Normal-approximation-based test for GLHT problem proposed by Fujikoshi et al. (2004)

Fujikoshi et al. (2004)'s test for general linear hypothesis testing (GLHT) problem for high-dimensional data with assuming that underlying covariance matrices are the same.

Usage

FHW2004.GLHT.NABT(Y,X,C,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.
X: A known $n\times k$ full-rank design matrix with $\operatorname{rank}(\boldsymbol{X})=k<n$.
C: A known matrix of size $q\times k$ with $\operatorname{rank}(\boldsymbol{C})=q<k$.
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

A high-dimensional linear regression model can be expressed as $$\boldsymbol{Y}=\boldsymbol{X\Theta}+\boldsymbol{\epsilon},$$ where $\Theta$ is a $k\times p$ unknown parameter matrix and $\boldsymbol{\epsilon}$ is an $n\times p$ error matrix.

It is of interest to test the following GLHT problem $$H_0: \boldsymbol{C\Theta}=\boldsymbol{0}, \quad \text { vs. } \quad H_1: \boldsymbol{C\Theta} \neq \boldsymbol{0}.$$

Fujikoshi et al. (2004) proposed the following test statistic: $$T_{FHW}=\sqrt{p}\left[(n-k)\frac{\operatorname{tr}(\boldsymbol{S}_h)}{\operatorname{tr}(\boldsymbol{S}_e)}-q\right],$$ where $\boldsymbol{S}_h$ and $\boldsymbol{S}_e$ are the matrices of sums of squares and products due to the hypothesis and the error, respecitively.

They showed that under the null hypothesis, $T_{FHW}$ is asymptotically normally distributed.

References

Fujikoshi Y, Himeno T, Wakaki H (2004). “Asymptotic results of a high dimensional MANOVA test and power comparison when the dimension is large compared to the sample size.” Journal of the Japan Statistical Society, 34(1), 19--26. doi:10.14490/jjss.34.19 .

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()
k <- 4
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]]))
X <- matrix(c(rep(1,n[1]),rep(0,sum(n)),rep(1,n[2]), rep(0,sum(n)),
            rep(1,n[3]),rep(0,sum(n)),rep(1,n[4])),ncol=k,nrow=sum(n))
q <- k-1
C <- cbind(diag(q),-rep(1,q))
FHW2004.GLHT.NABT(Y,X,C,n,p)
#> 
#> Results of Hypothesis Test
#> --------------------------
#> 
#> Test name:                       Fujikoshi et al. (2004)'s test
#> 
#> Null Hypothesis:                 The general linear hypothesis is true
#> 
#> Alternative Hypothesis:          The general linear hypothesis is not true
#> 
#> Data:                            Y
#> 
#> Sample Sizes:                    n1 = 43
#>                                  n2 = 14
#>                                  n3 = 21
#>                                  n4 = 72
#> 
#> Sample Dimension:                2000
#> 
#> Test Statistic:                  T[FHW] = 6.4015
#> 
#> Approximation method to the      Normal approximation
#> null distribution of T[FHW]: 
#> 
#> P-value:                         7.694084e-11
#>