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).

1 2 3 | ```
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 |

`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 |

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 *i*th 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 |

`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. |

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.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 | ```
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)
``` |

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