Normal-approximation-based test for GLHT problem proposed by Srivastava and Fujikoshi (2006)

Srivastava and Fujikoshi (2006)'s test for general linear hypothesis testing (GLHT) problem for high-dimensional data with assuming that underlying covariance matrices are the same.

Usage

SF2006.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 $$H_0: \boldsymbol{C\Theta}=\boldsymbol{0}, \quad \text { vs. } \quad H_1: \boldsymbol{C\Theta} \neq \boldsymbol{0}.$$

Srivastava and Fujikoshi (2006) proposed the following test statistic: $$T_{SF}=\left[2q\hat{a}_2(1+(n-k)^{-1}q)\right]^{-1/2}\left[\frac{\operatorname{tr}(\boldsymbol{B})}{\sqrt{p}}-\frac{q}{\sqrt{n-k}}\frac{\operatorname{tr}(\boldsymbol{W})}{\sqrt{(n-k)p}}\right].$$ where \(\boldsymbol{W}\) and \(\boldsymbol{B}\) are the matrix of sum of squares and products due to error and the error, respectively, and \(\hat{a}_2=[\operatorname{tr}(\boldsymbol{W}^2)-\operatorname{tr}^2(\boldsymbol{W})/(n-k)]/[(n-k-1)(n-k+2)p]\). They showed that under the null hypothesis, \(T_{SF}\) is asymptotically normally distributed.

References

srivastava2006multivariateHDNRA

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))
SF2006.GLHT.NABT(Y,X,C,n,p)
#> 
#> Results of Hypothesis Test
#> --------------------------
#> 
#> Test name:                       Srivastava and Fujikoshi (2006)'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[SF] = 6.4231
#> 
#> Approximation method to the      Normal approximation
#> null distribution of T[SF]: 
#> 
#> P-value:                         6.677982e-11
#>