S2007.ks.NABT {HDNRA} | R Documentation |
Normal-approximation-based test for one-way MANOVA problem proposed by Schott (2007)
Description
Schott, J. R. (2007)'s test for one-way MANOVA problem for high-dimensional data with assuming that underlying covariance matrices are the same.
Usage
S2007.ks.NABT(Y, n, p)
Arguments
Y |
A list of |
n |
A vector of |
p |
The dimension of data. |
Details
Suppose we have the following k
independent high-dimensional samples:
\boldsymbol{y}_{i1},\ldots,\boldsymbol{y}_{in_i}, \;\operatorname{are \; i.i.d. \; with}\; \operatorname{E}(\boldsymbol{y}_{i1})=\boldsymbol{\mu}_i,\; \operatorname{Cov}(\boldsymbol{y}_{i1})=\boldsymbol{\Sigma},i=1,\ldots,k.
It is of interest to test the following one-way MANOVA problem:
H_0: \boldsymbol{\mu}_1=\cdots=\boldsymbol{\mu}_k, \quad \text { vs. }\; H_1: H_0 \;\operatorname{is \; not\; ture}.
Schott (2007) proposed the following test statistic:
T_{S}=[\operatorname{tr}(\boldsymbol{H})/h-\operatorname{tr}(\boldsymbol{E})/e]/\sqrt{N-1},
where \boldsymbol{H}=\sum_{i=1}^kn_i(\bar{\boldsymbol{y}}_i-\bar{\boldsymbol{y}})(\bar{\boldsymbol{y}}_i-\bar{\boldsymbol{y}})^\top
, \boldsymbol{E}=\sum_{i=1}^k\sum_{j=1}^{n_i}(\boldsymbol{y}_{ij}-\bar{\boldsymbol{y}}_{i})(\boldsymbol{y}_{ij}-\bar{\boldsymbol{y}}_{i})^\top
, h=k-1
, and e=N-k
, with N=n_1+\cdots+n_k
.
They showed that under the null hypothesis, T_{S}
is asymptotically normally distributed.
Value
A list of class "NRtest"
containing the results of the hypothesis test. See the help file for NRtest.object
for details.
References
Schott JR (2007). “Some high-dimensional tests for a one-way MANOVA.” Journal of Multivariate Analysis, 98(9), 1825–1839. doi:10.1016/j.jmva.2006.11.007.
Examples
library("HDNRA")
data("corneal")
dim(corneal)
group1 <- as.matrix(corneal[1:43, ]) ## normal group
group2 <- as.matrix(corneal[44:57, ]) ## unilateral suspect group
group3 <- as.matrix(corneal[58:78, ]) ## suspect map group
group4 <- as.matrix(corneal[79:150, ]) ## clinical keratoconus group
p <- dim(corneal)[2]
Y <- list()
Y[[1]] <- group1
Y[[2]] <- group2
Y[[3]] <- group3
Y[[4]] <- group4
n <- c(nrow(Y[[1]]),nrow(Y[[2]]),nrow(Y[[3]]),nrow(Y[[4]]))
S2007.ks.NABT(Y, n, p)