Normal-approximation-based test for one-way MANOVA problem proposed by Schott (2007)

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 \(i\)th element represents the data matrix (\(n_i \times p\)) from the \(i\)th population with each row representing a \(p\)-dimensional observation.

n

A vector of \(k\) sample sizes. The \(i\)th element represents the sample size of group \(i\), \(n_i\).

p

The dimension of data.

Value

A list of class "NRtest" containing the results of the hypothesis test. See the help file for NRtest.object for details.

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.

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)
#> [1]  150 2000
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)
#> 
#> Results of Hypothesis Test
#> --------------------------
#> 
#> Test name:                       Schott (2007)'s test
#> 
#> Null Hypothesis:                 Difference between k mean vectors is 0
#> 
#> Alternative Hypothesis:          Difference between k mean vectors is not 0
#> 
#> Data:                            Y
#> 
#> Sample Sizes:                    n1 = 43
#>                                  n2 = 14
#>                                  n3 = 21
#>                                  n4 = 72
#> 
#> Sample Dimension:                2000
#> 
#> Test Statistic:                  T[S] = 6.3581
#> 
#> Approximation method to the      Normal approximation
#> null distribution of T[S]: 
#> 
#> P-value:                         1.021509e-10
#>