Package 'SHT'

Description

Given a multivariate sample $X$ and hypothesized covariance matrix $\Sigma_0$, it tests

$H_0 : \Sigma_x = \Sigma_0\quad vs\quad H_1 : \Sigma_x \neq \Sigma_0$

using the procedure by Fisher (2012). This method utilizes the generalized form of the inequality

$\frac{1}{p} \sum_{i=1}^p (\lambda_i^r - 1)^{2s} \ge 0$

and offers two types of test statistics $T_1$ and $T_2$ corresponding to the case $(r,s)=(1,2)$ and $(2,1)$ respectively.

Usage

cov1.2012Fisher(X, Sigma0 = diag(ncol(X)), type)

Arguments

Value

a (list) object of S3 class htest containing:

statistic

a test statistic.

p.value

$p$-value under $H_0$.

alternative

alternative hypothesis.

method

name of the test.

data.name

name(s) of provided sample data.

References

Fisher TJ (2012). “On testing for an identity covariance matrix when the dimensionality equals or exceeds the sample size.” Journal of Statistical Planning and Inference, 142(1), 312–326. ISSN 03783758.

Examples

## CRAN-purpose small example
smallX = matrix(rnorm(10*3),ncol=3)
cov1.2012Fisher(smallX) # run the test


## empirical Type 1 error 
niter   = 1000
counter1 = rep(0,niter)  # p-values of the type 1
counter2 = rep(0,niter)  # p-values of the type 2
for (i in 1:niter){
  X = matrix(rnorm(50*5), ncol=50) # (n,p) = (5,50)
  counter1[i] = ifelse(cov1.2012Fisher(X, type=1)$p.value < 0.05, 1, 0)
  counter2[i] = ifelse(cov1.2012Fisher(X, type=2)$p.value < 0.05, 1, 0)
}

## print the result
cat(paste("\n* Example for 'cov1.2012Fisher' \n","*\n",
"* empirical error with statistic 1 : ", round(sum(counter1/niter),5),"\n",
"* empirical error with statistic 2 : ", round(sum(counter2/niter),5),"\n",sep=""))

Description

Given a multivariate sample $X$ and hypothesized covariance matrix $\Sigma_0$, it tests

$H_0 : \Sigma_x = \Sigma_0\quad vs\quad H_1 : \Sigma_x \neq \Sigma_0$

using the procedure by Wu and Li (2015). They proposed to use $m$ number of multiple random projections since only a single operation might attenuate the efficacy of the test.

Usage

cov1.2015WL(X, Sigma0 = diag(ncol(X)), m = 25)

Arguments

Value

a (list) object of S3 class htest containing:

statistic

a test statistic.

p.value

$p$-value under $H_0$.

alternative

alternative hypothesis.

method

name of the test.

data.name

name(s) of provided sample data.

References

Wu T, Li P (2015). “Tests for High-Dimensional Covariance Matrices Using Random Matrix Projection.” arXiv:1511.01611 [stat].

Examples

## CRAN-purpose small example
smallX = matrix(rnorm(10*3),ncol=3)
cov1.2015WL(smallX) # run the test


## empirical Type 1 error 
##   compare effects of m=5, 10, 50
niter = 1000
rec1  = rep(0,niter) # for m=5
rec2  = rep(0,niter) #     m=10
rec3  = rep(0,niter) #     m=50
for (i in 1:niter){
  X = matrix(rnorm(50*10), ncol=50) # (n,p) = (10,50)
  rec1[i] = ifelse(cov1.2015WL(X, m=5)$p.value < 0.05, 1, 0)
  rec2[i] = ifelse(cov1.2015WL(X, m=10)$p.value < 0.05, 1, 0)
  rec3[i] = ifelse(cov1.2015WL(X, m=50)$p.value < 0.05, 1, 0)
}

## print the result
cat(paste("\n* Example for 'cov1.2015WL'\n","*\n",
"* Type 1 error with m=5   : ",round(sum(rec1/niter),5),"\n",
"* Type 1 error with m=10  : ",round(sum(rec2/niter),5),"\n",
"* Type 1 error with m=50  : ",round(sum(rec3/niter),5),"\n",sep=""))

Description

Given two multivariate data $X$ and $Y$ of same dimension, it tests

$H_0 : \Sigma_x = \Sigma_y\quad vs\quad H_1 : \Sigma_x \neq \Sigma_y$

using the procedure by Li and Chen (2012).

Usage

cov2.2012LC(X, Y, use.unbiased = TRUE)

Arguments

Value

a (list) object of S3 class htest containing:

statistic

a test statistic.

p.value

$p$-value under $H_0$.

alternative

alternative hypothesis.

method

name of the test.

data.name

name(s) of provided sample data.

References

Li J, Chen SX (2012). “Two sample tests for high-dimensional covariance matrices.” The Annals of Statistics, 40(2), 908–940. ISSN 0090-5364.

Examples

## CRAN-purpose small example
smallX = matrix(rnorm(10*4),ncol=5)
smallY = matrix(rnorm(10*4),ncol=5)
cov2.2012LC(smallX, smallY) # run the test

## Not run: 
## empirical Type 1 error : use 'biased' version for faster computation
niter   = 1000
counter = rep(0,niter)
for (i in 1:niter){
  X = matrix(rnorm(500*25), ncol=10)
  Y = matrix(rnorm(500*25), ncol=10)
  
  counter[i] = ifelse(cov2.2012LC(X,Y,use.unbiased=FALSE)$p.value  < 0.05,1,0)
  print(paste0("iteration ",i,"/1000 complete.."))
}

## print the result
cat(paste("\n* Example for 'cov2.2012LC'\n","*\n",
"* number of rejections   : ", sum(counter),"\n",
"* total number of trials : ", niter,"\n",
"* empirical Type 1 error : ",round(sum(counter/niter),5),"\n",sep=""))

## End(Not run)

Description

Given two multivariate data $X$ and $Y$ of same dimension, it tests

$H_0 : \Sigma_x = \Sigma_y\quad vs\quad H_1 : \Sigma_x \neq \Sigma_y$

using the procedure by Cai, Liu, and Xia (2013).

Usage

cov2.2013CLX(X, Y)

Arguments

Value

a (list) object of S3 class htest containing:

statistic

a test statistic.

p.value

$p$-value under $H_0$.

alternative

alternative hypothesis.

method

name of the test.

data.name

name(s) of provided sample data.

References

Cai T, Liu W, Xia Y (2013). “Two-Sample Covariance Matrix Testing and Support Recovery in High-Dimensional and Sparse Settings.” Journal of the American Statistical Association, 108(501), 265–277. ISSN 0162-1459, 1537-274X.

Examples

## CRAN-purpose small example
smallX = matrix(rnorm(10*3),ncol=3)
smallY = matrix(rnorm(10*3),ncol=3)
cov2.2013CLX(smallX, smallY) # run the test


## empirical Type 1 error 
niter   = 1000
counter = rep(0,niter)  # record p-values
for (i in 1:niter){
  X = matrix(rnorm(50*5), ncol=10)
  Y = matrix(rnorm(50*5), ncol=10)
  
  counter[i] = ifelse(cov2.2013CLX(X, Y)$p.value < 0.05, 1, 0)
}

## print the result
cat(paste("\n* Example for 'cov2.2013CLX'\n","*\n",
"* number of rejections   : ", sum(counter),"\n",
"* total number of trials : ", niter,"\n",
"* empirical Type 1 error : ",round(sum(counter/niter),5),"\n",sep=""))

Description

Given two multivariate data $X$ and $Y$ of same dimension, it tests

$H_0 : \Sigma_x = \Sigma_y\quad vs\quad H_1 : \Sigma_x \neq \Sigma_y$

using the procedure by Wu and Li (2015).

Usage

cov2.2015WL(X, Y, m = 50)

Arguments

Value

a (list) object of S3 class htest containing:

statistic

a test statistic.

p.value

$p$-value under $H_0$.

alternative

alternative hypothesis.

method

name of the test.

data.name

name(s) of provided sample data.

References

Wu T, Li P (2015). “Tests for High-Dimensional Covariance Matrices Using Random Matrix Projection.” arXiv:1511.01611 [stat].

Examples

## CRAN-purpose small example
smallX = matrix(rnorm(10*3),ncol=3)
smallY = matrix(rnorm(10*3),ncol=3)
cov2.2015WL(smallX, smallY) # run the test


## empirical Type 1 error 
niter   = 1000
counter = rep(0,niter)  # record p-values
for (i in 1:niter){
  X = matrix(rnorm(50*5), ncol=10)
  Y = matrix(rnorm(50*5), ncol=10)
  
  counter[i] = ifelse(cov2.2015WL(X, Y)$p.value < 0.05, 1, 0)
}

## print the result
cat(paste("\n* Example for 'cov2.2015WL'\n","*\n",
"* number of rejections   : ", sum(counter),"\n",
"* total number of trials : ", niter,"\n",
"* empirical Type 1 error : ",round(sum(counter/niter),5),"\n",sep=""))

Description

Given univariate samples $X_1~,\ldots,~X_k$, it tests

$H_0 : \Sigma_1 = \cdots \Sigma_k\quad vs\quad H_1 : \textrm{at least one equality does not hold}$

using the procedure by Schott (2001) using Wald statistics. In the original paper, it provides 4 different test statistics for general elliptical distribution cases. However, we only deliver the first one with an assumption of multivariate normal population.

Usage

covk.2001Schott(dlist)

Arguments

Value

a (list) object of S3 class htest containing:

statistic

a test statistic.

p.value

$p$-value under $H_0$.

alternative

alternative hypothesis.

method

name of the test.

data.name

name(s) of provided sample data.

References

Schott JR (2001). “Some tests for the equality of covariance matrices.” Journal of Statistical Planning and Inference, 94(1), 25–36. ISSN 03783758.

Examples

## CRAN-purpose small example
tinylist = list()
for (i in 1:3){ # consider 3-sample case
  tinylist[[i]] = matrix(rnorm(10*3),ncol=3)
}
covk.2001Schott(tinylist) # run the test

## Not run: 
## test when k=5 samples with (n,p) = (100,20)
## empirical Type 1 error 
niter   = 1000
counter = rep(0,niter)  # record p-values
for (i in 1:niter){
  mylist = list()
  for (j in 1:5){
     mylist[[j]] = matrix(rnorm(100*20),ncol=20)
  }
  
  counter[i] = ifelse(covk.2001Schott(mylist)$p.value < 0.05, 1, 0)
}

## print the result
cat(paste("\n* Example for 'covk.2001Schott'\n","*\n",
"* number of rejections   : ", sum(counter),"\n",
"* total number of trials : ", niter,"\n",
"* empirical Type 1 error : ",round(sum(counter/niter),5),"\n",sep=""))

## End(Not run)

Description

Given univariate samples $X_1~,\ldots,~X_k$, it tests

$H_0 : \Sigma_1 = \cdots \Sigma_k\quad vs\quad H_1 : \textrm{at least one equality does not hold}$

using the procedure by Schott (2007).

Usage

covk.2007Schott(dlist)

Arguments

Value

a (list) object of S3 class htest containing:

statistic

a test statistic.

p.value

$p$-value under $H_0$.

alternative

alternative hypothesis.

method

name of the test.

data.name

name(s) of provided sample data.

References

Schott JR (2007). “A test for the equality of covariance matrices when the dimension is large relative to the sample sizes.” Computational Statistics & Data Analysis, 51(12), 6535–6542. ISSN 01679473.

Examples

## CRAN-purpose small example
tinylist = list()
for (i in 1:3){ # consider 3-sample case
  tinylist[[i]] = matrix(rnorm(10*3),ncol=3)
}
covk.2007Schott(tinylist) # run the test


## test when k=4 samples with (n,p) = (100,20)
## empirical Type 1 error 
niter   = 1234
counter = rep(0,niter)  # record p-values
for (i in 1:niter){
  mylist = list()
  for (j in 1:4){
     mylist[[j]] = matrix(rnorm(100*20),ncol=20)
  }
  
  counter[i] = ifelse(covk.2007Schott(mylist)$p.value < 0.05, 1, 0)
}

## print the result
cat(paste("\n* Example for 'covk.2007Schott'\n","*\n",
"* number of rejections   : ", sum(counter),"\n",
"* total number of trials : ", niter,"\n",
"* empirical Type 1 error : ",round(sum(counter/niter),5),"\n",sep=""))

Description

Given two samples (either univariate or multivariate) $X$ and $Y$ of same dimension, it tests

$H_0 : F_X = F_Y\quad vs\quad H_1 : F_X \neq F_Y$

using the procedure by Biswas and Ghosh (2014) in a nonparametric way based on pairwise distance measures. Both asymptotic and permutation-based determination of $p$-values are supported.

Usage

eqdist.2014BG(X, Y, method = c("permutation", "asymptotic"), nreps = 999)

Arguments

Value

a (list) object of S3 class htest containing:

statistic

a test statistic.

p.value

$p$-value under $H_0$.

alternative

alternative hypothesis.

method

name of the test.

data.name

name(s) of provided sample data.

References

Biswas M, Ghosh AK (2014). “A nonparametric two-sample test applicable to high dimensional data.” Journal of Multivariate Analysis, 123, 160–171. ISSN 0047259X.

Examples

## CRAN-purpose small example
smallX = matrix(rnorm(10*3),ncol=3)
smallY = matrix(rnorm(10*3),ncol=3)
eqdist.2014BG(smallX, smallY) # run the test

## Not run: 
## compare asymptotic and permutation-based powers
set.seed(777)
ntest  = 1000
pval.a = rep(0,ntest)
pval.p = rep(0,ntest)

for (i in 1:ntest){
  x = matrix(rnorm(100), nrow=5)
  y = matrix(rnorm(100), nrow=5)
  
  pval.a[i] = ifelse(eqdist.2014BG(x,y,method="a")$p.value<0.05,1,0)
  pval.p[i] = ifelse(eqdist.2014BG(x,y,method="p",nreps=100)$p.value <0.05,1,0)
}

## print the result
cat(paste("\n* EMPIRICAL TYPE 1 ERROR COMPARISON \n","*\n",
"* Asymptotics : ", round(sum(pval.a/ntest),5),"\n",
"* Permutation : ", round(sum(pval.p/ntest),5),"\n",sep=""))

## End(Not run)

Description

Given a multivariate sample $X$ and hypothesized mean $\mu_0$, it tests

$H_0 : \mu_x = \mu_0\quad vs\quad H_1 : \mu_x \neq \mu_0$

using the procedure by Hotelling (1931).

Usage

mean1.1931Hotelling(X, mu0 = rep(0, ncol(X)))

Arguments

Value

a (list) object of S3 class htest containing:

statistic

a test statistic.

p.value

$p$-value under $H_0$.

alternative

alternative hypothesis.

method

name of the test.

data.name

name(s) of provided sample data.

References

Hotelling H (1931). “The Generalization of Student's Ratio.” The Annals of Mathematical Statistics, 2(3), 360–378. ISSN 0003-4851.

Examples

## CRAN-purpose small example
smallX = matrix(rnorm(10*3),ncol=3)
mean1.1931Hotelling(smallX) # run the test

## Not run: 
## empirical Type 1 error 
niter   = 1000
counter = rep(0,niter)  # record p-values
for (i in 1:niter){
  X = matrix(rnorm(50*5), ncol=5)
  counter[i] = ifelse(mean1.1931Hotelling(X)$p.value < 0.05, 1, 0)
}

## print the result
cat(paste("\n* Example for 'mean1.1931Hotelling'\n","*\n",
"* number of rejections   : ", sum(counter),"\n",
"* total number of trials : ", niter,"\n",
"* empirical Type 1 error : ",round(sum(counter/niter),5),"\n",sep=""))

## End(Not run)

Description

Given a multivariate sample $X$ and hypothesized mean $\mu_0$, it tests

$H_0 : \mu_x = \mu_0\quad vs\quad H_1 : \mu_x \neq \mu_0$

using the procedure by Dempster (1958, 1960).

Usage

mean1.1958Dempster(X, mu0 = rep(0, ncol(X)))

Arguments

Value

a (list) object of S3 class htest containing:

statistic

a test statistic.

p.value

$p$-value under $H_0$.

alternative

alternative hypothesis.

method

name of the test.

data.name

name(s) of provided sample data.

References

Dempster AP (1958). “A High Dimensional Two Sample Significance Test.” The Annals of Mathematical Statistics, 29(4), 995–1010. ISSN 0003-4851.

Dempster AP (1960). “A Significance Test for the Separation of Two Highly Multivariate Small Samples.” Biometrics, 16(1), 41. ISSN 0006341X.

Examples

## CRAN-purpose small example
smallX = matrix(rnorm(10*3),ncol=3)
mean1.1958Dempster(smallX) # run the test


## empirical Type 1 error 
niter   = 1000
counter = rep(0,niter)  # record p-values
for (i in 1:niter){
  X = matrix(rnorm(50*5), ncol=50)
  counter[i] = ifelse(mean1.1958Dempster(X)$p.value < 0.05, 1, 0)
}

## print the result
cat(paste("\n* Example for 'mean1.1958Dempster'\n","*\n",
"* number of rejections   : ", sum(counter),"\n",
"* total number of trials : ", niter,"\n",
"* empirical Type 1 error : ",round(sum(counter/niter),5),"\n",sep=""))

Description

Given a multivariate sample $X$ and hypothesized mean $\mu_0$, it tests

$H_0 : \mu_x = \mu_0\quad vs\quad H_1 : \mu_x \neq \mu_0$

using the procedure by Bai and Saranadasa (1996).

Usage

mean1.1996BS(X, mu0 = rep(0, ncol(X)))

Arguments

Value

a (list) object of S3 class htest containing:

statistic

a test statistic.

p.value

$p$-value under $H_0$.

alternative

alternative hypothesis.

method

name of the test.

data.name

name(s) of provided sample data.

References

Bai Z, Saranadasa H (1996). “HIGH DIMENSION: BY AN EXAMPLE OF A TWO SAMPLE PROBLEM.” Statistica Sinica, 6(2), 311–329. ISSN 10170405, 19968507.

Examples

## CRAN-purpose small example
smallX = matrix(rnorm(10*3),ncol=3)
mean1.1996BS(smallX) # run the test


## empirical Type 1 error 
niter   = 1000
counter = rep(0,niter)  # record p-values
for (i in 1:niter){
  X = matrix(rnorm(50*5), ncol=25)
  counter[i] = ifelse(mean1.1996BS(X)$p.value < 0.05, 1, 0)
}

## print the result
cat(paste("\n* Example for 'mean1.1996BS'\n","*\n",
"* number of rejections   : ", sum(counter),"\n",
"* total number of trials : ", niter,"\n",
"* empirical Type 1 error : ",round(sum(counter/niter),5),"\n",sep=""))

Description

Given a multivariate sample $X$ and hypothesized mean $\mu_0$, it tests

$H_0 : \mu_x = \mu_0\quad vs\quad H_1 : \mu_x \neq \mu_0$

using the procedure by Srivastava and Du (2008).

Usage

mean1.2008SD(X, mu0 = rep(0, ncol(X)))

Arguments

Value

a (list) object of S3 class htest containing:

statistic

a test statistic.

p.value

$p$-value under $H_0$.

alternative

alternative hypothesis.

method

name of the test.

data.name

name(s) of provided sample data.

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. ISSN 0047259X.

Examples

## CRAN-purpose small example
smallX = matrix(rnorm(10*3),ncol=3)
mean1.2008SD(smallX) # run the test


## empirical Type 1 error 
niter   = 1000
counter = rep(0,niter)  # record p-values
for (i in 1:niter){
  X = matrix(rnorm(50*5), ncol=5)
  counter[i] = ifelse(mean1.2008SD(X)$p.value < 0.05, 1, 0)
}

## print the result
cat(paste("\n* Example for 'mean1.2008SD'\n","*\n",
"* number of rejections   : ", sum(counter),"\n",
"* total number of trials : ", niter,"\n",
"* empirical Type 1 error : ",round(sum(counter/niter),5),"\n",sep=""))

Description

Given an univariate sample $x$, it tests

$H_0 : \mu_x = \mu_0\quad vs\quad H_1 : \mu_x \neq \mu_0$

using the procedure by Student (1908).

Usage

mean1.ttest(x, mu0 = 0, alternative = c("two.sided", "less", "greater"))

Arguments

Value

a (list) object of S3 class htest containing:

statistic

a test statistic.

p.value

$p$-value under $H_0$.

alternative

alternative hypothesis.

method

name of the test.

data.name

name(s) of provided sample data.

References

Student (1908). “The Probable Error of a Mean.” Biometrika, 6(1), 1. ISSN 00063444.

Student (1908). “Probable Error of a Correlation Coefficient.” Biometrika, 6(2-3), 302–310. ISSN 0006-3444, 1464-3510.

Examples

## empirical Type 1 error 
niter   = 1000
counter = rep(0,niter)  # record p-values
for (i in 1:niter){
  x = rnorm(10)         # sample from N(0,1)
  counter[i] = ifelse(mean1.ttest(x)$p.value < 0.05, 1, 0)
}

## print the result
cat(paste("\n* Example for 'mean1.ttest'\n","*\n",
"* number of rejections   : ", sum(counter),"\n",
"* total number of trials : ", niter,"\n",
"* empirical Type 1 error : ",round(sum(counter/niter),5),"\n",sep=""))

Description

Given two multivariate data $X$ and $Y$ of same dimension, it tests

$H_0 : \mu_x = \mu_y\quad vs\quad H_1 : \mu_x \neq \mu_y$

using the procedure by Hotelling (1931).

Usage

mean2.1931Hotelling(X, Y, paired = FALSE, var.equal = TRUE)

Arguments

Value

a (list) object of S3 class htest containing:

statistic

a test statistic.

p.value

$p$-value under $H_0$.

alternative

alternative hypothesis.

method

name of the test.

data.name

name(s) of provided sample data.

References

Hotelling H (1931). “The Generalization of Student's Ratio.” The Annals of Mathematical Statistics, 2(3), 360–378. ISSN 0003-4851.

Examples

## CRAN-purpose small example
smallX = matrix(rnorm(10*3),ncol=3)
smallY = matrix(rnorm(10*3),ncol=3)
mean2.1931Hotelling(smallX, smallY) # run the test


## generate two samples from standard normal distributions.
X = matrix(rnorm(50*5), ncol=5)
Y = matrix(rnorm(77*5), ncol=5)

## run single test
print(mean2.1931Hotelling(X,Y))

## empirical Type 1 error 
niter   = 1000
counter = rep(0,niter)  # record p-values
for (i in 1:niter){
  X = matrix(rnorm(50*5), ncol=5)
  Y = matrix(rnorm(77*5), ncol=5)
  
  counter[i] = ifelse(mean2.1931Hotelling(X,Y)$p.value < 0.05, 1, 0)
}

## print the result
cat(paste("\n* Example for 'mean2.1931Hotelling'\n","*\n",
"* number of rejections   : ", sum(counter),"\n",
"* total number of trials : ", niter,"\n",
"* empirical Type 1 error : ",round(sum(counter/niter),5),"\n",sep=""))

Description

Given two multivariate data $X$ and $Y$ of same dimension, it tests

$H_0 : \mu_x = \mu_y\quad vs\quad H_1 : \mu_x \neq \mu_y$

using the procedure by Dempster (1958, 1960).

Usage

mean2.1958Dempster(X, Y)

Arguments

Value

a (list) object of S3 class htest containing:

statistic

a test statistic.

p.value

$p$-value under $H_0$.

alternative

alternative hypothesis.

method

name of the test.

data.name

name(s) of provided sample data.

References

Dempster AP (1958). “A High Dimensional Two Sample Significance Test.” The Annals of Mathematical Statistics, 29(4), 995–1010. ISSN 0003-4851.

Dempster AP (1960). “A Significance Test for the Separation of Two Highly Multivariate Small Samples.” Biometrics, 16(1), 41. ISSN 0006341X.

Examples

## CRAN-purpose small example
smallX = matrix(rnorm(10*3),ncol=3)
smallY = matrix(rnorm(10*3),ncol=3)
mean2.1958Dempster(smallX, smallY) # run the test


## empirical Type 1 error 
niter   = 1000
counter = rep(0,niter)  # record p-values
for (i in 1:niter){
  X = matrix(rnorm(50*5), ncol=10)
  Y = matrix(rnorm(50*5), ncol=10)
  
  counter[i] = ifelse(mean2.1958Dempster(X,Y)$p.value < 0.05, 1, 0)
}

## print the result
cat(paste("\n* Example for 'mean2.1958Dempster'\n","*\n",
"* number of rejections   : ", sum(counter),"\n",
"* total number of trials : ", niter,"\n",
"* empirical Type 1 error : ",round(sum(counter/niter),5),"\n",sep=""))

Description

Given two multivariate data $X$ and $Y$ of same dimension, it tests

$H_0 : \mu_x = \mu_y\quad vs\quad H_1 : \mu_x \neq \mu_y$

using the procedure by Yao (1965) via multivariate modification of Welch's approximation of degrees of freedoms.

Usage

mean2.1965Yao(X, Y)

Arguments

Value

a (list) object of S3 class htest containing:

statistic

a test statistic.

p.value

$p$-value under $H_0$.

alternative

alternative hypothesis.

method

name of the test.

data.name

name(s) of provided sample data.

References

Yao Y (1965). “An Approximate Degrees of Freedom Solution to the Multivariate Behrens Fisher Problem.” Biometrika, 52(1/2), 139. ISSN 00063444.

Examples

## CRAN-purpose small example
smallX = matrix(rnorm(10*3),ncol=3)
smallY = matrix(rnorm(10*3),ncol=3)
mean2.1965Yao(smallX, smallY) # run the test


## empirical Type 1 error 
niter   = 1000
counter = rep(0,niter)  # record p-values
for (i in 1:niter){
  X = matrix(rnorm(50*5), ncol=10)
  Y = matrix(rnorm(50*5), ncol=10)
  
  counter[i] = ifelse(mean2.1965Yao(X,Y)$p.value < 0.05, 1, 0)
}

## print the result
cat(paste("\n* Example for 'mean2.1965Yao'\n","*\n",
"* number of rejections   : ", sum(counter),"\n",
"* total number of trials : ", niter,"\n",
"* empirical Type 1 error : ",round(sum(counter/niter),5),"\n",sep=""))

Description

Given two multivariate data $X$ and $Y$ of same dimension, it tests

$H_0 : \mu_x = \mu_y\quad vs\quad H_1 : \mu_x \neq \mu_y$

using the procedure by Johansen (1980) by adapting Welch-James approximation of the degree of freedom for Hotelling's $T^2$ test.

Usage

mean2.1980Johansen(X, Y)

Arguments

Value

a (list) object of S3 class htest containing:

statistic

a test statistic.

p.value

$p$-value under $H_0$.

alternative

alternative hypothesis.

method

name of the test.

data.name

name(s) of provided sample data.

References

Johansen S (1980). “The Welch-James Approximation to the Distribution of the Residual Sum of Squares in a Weighted Linear Regression.” Biometrika, 67(1), 85. ISSN 00063444.

Examples

## CRAN-purpose small example
smallX = matrix(rnorm(10*3),ncol=3)
smallY = matrix(rnorm(10*3),ncol=3)
mean2.1980Johansen(smallX, smallY) # run the test

## Not run: 
## empirical Type 1 error 
niter   = 1000
counter = rep(0,niter)  # record p-values
for (i in 1:niter){
  X = matrix(rnorm(50*5), ncol=10)
  Y = matrix(rnorm(50*5), ncol=10)
  
  counter[i] = ifelse(mean2.1980Johansen(X,Y)$p.value < 0.05, 1, 0)
}

## print the result
cat(paste("\n* Example for 'mean2.1980Johansen'\n","*\n",
"* number of rejections   : ", sum(counter),"\n",
"* total number of trials : ", niter,"\n",
"* empirical Type 1 error : ",round(sum(counter/niter),5),"\n",sep=""))

## End(Not run)

Description

Given two multivariate data $X$ and $Y$ of same dimension, it tests

$H_0 : \mu_x = \mu_y\quad vs\quad H_1 : \mu_x \neq \mu_y$

using the procedure by Nel and Van der Merwe (1986).

Usage

mean2.1986NVM(X, Y)

Arguments

Value

a (list) object of S3 class htest containing:

statistic

a test statistic.

p.value

$p$-value under $H_0$.

alternative

alternative hypothesis.

method

name of the test.

data.name

name(s) of provided sample data.

References

Nel DG, Van Der Merwe CA (1986). “A solution to the multivariate behrens-fisher problem.” Communications in Statistics - Theory and Methods, 15(12), 3719–3735. ISSN 0361-0926, 1532-415X.

Examples

## CRAN-purpose small example
smallX = matrix(rnorm(10*3),ncol=3)
smallY = matrix(rnorm(10*3),ncol=3)
mean2.1986NVM(smallX, smallY) # run the test


## empirical Type 1 error 
niter   = 1000
counter = rep(0,niter)  # record p-values
for (i in 1:niter){
  X = matrix(rnorm(50*5), ncol=10)
  Y = matrix(rnorm(50*5), ncol=10)
  
  counter[i] = ifelse(mean2.1986NVM(X,Y)$p.value < 0.05, 1, 0)
}

## print the result
cat(paste("\n* Example for 'mean2.1986NVM'\n","*\n",
"* number of rejections   : ", sum(counter),"\n",
"* total number of trials : ", niter,"\n",
"* empirical Type 1 error : ",round(sum(counter/niter),5),"\n",sep=""))

Description

Given two multivariate data $X$ and $Y$ of same dimension, it tests

$H_0 : \mu_x = \mu_y\quad vs\quad H_1 : \mu_x \neq \mu_y$

using the procedure by Bai and Saranadasa (1996).

Usage

mean2.1996BS(X, Y)

Arguments

Value

a (list) object of S3 class htest containing:

statistic

a test statistic.

p.value

$p$-value under $H_0$.

alternative

alternative hypothesis.

method

name of the test.

data.name

name(s) of provided sample data.

References

Bai Z, Saranadasa H (1996). “HIGH DIMENSION: BY AN EXAMPLE OF A TWO SAMPLE PROBLEM.” Statistica Sinica, 6(2), 311–329. ISSN 10170405, 19968507.

Examples

## CRAN-purpose small example
smallX = matrix(rnorm(10*3),ncol=3)
smallY = matrix(rnorm(10*3),ncol=3)
mean2.1996BS(smallX, smallY) # run the test


## empirical Type 1 error 
niter   = 1000
counter = rep(0,niter)  # record p-values
for (i in 1:niter){
  X = matrix(rnorm(50*5), ncol=10)
  Y = matrix(rnorm(50*5), ncol=10)
  
  counter[i] = ifelse(mean2.1996BS(X,Y)$p.value < 0.05, 1, 0)
}

## print the result
cat(paste("\n* Example for 'mean2.1996BS'\n","*\n",
"* number of rejections   : ", sum(counter),"\n",
"* total number of trials : ", niter,"\n",
"* empirical Type 1 error : ",round(sum(counter/niter),5),"\n",sep=""))

Description

Given two multivariate data $X$ and $Y$ of same dimension, it tests

$H_0 : \mu_x = \mu_y\quad vs\quad H_1 : \mu_x \neq \mu_y$

using the procedure by Krishnamoorthy and Yu (2004), which is a modified version of Nel and Van der Merwe (1986).

Usage

mean2.2004KY(X, Y)

Arguments

Value

a (list) object of S3 class htest containing:

statistic

a test statistic.

p.value

$p$-value under $H_0$.

alternative

alternative hypothesis.

method

name of the test.

data.name

name(s) of provided sample data.

References

Krishnamoorthy K, Yu J (2004). “Modified Nel and Van der Merwe test for the multivariate Behrens–Fisher problem.” Statistics & Probability Letters, 66(2), 161–169. ISSN 01677152.

Examples

## CRAN-purpose small example
smallX = matrix(rnorm(10*3),ncol=3)
smallY = matrix(rnorm(10*3),ncol=3)
mean2.2004KY(smallX, smallY) # run the test

## Not run: 
## empirical Type 1 error 
niter   = 1000
counter = rep(0,niter)  # record p-values
for (i in 1:niter){
  X = matrix(rnorm(50*5), ncol=10)
  Y = matrix(rnorm(50*5), ncol=10)
  
  counter[i] = ifelse(mean2.2004KY(X,Y)$p.value < 0.05, 1, 0)
}

## print the result
cat(paste("\n* Example for 'mean2.2004KY'\n","*\n",
"* number of rejections   : ", sum(counter),"\n",
"* total number of trials : ", niter,"\n",
"* empirical Type 1 error : ",round(sum(counter/niter),5),"\n",sep=""))

## End(Not run)

Description

Given two multivariate data $X$ and $Y$ of same dimension, it tests

$H_0 : \mu_x = \mu_y\quad vs\quad H_1 : \mu_x \neq \mu_y$

using the procedure by Srivastava and Du (2008).

Usage

mean2.2008SD(X, Y)

Arguments

Value

a (list) object of S3 class htest containing:

statistic

a test statistic.

p.value

$p$-value under $H_0$.

alternative

alternative hypothesis.

method

name of the test.

data.name

name(s) of provided sample data.

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. ISSN 0047259X.

Examples

## CRAN-purpose small example
smallX = matrix(rnorm(10*3),ncol=3)
smallY = matrix(rnorm(10*3),ncol=3)
mean2.2008SD(smallX, smallY) # run the test


## empirical Type 1 error 
niter   = 1000
counter = rep(0,niter)  # record p-values
for (i in 1:niter){
  X = matrix(rnorm(50*5), ncol=10)
  Y = matrix(rnorm(50*5), ncol=10)
  
  counter[i] = ifelse(mean2.2008SD(X,Y)$p.value < 0.05, 1, 0)
}

## print the result
cat(paste("\n* Example for 'mean2.2008SD'\n","*\n",
"* number of rejections   : ", sum(counter),"\n",
"* total number of trials : ", niter,"\n",
"* empirical Type 1 error : ",round(sum(counter/niter),5),"\n",sep=""))

Description

Given two multivariate data $X$ and $Y$ of same dimension, it tests

$H_0 : \mu_x = \mu_y\quad vs\quad H_1 : \mu_x \neq \mu_y$

using the procedure by Lopes, Jacob, and Wainwright (2011) using random projection. Due to solving system of linear equations, we suggest you to opt for asymptotic-based $p$-value computation unless truly necessary for random permutation tests.

Usage

mean2.2011LJW(X, Y, method = c("asymptotic", "MC"), nreps = 1000)

Arguments

Value

a (list) object of S3 class htest containing:

statistic

a test statistic.

p.value

$p$-value under $H_0$.

alternative

alternative hypothesis.

method

name of the test.

data.name

name(s) of provided sample data.

References

Lopes ME, Jacob L, Wainwright MJ (2011). “A More Powerful Two-sample Test in High Dimensions Using Random Projection.” In Proceedings of the 24th International Conference on Neural Information Processing Systems, NIPS'11, 1206–1214. ISBN 978-1-61839-599-3.

Examples

## CRAN-purpose small example
smallX = matrix(rnorm(10*3),ncol=10)
smallY = matrix(rnorm(10*3),ncol=10)
mean2.2011LJW(smallX, smallY) # run the test


## empirical Type 1 error 
niter   = 1000
counter = rep(0,niter)  # record p-values
for (i in 1:niter){
  X = matrix(rnorm(10*20), ncol=20)
  Y = matrix(rnorm(10*20), ncol=20)
  
  counter[i] = ifelse(mean2.2011LJW(X,Y)$p.value < 0.05, 1, 0)
}

## print the result
cat(paste("\n* Example for 'mean2.2011LJW'\n","*\n",
"* number of rejections   : ", sum(counter),"\n",
"* total number of trials : ", niter,"\n",
"* empirical Type 1 error : ",round(sum(counter/niter),5),"\n",sep=""))

Description

Given two multivariate data $X$ and $Y$ of same dimension, it tests

$H_0 : \mu_x = \mu_y\quad vs\quad H_1 : \mu_x \neq \mu_y$

using the procedure by Cai, Liu, and Xia (2014) which is equivalent to test

$H_0 : \Omega(\mu_x - \mu_y)=0$

for an inverse covariance (or precision) $\Omega$. When $\Omega$ is not given and known to be sparse, it is first estimated with CLIME estimator. Otherwise, adaptive thresholding estimator is used. Also, if two samples are assumed to have different covariance structure, it uses weighting scheme for adjustment.

Usage

mean2.2014CLX(
  X,
  Y,
  precision = c("sparse", "unknown"),
  delta = 2,
  Omega = NULL,
  cov.equal = TRUE
)

Arguments

Value

a (list) object of S3 class htest containing:

statistic

a test statistic.

p.value

$p$-value under $H_0$.

alternative

alternative hypothesis.

method

name of the test.

data.name

name(s) of provided sample data.

References

Cai TT, Liu W, Xia Y (2014). “Two-sample test of high dimensional means under dependence.” Journal of the Royal Statistical Society: Series B (Statistical Methodology), 76(2), 349–372. ISSN 13697412.

Examples

## CRAN-purpose small example
smallX = matrix(rnorm(10*3),ncol=3)
smallY = matrix(rnorm(10*3),ncol=3)
mean2.2014CLX(smallX, smallY, precision="unknown")
mean2.2014CLX(smallX, smallY, precision="sparse")

## Not run: 
## empirical Type 1 error 
niter   = 100
counter = rep(0,niter)  # record p-values
for (i in 1:niter){
  X = matrix(rnorm(50*5), ncol=10)
  Y = matrix(rnorm(50*5), ncol=10)
  
  counter[i] = ifelse(mean2.2014CLX(X, Y)$p.value < 0.05, 1, 0)
}

## print the result
cat(paste("\n* Example for 'mean2.2014CLX'\n","*\n",
"* number of rejections   : ", sum(counter),"\n",
"* total number of trials : ", niter,"\n",
"* empirical Type 1 error : ",round(sum(counter/niter),5),"\n",sep=""))

## End(Not run)

Description

Given two multivariate data $X$ and $Y$ of same dimension, it tests

$H_0 : \mu_x = \mu_y\quad vs\quad H_1 : \mu_x \neq \mu_y$

using the procedure by Thulin (2014) using random subspace methods. We did not enable parallel computing schemes for this in that it might incur huge computational burden since it entirely depends on random permutation scheme.

Usage

mean2.2014Thulin(X, Y, B = 100, nreps = 1000)

Arguments

Value

a (list) object of S3 class htest containing:

statistic

a test statistic.

p.value

$p$-value under $H_0$.

alternative

alternative hypothesis.

method

name of the test.

data.name

name(s) of provided sample data.

References

Thulin M (2014). “A high-dimensional two-sample test for the mean using random subspaces.” Computational Statistics & Data Analysis, 74, 26–38. ISSN 01679473.

Examples

## CRAN-purpose small example
smallX = matrix(rnorm(10*3),ncol=10)
smallY = matrix(rnorm(10*3),ncol=10)
mean2.2014Thulin(smallX, smallY, B=10, nreps=10) # run the test


## Compare with 'mean2.2011LJW' 
## which is based on random projection.
n = 33    # number of observations for each sample
p = 100   # dimensionality

X = matrix(rnorm(n*p), ncol=p)
Y = matrix(rnorm(n*p), ncol=p)

## run both methods with 100 permutations
mean2.2011LJW(X,Y,nreps=100,method="m")  # 2011LJW requires 'm' to be set.
mean2.2014Thulin(X,Y,nreps=100)

Description

Not Written Here - No Reference Yet.

Usage

mean2.mxPBF(X, Y, a0 = 0, b0 = 0, gamma = 1, nthreads = 1)

Arguments

Value

a (list) object of S3 class htest containing:

statistic

maximum of pairwise Bayes factor.

alternative

alternative hypothesis.

method

name of the test.

data.name

name(s) of provided sample data.

log.BF.vec

vector of pairwise Bayes factors in natural log.

Examples

## Not run: 
## empirical Type 1 error with BF threshold = 10
niter   = 1000
counter = rep(0,niter)  # record p-values
for (i in 1:niter){
  X = matrix(rnorm(100*10), ncol=10)
  Y = matrix(rnorm(200*10), ncol=10)
  
  counter[i] = ifelse(mean2.mxPBF(X,Y)$statistic > 10, 1, 0)
}

## print the result
cat(paste("\n* Example for 'mean2.mxPBF'\n","*\n",
"* number of rejections   : ", sum(counter),"\n",
"* total number of trials : ", niter,"\n",
"* empirical Type 1 error : ",round(sum(counter/niter),5),"\n",sep=""))

## End(Not run)

Description

Given two univariate samples $x$ and $y$, it tests

$H_0 : \mu_x^2 \left\lbrace =,\geq,\leq \right\rbrace \mu_y^2\quad vs\quad H_1 : \mu_x^2 \left\lbrace \neq,<,>\right\rbrace \mu_y^2$

using the procedure by Student (1908) and Welch (1947).

Usage

mean2.ttest(
  x,
  y,
  alternative = c("two.sided", "less", "greater"),
  paired = FALSE,
  var.equal = FALSE
)

Arguments

Value

a (list) object of S3 class htest containing:

statistic

a test statistic.

p.value

$p$-value under $H_0$.

alternative

alternative hypothesis.

method

name of the test.

data.name

name(s) of provided sample data.

References

Student (1908). “The Probable Error of a Mean.” Biometrika, 6(1), 1. ISSN 00063444.

Student (1908). “Probable Error of a Correlation Coefficient.” Biometrika, 6(2-3), 302–310. ISSN 0006-3444, 1464-3510.

Welch BL (1947). “The Generalization of ‘Student’s' Problem when Several Different Population Variances are Involved.” Biometrika, 34(1/2), 28. ISSN 00063444.

Examples

## empirical Type 1 error 
niter   = 1000
counter = rep(0,niter)  # record p-values
for (i in 1:niter){
  x = rnorm(57)  # sample x from N(0,1)
  y = rnorm(89)  # sample y from N(0,1)
  
  counter[i] = ifelse(mean2.ttest(x,y)$p.value < 0.05, 1, 0)
}

## print the result
cat(paste("\n* Example for 'mean2.ttest'\n","*\n",
"* number of rejections   : ", sum(counter),"\n",
"* total number of trials : ", niter,"\n",
"* empirical Type 1 error : ",round(sum(counter/niter),5),"\n",sep=""))

Description

Given univariate samples $X_1~,\ldots,~X_k$, it tests

$H_0 : \mu_1 = \cdots \mu_k\quad vs\quad H_1 : \textrm{at least one equality does not hold}$

using the procedure by Schott (2007). It can be considered as a generalization of two-sample testing procedure proposed by Bai and Saranadasa (1996).

Usage

meank.2007Schott(dlist)

Arguments

Value

a (list) object of S3 class htest containing:

statistic

a test statistic.

p.value

$p$-value under $H_0$.

alternative

alternative hypothesis.

method

name of the test.

data.name

name(s) of provided sample data.

References

Schott JR (2007). “Some high-dimensional tests for a one-way MANOVA.” Journal of Multivariate Analysis, 98(9), 1825–1839. ISSN 0047259X.

Examples

## CRAN-purpose small example
tinylist = list()
for (i in 1:3){ # consider 3-sample case
  tinylist[[i]] = matrix(rnorm(10*3),ncol=3)
}
meank.2007Schott(tinylist)


## test when k=5 samples with (n,p) = (10,50)
## empirical Type 1 error 
niter   = 1000
counter = rep(0,niter)  # record p-values
for (i in 1:niter){
  mylist = list()
  for (j in 1:5){
     mylist[[j]] = matrix(rnorm(10*5),ncol=5)
  }
  
  counter[i] = ifelse(meank.2007Schott(mylist)$p.value < 0.05, 1, 0)
}

## print the result
cat(paste("\n* Example for 'meank.2007Schott'\n","*\n",
"* number of rejections   : ", sum(counter),"\n",
"* total number of trials : ", niter,"\n",
"* empirical Type 1 error : ",round(sum(counter/niter),5),"\n",sep=""))

Description

Given univariate samples $X_1~,\ldots,~X_k$, it tests

$H_0 : \mu_1 = \cdots \mu_k\quad vs\quad H_1 : \textrm{at least one equality does not hold}$

using the procedure by Zhang and Xu (2009) by applying multivariate extension of Scheffe's method of transformation.

Usage

meank.2009ZX(dlist, method = c("L", "T"))

Arguments

Value

a (list) object of S3 class htest containing:

statistic

a test statistic.

p.value

$p$-value under $H_0$.

alternative

alternative hypothesis.

method

name of the test.

data.name

name(s) of provided sample data.

References

Zhang J, Xu J (2009). “On the k-sample Behrens-Fisher problem for high-dimensional data.” Science in China Series A: Mathematics, 52(6), 1285–1304. ISSN 1862-2763.

Examples

## CRAN-purpose small example
tinylist = list()
for (i in 1:3){ # consider 3-sample case
  tinylist[[i]] = matrix(rnorm(10*3),ncol=3)
}
meank.2009ZX(tinylist) # run the test


## test when k=5 samples with (n,p) = (100,20)
## empirical Type 1 error 
niter   = 1000
counter = rep(0,niter)  # record p-values
for (i in 1:niter){
  mylist = list()
  for (j in 1:5){
     mylist[[j]] = matrix(rnorm(100*10),ncol=10)
  }
  
  counter[i] = ifelse(meank.2009ZX(mylist, method="L")$p.value < 0.05, 1, 0)
}

## print the result
cat(paste("\n* Example for 'meank.2009ZX'\n","*\n",
"* number of rejections   : ", sum(counter),"\n",
"* total number of trials : ", niter,"\n",
"* empirical Type 1 error : ",round(sum(counter/niter),5),"\n",sep=""))

Description

Given univariate samples $X_1~,\ldots,~X_k$, it tests

$H_0 : \mu_1 = \cdots \mu_k\quad vs\quad H_1 : \textrm{at least one equality does not hold}$

using the procedure by Cao, Park, and He (2019).

Usage

meank.2019CPH(dlist, method = c("original", "Hu"))

Arguments