apval_aSPU {highmean} | R Documentation |
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
sam1 |
an n1 by p matrix from sample population 1. Each row represents a |
sam2 |
an n2 by p matrix from sample population 2. Each row represents a |
pow |
a numeric vector indicating the candidate powers |
eq.cov |
a logical value. The default is |
cov.est |
a consistent estimate of the common covariance matrix when |
cov1.est |
a consistent estimate of the covariance matrix of sample population 1 when |
cov2.est |
a consistent estimate of the covariance matrix of sample population 2 when |
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 |
bandwidth1 |
similar with the argument |
bandwidth2 |
similar with the argument |
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 |
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).
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 |
opt.bw |
the optimal bandwidth determined by the cross-validation when |
opt.bw1 |
the optimal bandwidth determined by the cross-validation when |
opt.bw2 |
the optimal bandwidth determined by the cross-validation when |
spu.stat |
the observed SPU test statistics. |
spu.e |
the asymptotic means of SPU test statistics with finite |
spu.var |
the asymptotic variances of SPU test statistics with finite |
spu.corr.odd |
the asymptotic correlations between SPU test statistics with odd |
spu.corr.even |
the asymptotic correlations between SPU test statistics with even |
cov.assumption |
the equality assumption on the covariances of the two sample populations; this was specified by the argument |
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
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)