Package 'HDNRA'

Normal-approximation-based test for two-sample problem proposed by Bai and Saranadasa (1996)

Description

Bai and Saranadasa (1996)'s test for testing equality of two-sample high-dimensional mean vectors with assuming that two covariance matrices are the same.

Usage

BS1996.TS.NABT(y1, y2)

Arguments

y1	The data matrix ($n_1 \times p$) from the first population. Each row represents a $p$-dimensional observation.
y2	The data matrix ($n_2 \times p$) from the second population. Each row represents a $p$-dimensional observation.

Details

Suppose we have two independent high-dimensional samples:

$\boldsymbol{y}_{i1},\ldots,\boldsymbol{y}_{in_i}, \;\operatorname{are \; i.i.d. \; with}\; \operatorname{E}(\boldsymbol{y}_{i1})=\boldsymbol{\mu}_i,\; \operatorname{Cov}(\boldsymbol{y}_{i1})=\boldsymbol{\Sigma},i=1,2.$

The primary object is to test

$H_{0}: \boldsymbol{\mu}_1 = \boldsymbol{\mu}_2\; \operatorname{versus}\; H_{1}: \boldsymbol{\mu}_1 \neq \boldsymbol{\mu}_2.$

Bai and Saranadasa (1996) proposed the following centralised $L^2$-norm-based test statistic:

$T_{BS} = \frac{n_1n_2}{n} \|\bar{\boldsymbol{y}}_1 - \bar{\boldsymbol{y}}_2\|^2-\operatorname{tr}(\hat{\boldsymbol{\Sigma}}),$

where $\bar{\boldsymbol{y}}_{i},i=1,2$ are the sample mean vectors and $\hat{\boldsymbol{\Sigma}}$ is the pooled sample covariance matrix. They showed that under the null hypothesis, $T_{BS}$ 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

Bai Z, Saranadasa H (1996). “Effect of high dimension: by an example of a two sample problem.” Statistica Sinica, 311–329. https://www.jstor.org/stable/24306018.

Examples

library("HDNRA")
data("COVID19")
dim(COVID19)
group1 <- as.matrix(COVID19[c(2:19, 82:87), ]) ## healthy group
group2 <- as.matrix(COVID19[-c(1:19, 82:87), ]) ## COVID-19 patients
BS1996.TS.NABT(group1,group2)

---

HDNRA_data corneal

Description

This dataset was acquired during a keratoconus study, a collaborative project involving Ms.Nancy Tripoli and Dr.Kenneth L.Cohen of Department of Ophthalmology at the University of North Carolina, Chapel Hill. The fitted feature vectors for the complete corneal surface dataset collectively into a feature matrix with dimensions of 150 × 2000.

Usage

data(corneal)

Format

'corneal'

A data frame with 150 observations on the following 4 groups.

normal group1

row 1 to row 43 in total 43 rows of the feature matrix correspond to observations from the normal group

unilateral suspect group2

row 44 to row 57 in total 14 rows of the feature matrix correspond to observations from the unilateral suspect group

suspect map group3

row 58 to row 78 in total 21 of the feature matrix correspond to observations from the suspect map group

clinical keratoconus group4

row 79 to row 150 in total 72 of the feature matrix correspond to observations from the clinical keratoconus group

References

Smaga Ł, Zhang J (2019). “Linear hypothesis testing with functional data.” Technometrics, 61(1), 99–110. doi:10.1080/00401706.2018.1456976.

Examples

library(HDNRA)
data(corneal)
dim(corneal)
group1 <- as.matrix(corneal[1:43, ]) ## normal group
dim(group1)
group2 <- as.matrix(corneal[44:57, ]) ## unilateral suspect group
dim(group2)
group3 <- as.matrix(corneal[58:78, ]) ## suspect map group
dim(group3)
group4 <- as.matrix(corneal[79:150, ]) ## clinical keratoconus group
dim(group4)

---

HDNRA_data COVID19

Description

A COVID19 data set from NCBI with ID GSE152641. The data set profiled peripheral blood from 24 healthy controls and 62 prospectively enrolled patients with community-acquired lower respiratory tract infection by SARS-COV-2 within the first 24 hours of hospital admission using RNA sequencing.

Usage

data(COVID19)

Format

'COVID19'

A data frame with 86 observations on the following 2 groups.

healthy group1

row 2 to row 19, and row 82 to 87, in total 24 healthy controls

patients group2

row 20 to 81, in total 62 prospectively enrolled patients

References

Thair SA, He YD, Hasin-Brumshtein Y, Sakaram S, Pandya R, Toh J, Rawling D, Remmel M, Coyle S, Dalekos GN, others (2021). “Transcriptomic similarities and differences in host response between SARS-CoV-2 and other viral infections.” Iscience, 24(1). doi:10.1016/j.isci.2020.101947.

Examples

library(HDNRA)
data(COVID19)
dim(COVID19)
group1 <- as.matrix(COVID19[c(2:19, 82:87), ]) ## healthy group
dim(group1)
group2 <- as.matrix(COVID19[-c(1:19, 82:87), ]) ## COVID-19 patients
dim(group2)

---

Normal-approximation-based test for two-sample BF problem proposed by Chen and Qin (2010)

Description

Chen and Qin (2010)'s test for testing equality of two-sample high-dimensional mean vectors without assuming that two covariance matrices are the same.

Usage

CQ2010.TSBF.NABT(y1, y2)

Arguments

y1	The data matrix ($n_1 \times p$) from the first population. Each row represents a $p$-dimensional observation.
y2	The data matrix ($n_2 \times p$) from the second population. Each row represents a $p$-dimensional observation.

Details

Suppose we have two independent high-dimensional samples:

$\boldsymbol{y}_{i1},\ldots,\boldsymbol{y}_{in_i}, \;\operatorname{are \; i.i.d. \; with}\; \operatorname{E}(\boldsymbol{y}_{i1})=\boldsymbol{\mu}_i,\; \operatorname{Cov}(\boldsymbol{y}_{i1})=\boldsymbol{\Sigma}_i,i=1,2.$

The primary object is to test

$H_{0}: \boldsymbol{\mu}_1 = \boldsymbol{\mu}_2\; \operatorname{versus}\; H_{1}: \boldsymbol{\mu}_1 \neq \boldsymbol{\mu}_2.$

Chen and Qin (2010) proposed the following test statistic:

$T_{CQ} = \frac{\sum_{i \neq j}^{n_1} \boldsymbol{y}_{1i}^\top \boldsymbol{y}_{1j}}{n_1 (n_1 - 1)} + \frac{\sum_{i \neq j}^{n_2} \boldsymbol{y}_{2i}^\top \boldsymbol{y}_{2j}}{n_2 (n_2 - 1)} - 2 \frac{\sum_{i = 1}^{n_1} \sum_{j = 1}^{n_2} \boldsymbol{y}_{1i}^\top \boldsymbol{y}_{2j}}{n_1 n_2}.$

They showed that under the null hypothesis, $T_{CQ}$ 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

Chen SX, Qin Y (2010). “A two-sample test for high-dimensional data with applications to gene-set testing.” The Annals of Statistics, 38(2). doi:10.1214/09-aos716.

Examples

library("HDNRA")
data("COVID19")
dim(COVID19)
group1 <- as.matrix(COVID19[c(2:19, 82:87), ]) ## healthy group
group2 <- as.matrix(COVID19[-c(1:19, 82:87), ]) ## COVID-19 patients
CQ2010.TSBF.NABT(group1,group2)

---

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

Description

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.

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.

Value

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

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)
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)

---

S3 Class "NRtest"

Description

The "NRtest" objects provide a comprehensive summary of hypothesis test outcomes, including test statistics, p-values, parameter estimates, and confidence intervals, if applicable.

Usage

NRtest.object(
  statistic,
  p.value,
  method,
  null.value,
  alternative,
  parameter = NULL,
  sample.size = NULL,
  sample.dimension = NULL,
  estimation.method = NULL,
  data.name = NULL,
  ...
)

Arguments

statistic	Numeric scalar containing the value of the test statistic, with a names attribute indicating the name of the test statistic.
p.value	Numeric scalar containing the p-value for the test.
method	Character string giving the name of the test.
null.value	Character string indicating the null hypothesis.
alternative	Character string indicating the alternative hypothesis.
parameter	Numeric vector containing the estimated approximation parameter(s) associated with the approximation method. This vector has a names attribute describing its element(s).
sample.size	Numeric vector containing the number of observations in each group used for the hypothesis test.
sample.dimension	Numeric scalar containing the dimension of the dataset used for the hypothesis test.
estimation.method	Character string giving the name of the approximation approach used to approximate the null distribution of the test statistic.
data.name	Character string describing the data set used in the hypothesis test.
...	Additional optional arguments.

Details

A class of objects returned by high-dimensional hypothesis testing functions in the HDNRA package, designed to encapsulate detailed results from statistical hypothesis tests. These objects are structured similarly to htest objects in the package EnvStats but are tailored to the needs of the HDNRA package.

Value

An object of class "NRtest" containing both required and optional components depending on the specifics of the hypothesis test, shown as follows:

Required Components

These components must be present in every "NRtest" object:

statistic

Must e present.

p.value

Must e present.

null.value

Must e present.

alternative

Must e present.

method

Must e present.

Optional Components

These components are included depending on the specifics of the hypothesis test performed:

parameter

May be present.

sample.size

May be present.

sample.dimension

May be present.

estimation.method

May be present.

data.name

May be present.

Methods

The class has the following methods:

print.NRtest

Printing the contents of the NRtest object in a human-readable form.

Examples

# Example 1: Using Bai and Saranadasa (1996)'s test (two-sample problem)
NRtest.obj1 <- NRtest.object(
  statistic = c("T[BS]" = 2.208),
  p.value = 0.0136,
  method = "Bai and Saranadasa (1996)'s test",
  data.name = "group1 and group2",
  null.value = c("Difference between two mean vectors is o"),
  alternative = "Difference between two mean vectors is not 0",
  parameter = NULL,
  sample.size = c(n1 = 24, n2 = 26),
  sample.dimension = 20460,
  estimation.method = "Normal approximation"
)
print(NRtest.obj1)

# Example 2: Using Fujikoshi et al. (2004)'s test (GLHT problem)
NRtest.obj2 <- NRtest.object(
  statistic = c("T[FHW]" = 6.4015),
  p.value = 0,
  method = "Fujikoshi et al. (2004)'s test",
  data.name = "Y",
  null.value  = "The general linear hypothesis is true",
  alternative = "The general linear hypothesis is not true",
  parameter = NULL,
  sample.size = c(n1 = 43, n2 = 14, n3 = 21, n4 = 72),
  sample.dimension = 2000,
  estimation.method = "Normal approximation"
)
print(NRtest.obj2)

---

Print Method for S3 Class "NRtest"

Description

Prints the details of the NRtest object in a user-friendly manner. This method provides a clear and concise presentation of the test results contained within the NRtest object, including all relevant statistical metrics and test details.

Usage

## S3 method for class \pkg{NRtest}
## S3 method for class 'NRtest'
print(x, ...)

Arguments

x	an NRtest object.
...	further arguments passed to or from other methods.

Details

The print.NRtest function formats and presents the contents of the NRtest object, which includes statistical test results and related parameters. This function is designed to provide a user-friendly display of the object's contents, making it easier to understand the results of the analysis.

Value

Invisibly returns the input x.

Author(s)

Pengfei Wang [email protected]

See Also

NRtest.object

---

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 $i$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.

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

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)
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)

---

Normal-approximation-based test for two-sample problem proposed by Srivastava and Du (2008)

Description

Srivastava and Du (2008)'s test for testing equality of two-sample high-dimensional mean vectors with assuming that two covariance matrices are the same.

Usage

SD2008.TS.NABT(y1, y2)

Arguments

y1	The data matrix ($n_1 \times p$) from the first population. Each row represents a $p$-dimensional observation.
y2	The data matrix ($n_2 \times p$) from the second population. Each row represents a $p$-dimensional observation.

Details

Suppose we have two independent high-dimensional samples:

$\boldsymbol{y}_{i1},\ldots,\boldsymbol{y}_{in_i}, \;\operatorname{are \; i.i.d. \; with}\; \operatorname{E}(\boldsymbol{y}_{i1})=\boldsymbol{\mu}_i,\; \operatorname{Cov}(\boldsymbol{y}_{i1})=\boldsymbol{\Sigma},i=1,2.$

The primary object is to test

$H_{0}: \boldsymbol{\mu}_1 = \boldsymbol{\mu}_2\; \operatorname{versus}\; H_{1}: \boldsymbol{\mu}_1 \neq \boldsymbol{\mu}_2.$

Srivastava and Du (2008) proposed the following test statistic:

$T_{SD} = \frac{n^{-1}n_1n_2(\bar{\boldsymbol{y}}_1 - \bar{\boldsymbol{y}}_2)^\top \boldsymbol{D}_S^{-1}(\bar{\boldsymbol{y}}_1 - \bar{\boldsymbol{y}}_2) - \frac{(n-2)p}{n-4}}{\sqrt{2 \left[\operatorname{tr}(\boldsymbol{R}^2) - \frac{p^2}{n-2}\right] c_{p, n}}},$

where $\bar{\boldsymbol{y}}_{i},i=1,2$ are the sample mean vectors, $\boldsymbol{D}_S$ is the diagonal matrix of sample variance, $\boldsymbol{R}$ is the sample correlation matrix and $c_{p, n}$ is the adjustment coefficient proposed by Srivastava and Du (2008). They showed that under the null hypothesis, $T_{SD}$ 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

Srivastava MS, Du M (2008). “A test for the mean vector with fewer observations than the dimension.” Journal of Multivariate Analysis, 99(3), 386–402. doi:10.1016/j.jmva.2006.11.002.

Examples

library("HDNRA")
data("COVID19")
dim(COVID19)
group1 <- as.matrix(COVID19[c(2:19, 82:87), ]) ## healthy group
group2 <- as.matrix(COVID19[-c(1:19, 82:87), ]) ## COVID-19 patients
SD2008.TS.NABT(group1,group2)

---

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

Description

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.

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.

Value

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

References

Srivastava MS, Fujikoshi Y (2006). “Multivariate analysis of variance with fewer observations than the dimension.” Journal of Multivariate Analysis, 97(9), 1927–1940. doi:10.1016/j.jmva.2005.08.010.

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()
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)

---

Normal-approximation-based test for two-sample BF problem proposed by Srivastava et al. (2013)

Description

Srivastava et al. (2013)'s test for testing equality of two-sample high-dimensional mean vectors without assuming that two covariance matrices are the same.

Usage

SKK2013.TSBF.NABT(y1, y2)

Arguments

y1	The data matrix ($n_1 \times p$) from the first population. Each row represents a $p$-dimensional observation.
y2	The data matrix ($n_2 \times p$ from the second population. Each row represents a $p$-dimensional observation.

Details

Suppose we have two independent high-dimensional samples:

$\boldsymbol{y}_{i1},\ldots,\boldsymbol{y}_{in_i}, \;\operatorname{are \; i.i.d. \; with}\; \operatorname{E}(\boldsymbol{y}_{i1})=\boldsymbol{\mu}_i,\; \operatorname{Cov}(\boldsymbol{y}_{i1})=\boldsymbol{\Sigma}_i,i=1,2.$

The primary object is to test

$H_{0}: \boldsymbol{\mu}_1 = \boldsymbol{\mu}_2\; \operatorname{versus}\; H_{1}: \boldsymbol{\mu}_1 \neq \boldsymbol{\mu}_2.$

Srivastava et al. (2013) proposed the following test statistic:

$T_{SKK} = \frac{(\bar{\boldsymbol{y}}_1 - \bar{\boldsymbol{y}}_2)^\top \hat{\boldsymbol{D}}^{-1}(\bar{\boldsymbol{y}}_1 - \bar{\boldsymbol{y}}_2) - p}{\sqrt{2 \widehat{\operatorname{Var}}(\hat{q}_n) c_{p,n}}},$

where $\bar{\boldsymbol{y}}_{i},i=1,2$ are the sample mean vectors, $\hat{\boldsymbol{D}}=\hat{\boldsymbol{D}}_1/n_1+\hat{\boldsymbol{D}}_2/n_2$ with $\hat{\boldsymbol{D}}_i,i=1,2$ being the diagonal matrices consisting of only the diagonal elements of the sample covariance matrices. $\widehat{\operatorname{Var}}(\hat{q}_n)$ is given by equation (1.18) in Srivastava et al. (2013), and $c_{p, n}$ is the adjustment coefficient proposed by Srivastava et al. (2013). They showed that under the null hypothesis, $T_{SKK}$ 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

Srivastava MS, Katayama S, Kano Y (2013). “A two sample test in high dimensional data.” Journal of Multivariate Analysis, 114, 349–358. doi:10.1016/j.jmva.2012.08.014.

Examples

library("HDNRA")
data("COVID19")
dim(COVID19)
group1 <- as.matrix(COVID19[c(2:19, 82:87), ]) ## healthy group
group2 <- as.matrix(COVID19[-c(1:19, 82:87), ]) ## COVID-19 patients
SKK2013.TSBF.NABT(group1,group2)

---

Normal-approximation-based test for GLHT problem proposed by Yamada and Srivastava (2012)

Description

Yamada and Srivastava (2012)'test for general linear hypothesis testing (GLHT) problem for high-dimensional data with assuming that underlying covariance matrices are the same.

Usage

YS2012.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.

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

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

Yamada and Srivastava (2012) proposed the following test statistic:

$T_{YS}=\frac{(n-k)\operatorname{tr}(\boldsymbol{S}_h\boldsymbol{D}_{\boldsymbol{S}_e}^{-1})-(n-k)pq/(n-k-2)}{\sqrt{2q[\operatorname{tr}(\boldsymbol{R}^2)-p^2/(n-k)]c_{p,n}}},$

where $\boldsymbol{S}_h$ and $\boldsymbol{S}_e$ are the variation matrices due to the hypothesis and error, respectively, and $\boldsymbol{D}_{\boldsymbol{S}_e}$ and $\boldsymbol{R}$ are diagonal matrix with the diagonal elements of $\boldsymbol{S}_e$ and the sample correlation matrix, respectively. $c_{p, n}$ is the adjustment coefficient proposed by Yamada and Srivastava (2012). They showed that under the null hypothesis, $T_{YS}$ 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

Yamada T, Srivastava MS (2012). “A test for multivariate analysis of variance in high dimension.” Communications in Statistics-Theory and Methods, 41(13-14), 2602–2615. doi:10.1080/03610926.2011.581786.

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()
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))
YS2012.GLHT.NABT(Y,X,C,n,p)

---

Normal-reference-test with two-cumulant (2-c) matched $\chi^2$-approximation for GLHT problem proposed Zhang et al. (2017)

Description

Zhang et al. (2017)'s test for general linear hypothesis testing (GLHT) problem for high-dimensional data with assuming that underlying covariance matrices are the same.

Usage

ZGZ2017.GLHT.2cNRT(Y,G,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.
G	A known full-rank coefficient matrix ($q\times k$) with $\operatorname{rank}(\boldsymbol{G})<k$.
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.

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 GLHT problem:

$H_0: \boldsymbol{G M}=\boldsymbol{0}, \quad \text { vs. } \quad H_1: \boldsymbol{G M} \neq \boldsymbol{0},$

where $\boldsymbol{M}=(\boldsymbol{\mu}_1,\ldots,\boldsymbol{\mu}_k)^\top$ is a $k\times p$ matrix collecting $k$ mean vectors and $\boldsymbol{G}:q\times k$ is a known full-rank coefficient matrix with $\operatorname{rank}(\boldsymbol{G})<k$.

Zhang et al. (2017) proposed the following test statistic:

$T_{ZGZ}=\|\boldsymbol{C \hat{\mu}}\|^2,$

where $\boldsymbol{C}=[(\boldsymbol{G D G}^\top)^{-1/2}\boldsymbol{G}]\otimes\boldsymbol{I}_p$, and $\hat{\boldsymbol{\mu}}=(\bar{\boldsymbol{y}}_1^\top,\ldots,\bar{\boldsymbol{y}}_k^\top)^\top$, with $\bar{\boldsymbol{y}}_{i},i=1,\ldots,k$ being the sample mean vectors and $\boldsymbol{D}=\operatorname{diag}(1/n_1,\ldots,1/n_k)$.

They showed that under the null hypothesis, $T_{ZGZ}$ and a chi-squared-type mixture have the same normal or non-normal limiting distribution.

Value

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

References

Zhang J, Guo J, Zhou B (2017). “Linear hypothesis testing in high-dimensional one-way MANOVA.” Journal of Multivariate Analysis, 155, 200–216. doi:10.1016/j.jmva.2017.01.002.

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()
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]]))
G <- cbind(diag(k-1),rep(-1,k-1))
ZGZ2017.GLHT.2cNRT(Y,G,n,p)

---

Normal-approximation-based test for GLHT problem under heteroscedasticity proposed by Zhou et al. (2017)

Description

Zhou et al. (2017)'s test for general linear hypothesis testing (GLHT) problem for high-dimensional data under heteroscedasticity.

Usage

ZGZ2017.GLHTBF.NABT(Y,G,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.
G	A known full-rank coefficient matrix ($q\times k$) with $\operatorname{rank}(\boldsymbol{G})< k$.
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.

Details

Suppose we have the following $k$ independent high-dimensional samples:

$\boldsymbol{y}_{i1},\ldots,\boldsymbol{y}_{in_i}, \;\operatorname{are \; i.i.d. \; with}\; \operatorname{E}(\boldsymbol{y}_{i1})=\boldsymbol{\mu}_i,\; \operatorname{Cov}(\boldsymbol{y}_{i1})=\boldsymbol{\Sigma}_i,i=1,\ldots,k.$

It is of interest to test the following GLHT problem:

$H_0: \boldsymbol{G M}=\boldsymbol{0}, \quad \text { vs. } H_1: \boldsymbol{G M} \neq \boldsymbol{0},$

where $\boldsymbol{M}=(\boldsymbol{\mu}_1,\ldots,\boldsymbol{\mu}_k)^\top$ is a $k\times p$ matrix collecting $k$ mean vectors and $\boldsymbol{G}:q\times k$ is a known full-rank coefficient matrix with $\operatorname{rank}(\boldsymbol{G})<k$.

Let $\bar{\boldsymbol{y}}_{i},i=1,\ldots,k$ be the sample mean vectors and $\hat{\boldsymbol{\Sigma}}_i,i=1,\ldots,k$ be the sample covariance matrices.

Zhou et al. (2017) proposed the following U-statistic based test statistic:

$T_{ZGZ}=\|\boldsymbol{C \hat{\mu}}\|^2-\sum_{i=1}^k h_{ii}\operatorname{tr}(\hat{\boldsymbol{\Sigma}}_i)/n_i,$

where $\boldsymbol{C}=[(\boldsymbol{G D G}^\top)^{-1/2}\boldsymbol{G}]\otimes\boldsymbol{I}_p$, $\boldsymbol{D}=\operatorname{diag}(1/n_1,\ldots,1/n_k)$, and $h_{ij}$ is the $(i,j)$th entry of the $k\times k$ matrix $\boldsymbol{H}=\boldsymbol{G}^\top(\boldsymbol{G}\boldsymbol{D}\boldsymbol{G}^\top)^{-1}\boldsymbol{G}$.

They showed that under the null hypothesis, $T_{ZGZ}$ 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

Zhou B, Guo J, Zhang J (2017). “High-dimensional general linear hypothesis testing under heteroscedasticity.” Journal of Statistical Planning and Inference, 188, 36–54. doi:10.1016/j.jspi.2017.03.005.

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()
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]]))
G <- cbind(diag(k-1),rep(-1,k-1))
ZGZ2017.GLHTBF.NABT(Y,G,n,p)

---

Normal-reference-test with two-cumulant (2-c) matched $\chi^2$-approximation for two-sample problem proposed by Zhang et al. (2020)

Description

Zhang et al. (2020)'s test for testing equality of two-sample high-dimensional mean vectors with assuming that two covariance matrices are the same.

Usage

ZGZC2020.TS.2cNRT(y1, y2)

Arguments

y1	The data matrix ($n_1 \times p$) from the first population. Each row represents a $p$-dimensional observation.
y2	The data matrix ($n_2 \times p$) from the second population. Each row represents a $p$-dimensional observation.

Details

Suppose we have two independent high-dimensional samples:

$\boldsymbol{y}_{i1},\ldots,\boldsymbol{y}_{in_i}, \;\operatorname{are \; i.i.d. \; with}\; \operatorname{E}(\boldsymbol{y}_{i1})=\boldsymbol{\mu}_i,\; \operatorname{Cov}(\boldsymbol{y}_{i1})=\boldsymbol{\Sigma},i=1,2.$

The primary object is to test

$H_{0}: \boldsymbol{\mu}_1 = \boldsymbol{\mu}_2\; \operatorname{versus}\; H_{1}: \boldsymbol{\mu}_1 \neq \boldsymbol{\mu}_2.$

Zhang et al.(2020) proposed the following test statistic:

$T_{ZGZC} = \frac{n_1n_2}{n} \|\bar{\boldsymbol{y}}_1 - \bar{\boldsymbol{y}}_2\|^2,$

where $\bar{\boldsymbol{y}}_{i},i=1,2$ are the sample mean vectors. They showed that under the null hypothesis, $T_{ZGZC}$ and a chi-squared-type mixture have the same normal or non-normal limiting distribution.

Value

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

References

Zhang J, Guo J, Zhou B, Cheng M (2020). “A simple two-sample test in high dimensions based on L 2-norm.” Journal of the American Statistical Association, 115(530), 1011–1027. doi:10.1080/01621459.2019.1604366.

Examples

library("HDNRA")
data("COVID19")
dim(COVID19)
group1 <- as.matrix(COVID19[c(2:19, 82:87), ]) ## healthy group
group2 <- as.matrix(COVID19[-c(1:19, 82:87), ]) ## COVID-19 patients
ZGZC2020.TS.2cNRT(group1, group2)

---

Normal-reference-test with two-cumulant (2-c) matched $\chi^2$-approximation for two-sample BF problem proposed by Zhu et al. (2023)

Description

Zhu et al. (2023)'s test for testing equality of two-sample high-dimensional mean vectors without assuming that two covariance matrices are the same.

Usage

ZWZ2023.TSBF.2cNRT(y1, y2)

Arguments

y1	The data matrix ($n_1 \times p$) from the first population. Each row represents a $p$-dimensional observation.
y2	The data matrix ($n_2 \times p$) from the second population. Each row represents a $p$-dimensional observation.

Details

Suppose we have two independent high-dimensional samples:

$\boldsymbol{y}_{i1},\ldots,\boldsymbol{y}_{in_i}, \;\operatorname{are \; i.i.d. \; with}\; \operatorname{E}(\boldsymbol{y}_{i1})=\boldsymbol{\mu}_i,\; \operatorname{Cov}(\boldsymbol{y}_{i1})=\boldsymbol{\Sigma}_i,\; i=1,2.$

The primary object is to test

$H_{0}: \boldsymbol{\mu}_1 = \boldsymbol{\mu}_2\; \operatorname{versus}\; H_{1}: \boldsymbol{\mu}_1 \neq \boldsymbol{\mu}_2.$

Zhu et al. (2023) proposed the following test statistic:

$T_{ZWZ}=\frac{n_1n_2n^{-1}\|\bar{\boldsymbol{y}}_1-\bar{\boldsymbol{y}}_2\|^2}{\operatorname{tr}(\hat{\boldsymbol{\Omega}}_n)},$

where $\bar{\boldsymbol{y}}_{i},i=1,2$ are the sample mean vectors and $\hat{\boldsymbol{\Omega}}_n$ is the estimator of $\operatorname{Cov}[(n_1n_2/n)^{1/2}(\bar{\boldsymbol{y}}_1-\bar{\boldsymbol{y}}_2)]$. They showed that under the null hypothesis, $T_{ZWZ}$ and an F-type mixture have the same normal or non-normal limiting distribution.

Value

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

References

Zhu T, Wang P, Zhang J (2023). “Two-sample Behrens–Fisher problems for high-dimensional data: a normal reference F-type test.” Computational Statistics, 1–24. doi:10.1007/s00180-023-01433-6.

Examples

library("HDNRA")
data("COVID19")
dim(COVID19)
group1 <- as.matrix(COVID19[c(2:19, 82:87), ]) ## healthy group
group2 <- as.matrix(COVID19[-c(1:19, 82:87), ]) ## COVID-19 patients
ZWZ2023.TSBF.2cNRT(group1, group2)

---

Normal-reference-test with three-cumulant (3-c) matched $\chi^2$-approximation for GLHT problem proposed by Zhu and Zhang (2022)

Description

Zhu and Zhang (2022)'s test for general linear hypothesis testing (GLHT) problem for high-dimensional data with assuming that underlying covariance matrices are the same.

Usage

ZZ2022.GLHT.3cNRT(Y,G,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.
G	A known full-rank coefficient matrix ($q\times k$) with $\operatorname{rank}(\boldsymbol{G})<k$.
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.

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 GLHT problem:

$H_0: \boldsymbol{G M}=\boldsymbol{0}, \quad \text { vs. } \quad H_1: \boldsymbol{G M} \neq \boldsymbol{0},$

where $\boldsymbol{M}=(\boldsymbol{\mu}_1,\ldots,\boldsymbol{\mu}_k)^\top$ is a $k\times p$ matrix collecting $k$ mean vectors and $\boldsymbol{G}:q\times k$ is a known full-rank coefficient matrix with $\operatorname{rank}(\boldsymbol{G})<k$.

Zhu and Zhang (2022) proposed the following test statistic:

$T_{ZZ}=\|\boldsymbol{C} \hat{\boldsymbol{\mu}}\|^2-q \operatorname{tr}(\hat{\boldsymbol{\Sigma}}),$

where $\boldsymbol{C}=[(\boldsymbol{G D G}^\top)^{-1/2}\boldsymbol{G}]\otimes\boldsymbol{I}_p$, and $\hat{\boldsymbol{\mu}}=(\bar{\boldsymbol{y}}_1^\top,\ldots,\bar{\boldsymbol{y}}_k^\top)^\top$, with $\bar{\boldsymbol{y}}_{i},i=1,\ldots,k$ being the sample mean vectors and $\hat{\boldsymbol{\Sigma}}$ being the usual pooled sample covariance matrix of the $k$ samples.

They showed that under the null hypothesis, $T_{ZZ}$ and a chi-squared-type mixture have the same normal or non-normal limiting distribution.

Value

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

References

Zhu T, Zhang J (2022). “Linear hypothesis testing in high-dimensional one-way MANOVA: a new normal reference approach.” Computational Statistics, 37(1), 1–27. doi:10.1007/s00180-021-01110-6.

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()
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]]))
G <- cbind(diag(k-1),rep(-1,k-1))
ZZ2022.GLHT.3cNRT(Y,G,n,p)

---

Normal-reference-test with three-cumulant (3-c) matched $\chi^2$-approximation for GLHT problem under heteroscedasticity proposed by Zhang and Zhu (2022)

Description

Zhang and Zhu (2022)'s test for general linear hypothesis testing (GLHT) problem for high-dimensional data under heteroscedasticity.

Usage

ZZ2022.GLHTBF.3cNRT(Y,G,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.
G	A known full-rank coefficient matrix ($q\times k$) with $\operatorname{rank}(\boldsymbol{G})< k$.
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.

Details

Suppose we have the following $k$ independent high-dimensional samples:

$\boldsymbol{y}_{i1},\ldots,\boldsymbol{y}_{in_i}, \;\operatorname{are \; i.i.d. \; with}\; \operatorname{E}(\boldsymbol{y}_{i1})=\boldsymbol{\mu}_i,\; \operatorname{Cov}(\boldsymbol{y}_{i1})=\boldsymbol{\Sigma}_i,i=1,\ldots,k.$

It is of interest to test the following GLHT problem:

$H_0: \boldsymbol{G M}=\boldsymbol{0}, \quad \text { vs. } H_1: \boldsymbol{G M} \neq \boldsymbol{0},$

where $\boldsymbol{M}=(\boldsymbol{\mu}_1,\ldots,\boldsymbol{\mu}_k)^\top$ is a $k\times p$ matrix collecting $k$ mean vectors and $\boldsymbol{G}:q\times k$ is a known full-rank coefficient matrix with $\operatorname{rank}(\boldsymbol{G})<k$.

Let $\bar{\boldsymbol{y}}_{i},i=1,\ldots,k$ be the sample mean vectors and $\hat{\boldsymbol{\Sigma}}_i,i=1,\ldots,k$ be the sample covariance matrices.

Zhang and Zhu (2022) proposed the following U-statistic based test statistic:

$T_{ZZ}=\|\boldsymbol{C \hat{\mu}}\|^2-\sum_{i=1}^kh_{ii}\operatorname{tr}(\hat{\boldsymbol{\Sigma}}_i)/n_i,$

where $\boldsymbol{C}=[(\boldsymbol{G D G}^\top)^{-1/2}\boldsymbol{G}]\otimes\boldsymbol{I}_p$, $\boldsymbol{D}=\operatorname{diag}(1/n_1,\ldots,1/n_k)$, and $h_{ij}$ is the $(i,j)$th entry of the $k\times k$ matrix $\boldsymbol{H}=\boldsymbol{G}^\top(\boldsymbol{G}\boldsymbol{D}\boldsymbol{G}^\top)^{-1}\boldsymbol{G}$.

Value

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

References

Zhang J, Zhu T (2022). “A new normal reference test for linear hypothesis testing in high-dimensional heteroscedastic one-way MANOVA.” Computational Statistics & Data Analysis, 168, 107385. doi:10.1016/j.csda.2021.107385.

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()
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]]))
G <- cbind(diag(k-1),rep(-1,k-1))
ZZ2022.GLHTBF.3cNRT(Y,G,n,p)

---

Normal-reference-test with three-cumulant (3-c) matched $\chi^2$-approximation for two-sample problem proposed by Zhang and Zhu (2022)

Description

Zhang and Zhu (2022)'s test for testing equality of two-sample high-dimensional mean vectors with assuming that two covariance matrices are the same.

Usage

ZZ2022.TS.3cNRT(y1, y2)

Arguments

y1	The data matrix ($n_1 \times p$) from the first population. Each row represents a $p$-dimensional observation.
y2	The data matrix ($n_2 \times p$) from the second population. Each row represents a $p$-dimensional observation.

Details

Suppose we have two independent high-dimensional samples:

$\boldsymbol{y}_{i1},\ldots,\boldsymbol{y}_{in_i}, \;\operatorname{are \; i.i.d. \; with}\; \operatorname{E}(\boldsymbol{y}_{i1})=\boldsymbol{\mu}_i,\; \operatorname{Cov}(\boldsymbol{y}_{i1})=\boldsymbol{\Sigma},i=1,2.$

The primary object is to test

$H_{0}: \boldsymbol{\mu}_1 = \boldsymbol{\mu}_2\; \operatorname{versus}\; H_{1}: \boldsymbol{\mu}_1 \neq \boldsymbol{\mu}_2.$

Zhang et al.(2022) proposed the following test statistic:

$T_{ZZ} = \frac{n_1n_2}{n} \|\bar{\boldsymbol{y}}_1 - \bar{\boldsymbol{y}}_2\|^2-\operatorname{tr}(\hat{\boldsymbol{\Sigma}}),$

where $\bar{\boldsymbol{y}}_{i},i=1,2$ are the sample mean vectors and $\hat{\boldsymbol{\Sigma}}$ is the pooled sample covariance matrix. They showed that under the null hypothesis, $T_{ZZ}$ and a chi-squared-type mixture have the same normal or non-normal limiting distribution.

Value

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

References

Zhang J, Zhu T (2022). “A revisit to Bai–Saranadasa's two-sample test.” Journal of Nonparametric Statistics, 34(1), 58–76. doi:10.1080/10485252.2021.2015768.

Examples

library("HDNRA")
data("COVID19")
dim(COVID19)
group1 <- as.matrix(COVID19[c(2:19, 82:87), ]) ## healthy group
group2 <- as.matrix(COVID19[-c(1:19, 82:87), ]) ## COVID-19 patients
ZZ2022.TS.3cNRT(group1, group2)

---