S2007.ks.NABT: Normal-approximation-based test for one-way MANOVA problem...

In HDNRA: High-Dimensional Location Testing with Normal-Reference Approaches

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 k data matrices. The ith element represents the data matrix (n_i \times p) from the ith population with each row representing a p-dimensional observation.
n	A vector of k sample sizes. The ith element represents the sample size of group i, n_i.
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

\insertRef

schott2007someHDNRA

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)

HDNRA documentation built on Oct. 30, 2024, 9:28 a.m.