# 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 asymptotic distributions of the test statistics (Xu et al, 2016). ```r 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 - `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)`. - `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"`). ## 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). ## 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. - **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 $\gamma$ 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 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. ## 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 ```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) # 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) ```

apval_aSPU function