Inference for four multivariate coefficients of variation

Description

The function GFDmcv() calculates the Wald-type statistic for global null hypotheses and max-type statistics for multiple local null hypotheses, both in terms of the four variants of the multivariate coefficient of variation. Respective p-values are obtained by a \chi^2-approximation, a pooled bootstrap strategy and a pooled permutation approach (only for the Wald-type statistic), respectively.

Usage

GFDmcv(
  x,
  h_mct,
  h_wald,
  alpha = 0.05,
  n_perm = 1000,
  n_boot = 1000,
  parallel = FALSE,
  n_cores = NULL
)

Arguments

Details

The function GFDmcv() calculates the Wald-type statistic for global null hypotheses of the form

\mathcal H_0: \mathbf{H} (C_1,\ldots,C_k)^\top = \mathbf{0}\,\,\text{and}\,\,\mathcal H_0: \mathbf{H} (B_1,\ldots,B_k)^\top = \mathbf{0},

where \mathbf{H} is a contrast matrix reflecting the research question of interest and C_i (B_i) are the subgroup-specific MCVs (and their reciprocal) by Reyment (1960, RR), Van Valen (1974, VV), Voinov and Nikulin (1996, VN) or Albert and Zhang (2010, AZ), respectively. We refer to the function e_mcv() for the detailed definitions of the different variants. The p-value of the Wald-type statistic relies on a \chi^2-approximation, a (pooled) bootstrap or permutation approach.

Furthermore, the function GFDmcv() calculates a max-type test statistic for the multiple comparison of q local null hypotheses:

\mathcal H_{0,\ell}: \mathbf{h_\ell}^\top \mathbf{C} = \mathbf{0}\,\, \text{or}\,\,\mathcal H_{0,\ell}: \mathbf{h_\ell}^\top \mathbf{B} = \mathbf{0}, \,\,\ell=1,\ldots,q,

where \mathbf{C}=(C_1,\ldots,C_k)^\top and \mathbf{B}=(B_1,\ldots,B_k)^\top. The p-values are determined by a Gaussian approximation and a bootstrap approach, respectively. In addition to the local test decisions, multiple adjusted confidence intervals for the contrasts \mathbf{h_{\ell}^{\top}\pmb{C}} and \mathbf{h_{\ell}^{\top}\pmb{B}}, respectively, are calculated.

Please have a look on the plot and summary functions designed for this package. They can be used to simplify the output of GFDmcv().

Value

A list of class gfdmcv containing the following components:

References

Albert A., Zhang L. (2010) A novel definition of the multivariate coefficient of variation. Biometrical Journal 52:667-675.

Ditzhaus M., Smaga L. (2022) Permutation test for the multivariate coefficient of variation in factorial designs. Journal of Multivariate Analysis 187, 104848.

Ditzhaus M., Smaga L. (2023) Inference for all variants of the multivariate coefficient of variation in factorial designs. Preprint https://arxiv.org/abs/2301.12009.

Reyment R.A. (1960) Studies on Nigerian Upper Cretaceous and Lower Tertiary Ostracoda: part 1. Senonian and Maastrichtian Ostracoda, Stockholm Contributions in Geology, vol 7.

Van Valen L. (1974) Multivariate structural statistics in natural history. Journal of Theoretical Biology 45:235-247.

Voinov V., Nikulin M. (1996) Unbiased Estimators and Their Applications, Vol. 2, Multivariate Case. Kluwer, Dordrecht.

Examples

# Some of the examples may run some time.

# one-way analysis for MCV and CV
# d > 1 (MCV)
data_set <- lapply(list(iris[iris$Species == "setosa", 1:3],
                        iris[iris$Species == "versicolor", 1:3],
                        iris[iris$Species == "virginica", 1:3]),
                   as.matrix)
# estimators and confidence intervals of MCVs and their reciprocals
lapply(data_set, e_mcv)
# contrast matrices
k <- length(data_set)
# Tukey's contrast matrix
h_mct <- contr_mat(k, type = "Tukey")
# centering matrix P_k
h_wald <- contr_mat(k, type = "center")

# testing without parallel computing
res <- GFDmcv(data_set, h_mct, h_wald)
summary(res, digits = 3)
oldpar <- par(mar = c(4, 5, 2, 0.3))
plot(res)
par(oldpar)

# testing with parallel computing
library(doParallel)
res <- GFDmcv(data_set, h_mct, h_wald, parallel = TRUE, n_cores = 2)
summary(res, digits = 3)
oldpar <- par(mar = c(4, 5, 2, 0.3))
plot(res)
par(oldpar)

# two-way analysis for CV (based on the example in Ditzhaus and Smaga, 2022)
library(HSAUR)
data_set <- lapply(list(BtheB$bdi.pre[BtheB$drug == "No" & BtheB$length == "<6m"],
                        BtheB$bdi.pre[BtheB$drug == "No" & BtheB$length == ">6m"],
                        BtheB$bdi.pre[BtheB$drug == "Yes" & BtheB$length == "<6m"],
                        BtheB$bdi.pre[BtheB$drug == "Yes" & BtheB$length == ">6m"]), 
                   as.matrix)
# estimators and confidence intervals of CV and its reciprocal
lapply(data_set, e_mcv)

# interaction
h_mct <- contr_mat(4, type = "Tukey")
h_wald <- kronecker(contr_mat(2, type = "center"), 
                    contr_mat(2, type = "center"))
res <- GFDmcv(data_set, h_mct, h_wald)
summary(res, digits = 3)
oldpar <- par(mar = c(4, 6, 2, 0.1))
plot(res)
par(oldpar)

# main effect drug
h_mct <- matrix(c(1, 1, -1, -1), nrow = 1)
h_wald <- kronecker(contr_mat(2, type = "center"), 0.5 * matrix(1, 1, 2))
res <- GFDmcv(data_set, h_mct, h_wald)
summary(res, digits = 3)
oldpar <- par(mar = c(4, 6, 2, 0.1))
plot(res)
par(oldpar)

# main effect length
h_mct <- matrix(c(1, -1, 1, -1), nrow = 1)
h_wald <- kronecker(0.5 * matrix(1, 1, 2), contr_mat(2, type = "center"))
res <- GFDmcv(data_set, h_mct, h_wald)
summary(res, digits = 3)
oldpar <- par(mar = c(4, 6, 2, 0.1))
plot(res)
par(oldpar)

Contrasts' matrices

Description

The output are different contrast matrices, namely the centering matrix as well as the matrices of Dunnett's and Tukey's contrasts for given number of groups.

Usage

contr_mat(k, type = c("center", "Dunnett", "Tukey"))

Arguments

Details

The centering matrix is \mathbf{P}_k = \mathbf{I}_k - \mathbf{J}_k/k, where \mathbf{I}_k is the unity matrix and \mathbf{J}_k=\mathbf{1}\mathbf{1}^{\top} \in R^{k\times k} consisting of 1's only. The matrix of Dunnett's contrasts:

\left(\begin{array}{rrrrrr} -1 & 1 & 0 & \ldots & \ldots & 0\\ -1 & 0 & 1 & 0 & \ldots & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ -1 & 0 & \ldots & \ldots & 0 & 1\\ \end{array}\right)\in R^{(k-1)\times k}.

The matrix of Tukey's contrasts:

\left(\begin{array}{rrrrrrr} -1 & 1 & 0 & \ldots & \ldots & 0 & 0\\ -1 & 0 & 1 & 0 & \ldots & \ldots & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ -1 & 0 & 0 & 0 & \ldots & \ldots & 1 \\ 0 & -1 & 1 & 0 &\ldots & \ldots & 0 \\ \vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\ 0 & \ldots & \ldots & \ldots & 0 & -1 & 1\\ \end{array}\right)\in R^{{k \choose 2}\times k}.

Value

The matrix of contrasts.

References

Ditzhaus M., Smaga L. (2022) Permutation test for the multivariate coefficient of variation in factorial designs. Journal of Multivariate Analysis 187, 104848.

Ditzhaus M., Smaga L. (2023) Inference for all variants of the multivariate coefficient of variation in factorial designs. Preprint https://arxiv.org/abs/2301.12009.

Dunnett C. (1955) A multiple comparison procedure for comparing several treatments with a control. Journal of the American Statistical Association 50, 1096-1121.

Tukey J.W. (1953) The problem of multiple comparisons. Princeton University.

Examples

contr_mat(4, type = "center")
contr_mat(4, type = "Dunnett")
contr_mat(4, type = "Tukey")

Estimators and confidence intervals of four multivariate coefficients of variation and their reciprocals

Description

Calculates the estimators with respective (1-\alpha)-confidence intervals for the four different variants of the multivariate coefficients (MCV) and their reciprocals by Reyment (1960), Van Valen (1974), Voinov and Nikulin (1996) and Albert and Zhang (2010).

Usage

e_mcv(x, conf_level = 0.95)

Arguments

Details

The function e_mcv() calculates four different variants of multivariate coefficient of variation for d-dimensional data. These variant were introduced by by Reyment (1960, RR), Van Valen (1974, VV), Voinov and Nikulin (1996, VN) and Albert and Zhang (2010, AZ):

{\widehat C}^{RR}=\sqrt{\frac{(\det\mathbf{\widehat\Sigma})^{1/d}}{\boldsymbol{\widehat\mu}^{\top}\boldsymbol{\widehat\mu}}},\ {\widehat C}^{VV}=\sqrt{\frac{\mathrm{tr}\mathbf{\widehat\Sigma}}{\boldsymbol{\widehat\mu}^{\top}\boldsymbol{\widehat\mu}}},\ {\widehat C}^{VN}=\sqrt{\frac{1}{\boldsymbol{\widehat\mu}^{\top}\mathbf{\widehat\Sigma}^{-1}\boldsymbol{\widehat\mu}}},\ {\widehat C}^{AZ}=\sqrt{\frac{\boldsymbol{\widehat\mu}^{\top}\mathbf{\widehat\Sigma}\boldsymbol{\widehat\mu}}{(\boldsymbol{\widehat\mu}^{\top}\boldsymbol{\widehat\mu})^2}},

where n is the sample size, \boldsymbol{\widehat\mu} is the empirical mean vector and \mathbf{\widehat \Sigma} is the empirical covariance matrix:

\boldsymbol{\widehat\mu}_i = \frac{1}{n}\sum_{j=1}^{n} \mathbf{X}_{j},\; \mathbf{\widehat \Sigma} =\frac{1}{n}\sum_{j=1}^{n} (\mathbf{X}_{j} - \boldsymbol{\widehat \mu})(\mathbf{X}_{j} - \boldsymbol{\widehat \mu})^{\top}.

In the univariate case (d=1), all four variants reduce to coefficient of variation. Furthermore, their reciprocals, the so-called standardized means, are determined:

{\widehat B}^{RR}=\sqrt{\frac{\boldsymbol{\widehat\mu}^{\top}\boldsymbol{\widehat\mu}}{(\det\mathbf{\widehat\Sigma})^{1/d}}},\ {\widehat B}^{VV}=\sqrt{\frac{\boldsymbol{\widehat\mu}^{\top}\boldsymbol{\widehat\mu}}{\mathrm{tr}\mathbf{\widehat\Sigma}}},\ {\widehat B}^{VN}=\sqrt{\boldsymbol{\widehat\mu}^{\top}\mathbf{\widehat\Sigma}^{-1}\boldsymbol{\widehat\mu}},\ {\widehat B}^{AZ}=\sqrt{\frac{(\boldsymbol{\widehat\mu}^{\top}\boldsymbol{\widehat\mu})^2}{\boldsymbol{\widehat\mu}^{\top}\mathbf{\widehat\Sigma}\boldsymbol{\widehat\mu}}}.

In addition to the estimators, the respective confidence intervals [C_lwr, C_upr] for a given confidence level 1-\alpha are calculated by the e_mcv() function. These confidence intervals are based on an asymptotic approximation by a normal distribution, see Ditzhaus and Smaga (2023) for the technical details. These approximations do not rely on any specific (semi-)parametric assumption on the distribution and are valid nonparametrically, even for tied data.

Value

When d>1 (respectively d=1) a data frame with four rows (one row) corresponding to the four MCVs (the univariate CV) and six columns containing the estimators C_est for the MCV (CV) and the estimators B_est for their reciprocals as well as the upper and lower bounds of the corresponding confidence intervals [C_lwr, C_upr] and [B_lwr, B_upr].

References

Albert A., Zhang L. (2010) A novel definition of the multivariate coefficient of variation. Biometrical Journal 52:667-675.

Ditzhaus M., Smaga L. (2023) Inference for all variants of the multivariate coefficient of variation in factorial designs. Preprint https://arxiv.org/abs/2301.12009.

Reyment R.A. (1960) Studies on Nigerian Upper Cretaceous and Lower Tertiary Ostracoda: part 1. Senonian and Maastrichtian Ostracoda, Stockholm Contributions in Geology, vol 7.

Van Valen L. (1974) Multivariate structural statistics in natural history. Journal of Theoretical Biology 45:235-247.

Voinov V., Nikulin M. (1996) Unbiased Estimators and Their Applications, Vol. 2, Multivariate Case. Kluwer, Dordrecht.

Examples

# d > 1 (MCVs)
data_set <- lapply(list(iris[iris$Species == "setosa", 1:3],
                        iris[iris$Species == "versicolor", 1:3],
                        iris[iris$Species == "virginica", 1:3]),
                   as.matrix)
lapply(data_set, e_mcv)
# d = 1 (CV)
data_set <- lapply(list(iris[iris$Species == "setosa", 1],
                        iris[iris$Species == "versicolor", 1],
                        iris[iris$Species == "virginica", 1]),
                   as.matrix)
lapply(data_set, e_mcv)

Plot simultaneous confidence intervals for contrasts

Description

Simultaneous confidence intervals for contrasts for CV and MCVs and their reciprocals are plotted.

Usage

## S3 method for class 'gfdmcv'
plot(x, ...)

Arguments

Value

No return value, called for side effects.

Examples

# Some of the examples may run some time. 
# For more examples, see the documentation of the GFDmcv() function.
data_set <- lapply(list(iris[iris$Species == "setosa", 1:3],
                        iris[iris$Species == "versicolor", 1:3],
                        iris[iris$Species == "virginica", 1:3]),
                   as.matrix)
# estimators and confidence intervals of MCVs and their reciprocals
lapply(data_set, e_mcv)
# contrast matrices
k <- length(data_set)
# Tukey's contrast matrix
h_mct <- contr_mat(k, type = "Tukey")
# centering matrix P_k
h_wald <- contr_mat(k, type = "center") 

# testing without parallel computing
res <- GFDmcv(data_set, h_mct, h_wald)
oldpar <- par(mar = c(4, 5, 2, 0.3))
plot(res)
par(oldpar)

Print "gfdmcv" object

Description

Prints the summary of the inference methods for CV and MCVs.

Usage

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

Arguments

Details

The function prints out the information about the significance level, constrast matrices, test statistics, p-values, critical values, simultaneous confidence intervals for contrasts performed by the GFDmcv() function.

Value

No return value, called for side effects.

Examples

# Some of the examples may run some time. 
# For more examples, see the documentation of the GFDmcv() function.
data_set <- lapply(list(iris[iris$Species == "setosa", 1:3],
                        iris[iris$Species == "versicolor", 1:3],
                        iris[iris$Species == "virginica", 1:3]),
                   as.matrix)
# estimators and confidence intervals of MCVs and their reciprocals
lapply(data_set, e_mcv)
# contrast matrices
k <- length(data_set)
# Tukey's contrast matrix
h_mct <- contr_mat(k, type = "Tukey")
# centering matrix P_k
h_wald <- contr_mat(k, type = "center") 

# testing without parallel computing
res <- GFDmcv(data_set, h_mct, h_wald)
summary(res, digits = 3)