1000, seeds)

Arguments

Details

Suppose that the two groups of p-dimensional independent and identically distributed samples \{X_{1i}\}_{i=1}^{n_1} and \{X_{2j}\}_{j=1}^{n_2} are observed; we consider high-dimensional data with p \gg n := n_1 + n_2 - 2. Assume that the covariances of the two sample populations are \Sigma_1 = (\sigma_{1, ij}) and \Sigma_2 = (\sigma_{2, ij}). The primary object is to test H_{0}: \mu_1 = \mu_2 versus H_{A}: \mu_1 \neq \mu_2. Let \bar{X}_{k} be the sample mean for group k = 1, 2. For a vector v, we denote v^{(i)} as its ith element.

For any 1 \le \gamma < \infty, the sum-of-powers (SPU) test statistic is defined as:

L(\gamma) = \sum_{i = 1}^{p} (\bar{X}_1^{(i)} - \bar{X}_2^{(i)})^\gamma.

For \gamma = \infty,

L (\infty) = \max_{i = 1, \ldots, p} (\bar{X}_1^{(i)} - \bar{X}_2^{(i)})^2/(\sigma_{1,ii}/n_1 + \sigma_{2,ii}/n_2).

The adaptive SPU (aSPU) test combines the SPU tests and improve the test power:

T_{aSPU} = \min_{\gamma \in \Gamma} P_{SPU(\gamma)},

where P_{SPU(\gamma)} is the p-value of SPU(\gamma) test, and \Gamma is a candidate set of \gamma's. Note that T_{aSPU} is no longer a genuine p-value.

The asymptotic properties of the SPU and aSPU tests are studied in Xu et al (2016). When using the theoretical means, variances, and covarainces of L (\gamma) to calculate the p-values of SPU and aSPU tests (1 \le \gamma < \infty), the high-dimensional covariance matrix of the samples needs to be consistently estimated; such estimation is usually time-consuming.

Alternatively, assuming that the two sample groups have same covariance, the permutation method can be applied to efficiently estimate the means, variances, and covarainces of L (\gamma)'s asymptotic distributions, which then yield the p-values of SPU and aSPU tests based on the combination of permutation method and asymptotic distributions.

Value

A list including the following elements:

Note

The permutation technique assumes that the distributions of the two sample populations are the same under the null hypothesis.

References

Bickel PJ and Levina E (2008). "Regularized estimation of large covariance matrices." The Annals of Statistics, 36(1), 199–227.

Pan W, Kim J, Zhang Y, Shen X, and Wei P (2014). "A powerful and adaptive association test for rare variants." Genetics, 197(4), 1081–1095.

Pourahmadi M (2013). High-Dimensional Covariance Estimation. John Wiley & Sons, Hoboken, NJ.

Xu G, Lin L, Wei P, and Pan W (2016). "An adaptive two-sample test for high-dimensional means." Biometrika, 103(3), 609–624.

See Also

apval_aSPU, epval_aSPU

Examples

library(MASS)
set.seed(1234)
n1 <- n2 <- 50
p <- 200
mu1 <- rep(0, p)
mu2 <- mu1
mu2[1:10] <- 0.2
true.cov <- 0.4^(abs(outer(1:p, 1:p, "-"))) # AR1 covariance
sam1 <- mvrnorm(n = n1, mu = mu1, Sigma = true.cov)
sam2 <- mvrnorm(n = n2, mu = mu2, Sigma = true.cov)
cpval_aSPU(sam1, sam2, n.iter = 100)

Empirical Permutation-Based p-value of the Test Proposed by Bai and Saranadasa (1996)

Description

Calculates p-value of the test for testing equality of two-sample high-dimensional mean vectors proposed by Bai and Saranadasa (1996) based on permutation.

Usage

epval_Bai1996(sam1, sam2, n.iter = 1000, seeds)

Arguments

Details

See the details in apval_Bai1996.

Value

A list including the following elements:

Note

The permutation technique assumes that the distributions of the two sample populations are the same under the null hypothesis.

References

Bai ZD and Saranadasa H (1996). "Effect of high dimension: by an example of a two sample problem." Statistica Sinica, 6(2), 311–329.

See Also

apval_Bai1996

Examples

#library(MASS)
#set.seed(1234)
#n1 <- n2 <- 50
#p <- 200
#mu1 <- rep(0, p)
#mu2 <- mu1
#mu2[1:10] <- 0.2
#true.cov <- 0.4^(abs(outer(1:p, 1:p, "-"))) # AR1 covariance
#sam1 <- mvrnorm(n = n1, mu = mu1, Sigma = true.cov)
#sam2 <- mvrnorm(n = n2, mu = mu2, Sigma = true.cov)
# increase n.iter to reduce Monte Carlo error.
#epval_Bai1996(sam1, sam2, n.iter = 10)

Empirical Permutation- or Resampling-Based p-value of the Test Proposed by Cai et al (2014)

Description

Calculates p-value of the test for testing equality of two-sample high-dimensional mean vectors proposed by Cai et al (2014) based on permutation or parametric bootstrap resampling.

Usage

epval_Cai2014(sam1, sam2, eq.cov = TRUE, n.iter = 1000, cov1.est, cov2.est,
              bandwidth1, bandwidth2, cv.fold = 5, norm = "F", seeds)

Arguments

Details

See the details in apval_Cai2014.

Value

A list including the following elements:

Note

This function does not transform the data with their precision matrix (see Cai et al, 2014). To calculate the p-value of the test statisic with transformation, users can input transformed samples to sam1 and sam2.

References

Bickel PJ and Levina E (2008). "Regularized estimation of large covariance matrices." The Annals of Statistics, 36(1), 199–227.

Cai TT, 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.

Pourahmadi M (2013). High-Dimensional Covariance Estimation. John Wiley & Sons, Hoboken, NJ.

See Also

apval_Cai2014

Examples

#library(MASS)
#set.seed(1234)
#n1 <- n2 <- 50
#p <- 200
#mu1 <- rep(0, p)
#mu2 <- mu1
#mu2[1:10] <- 0.2
#true.cov <- 0.4^(abs(outer(1:p, 1:p, "-"))) # AR1 covariance
#sam1 <- mvrnorm(n = n1, mu = mu1, Sigma = true.cov)
#sam2 <- mvrnorm(n = n2, mu = mu2, Sigma = true.cov)
# increase n.iter to reduce Monte Carlo error
#epval_Cai2014(sam1, sam2, n.iter = 10)

# the two sample populations have different covariances
#true.cov1 <- 0.2^(abs(outer(1:p, 1:p, "-")))
#true.cov2 <- 0.6^(abs(outer(1:p, 1:p, "-")))
#sam1 <- mvrnorm(n = n1, mu = mu1, Sigma = true.cov1)
#sam2 <- mvrnorm(n = n2, mu = mu2, Sigma = true.cov2)
# increase n.iter to reduce Monte Carlo error
#epval_Cai2014(sam1, sam2, eq.cov = FALSE, n.iter = 10,
#	bandwidth1 = 10, bandwidth2 = 10)

Empirical Permutation- or Resampling-Based p-value of the Test Proposed by Chen and Qin (2010)

Description

Calculates p-value of the test for testing equality of two-sample high-dimensional mean vectors proposed by Chen and Qin (2010) based on permutation or parametric bootstrap resampling.

Usage

epval_Chen2010(sam1, sam2, eq.cov = TRUE, n.iter = 1000, cov1.est, cov2.est,
               bandwidth1, bandwidth2, cv.fold = 5, norm = "F", seeds)

Arguments

Details

See the details in apval_Chen2010.

Value

A list including the following elements:

References

Bickel PJ and Levina E (2008). "Regularized estimation of large covariance matrices." The Annals of Statistics, 36(1), 199–227.

Chen SX and Qin YL (2010). "A two-sample test for high-dimensional data with applications to gene-set testing." The Annals of Statistics, 38(2), 808–835.

Pourahmadi M (2013). High-Dimensional Covariance Estimation. John Wiley & Sons, Hoboken, NJ.

See Also

apval_Chen2010

Examples

#library(MASS)
#set.seed(1234)
#n1 <- n2 <- 50
#p <- 200
#mu1 <- rep(0, p)
#mu2 <- mu1
#mu2[1:10] <- 0.2
#true.cov <- 0.4^(abs(outer(1:p, 1:p, "-"))) # AR1 covariance
#sam1 <- mvrnorm(n = n1, mu = mu1, Sigma = true.cov)
#sam2 <- mvrnorm(n = n2, mu = mu2, Sigma = true.cov)
# increase n.iter to reduce Monte Carlo error.
#epval_Chen2010(sam1, sam2, n.iter = 10)

# the two sample populations have different covariances
#true.cov1 <- 0.2^(abs(outer(1:p, 1:p, "-")))
#true.cov2 <- 0.6^(abs(outer(1:p, 1:p, "-")))
#sam1 <- mvrnorm(n = n1, mu = mu1, Sigma = true.cov1)
#sam2 <- mvrnorm(n = n2, mu = mu2, Sigma = true.cov2)
# increase n.iter to reduce Monte Carlo error
#epval_Chen2010(sam1, sam2, eq.cov = FALSE, n.iter = 10,
#	bandwidth1 = 10, bandwidth2 = 10)

Empirical Permutation- or Resampling-Based p-value of the Test Proposed by Chen et al (2014)

Description

Calculates p-value of the test for testing equality of two-sample high-dimensional mean vectors proposed by Chen et al (2014) based on permutation or parametric bootstrap resampling.

Usage

epval_Chen2014(sam1, sam2, eq.cov = TRUE, n.iter = 1000, cov1.est, cov2.est,
               bandwidth1, bandwidth2, cv.fold = 5, norm = "F", seeds)

Arguments

Details

See the details in apval_Chen2014.

Value

A list including the following elements:

Note

This function does not transform the data with their precision matrix (see Chen et al, 2014). To calculate the p-value of the test statisic with transformation, users can input transformed samples to sam1 and sam2.

References

Bickel PJ and Levina E (2008). "Regularized estimation of large covariance matrices." The Annals of Statistics, 36(1), 199–227.

Chen SX, Li J, and Zhong PS (2014). "Two-Sample Tests for High Dimensional Means with Thresholding and Data Transformation." arXiv preprint arXiv:1410.2848.

Pourahmadi M (2013). High-Dimensional Covariance Estimation. John Wiley & Sons, Hoboken, NJ.

See Also

apval_Chen2014

Examples

#library(MASS)
#set.seed(1234)
#n1 <- n2 <- 50
#p <- 200
#mu1 <- rep(0, p)
#mu2 <- mu1
#mu2[1:10] <- 0.2
#true.cov <- 0.4^(abs(outer(1:p, 1:p, "-"))) # AR1 covariance
#sam1 <- mvrnorm(n = n1, mu = mu1, Sigma = true.cov)
#sam2 <- mvrnorm(n = n2, mu = mu2, Sigma = true.cov)
# increase n.iter to reduce Monte Carlo error
#epval_Chen2014(sam1, sam2, n.iter = 10)

# the two sample populations have different covariances
#true.cov1 <- 0.2^(abs(outer(1:p, 1:p, "-")))
#true.cov2 <- 0.6^(abs(outer(1:p, 1:p, "-")))
#sam1 <- mvrnorm(n = n1, mu = mu1, Sigma = true.cov1)
#sam2 <- mvrnorm(n = n2, mu = mu2, Sigma = true.cov2)
# increase n.iter to reduce Monte Carlo error
#epval_Chen2014(sam1, sam2, eq.cov = FALSE, n.iter = 10,
#	bandwidth1 = 10, bandwidth2 = 10)

Empirical Permutation-Based p-value of the Test Proposed by Srivastava and Du (2008)

Description

Calculates p-value of the test for testing equality of two-sample high-dimensional mean vectors proposed by Srivastava and Du (1996) based on permutation.

Usage

epval_Sri2008(sam1, sam2, n.iter = 1000, seeds)

Arguments

Details

See the details in apval_Sri2008.

Value

A list including the following elements:

Note

The permutation technique assumes that the distributions of the two sample populations are the same under the null hypothesis.

References

Srivastava MS and Du M (2008). "A test for the mean vector with fewer observations than the dimension." Journal of Multivariate Analysis, 99(3), 386–402.

See Also

apval_Sri2008

Examples

#library(MASS)
#set.seed(1234)
#n1 <- n2 <- 50
#p <- 200
#mu1 <- rep(0, p)
#mu2 <- mu1
#mu2[1:10] <- 0.2
#true.cov <- 0.4^(abs(outer(1:p, 1:p, "-"))) # AR1 covariance
#sam1 <- mvrnorm(n = n1, mu = mu1, Sigma = true.cov)
#sam2 <- mvrnorm(n = n2, mu = mu2, Sigma = true.cov)
# increase n.iter to reduce Monte Carlo error.
#epval_Sri2008(sam1, sam2, n.iter = 10)

Empirical Permutation- or Resampling-Based p-values of the SPU and aSPU Tests

Description

Calculates p-values of the sum-of-powers (SPU) and adaptive SPU (aSPU) tests based on permutation or parametric bootstrap resampling.

Usage

epval_aSPU(sam1, sam2, pow = c(1:6, Inf), eq.cov = TRUE, n.iter = 1000, cov1.est,
           cov2.est, bandwidth1, bandwidth2, cv.fold = 5, norm = "F", seeds)

Arguments

Details

See the details in apval_aSPU.

Value

A list including the following elements:

References

Bickel PJ and Levina E (2008). "Regularized estimation of large covariance matrices." The Annals of Statistics, 36(1), 199–227.

Pan W, Kim J, Zhang Y, Shen X, and Wei P (2014). "A powerful and adaptive association test for rare variants." Genetics, 197(4), 1081–1095.

Pourahmadi M (2013). High-Dimensional Covariance Estimation. John Wiley & Sons, Hoboken, NJ.

Xu G, Lin L, Wei P, and Pan W (2016). "An adaptive two-sample test for high-dimensional means." Biometrika, 103(3), 609–624.

See Also

apval_aSPU, cpval_aSPU

Examples

library(MASS)
set.seed(1234)
n1 <- n2 <- 50
p <- 200
mu1 <- rep(0, p)
mu2 <- mu1
mu2[1:10] <- 0.2
true.cov <- 0.4^(abs(outer(1:p, 1:p, "-"))) # AR1 covariance
sam1 <- mvrnorm(n = n1, mu = mu1, Sigma = true.cov)
sam2 <- mvrnorm(n = n2, mu = mu2, Sigma = true.cov)
# increase n.iter to reduce Monte Carlo error
epval_aSPU(sam1, sam2, n.iter = 10)

# the two sample populations have different covariances
#true.cov1 <- 0.2^(abs(outer(1:p, 1:p, "-")))
#true.cov2 <- 0.6^(abs(outer(1:p, 1:p, "-")))
#sam1 <- mvrnorm(n = n1, mu = mu1, Sigma = true.cov1)
#sam2 <- mvrnorm(n = n2, mu = mu2, Sigma = true.cov2)
# increase n.iter to reduce Monte Carlo error
#epval_aSPU(sam1, sam2, eq.cov = FALSE, n.iter = 10,
#	bandwidth1 = 10, bandwidth2 = 10)