apval_aSPU: Asymptotics-Based p-values of the SPU and aSPU Tests
In highmean: Two-Sample Tests for High-Dimensional Mean Vectors

Description Usage Arguments Details Value References See Also Examples

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).

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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")

`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 powers γ in the 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.
`cov.est`	a consistent estimate of the common covariance matrix when `eq.cov` is `TRUE`. 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.
`cov1.est`	a consistent estimate of the covariance matrix of sample population 1 when `eq.cov` is `FALSE`. It is similar with the argument `cov.est`.
`cov2.est`	a consistent estimate of the covariance matrix of sample population 2 when `eq.cov` is `FALSE`. It is similar with the argument `cov.est`.
`bandwidth`	a vector of nonnegative integers indicating the candidate bandwidths to be used in the banding approach (Bickel and Levina, 2008) for estimating the common covariance when `eq.cov` is `TRUE`. This argument is effective only if `cov.est` is not provided. The default is a vector containing 50 candidate bandwidths chosen from {0, 1, 2, ..., p}.
`bandwidth1`	similar with the argument `bandwidth`; it is used to specify candidate bandwidths for estimating the covariance of sample population 1 when `eq.cov` is `FALSE`.
`bandwidth2`	similar with the argument `bandwidth`; 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"`).

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 Σ_1 = (σ_{1, ij}) and Σ_2 = (σ_{2, ij}). The primary object is to test H_{0}: μ_1 = μ_2 versus H_{A}: μ_1 \neq μ_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 ≤ γ < ∞, the sum-of-powers (SPU) test statistic is defined as:

L(γ) = ∑_{i = 1}^{p} (\bar{X}_1^{(i)} - \bar{X}_2^{(i)})^γ.

For γ = ∞,

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

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

T_{aSPU} = \min_{γ \in Γ} P_{SPU(γ)},

where P_{SPU(γ)} is the p-value of SPU(γ) test, and Γ is a candidate set of γ'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).

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 γ used for the SPU tests.
`opt.bw`	the optimal bandwidth determined by the cross-validation when `eq.cov` was `TRUE` and `cov.est` was not specified.
`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.
`spu.stat`	the observed SPU test statistics.
`spu.e`	the asymptotic means of SPU test statistics with finite γ under the null hypothesis.
`spu.var`	the asymptotic variances of SPU test statistics with finite γ under the null hypothesis.
`spu.corr.odd`	the asymptotic correlations between SPU test statistics with odd γ.
`spu.corr.even`	the asymptotic correlations between SPU test statistics with even γ.
`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 the asymptotic distributions of test statistics.
`pval`	the p-values of the SPU tests and the aSPU test.

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.

cpval_aSPU, epval_aSPU

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)