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 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 γ in the SPU tests. It should contain |
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 ith 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 γ used for the SPU tests. |
spu.stat |
the observed SPU test statistics. |
spu.e |
the asymptotic means of SPU test statistics with finite γ under the null hypothesis. |
spu.var |
the asymptotic variances of SPU test statistics with finite γ under the null hypothesis. |
spu.corr.odd |
the asymptotic correlations between SPU test statistics with odd γ. |
spu.corr.even |
the asymptotic correlations between SPU test statistics with even γ. |
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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