Description Usage Arguments Details Value Note References See Also Examples

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).

1 | ```
cpval_aSPU(sam1, sam2, pow = c(1:6, Inf), n.iter = 1000, seeds)
``` |

`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 |

`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. |

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). When using the theoretical means, variances, and covarainces of *L (γ)* to calculate the p-values of SPU and aSPU tests (*1 ≤ γ < ∞*), 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 (γ)*'s asymptotic distributions, which then yield the p-values of SPU and aSPU tests based on the combination of permutation method and asymptotic distributions.

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 |

`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 reminders users that |

`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. |

The permutation technique assumes that the distributions of the two sample populations are the same under the null hypothesis.

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 |

```
Loading required package: mvtnorm
$sam.info
n1 n2 p
50 50 200
$pow
[1] 1 2 3 4 5 6 Inf
$spu.stat
1 2 3 4 5 6 Inf
4.6822264 7.8050779 0.4105027 0.9004957 0.1007508 0.1609131 0.3382633
$spu.e
1 2 3 4 5 6
-0.23923112 7.99985462 -0.09213889 0.95736267 -0.02843553 0.18780008
$spu.var
1 2 3 4 5 6
16.266681073 0.917319659 0.344448920 0.056393198 0.024055848 0.007403876
$spu.corr.odd
1 3 5
1 1.0000000 0.8795804 0.6886379
3 0.8795804 1.0000000 0.9211705
5 0.6886379 0.9211705 1.0000000
$spu.corr.even
2 4 6
2 1.0000000 0.8736426 0.6432675
4 0.8736426 1.0000000 0.9226829
6 0.6432675 0.9226829 1.0000000
$cov.assumption
[1] "the two groups have same covariance"
$method
[1] "combination of permutation method and asymptotic distributions"
$pval
SPU_1 SPU_2 SPU_3 SPU_4 SPU_5 SPU_6 SPU_Inf aSPU
0.2223750 0.5805752 0.3917555 0.5946286 0.4048868 0.6226595 0.5106119 0.7300268
```

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