| meantest.cq {PEtests} | R Documentation |
Two-sample high-dimensional mean test (Chen and Qin, 2010)
Description
This function implements the two-sample l_2-norm-based high-dimensional
mean test proposed by Chen and Qin (2010).
Suppose \{\mathbf{X}_1, \ldots, \mathbf{X}_{n_1}\} are i.i.d.
copies of \mathbf{X}, and \{\mathbf{Y}_1, \ldots, \mathbf{Y}_{n_2}\}
are i.i.d. copies of \mathbf{Y}.
The test statistic M_{CQ} is defined as
M_{CQ} = \frac{1}{n_1(n_1-1)}\sum_{u\neq v}^{n_1} \mathbf{X}_{u}'\mathbf{X}_{v}
+\frac{1}{n_2(n_2-1)}\sum_{u\neq v}^{n_2} \mathbf{Y}_{u}'\mathbf{Y}_{v}
-\frac{2}{n_1n_2}\sum_u^{n_1}\sum_v^{n_2} \mathbf{X}_{u}'\mathbf{Y}_{v}.
Under the null hypothesis H_{0m}: \boldsymbol{\mu}_1 = \boldsymbol{\mu}_2,
the leading variance of M_{CQ} is
\sigma^2_{M_{CQ}}=\frac{2}{n_1(n_1-1)}\text{tr}(\mathbf{\Sigma}_1^2)+
\frac{2}{n_2(n_2-1)}\text{tr}(\mathbf{\Sigma}_2^2)+
\frac{4}{n_1n_2}\text{tr}(\mathbf{\Sigma}_1\mathbf{\Sigma}_2),
which can be consistently estimated by \widehat\sigma^2_{M_{CQ}}=
\frac{2}{n_1(n_1-1)}\widehat{\text{tr}(\mathbf{\Sigma}_1^2)}+
\frac{2}{n_2(n_2-1)}\widehat{\text{tr}(\mathbf{\Sigma}_2^2)}+
\frac{4}{n_1n_2}\widehat{\text{tr}(\mathbf{\Sigma}_1\mathbf{\Sigma}_2)}.
The explicit formulas of \widehat{\text{tr}(\mathbf{\Sigma}_1^2)},
\widehat{\text{tr}(\mathbf{\Sigma}_2^2)}, and
\widehat{\text{tr}(\mathbf{\Sigma}_1\mathbf{\Sigma}_2)}
can be found in Section 3 of Chen and Qin (2010).
With some regularity conditions, under the null hypothesis
H_{0m}: \boldsymbol{\mu}_1 = \boldsymbol{\mu}_2,
the test statistic M_{CQ} converges in distribution to a standard normal distribution
as n_1, n_2, p \rightarrow \infty.
The asymptotic p-value is obtained by
p_{CQ} = 1-\Phi(M_{CQ}/\hat\sigma_{M_{CQ}}),
where \Phi(\cdot) is the cdf of the standard normal distribution.
Usage
meantest.cq(dataX,dataY)
Arguments
dataX |
an |
dataY |
an |
Value
stat the value of test statistic
pval the p-value for the test.
References
Chen, S. X. and Qin, Y. L. (2010). A two-sample test for high-dimensional data with applications to gene-set testing. Annals of Statistics, 38(2):808–835.
Examples
n1 = 100; n2 = 100; pp = 500
set.seed(1)
X = matrix(rnorm(n1*pp), nrow=n1, ncol=pp)
Y = matrix(rnorm(n2*pp), nrow=n2, ncol=pp)
meantest.cq(X,Y)