# Permutation-And-Asymptotics-Based p-values of the SPU and aSPU Tests 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). ```r cpval_aSPU(sam1, sam2, pow = c(1:6, Inf), n.iter = 1000, 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 powers $\gamma$ in the SPU tests. It should contain `Inf` and both odd and even integers. The default is `c(1:6, Inf)`. - `n.iter`: a numeric integer indicating the number of permutation iterations for calculating the means, variances, covariances of SPU test statistics' asymptotic distributions. The default is 1,000. - `seeds`: a vector of seeds for each permutation iteration; this is optional. ## 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 $i$th 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. ## Returns 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. - **spu.stat**: the observed SPU test statistics. - **spu.e**: the asymptotic means of SPU test statistics with finite $\gamma$ under the null hypothesis. - **spu.var**: the asymptotic variances of SPU test statistics with finite $\gamma$ under the null hypothesis. - **spu.corr.odd**: the asymptotic correlations between SPU test statistics with odd $\gamma$. - **spu.corr.even**: the asymptotic correlations between SPU test statistics with even $\gamma$. - **cov.assumption**: the equality assumption on the covariances of the two sample populations; this reminders users that `cpval_aSPU()` assumes that the two sample groups have same covariance. - **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. ## 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. ## Note The permutation technique assumes that the distributions of the two sample populations are the same under the null hypothesis. ## See Also `apval_aSPU`, `epval_aSPU` ## Examples ```r 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) ```

cpval_aSPU function