Package {CovCorTest}

Title:	Statistical Tests for Covariance and Correlation Matrices and their Structures
Version:	1.2.0
Maintainer:	Svenja Jedhoff <jedhoff@statistik.tu-dortmund.de>
Description:	A compilation of tests for hypotheses regarding covariance and correlation matrices for one or more groups. The hypothesis can be specified through a corresponding hypothesis matrix and a vector or by choosing one of the basic hypotheses, while for the structure test, only the latter works. Thereby Monte-Carlo and Bootstrap-techniques are used, and the respective method must be chosen, and the functions provide p-values and mostly also estimators of calculated covariance matrices of test statistics. For more details on the methodology, see Sattler et al. (2022) <doi:10.1016/j.jspi.2021.12.001>, Sattler and Pauly (2024) <doi:10.1007/s11749-023-00906-6>, Sattler and Dobler (2026) <doi:10.1016/j.jmva.2025.105517>, and Sattler and Jedhoff (2025) <doi:10.48550/arXiv.2507.03406>.
License:	GPL (≥ 3)
Imports:	MANOVA.RM, matrixcalc, Rdpack
Suggests:	testthat (≥ 3.0.0)
RdMacros:	Rdpack
Config/testthat/edition:	3
Encoding:	UTF-8
URL:	https://github.com/sjedhoff/CovCorTest
BugReports:	https://github.com/sjedhoff/CovCorTest/issues
Config/roxygen2/version:	8.0.0
RoxygenNote:	7.3.3
NeedsCompilation:	no
Packaged:	2026-07-30 07:49:55 UTC; jedhoff
Author:	Paavo Sattler [aut], Svenja Jedhoff [cre, aut], Markus Pauly [ctb], Arne C Bathke [ctb], Dennis Dobler [ctb]
Repository:	CRAN
Date/Publication:	2026-07-30 13:20:02 UTC

CovCorTest: Tests for covariance and correlation matrices

Description

The CovCorTest package provides statistical tests for hypotheses about covariance matrices, correlation matrices, and structured covariance or correlation matrices. It also provides a combined procedure for comparing the covariance and correlation matrices of two groups.

The procedures are based on ANOVA-type statistics and use bootstrap, Monte Carlo, or Taylor-based Monte Carlo approximations, depending on the selected test. The methods require comparatively weak distributional assumptions and support both one-sample and multiple-group designs.

Data format

The main testing functions accept observations in either of two forms:

• A numeric matrix or data frame with observations in rows and variables in columns.

• A list of numeric matrices or data frames, with one element per group. Within every element, observations are stored in rows and variables in columns.

If multiple groups are combined into one matrix, the argument nv specifies the sample size of each group. When a list is supplied, the group sizes can be inferred from the numbers of rows.

All groups must contain the same variables. Observations containing missing values are removed before the test is calculated.

Tests for covariance matrices

test_covariance() tests hypotheses about one or more covariance matrices. A predefined hypothesis can be selected using the hypothesis argument. The available hypotheses are:

"equal"

Tests equality of covariance matrices across groups. For a single group, it tests equality of the diagonal elements.

"equal-trace"

Tests equality of covariance-matrix traces across multiple groups.

"equal-diagonals"

Tests equality of the diagonal elements across multiple groups.

"given-trace"

Tests whether the trace of a single covariance matrix is equal to a specified scalar.

"given-matrix"

Tests whether a single covariance matrix is equal to a specified matrix.

"uncorrelated"

Tests whether all off-diagonal elements of a single covariance matrix are zero.

Custom linear hypotheses can be specified using a hypothesis matrix C and a corresponding null vector Xi. The underlying covariance-testing methodology is described by Sattler et al. (2022).

Tests for correlation matrices

test_correlation() tests hypotheses about one or more correlation matrices. The predefined hypotheses are:

"equal-correlated"

Tests equality of correlation matrices across groups.

"uncorrelated"

Tests whether all correlations in a single group are zero.

As with test_covariance(), custom linear hypotheses can be supplied using C and Xi. The correlation-testing methodology is described by Sattler and Pauly (2024).

Covariance-structure tests

test_covariance_structure() tests whether the covariance matrix of one group has a specified structure. The implemented structures include:

• autoregressive, "autoregressive" or "ar";

• first-order autoregressive, "FO-autoregressive" or "FO-ar";

• diagonal, "diagonal" or "diag";

• sphericity, "sphericity" or "spher";

• compound symmetry, "compoundsymmetry" or "cs";

• Toeplitz, "toeplitz" or "toep";

• constant off-diagonal, "constant-offdiagonal" or "const-offdiag";

• standard Toeplitz, "standard-toeplitz" or "std-toep";

• banded, "banded" or "band";

• banded Toeplitz, "banded-toeplitz" or "band-toep".

For banded structures, bandwidth specifies the number of off-diagonals that may be nonzero.

Correlation-structure tests

test_correlation_structure() tests whether the correlation matrix of one group has a specified structure. The implemented structures include:

• heterogeneous autoregressive, "Hautoregressive" or "Har";

• diagonal, "diagonal" or "diag";

• heterogeneous compound symmetry, "Hcompoundsymmetry" or "Hcs";

• heterogeneous Toeplitz, "Htoeplitz" or "Htoep";

• banded, "banded" or "band";

• banded Toeplitz, "banded-toeplitz" or "band-toep".

The covariance- and correlation-structure procedures are described by Sattler and Dobler (2026).

Combined test

test_combined() simultaneously compares the variances and correlations of exactly two groups. It returns separate p-values for equality of the variances and equality of the correlations, together with a p-value for the global combined hypothesis.

The combined procedure is based on the correlation-testing methodology of Sattler and Pauly (2024).

Resampling methods

Depending on the function, the following methods are available:

"BT"

A parametric bootstrap approximation.

"MC"

A Monte Carlo approximation based on the estimated asymptotic distribution.

"TAY"

A Taylor-based Monte Carlo approximation available for correlation procedures.

The repetitions argument controls the number of generated resampling observations. At least 500 repetitions are recommended. Larger values generally provide more precise p-values but require more computation time.

Alternative hypothesis matrices

By default, the main procedures use AM = 1. This replaces the original hypothesis matrix with a lower-dimensional companion matrix that yields the same ANOVA-type test statistic. The original and transformed hypothesis matrices are retained in the returned object. This approach is described by Sattler and Rosenbaum (2025).

Setting AM = 0 disables this transformation.

Constructing custom hypotheses

get_hypothesis() constructs a hypothesis matrix and vector for a linear covariance model. It can be used together with test_covariance() when the required hypothesis is not among the predefined alternatives.

Returned objects

Covariance, correlation, and structure tests return an object of class CovTest. Important components include:

• pvalue: the estimated p-value;

• Teststatistic: the observed ANOVA-type statistic;

• CovarianceMatrix: the estimated covariance matrix of the statistic;

• C and Xi: the hypothesis used in the calculation;

• C_original and Xi_original: the hypothesis before the optional companion-matrix transformation;

• resampling_method: the selected approximation;

• repetitions: the number of resampling repetitions;

• nv: the sample sizes.

The combined test returns an object of class CombTest containing separate variance, correlation, and global p-values.

Citation

When using CovCorTest, cite the package paper (Sattler and Jedhoff 2025).

Author(s)

Maintainer: Svenja Jedhoff jedhoff@statistik.tu-dortmund.de (ORCID)

Authors:

• Paavo Sattler paavo.sattler@tu-dortmund.de (ORCID)

Other contributors:

• Markus Pauly (ORCID) [contributor]

• Arne C Bathke (ORCID) [contributor]

• Dennis Dobler (ORCID) [contributor]

References

Sattler P, Bathke AC, Pauly M (2022). “Testing Hypotheses about Covariance Matrices in General MANOVA Designs.” Journal of Statistical Planning and Inference, 219, 134–146. doi:10.1016/j.jspi.2021.12.001.

Sattler P, Dobler D (2026). “Testing for Patterns and Structures in Covariance and Correlation Matrices.” Journal of Multivariate Analysis, 211, 105517. doi:10.1016/j.jmva.2025.105517.

Sattler P, Jedhoff S (2025). “Testing Hypotheses Regarding Covariance and Correlation Matrices with the R Package CovCorTest.” arXiv. doi:10.48550/arXiv.2507.03406, arXiv:2507.03406, https://arxiv.org/abs/2507.03406.

Sattler P, Pauly M (2024). “Testing hypotheses about correlation matrices in general MANOVA designs.” TEST, 33(2), 496–516. ISSN 1863-8260, doi:10.1007/s11749-023-00906-6.

Sattler P, Rosenbaum M (2025). “Choice of the hypothesis matrix for using the Anova-type-statistic.” Statistics & Probability Letters, 219, 110356. doi:10.1016/j.spl.2025.110356.

See Also

test_covariance(), test_correlation(), test_covariance_structure(), test_correlation_structure(), test_combined(), and get_hypothesis().

Anova-Type-statistic

Description

Anova-Type-statistic

Usage

ATS(N, vVarData, C, HatCov, Xi = 0)

Arguments

Value

a vector

ATS for transformed vectors

Description

A function which calculates the Anova-type-statistic based on a transformation function

Usage

ATS_fun(N, X, C, v, a, d, p, fun)

Arguments

Value

a scalar, the value of the ATS

Anova-Type-Statistic with weighted sum

Description

Calculation of a Anova-Type-Statistic using

Usage

ATSwS(A, repetitions)

Arguments

Value

a vector of the length of repetitions

Bootstrap for one and multiple groups

Description

This function generates normal distributed random vectors. For one group, nv random vectors with covariance matrix HatCov are generated and the corresponding value of the ATS is generated. For multiple groups the corresponding sample sizes from nv are used. The weighted sum of covariance matrices is calculated and used to calculate the value of the ATS.

Usage

Bootstrap(N.sim, nv, C, MSrootHatCov)

Arguments

Value

a scalar, the value of the ATS

Bootstrap using transformation for one group

Description

This function generates n1 normal distributed random vectors with covariance matrix HatCov, which matrix root MSrootHatCov is given and expectation vector vX. For the generated bootstrap sample the value of the ATS based on a transformation is calculated

Usage

Bootstrap_trans(N.sim, n1, a, d, p, C, MSrootHatCov, vX, fun)

Arguments

Value

a scalar, the value of the ATS

Check bandwidth argument

Description

Checks whether the bandwidth argument is valid for banded covariance or correlation structures. The bandwidth specifies the number of off-diagonals that are allowed to be non-zero. It must be a positive integer and smaller than the maximal possible bandwidth, since bandwidth zero corresponds to a diagonal structure and the maximal bandwidth would impose no restriction.

Usage

CheckBandwidth(bandwidth, d)

Arguments

Value

The checked bandwidth as an integer.

Validate the number of resampling repetitions

Description

Checks that the requested number of repetitions is a single, finite, positive integer. A warning is issued when the number is below the recommended minimum because the resulting p-value may be imprecise.

Usage

CheckRepetitions(repetitions, minimum_recommended = 500L)

Arguments

Value

The validated number of repetitions as an integer.

Combined covariance and correlation test result

Description

Constructs an object of class CombTest. Objects of this class represent results from the combined test for equality of covariance and correlation matrices.

Usage

CombTest(
  pvalue_Variances = NA_real_,
  pvalue_Correlations = NA_real_,
  pvalue_Total = NA_real_,
  Teststatistic = NA_real_,
  repetitions = NA_integer_,
  nv = integer()
)

Arguments

Details

A CombTest object contains the following components:

pvalue_Variances

Numeric. The p-value for equality of the variances.

pvalue_Correlations

Numeric. The p-value for equality of the correlations.

pvalue_Total

Numeric. The p-value for the global combined hypothesis.

Teststatistic

Numeric. The value of the combined test statistic.

repetitions

Integer. The number of resampling repetitions.

nv

Integer vector. The sample sizes of the two groups.

Value

An object of class CombTest.

Examples

result <- CombTest(
  pvalue_Variances = 0.12,
  pvalue_Correlations = 0.08,
  pvalue_Total = 0.08,
  Teststatistic = 2.4,
  repetitions = 1000L,
  nv = c(20L, 25L)
)

CovTest Object

Description

This help page describes the structure of the CovTest class, which is used to represent the results of a covariance and correlation test.

Usage

CovTest()

Details

A CovTest object is a list with the following components:

method

Character. Either 'Covariance' or 'Correlation'.

pvalue

Numeric. The p-value of the test.

Teststatistic

Numeric. The test statistic.

CovarianceMatrix

Matrix. The covariance estimator for the teststatistic.

C

Numeric matrix. The hypothesis matrix used for the computation of the test statistic. If AM = 1, this may be the alternative hypothesis matrix used internally.

Xi

Numeric vector. The hypothesis vector used together with C. If AM = 1, this may be the transformed hypothesis vector used internally.

AM

Integer. Indicates whether the alternative hypothesis matrix was used.

C_original

Numeric matrix. The original hypothesis matrix before the optional transformation by AM, if applicable.

Xi_original

Numeric vector. The original hypothesis vector before the optional transformation by AM, if applicable.

resampling_method

Character. The resampling method used in the test.

repetitions

Integer. The number of repetitions used in resampling.

hypothesis

Character. The hypothesis being tested.

nv

Numeric. The sample size or the number of variables.

Value

An object of class CovTest.

Jacobian matrix for transformation functions

Description

A function which calculates the Jacobian matrix for a given transformation function subdiagonal_mean_ratio_cor or subdiagonal_mean_ratio_fct

Usage

Jacobian(X, a, d, p, fun)

Arguments

Value

the Jacobian matrix applied for the given vector

Function to transform the data into a list, if there are not already

Description

Function to transform the data into a list, if there are not already

Usage

Listcheck(X, nv = NULL)

Arguments

Value

A list with two components

Auxiliary function to calculate the covariance of the vectorized correlation matrix

Description

Auxiliary function to calculate the covariance of the vectorized correlation matrix

Usage

Qvech(X, n)

Arguments

Value

matrix

Function for the Taylor-based Monte-Carlo-approximation for one group

Description

An auxiliary function to calculate the values for the Taylor-based Monte-Carlo-approximation for one group. After receiving some auxiliary matrices and data, the Monte-Carlo observations are generated and different parts of the final sum are defined. Based on this a number of the Taylor-based ATS are calculated, where the number can be chosen.

Usage

Tayapp1G(
  repetitions,
  C,
  MSrootStUpsi,
  CorData,
  MvrH,
  Trace,
  M4,
  L,
  P,
  Q,
  n1,
  Atilde = NULL,
  Jacobi = NULL
)

Arguments

Value

a matrix containing the values of the Taylor ATS for a number of repetitions

Function for the Taylor-based Monte-Carlo-approximation for multiple groups

Description

An auxiliary function to calculate the values for the Taylor-based Monte-Carlo-approximation for multiple groups. After receiving some auxiliary matrices and data, the Monte-Carlo observations are generated and different parts of the final sum are defined. Based on this a number of the Taylor-based ATS are calculated, where the number can be chosen.

Usage

TayappMG(repetitions, C, MSrootStUpsi, CorData, MvrH, Trace, M4, L, P, Q, nv)

Arguments

Value

a matrix containing the values of the Taylor ATS for a number of repetitions

The Taylor-based Monte-Carlo-approximation for a combined test

Description

An auxiliary function to calculate the values for the Taylor-based Monte-Carlo-approximation for the combined test. After receiving some auxiliary matrices and data, the Monte-Carlo observations are generated, and different parts of the final sum are defined. Based on this, a number of the Taylor-based vectors are calculated, where the number can be chosen.

Usage

TaylorCombined(
  repetitions,
  MSrootHatCov,
  CorData,
  MvrH1,
  MvrH2,
  M4,
  L,
  P,
  Q,
  nv
)

Arguments

Value

a matrix containing the values of the test vector for a number of repetitions

Weighted direct sums for lists the corresponding components of a matrix w, containing the weights. These, now weighted matrices are put together to one larger block-diagonal matrix.

Description

Weighted direct sums for lists the corresponding components of a matrix w, containing the weights. These, now weighted matrices are put together to one larger block-diagonal matrix.

Usage

WDirect.sumL(X, w)

Arguments

Value

matrix

Diagonal vectorization

Description

Diagonal vectorization of the upper triangular matrix

Usage

dvech(X, a, d, p, inc_diag)

Arguments

Value

vector

Format a resampling-based p-value

Description

Formats a p-value obtained from a finite number of resampling repetitions. If none of the resampled statistics is at least as extreme as the observed statistic, the estimated p-value is zero and is displayed using the resolution bound 1 / repetitions.

Usage

format_resampling_pvalue(pvalue, repetitions, digits = 4L)

Arguments

Value

A character string containing the formatted p-value.

Function to generate bootstrap observations

Description

Function to generate bootstrap observations

Usage

generateData(WSigma, nv)

Arguments

Value

a matrix

Extend a matrix to full rank using identity columns

Description

This function extends a matrix V (assumed to have fewer than p columns) to full rank by iteratively appending standard basis vectors from the identity matrix, ensuring that each added column increases the rank.

Usage

get_extended_matrix(V, tol = sqrt(.Machine$double.eps))

Arguments

Details

The function assumes that nrow(V) equals the intended dimension p of the parameter space. It adds columns from the identity matrix (i.e., standard basis vectors) only when they increase the rank, and stops when full rank is achieved.

Value

A numeric matrix of dimension ⁠p x p⁠ with full column rank.

Construct hypothesis matrix and vector from linear covariance model structure

Description

Computes a hypothesis matrix C and hypothesis vector zeta based on a given parameter vector v0 and a matrix V representing the model structure (e.g., vectorized components of a linear covariance structure model).

Usage

get_hypothesis(v0, V)

Arguments

Details

The function extends V to full rank using get_extended_matrix, constructs a contrast matrix E for the complement of the model-implied space, and computes the corresponding hypothesis matrix C.

Value

A list with two elements:

hypothesis_matrix

A numeric matrix C such that the hypothesis can be written as C %*% theta = zeta

hypothesis_vector

The numeric vector zeta, computed as C %*% v0

References

Sattler P, Dobler D (2026). “Testing for Patterns and Structures in Covariance and Correlation Matrices.” Journal of Multivariate Analysis, 211, 105517. doi:10.1016/j.jmva.2025.105517.

Examples

# Load the data
data("EEGwide", package = "MANOVA.RM")

X <- as.matrix(EEGwide[EEGwide$sex == "W" & EEGwide$diagnosis == "AD",
                         c("brainrate_temporal", "brainrate_frontal","brainrate_central",
                            "complexity_temporal","complexity_frontal", "complexity_central")])
v0 <- rep(0,21)
v_auxiliary <- c(1, rep(0,5), 1, rep(0,4), 1, rep(0,3), 1, rep(0,2), 1, 0, 1)
V <- cbind(v_auxiliary, 1-v_auxiliary)
h <- get_hypothesis(v0,V)
set.seed(123)
test_covariance(X = X,C = h$hypothesis_matrix, Xi = h$hypothesis_vector,
                method = "MC", repetitions = 1000)

Print function for CombTest object

Description

Print function for CombTest object

Usage

## S3 method for class 'CombTest'
print(x, ...)

Arguments

Value

no return, just print

Print function for CovTest object

Description

Print function for CovTest object

Usage

## S3 method for class 'CovTest'
print(x, ...)

Arguments

Value

no return, just print

Root transformation of the vectorized correlation matrix

Description

A function calculating the roots of a vectorized covariance matrix. The roots increasing, so square root for the first secondary diagonals, third root for the second secondary diagonal and so on. For roots with even order the absolute value of the argument is used, since the arguments can be negative.

Usage

subdiagonal_mean_ratio_cor(v, a, d)

Arguments

Value

a transformed vector

Transformation of the vectorized covariance matrix by quotients of means

Description

A function which calculates the mean of the secondary diagonals and divide them through the next one. Since the elements can be negative, for the denominator absolute values are used.

Usage

subdiagonal_mean_ratio_fct(v, a, d)

Arguments

Value

a transformed vector

Combined test for equality of covariance matrices and correlation matrices

Description

For two groups a combined test for equality of covariance matrices and equality of correlation matrices between these groups is conducted. Both hypotheses can be rejected or only the larger one, the equality of the covariance matrices

Usage

test_combined(X, nv = NULL, repetitions = 1000)

Arguments

Value

an object of the class CovTest.

Examples

# Load the data
data("EEGwide", package = "MANOVA.RM")

vars <- colnames(EEGwide)[1:6]

# Part the data into two diagnosis groups
X_list <- list(EEGwide[EEGwide$sex=="M" & EEGwide$diagnosis=="AD",vars],
               EEGwide[EEGwide$sex=="M" & EEGwide$diagnosis=="MCI",vars])

nv <- unlist(lapply(X_list, nrow))
set.seed(31415)
test_combined(X_list, nv)

Test for Correlation Matrices

Description

This function conducts statistical tests for hypotheses regarding correlation matrices. Users can either select from predefined hypotheses or provide their own contrast matrix C and vector Xi for custom hypotheses. It supports both bootstrap and Monte Carlo resampling methods to obtain the p-value of the ANOVA-type statistic (ATS).

Usage

test_correlation(
  X,
  nv = NULL,
  C = NULL,
  Xi = NULL,
  AM = 1,
  hypothesis = NULL,
  method = "BT",
  repetitions = 1000
)

Arguments

Value

An object of class "CovTest".

References

Sattler, P. and Pauly, M. (2024). Testing hypotheses about correlation matrices in general MANOVA designs. TEST, 33(2), 496–516. doi:10.1007/s11749-023-00906-6

Examples

# Example with one group:
set.seed(31415)
X <- matrix(rnorm(5 * 100), ncol = 5)
test_correlation(X, hypothesis = "uncorrelated",
                  method = "BT", repetitions = 100)

Test for structure of data's correlation matrix

Description

With this function the correlation matrix of data can be checked for one of the predefined structures. Depending on the chosen method a bootstrap, the Taylor-based Monte-Carlo approach or Monte-Carlo-technique is used to calculate the p-value of the Anova-type-statistic(ATS) based on a specified number of runs.

Usage

test_correlation_structure(
  X,
  structure,
  AM = 1,
  method = "BT",
  repetitions = 1000,
  bandwidth = NA
)

Arguments

Value

an object of the class CovTest

References

Sattler P, Dobler D (2026). “Testing for Patterns and Structures in Covariance and Correlation Matrices.” Journal of Multivariate Analysis, 211, 105517. doi:10.1016/j.jmva.2025.105517.

Examples

# Load the data
data("EEGwide", package = "MANOVA.RM")

# Select only the females with the diagnosis AD
X <- as.matrix(EEGwide[EEGwide$sex == "W" & EEGwide$diagnosis == "AD",
             c("brainrate_temporal", "brainrate_frontal","brainrate_central",
             "complexity_temporal","complexity_frontal",
             "complexity_central")])

test_correlation_structure(X = X, structure = "diagonal", method = "MC")

Test for Covariance Matrices

Description

This function conducts statistical tests for hypotheses regarding covariance matrices. Users can either select from predefined hypotheses (e.g., equal covariance, equal trace, etc.) or provide their own contrast matrix C and vector Xi for custom hypotheses. It supports both bootstrap and Monte Carlo resampling methods to obtain the p-value of the ANOVA-type statistic (ATS).

Usage

test_covariance(
  X,
  nv = NULL,
  C = NULL,
  Xi = NULL,
  AM = 1,
  hypothesis = NULL,
  A = NULL,
  method = "MC",
  repetitions = 1000
)

Arguments

Value

An object of class CovTest.

References

Sattler P, Bathke AC, Pauly M (2022). “Testing Hypotheses about Covariance Matrices in General MANOVA Designs.” Journal of Statistical Planning and Inference, 219, 134–146. doi:10.1016/j.jspi.2021.12.001.

Examples

# Load the data
data("EEGwide", package = "MANOVA.RM")

vars <- colnames(EEGwide)[1:6]

X <- EEGwide[EEGwide$sex == "M" & EEGwide$diagnosis == "AD",vars]

# Testing the trace
C <- matrix(c(1,0,0,0,0,0,1,0,0,0,0,1,0,0,0,1,0,0,1,0,1),
            nrow = 1, ncol = 21)
Xi <- 2
set.seed(31415)
test_covariance(X = X, nv = NULL, C = C, Xi = Xi, AM = 1,
                method = "BT", repetitions = 1000)

Test for structure of data's covariance matrix

Description

This function conducts the test for the covariance matrix of data regarding structures. Depending on the chosen method a bootstrap or Monte-Carlo-technique is used to calculate p-value of the Anova-type-statistic(ATS) based on a specified number of runs.

Usage

test_covariance_structure(
  X,
  structure,
  AM = 1,
  method = "BT",
  repetitions = 1000,
  bandwidth = NA
)

Arguments

Value

an object of the class CovTest

References

Sattler P, Dobler D (2026). “Testing for Patterns and Structures in Covariance and Correlation Matrices.” Journal of Multivariate Analysis, 211, 105517. doi:10.1016/j.jmva.2025.105517.

Examples

# Load the data
data("EEGwide", package = "MANOVA.RM")

# Select only the females with the diagnosis AD
X <- as.matrix(EEGwide[EEGwide$sex == "W" & EEGwide$diagnosis == "AD",
                          c("brainrate_temporal", "brainrate_frontal",
                          "brainrate_central","complexity_temporal",
                          "complexity_frontal", "complexity_central")])

test_covariance_structure(X = X, structure = "diagonal", method = "MC")

Function to calculate dvech(X t(X))

Description

Function to calculate dvech(X t(X))

Usage

vdtcrossprod(X, a, d, p)

Arguments

Value

vector

Vectorization of the upper triangular part of the matrix

Description

Vectorization of the upper triangular part of the matrix

Usage

vechp(X)

Arguments

Value

A vector containing the upper triangular part of X.

Function to calculate vech(X t(X))

Description

Function to calculate vech(X t(X))

Usage

vtcrossprod(X)

Arguments

Value

vector