hdmv: High-dimensional mean vector(s) test

In stat-cz/hdmv: What the package does (short line)

High-dimensional mean vector(s) test

Description

hdmv function allows a high dimensional test for the one and two-sample mean vector(s) hypothesis problem, where the dimension p (number of variables) is large than the sample size n (number of individual).

Usage

hdmv(X, Y = NULL, mu = NULL, cov.equal = TRUE, l = NULL, kappa = NULL, type = "a3", linearcoef = NULL, na.action = na.fail, ...)

Arguments

X	a numerical n_X*p data frame or matrix.
Y	an optional numerical n_Y*p data frame or matrix,for the two-sample high dimensional mean vectors test. If NULL, a one-sample high dimensional mean vector test is supplied.
mu	a vector indicating the hypothesized value of the mean. If NULL the origin or a two-sample test is supplied.
cov.equal	logical. If TRUE (default) the statistic test using identical covariance matrix of two samples is supplied. Otherwise, the test for different covariance matrix of two samples are is used.
l	a positive constant. If NULL, the default value l=n^(5/4) log(n) is used or the two-sample mean vectors test is performed. See Details.
kappa	a positive constant. If NULL, the default value kappa = sqrt(p/log(p)) is used. See Details.
type	a different option of linear coefficient linearcoef. There are currently three options of linear coefficients linearcoef. The options are "a1", "a2" and "a3". If NULL the default optionl "a3" is performed. The specific values are given in Details.
linearcoef	an optional numeric p-unit vector. If NULL type is automatically supplied.
na.action	a generic function deals with NAs in e.g., data frame or matrix. If na.fail (default) the object if it does not contain any missing value is returned. Otherwise, an error is returned. If na.omit the object with incomplete data frame or matrix, which is removed the row with NAs value is returned.
...

Details

The powerful high dimensional test for testing the specific mean vector or for testing the equality of two mean vectors with identical or different covaraince matrices.Two high dimensional hypothesis testing are an empirical likelihood ratio test for one sample problem (Cui et al., 2020), and a projection test for two-sample problem (Huang et al. 2022). Two testing approaches are both more powerful than that of their competitors, such as Bai and Saradarara (1996) and Chen and Qin (2010).

l argument is constrained on the condition which l^(-1)*n^(5/4) tends to 0. See Cui et al. (2020) for more details.

kappa argument is constrained on the condition which kappa / sqrt(p) tends to 0. See Cui et al. (2020) and Huang et al. (2022) for more details.

The type of linear coefficient to be used is specified through the type argument (see hdmv for details).

The available options are:

• type = "a1": a unit p-vector with first element 1 and 0 otherwise, i.e. (1, 0_p-1), where 0_p-1 denotes a p-1 vector of zeroes;

• type = "a2": a unit p-vector with first ceiling half elements sqrt(ceiling(p/2))/ceiling(p/2) and 0 otherwise; i.e. ( sqrt(ceiling(p/2))/ceiling(p/2) * 1_(ceiling(p/2)), 0_(p-ceiling(p/2)) );

• type = "a3": a unit p-vector with each element sqrt(p)/p; i.e. sqrt(p)/p 1_p, where 1_p denotes a p vector of ones.

Value

A list with class hdmv includes the following components:

• method a character string indicating what type of test is performed.

• data.name a character string giving the name of the data.

• statistic the value of the empirical likelihood ratio statistic or projection statistic, with 5 digits.

• sample size the sample size(s) of the data.

• p-value the p-value for the test, with 5 digits.

• null.value the specified hypothesized value of the mean depending on whether it was a one-sample test.

• alternative a character string with the value "two.sided".

Author(s)

Caizhu Huang [aut, cre] [e-mail], Euloge Clovis Kenne Pagui [aut, cre] [e-mail], Xia Cui [aut] [e-mail]

References

Cui X., Li R., Yang G. and Zhou W. (2020), Empirical likelihood test for a large-dimensional mean vector, Biometrika, 107(3), 591-607, https://doi.org/10.1093/biomet/asaa005

Huang C., Kenne Pagui E.C. and Cui X. (2022). Two-sample mean vector projection test in high-dimensional data. Submitted paper.

Examples

## One-sample high dimensional mean vector test
library(mnormt)
p <- 100
n <- 50
varcov <- matrix(0.65, p, p)
diag(varcov) <- rep(1,p)
mu <- rnorm(p)
X <- rmnorm(n = n, mean = mu, varcov)
res1 <- hdmv(X = X) ## the test check if the true mean vector is zero 
res1$p.value
res2 <- hdmv(X = X, mu = mu) ## the test check if the true mean vector is equal to mu.
res2$p.value

# Two-sample high dimensional mean vector test
# Example in genetic data
library(simPATHy)
data(chimera)
group <- colnames(chimera)
X1 <- t(chimera[, group == 1])
X2 <- t(chimera[, group == 2])
n1 <- nrow(X1)
n2 <- nrow(X2)
p <- ncol(X1)


# Two-sample high dimensional mean vectors test
res_same <- hdmv(X = X1, Y = X2, cov.equal = FALSE)
res_diff <- hdmv(X = X1, Y = X2, cov.equal = TRUE)

# Using formula argument
g <- as.numeric(group)
df <- rbind(X1, X2)
res_same_f <- hdmv(df ~ g, cov.equal = FALSE)
res_diff_f <- hdmv(df ~ g, cov.equal = TRUE)

# Use one of the three default linear coefficients
res_same_a1 <- hdmv(X = X1, Y = X2, cov.equal = FALSE, type = "a1")
res_same_a2 <- hdmv(X = X1, Y = X2, cov.equal = FALSE, type = "a2")
res_same_a3 <- hdmv(X = X1, Y = X2, cov.equal = FALSE, type = "a3")

# Using a specific linear coefficient 
linear_coef <- c(rep(sqrt(ceiling(p/4))/ceiling(p/4), ceiling(p/4)), rep(0,p-ceiling(p/4)))
res_same_linear <- hdmv(X = X1, Y = X2, cov.equal = FALSE, linearcoef = linear_coef)

# Specific the constant kappa
kappa <- 2 * sqrt(p/log(p))
res_same_kappa <- hdmv(X = X1, Y = X2, cov.equal = FALSE, kappa = kappa)

stat-cz/hdmv documentation built on April 10, 2022, 12:59 a.m.