epval_aSPU {highmean}

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

sam1

an n1 by p matrix from sample population 1. Each row represents a p-dimensional sample.

sam2

an n2 by p matrix from sample population 2. Each row represents a p-dimensional sample.

pow

a numeric vector indicating the candidate values for the power \gamma in SPU tests. It should contain Inf and both odd and even integers. The default is c(1:6, Inf).

eq.cov

a logical value. The default is TRUE, indicating that the two sample populations have same covariance; otherwise, the covariances are assumed to be different. If eq.cov is TRUE, the permutation method is used to calculate p-values; otherwise, the parametric bootstrap resampling is used.

n.iter

a numeric integer indicating the number of permutation/resampling iterations. The default is 1,000.

cov1.est

This and the following arguments are only effective when eq.cov = FALSE and the parametric bootstrap resampling is used to calculate p-values. This argument specifies a consistent estimate of the covariance matrix of sample population 1 when eq.cov is FALSE. This can be obtained from various apporoaches (e.g., banding, tapering, and thresholding; see Pourahmadi 2013). If not specified, this function uses a banding approach proposed by Bickel and Levina (2008) to estimate the covariance matrix.

cov2.est

a consistent estimate of the covariance matrix of sample population 2 when eq.cov is FALSE. It is similar with the argument cov1.est.

bandwidth1

a vector of nonnegative integers indicating the candidate bandwidths to be used in the banding approach (Bickel and Levina, 2008) for estimating the covariance of sample population 1 when eq.cov is FALSE. This argument is effective when cov1.est is not provided. The default is a vector containing 50 candidate bandwidths chosen from {0, 1, 2, ..., p}.

bandwidth2

similar with the argument bandwidth1; it is used to specify candidate bandwidths for estimating the covariance of sample population 2 when eq.cov is FALSE.

cv.fold

an integer greater than or equal to 2 indicating the fold of cross-validation. The default is 5. See page 211 in Bickel and Levina (2008).

norm

a character string indicating the type of matrix norm for the calculation of risk function in cross-validation. This argument will be passed to the norm function. The default is the Frobenius norm ("F").

seeds

a vector of seeds for each permutation or parametric bootstrap resampling iteration; this is optional.

Details

See the details in apval_aSPU.

Value

A list including the following elements:

sam.info

the basic information about the two groups of samples, including the samples sizes and dimension.

pow

the powers \gamma used for the SPU tests.

opt.bw1

the optimal bandwidth determined by the cross-validation when eq.cov was FALSE and cov1.est was not specified.

opt.bw2

the optimal bandwidth determined by the cross-validation when eq.cov was FALSE and cov2.est was not specified.

cov.assumption

the equality assumption on the covariances of the two sample populations; this was specified by the argument eq.cov.

method

this output reminds users that the p-values are obtained using permutation or parametric bootstrap resampling.

pval

the p-values of the SPU tests and the aSPU test.

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)