Help for package MVTests

Type:

Package

Date:

2025-09-08

Title:

Multivariate Hypothesis Tests

Version:

2.3.1

Author:

Hasan Bulut [aut, cre]

Maintainer:

Hasan Bulut <hasan.bulut@omu.edu.tr>

Description:

Multivariate hypothesis tests and confidence intervals...

License:

GPL-2

Depends:

R (≥ 4.0)

Imports:

stats, mvtnorm

Suggests:

robustbase, rrcov

Encoding:

UTF-8

LazyData:

true

RoxygenNote:

7.3.2

NeedsCompilation:

Packaged:

2025-09-08 18:44:30 UTC; hsn

Repository:

CRAN

Date/Publication:

2025-09-08 21:20:17 UTC

Bartlett's Test for One Sample Covariance Matrix

Description

Bcov function tests whether the covariance matrix is equal to a given matrix or not.

Usage

Bcov(data, Sigma)

Arguments

data

a data frame.

Sigma

The covariance matrix in NULL hypothesis.

Details

This function computes Bartlett's test statistic for the covariance matrix of one sample.

Value

a list with 3 elements:

ChiSquare

The value of Test Statistic

df

The Chi-Square statistic's degree of freedom

p.value

p value

Author(s)

Hasan BULUT <hasan.bulut@omu.edu.tr>

References

Rencher, A. C. (2003). Methods of multivariate analysis (Vol. 492). John Wiley & Sons.

Examples


data(iris) 
S<-matrix(c(5.71,-0.8,-0.6,-0.5,-0.8,4.09,-0.74,-0.54,-0.6,
     -0.74,7.38,-0.18,-0.5,-0.54,-0.18,8.33),ncol=4,nrow=4)
result <- Bcov(data=iris[,1:4],Sigma=S)
summary(result)

Box's M Test

Description

BoxM function tests whether the covariance matrices of independent samples are equal or not.

Usage

BoxM(data, group)

Arguments

data

a data frame.

group

grouping vector.

Details

This function computes Box-M test statistic for the covariance matrices of independent samples. The hypotheses are defined as H0:The Covariance matrices are homogeneous and H1:The Covariance matrices are not homogeneous

Value

a list with 3 elements:

ChiSquare

The value of Test Statistic

df

The Chi-Square statistic's degree of freedom

p.value

p value

Author(s)

Hasan BULUT <hasan.bulut@omu.edu.tr>

References

Rencher, A. C. (2003). Methods of multivariate analysis (Vol. 492). John Wiley & Sons.

Examples


data(iris) 
results <- BoxM(data=iris[,1:4],group=iris[,5])
summary(results)

Bartlett's Sphericity Test

Description

Bsper function tests whether a correlation matrix is equal to the identity matrix or not.

Usage

Bsper(data)

Arguments

data

a data frame.

Details

This function computes Bartlett's test statistic for Sphericity Test. The hypotheses are H0:R is equal to I and H1:R is not equal to I.

Value

a list with 4 elements:

ChiSquare

The value of Test Statistic

df

The Chi-Square statistic's degree of freedom

p.value

p value

R

Correlation matrix

Author(s)

Hasan BULUT <hasan.bulut@omu.edu.tr>

References

Tatlidil, H. (1996). Uygulamali Cok Degiskenli Istatistiksel Yontemler. Cem Web.

Examples

data(iris) 
results <- Bsper(data=iris[,1:4])
summary(results)

Coated

Description

The data set is given in Table 5.3 in Rencher (2003). The data set consists of 2 variables (Depth and Number), 2 treatments and 15 observations. The first column of the data is Location numbers.

Usage

Coated

Format

A data frame with 15 rows and 5 columns. The columns are as follows:

Location: The location numbers of observations.
Coating1.Depth1: The Depth values in the first treatment
Coating1.Number1: The Number values in the first treatment
Coating2.Depth2: The Depth values in the second treatment
Coating2.Number2: The Number values in the second treatment

Source

The data set is used in the book entitled Methods of Multivariate Analysis (Rencher,2003).

References

Rencher, A. C. (2003). Methods of multivariate analysis (Vol. 492). John Wiley & Sons.

Pair-Wise comparison between hth and gth sample

Description

Pair-Wise comparison of covariance matrices between hth and gth sample

Usage

Mhg(Sh, Sg, S, nh, ng, n)

Arguments

Sh

the robust covariance matrix of the hth sample

Sg

the robust covariance matrix of the gth sample

S

the robust pooled covariance matrix.

nh

the sample size of the hth sample

ng

the sample size of the gth sample

n

the sample size of the full data

Details

Mhg function computes proposed Mgh values as defined in the paper.

Value

a list with 1 elements:

Mhg

Mgh value

Author(s)

Hasan BULUT <hasan.bulut@omu.edu.tr>

References

Bulut, H (2024). A robust permutational test to compare covariance matrices in high dimensional data. (Unpublished)

Examples


if (requireNamespace("rrcov", quietly=TRUE)) {
x1<-mvtnorm::rmvnorm(n = 10,mean = rep(0,20),sigma = diag(20))
x2<-mvtnorm::rmvnorm(n = 10,mean = rep(0,20),sigma = 2*diag(20))
x3<-mvtnorm::rmvnorm(n = 10,mean = rep(0,20),sigma = 3*diag(20))
data<-rbind(x1,x2,x3)
group_label<-c(rep(1,10),rep(2,10),rep(3,10))
n <- nrow(data)
p <- ncol(data)
nk <- table(group_label)
g <- length(nk)
Levels <- unique(group_label)
Si.matrices<-lapply(1:g, function(i) rrcov::CovMrcd(data[(group_label==Levels[i]),],
alpha=0.9)@cov)
Spool <- Reduce("+", Map("*", nk, Si.matrices)) / n
#for the first and second groups
Mhg(Sh = Si.matrices[[1]], Sg = Si.matrices[[2]],S = Spool, nh = nk[1], ng = nk[2], n = n)}

Multivariate Paired Test

Description

Mpaired function computes the value of test statistic based on Hotelling T Square approach in multivariate paired data sets.

Usage

Mpaired(T1, T2)

Arguments

T1

The first treatment data.

T2

The second treatment data.

Details

This function computes one sample Hotelling T^2 statistics for paired data sets.

Value

a list with 7 elements:

HT2

The value of Hotelling T^2 Test Statistic

F

The value of F Statistic

df

The F statistic's degree of freedom

p.value

p value

Descriptive1

The descriptive statistics of the first treatment

Descriptive2

The descriptive statistics of the second treatment

Descriptive.Difference

The descriptive statistics of the differences

Author(s)

Hasan BULUT <hasan.bulut@omu.edu.tr>

References

Rencher, A. C. (2003). Methods of multivariate analysis (Vol. 492). John Wiley & Sons.

Examples


data(Coated)
X<-Coated[,2:3]; Y<-Coated[,4:5]
result <- Mpaired(T1=X,T2=Y)
summary(result)

One Sample Hotelling T^2 Test

Description

OneSampleHT2 computes one sample Hotelling T^2 statistics and gives confidence intervals

Usage

OneSampleHT2(data, mu0, alpha = 0.05)

Arguments

data

a data frame.

mu0

mean vector that is used to test whether population mean parameter is equal to it.

alpha

Significance Level that will be used for confidence intervals. default alpha=0.05.

Details

This function computes one sample Hotelling T^2 statistics that is used to test whether population mean vector is equal to a vector given by a user. When H0 is rejected, this function computes confidence intervals for all variables.

Value

a list with 7 elements:

HT2

The value of Hotelling T^2 Test Statistic

F

The value of F Statistic

df

The F statistic's degree of freedom

p.value

p value

CI

The lower and upper limits of confidence intervals obtained for all variables

alpha

The alpha value using in confidence intervals

Descriptive

Descriptive Statistics

Author(s)

Hasan BULUT <hasan.bulut@omu.edu.tr>

References

Rencher, A. C. (2003). Methods of multivariate analysis (Vol. 492). John Wiley & Sons.

Tatlidil, H. (1996). Uygulamali Cok Degiskenli Istatistiksel Yontemler. Cem Web.

Examples

data(iris)

mean0<-c(6,3,1,0.25)
result <- OneSampleHT2(data=iris[1:50,-5],mu0=mean0,alpha=0.05)
summary(result)

Robust Hotelling T^2 Test for One Sample in High Dimensional Data

Description

Robust Hotelling T^2 Test for One Sample in high Dimensional Data

Usage

RHT2(data, mu0, alpha = 0.75, d, q)

Arguments

data

the data. It must be matrix or data.frame.

mu0

the mean vector which will be used to test the null hypothesis.

alpha

numeric parameter controlling the size of the subsets over which the determinant is minimized. Allowed values are between 0.5 and 1 and the default is 0.75.

d

the constant in Equation (11) in the study by Bulut (2021).

q

the second degree of freedom value of the approximate F distribution in Equation (11) in the study by Bulut (2021).

Details

RHT2 function performs a robust Hotelling T^2 test in high dimensional test based on the minimum regularized covariance determinant estimators. This function needs the q and d values. These values can be obtained simRHT2 function. For more detailed information, you can see the study by Bulut (2021).

Value

a list with 3 elements:

T2

The Robust Hotelling T^2 value in high dimensional data

Fval

The F value based on T2

pval

The p value based on the approximate F distribution

Author(s)

Hasan BULUT <hasan.bulut@omu.edu.tr>

References

Bulut, H (2021). A robust Hotelling test statistic for one sample case in high dimensional data, Communication in Statistics: Theory and Methods.

Examples


if (requireNamespace("rrcov", quietly = TRUE)) {
utils::data("octane", package = "rrcov")
mu.clean <- colMeans(octane[-c(25,26,36,37,38,39), ])
RHT2(data = octane, mu0 = mu.clean, alpha = 0.84, d = 1396.59, q = 1132.99)}

Robust CAT Algorithm

Description

RobCat computes p value based on robust CAT algorithm to compare two means vectors under multivariate Behrens-Fisher problem.

Usage

RobCat(X, Y, M = 1000, alpha = 0.75)

Arguments

X

a matrix or data frame for first group.

Y

a matrix or data frame for second group.

M

iteration number and the default is 1000.

alpha

numeric parameter controlling the size of the subsets over which the determinant is minimized; roughly alpha*n, observations are used for computing the determinant. Allowed values are between 0.5 and 1 and the default is 0.75.

Details

This function computes p value based on robust CAT algorithm to compare two means vectors under multivariate Behrens-Fisher problem. When p value<0.05, it means the difference of two mean vectors is significant statistically.

Value

a list with 2 elements:

Cstat

Calculated value of test statistic

pval

The p value

Author(s)

Hasan BULUT <hasan.bulut@omu.edu.tr>

Examples


data(iris)
if (requireNamespace("robustbase", quietly=TRUE)) {
RobCat(X=iris[1:20,-5],Y=iris[81:100,-5])}

Robust Permutation Test for Covariance Matrices

Description

Robust Permutation Test for Covariance Matrices in High Dimensional Data

Usage

RobPer_CovTest(x, group, N = 100, alpha = 0.75)

Arguments

x

the data matrix

group

the grouping vector. It must be factor.

N

the permutation number and the default value is 100.

alpha

numeric parameter controlling the size of the subsets over which the determinant is minimized. Allowed values are between 0.5 and 1 and the default is 0.75.

Details

RobPer_CovTest function calculates directly p-value based on the calculated value of test statistics and the permutational distribution of test statistics for covariance matrices of two or more independent samples in high dimensional data based on the minimum regularized covariance determinant estimators.

Value

a list with 3 elements:

pval

p-value of the robust permutation test process

TM

The calculated value of test statistics based on raw data

Permutations_TM

The calculated values of test statistics based on each permutational data

Author(s)

Hasan BULUT <hasan.bulut@omu.edu.tr>

References

Bulut, H (2024). A robust permutational test to compare covariance matrices in high dimensional data. (Unpublished)

Examples


if (requireNamespace("rrcov", quietly=TRUE)) {
x1<-mvtnorm::rmvnorm(n = 10,mean = rep(0,20),sigma = diag(20))
x2<-mvtnorm::rmvnorm(n = 10,mean = rep(0,20),sigma = 2*diag(20))
x3<-mvtnorm::rmvnorm(n = 10,mean = rep(0,20),sigma = 3*diag(20))
data<-rbind(x1,x2,x3)
group_label<-c(rep(1,10),rep(2,10),rep(3,10))
RobPer_CovTest(x=data, group=group_label)}

Robust Test for Covariance Matrices

Description

Robust Test for Covariance Matrices in High Dimensional Data

Usage

Rob_CovTest(x, group, alpha = 0.75)

Arguments

x

the data matrix

group

the grouping vector. It must be factor.

alpha

numeric parameter controlling the size of the subsets over which the determinant is minimized. Allowed values are between 0.5 and 1 and the default is 0.75.

Details

Rob_CovTest function computes the calculated value of the test statistic for covariance matrices of two or more independent samples in high dimensional data based on the minimum regularized covariance determinant estimators.

Value

a list with 1 elements:

TM

The calculated value of test statistics based on raw data

Author(s)

Hasan BULUT <hasan.bulut@omu.edu.tr>

References

Bulut, H (2024). A robust permutational test to compare covariance matrices in high dimensional data. (Unpublished)

Examples


if (requireNamespace("rrcov", quietly=TRUE)) {
x1<-mvtnorm::rmvnorm(n = 8,mean = rep(0,10),sigma = diag(10))
x2<-mvtnorm::rmvnorm(n = 8,mean = rep(0,10),sigma = 2*diag(10))
x3<-mvtnorm::rmvnorm(n = 8,mean = rep(0,10),sigma = 3*diag(10))
data<-rbind(x1,x2,x3)
group_label<-c(rep(1,8),rep(2,8),rep(3,8))
Rob_CovTest(x=data, group=group_label)}

Robust Permutation Hotelling T^2 Test in High Dimensional Data

Description

Robust Permutation Hotelling T^2 Test for Two Independent Samples in high Dimensional Data

Usage

RperT2(X1, X2, alpha = 0.75, N = 100)

Arguments

X1

the data matrix for the first group. It must be matrix or data.frame.

X2

the data matrix for the first group. It must be matrix or data.frame.

alpha

numeric parameter controlling the size of the subsets over which the determinant is minimized. Allowed values are between 0.5 and 1 and the default is 0.75.

N

the permutation number

Details

RperT2 function performs a robust permutation Hotelling T^2 test for two independent samples in high dimensional test based on the minimum regularized covariance determinant estimators.

Value

a list with 2 elements:

T2

The calculated value of Robust Hotelling T^2 statistic based on MRCD estimations

p.value

p value obtained from test process

Author(s)

Hasan BULUT <hasan.bulut@omu.edu.tr>

References

Bulut et al. (2024). A Robust High-Dimensional Test for Two-Sample Comparisons, Axioms.

Examples


if (requireNamespace("rrcov", quietly=TRUE)) {
x<-mvtnorm::rmvnorm(n=10,sigma=diag(20),mean=rep(0,20))
y<-mvtnorm::rmvnorm(n=10,sigma=diag(20),mean=rep(1,20))
RperT2(X1=x,X2=y)$p.value}

Robust Hotelling T^2 Test Statistic

Description

Robust Hotelling T^2 Test Statistic for Two Independent Samples in high Dimensional Data

Usage

TR2(x1, x2, alpha = 0.75)

Arguments

x1

the data matrix for the first group. It must be matrix or data.frame.

x2

the data matrix for the first group. It must be matrix or data.frame.

alpha

numeric parameter controlling the size of the subsets over which the determinant is minimized. Allowed values are between 0.5 and 1 and the default is 0.75.

Details

TR2 function calculates the robust Hotelling T^2 test statistic for two independent samples in high dimensional data based on the minimum regularized covariance determinant estimators.

Value

a list with 2 elements:

TR2

The calculated value of Robust Hotelling T^2 statistic based on MRCD estimations

Author(s)

Hasan BULUT <hasan.bulut@omu.edu.tr>

References

Bulut et al. (2024). A Robust High-Dimensional Test for Two-Sample Comparisons, Axioms

Examples


if (requireNamespace("rrcov", quietly=TRUE)) {
x<-mvtnorm::rmvnorm(n=10,sigma=diag(20),mean=rep(0,20))
y<-mvtnorm::rmvnorm(n=10,sigma=diag(20),mean=rep(1,20))
TR2(x1=x,x2=y)}

Two Independent Samples Hotelling T^2 Test

Description

TwoSamplesHT2 function computes Hotelling T^2 statistic for two independent samples and gives confidence intervals.

Usage

TwoSamplesHT2(data, group, alpha = 0.05, Homogenity = TRUE)

Arguments

data

a data frame.

group

a group vector consisting of 1 and 2 values.

alpha

Significance Level that will be used for confidence intervals. default=0.05

Homogenity

a logical argument. If sample covariance matrices are homogeneity,then Homogenity=TRUE. Otherwise Homogenity=FALSE The homogeneity of covariance matrices can be investigated with BoxM function.

Details

This function computes two independent samples Hotelling T^2 statistics that is used to test whether two population mean vectors are equal to each other. When H0 is rejected, this function computes confidence intervals for all variables to determine variable(s) affecting on rejection decision. Moreover, when covariance matrices are not homogeneity, the approach proposed by D. G. Nel and V. D. Merwe (1986) is used.

Value

a list with 8 elements:

HT2

The value of Hotelling T^2 Test Statistic

F

The value of F Statistic

df

The F statistic's degree of freedom

p.value

p value

CI

The lower and upper limits of confidence intervals obtained for all variables

alpha

The alpha value using in confidence intervals

Descriptive1

Descriptive Statistics for the first group

Descriptive2

Descriptive Statistics for the second group

Author(s)

Hasan BULUT <hasan.bulut@omu.edu.tr>

References

Rencher, A. C. (2003). Methods of multivariate analysis (Vol. 492). John Wiley & Sons.

Tatlidil, H. (1996). Uygulamali Cok Degiskenli Istatistiksel Yontemler. Cem Web.

D.G. Nel & C.A. Van Der Merwe (1986) A solution to the multivariate behrens fisher problem, Communications in Statistics:Theory and Methods, 15:12, 3719-3735

Examples


data(iris)
G<-c(rep(1,50),rep(2,50))
# When covariances matrices are homogeneity
results1 <- TwoSamplesHT2(data=iris[1:100,1:4],group=G,alpha=0.05)
summary(results1)
# When covariances matrices are not homogeneity
results2 <- TwoSamplesHT2(data=iris[1:100,1:4],group=G,Homogenity=FALSE)
summary(results2)

Concordance Correlation Coefficient

Description

Classical Concordance Correlation Coefficient

Usage

ccc(x, y)

Arguments

x

the vector which contains the first variable values

y

the vector which contains the second variable values

Details

ccc function calculates directly classical concordance correlation coefficient.

Value

a list with 1 elements:

coef

The value of concordance correlation coeffient

Author(s)

Hasan BULUT <hasan.bulut@omu.edu.tr>

References

Bulut, H (2025). A Robust Concordance Correlation Coefficient. (Unpublished)

Lin, L. I. "A Concordance Correlation-Coefficient to Evaluate Reproducibility." Biometrics 45, no. 1 (1989): 255-68.

Examples


x<-rnorm(50)
y<-2+3*x+rnorm(50,mean = 3)
ccc(x,y)

Iris Data

Description

The Iris dataset is consists of 4 variables, 3 groups and 150 observations. The last column of the data is Iris species.

Usage

iris

Format

A data frame with 150 rows and 5 columns. The columns are as follows:

Sepal.Length: The Sepal length values of iris flowers
Sepal.Width: The Sepal width values of iris flowers
Petal.Length: The Petal length values of iris flowers
Petal.Width: The Petal width values of iris flowers
Species: The species of iris flowers

Source

https://archive.ics.uci.edu/ml/datasets/Iris

Robust Concordance Correlation Coefficient (rCCC)

Description

Computes a robust concordance correlation coefficient using Minimum Covariance Determinant (MCD) estimates.

Usage

rccc(x, y, alpha = 0.75)

Arguments

x

Numeric vector; first variable.

y

Numeric vector; second variable.

alpha

Numeric in (0.5, 1]; MCD subset size proportion. Default 0.75.

Details

The rCCC replaces means and (co)variances in Lin's CCC with their MCD counterparts: \rho_c = \frac{2\sigma_{xy}}{\sigma_x^2+\sigma_y^2+(\mu_x-\mu_y)^2}.

Value

A list with one element:

coef

Robust concordance correlation coefficient

Author(s)

Hasan BULUT <hasan.bulut@omu.edu.tr>

References

Bulut, H. (2025). A Robust Concordance Correlation Coefficient. (Unpublished)

Examples

if (requireNamespace("robustbase", quietly = TRUE)) {
  set.seed(1)
  x <- rnorm(50)
  y <- 2 + 3*x + rnorm(50, mean = 3)
  rccc(x, y)
}

Monte Carlo Simulation to obtain d and q constants for RHT2 function

Description

Monte Carlo Simulation to obtain d and q constants for RHT2 function

Usage

simRHT2(n, p, nrep = 500, alpha = 0.75)

Arguments

n

the sample size

p

the number of variables

nrep

the number of iteration. The default value is 500.

alpha

numeric parameter controlling the size of the subsets over which the determinant is minimized. Allowed values are between 0.5 and 1 and the default is 0.75.

Details

simRHT2 function computes d and q constants to construct an approximate F distribution of robust Hotelling T^2 statistic in high dimensional data. These constants are used in RHT2 function. For more detailed information, you can see the study by Bulut (2021).

Value

a list with 2 elements:

q

The q value

d

The d value

Author(s)

Hasan BULUT <hasan.bulut@omu.edu.tr>

References

Bulut, H (2021). A robust Hotelling test statistic for one sample case in highdimensional data, Communication in Statistics: Theory and Methods.

Summarizing Results in MVTests Package

Description

summary.MVTests function summarizes of results of functions in this package.

Usage

## S3 method for class 'MVTests'
summary(object, ...)

Arguments

object

an object of class MVTests.

...

additional parameters.

Details

This function prints a summary of the results of multivariate hypothesis tests in the MVTests package.

Value

the input object is returned silently.

Author(s)

Hasan BULUT <hasan.bulut@omu.edu.tr>

Examples


# One Sample Hotelling T Square Test
data(iris)
X<-iris[1:50,1:4]
mean0<-c(6,3,1,0.25)
result.onesample <- OneSampleHT2(data=X,mu0=mean0,alpha=0.05)
summary(result.onesample)

#Two Independent Sample Hotelling T Square Test
data(iris)
G<-c(rep(1,50),rep(2,50))
result.twosamples <- TwoSamplesHT2(data=iris[1:100,1:4],group=G,alpha=0.05)
summary(result.twosamples)

#Box's M Test
data(iris)
result.BoxM <- BoxM(data=iris[,1:4],group=iris[,5])
summary(result.BoxM)

#Barlett's Test of Sphericity
data(iris)
result.Bsper <- Bsper(data=iris[,1:4])
summary(result.Bsper)

#Bartlett's Test for One Sample Covariance Matrix
data(iris) 
S<-matrix(c(5.71,-0.8,-0.6,-0.5,-0.8,4.09,-0.74,-0.54,-0.6,-0.74,
          7.38,-0.18,-0.5,-0.54,-0.18,8.33),ncol=4,nrow=4)
result.bcov<- Bcov(data=iris[,1:4],Sigma=S)
summary(result.bcov)