Package 'MNormTest' reference manual

Title:	Multivariate Normal Hypothesis Testing
Description:	Hypothesis testing of the parameters of multivariate normal distributions, including the testing of a single mean vector, two mean vectors, multiple mean vectors, a single covariance matrix, multiple covariance matrices, a mean and a covariance matrix simultaneously, and the testing of independence of multivariate normal random vectors. Huixuan, Gao (2005, ISBN:9787301078587), "Applied Multivariate Statistical Analysis".
Authors:	Xifeng Zhang [aut, cre]
Maintainer:	Xifeng Zhang <[email protected]>
License:	MIT + file LICENSE
Version:	1.1.1
Built:	2024-11-22 05:53:24 UTC
Source:	https://github.com/astringency/mnormtest

Multiple Covariance Matrix Hypothesis Testing

Description

Test whether the covariance matrices of multiple multivariate normal populations are all equal. Suppose we have k populations, the null hypothesis is "H0: Sigma1 = Sigma2 = ... = Sigmak".

Usage

covTest.multi(X, label, alpha = 0.05, verbose = TRUE)
covTest.multi(X, label, alpha = 0.05, verbose = TRUE)

Arguments

`X`	The data matrix which is a matrix or data frame.
`label`	A vector of group labels.
`alpha`	The significance level. Default is 0.05.
`verbose`	A boolean value. Default is TRUE. If TRUE, the null hypothesis will be displayed. If FALSE, the test will be carried out silently.

Value

An object of class "testResult", which is a list with the following elements:

`Conclusion`	The conclusion of the test.
`Stat`	A data frame containing the statistics, p value and critical value.
`SampMeanT`	The sample mean.
`SampMeanWithin`	The sample mean of each group.
`SdWithin`	The sample deviation of each group.
`SdWithinT`	The sample deviation within group.
`Df`	The degree of freedom.
`sampleSize`	The sample size of each group.
`d`	The Modified factor of the statistic.

Author(s)

Xifeng Zhang

References

Huixuan, Gao. Applied Multivariate Statistical Analysis. Peking University Press, 2005: pp.88-89.

Examples

data(iris)
chart <- iris[, 1:4]
species <- iris[, 5]
# carry out the test
test1 <- covTest.multi(chart, species)
test2 <- covTest.multi(chart, species, verbose = FALSE)
# get the elements
test1$Stat
test1$SampMeanT
test1$sampleSize
data(iris)
chart <- iris[, 1:4]
species <- iris[, 5]
# carry out the test
test1 <- covTest.multi(chart, species)
test2 <- covTest.multi(chart, species, verbose = FALSE)
# get the elements
test1$Stat
test1$SampMeanT
test1$sampleSize

Single Covariance Matrix Hypothesis Testing

Description

Test whether the covariance matrix is equal to a certain value. The null hypothesis is "H0: Sigma = Sigma0" or "H0: Sigma = sigma^2 * Sigma0".

Usage

covTest.single(data, Sigma0, ball = FALSE, alpha = 0.05, verbose = TRUE)
covTest.single(data, Sigma0, ball = FALSE, alpha = 0.05, verbose = TRUE)

Arguments

`data`	The data matrix which is a matrix or data frame.
`Sigma0`	The covariance matrix when the null hypothesis is true.
`ball`	A boolean value. Default is FALSE. If FALSE, test whether the covariance matrix is Sigma0 (known), which means the null hypothesis is "H0: Sigma = Sigma0". If TRUE and the Sigma0 is a unit matrix, the Mauchly's ball test will be performed. If TRUE but Sigma0 (known) is not a unit matrix, the covariance matrix will be tested to see if it is sigma^2Sigma0 (sigma^2 is unknown), which means the null hypothesis is "H0: Sigma = sigma^2 Sigma0".
`alpha`	The significance level. Default is 0.05.
`verbose`	A boolean value. Default is TRUE. If TRUE, the null hypothesis will be displayed. If FALSE, the test will be carried out silently.

Value

An object of class "testResult", which is a list with the following elements: Return when ball is FALSE.

`Conclusion`	The conclusion of the test.
`Stat`	A data frame containing the statistics, p value and critical value.
`SampMean`	The sample mean.
`SampA`	The sample deviation.
`Df`	The degree of freedom.

Return when ball is TRUE

`Conclusion`	The conclusion of the test.
`Stat`	A data frame containing the statistics, p value and critical value.
`SampMean`	The sample mean.
`SampA`	The sample deviation.
`sigma.hat`	The estimation of sigma^2.
`Df`	The degree of freedom.

Author(s)

Xifeng Zhang

References

Huixuan, Gao. Applied Multivariate Statistical Analysis. Peking University Press, 2005: pp.83-88.

Examples

data(iris)
X <- iris[, 1:4]
# carry out the test
test1 <- covTest.single(X, diag(1, 4))
test2 <- covTest.single(X, diag(1, 4), ball = TRUE)
test3 <- covTest.single(X, diag(2, 4), ball = TRUE)
test4 <- covTest.single(X, diag(1, 4), verbose = FALSE)
# get the elements
test1$Stat
test2$Df
test3$sigma.hat
data(iris)
X <- iris[, 1:4]
# carry out the test
test1 <- covTest.single(X, diag(1, 4))
test2 <- covTest.single(X, diag(1, 4), ball = TRUE)
test3 <- covTest.single(X, diag(2, 4), ball = TRUE)
test4 <- covTest.single(X, diag(1, 4), verbose = FALSE)
# get the elements
test1$Stat
test2$Df
test3$sigma.hat

Multivariate Normal Independence Test

Description

Test whether a set of multivariate normal random vectors are independent. The null hypothesis is "H0: The random vectors are independent of each other".

Usage

indTest.multi(data, subdim = FALSE, alpha = 0.05, verbose = TRUE)
indTest.multi(data, subdim = FALSE, alpha = 0.05, verbose = TRUE)

Arguments

`data`	The data matrix which is a matrix or data frame. Each column represents a random variable.
`subdim`	The dimensions of submatrices. The default is FALSE, which means the independence of all components of the random vector will be tested.
`alpha`	The significance level. Default is 0.05.
`verbose`	A boolean value. Default is TRUE. If TRUE, the null hypothesis will be displayed. If FALSE, the test will be carried out silently.

Value

An object of class "testResult", which is a list with the following elements:

`Conclusion`	The conclusion of the test.
`Stat`	A data frame containing the statistics, p value and critical value.
`SampMean`	The sample mean.
`SampA`	The sample deviation.
`SampAii`	The sample deviation of submatrices.
`Df`	The degree of freedom.
`b`	The Modified factor of the statistic.

Author(s)

Xifeng Zhang

References

Huixuan, Gao. Applied Multivariate Statistical Analysis. Peking University Press, 2005: pp.92-94.

Examples

data(iris)
chart <- iris[, 1:4]
# carry out the test
test1 <- indTest.multi(chart)
test2 <- indTest.multi(chart, subdim = c(2, 1, 1))
test3 <- indTest.multi(chart, verbose = FALSE)
# get the elements
test1$Stat
test1$SampMean
test2$SampAii
data(iris)
chart <- iris[, 1:4]
# carry out the test
test1 <- indTest.multi(chart)
test2 <- indTest.multi(chart, subdim = c(2, 1, 1))
test3 <- indTest.multi(chart, verbose = FALSE)
# get the elements
test1$Stat
test1$SampMean
test2$SampAii

Mean and Covariance Matrix Hypothesis Testing (Simultaneously)

Description

Test whether the mean vectors and covariance matrices of multiple multivariate normal populations are all equal simultaneously. Suppose we have k populations, the null hypothesis is "H0: mu1 = mu2 = ... = muk and Sigma1 = Sigma2 = ... = Sigmak".

Usage

meancov.Test(X, label, alpha = 0.05, verbose = TRUE)
meancov.Test(X, label, alpha = 0.05, verbose = TRUE)

Arguments

`X`	The data matrix which is a matrix or data frame.
`label`	A vector of group labels.
`alpha`	The significance level. Default is 0.05.
`verbose`	A boolean value. Default is TRUE. If TRUE, the null hypothesis will be displayed. If FALSE, the test will be carried out silently.

Value

An object of class "testResult", which is a list with the following elements:

`Conclusion`	The conclusion of the test.
`Stat`	A data frame containing the statistics, p value and critical value.
`SampMeanT`	The sample mean.
`SampMeanWithin`	The sample mean of each group.
`SdTotal`	The total sample deviation.
`SdWithin`	The sample deviation of each group.
`SdWithinT`	The sample deviation within group.
`Df`	The degree of freedom.
`sampleSize`	The sample size of each group.
`d`	The Modified factor of the statistic.

Author(s)

Xifeng Zhang

References

Huixuan, Gao. Applied Multivariate Statistical Analysis. Peking University Press, 2005: pp.90-91.

Examples

data(iris)
chart <- iris[, 1:4]
species <- iris[, 5]
# carry out the test
test1 <- meancov.Test(chart, species)
test2 <- meancov.Test(chart, species, verbose = FALSE)
# get the elements
test1$Stat
test1$SampMeanT
data(iris)
chart <- iris[, 1:4]
species <- iris[, 5]
# carry out the test
test1 <- meancov.Test(chart, species)
test2 <- meancov.Test(chart, species, verbose = FALSE)
# get the elements
test1$Stat
test1$SampMeanT

Multiple Mean Vectors Hypothesis Testing

Description

Test whether the mean vectors of multiple multivariate normal populations are all equal when the covariance matrices are equal. Suppose we have k populations, the null hypothesis is that "H0: mu1 = mu2 = ... = muk". There are two approximations (Bartlett's chi2 and Rao's F) to compute the p-value and the critical value. The realized value of the Wilks Lambda statistic and its degrees of freedom are also provided. If you want to perform an exact test, consult the Wilks Lambda statistic quantile table yourself, depending on the realized value of the statistic and its degrees of freedom.

Usage

meanTest.multi(X, label, alpha = 0.05, verbose = TRUE)
meanTest.multi(X, label, alpha = 0.05, verbose = TRUE)

Arguments

`X`	The data matrix which is a matrix or data frame.
`label`	A vector of group labels.
`alpha`	The significance level. Default is 0.05.
`verbose`	A boolean value. Default is TRUE. If TRUE, the null hypothesis will be displayed. If FALSE, the test will be carried out silently.

Value

An object of class "testResult", which is a list with the following elements:

`Conclusion`	The conclusion of the test.
`Stat`	A data frame containing the statistics, p value and critical value.
`SampMeanT`	The sample mean.
`SampMeanWithin`	The sample mean of each group.
`SdTotal`	The total sample deviation.
`SdBetween`	The sample deviation between group.
`SdWithin`	The sample deviation of each group.
`SdWithinT`	The sample deviation within group.
`Df`	The degree of freedom.
`sampleSize`	The sample size of each group.

Author(s)

Xifeng Zhang

References

Huixuan, Gao. Applied Multivariate Statistical Analysis. Peking University Press, 2005: pp.80-83.

Examples

data(iris)
chart <- iris[, 1:4]
species <- iris[, 5]
# carry out the test
test1 <- meanTest.multi(chart, species)
test2 <- meanTest.multi(chart, species, verbose = FALSE)
# get the elements
test1$Stat
test1$SampMeanT
test1$sampleSize
data(iris)
chart <- iris[, 1:4]
species <- iris[, 5]
# carry out the test
test1 <- meanTest.multi(chart, species)
test2 <- meanTest.multi(chart, species, verbose = FALSE)
# get the elements
test1$Stat
test1$SampMeanT
test1$sampleSize

Single Mean Vector Hypothesis Testing

Description

Test whether the mean vector of a single multivariate normal population is equal to a certain value when the population covariance matrix is known or unknown. The null hypothesis is that "H0: mu = mu0".

Usage

meanTest.single(data, mu0, Sigma0 = FALSE, alpha = 0.05, verbose = TRUE)
meanTest.single(data, mu0, Sigma0 = FALSE, alpha = 0.05, verbose = TRUE)

Arguments

`data`	The data matrix which is a matrix or data frame.
`mu0`	The mean vector when the null hypothesis is true.
`Sigma0`	The population covariance matrix. Default is FALSE which means the covariance matrix is unknown.
`alpha`	The significance level. Default is 0.05.
`verbose`	A boolean value. Default is TRUE. If TRUE, the null hypothesis will be displayed. If FALSE, the test will be carried out silently.

Value

An object of class "testResult", which is a list with the following elements:

`Conclusion`	The conclusion of the test.
`Stat`	A data frame containing the statistics, p value and critical value.
`SampMean`	The sample mean.
`SampA`	The sample deviation.
`Df`	The degree of freedom.

Author(s)

Xifeng Zhang

References

Huixuan, Gao. Applied Multivariate Statistical Analysis. Peking University Press, 2005: pp.66-68.

Examples

data(iris)
X <- iris[, 1:4]
mu0 <- c(5.8, 3.0, 4.3, 1.3)
# carry out the test
test1 <- meanTest.single(X, mu0)
test2 <- meanTest.single(X, mu0, Sigma0 = diag(1, 4))
test3 <- meanTest.single(X, mu0, verbose = FALSE)
# get the elements
test1$Stat
test1$SampMean
test1$SampA
test1$Df
data(iris)
X <- iris[, 1:4]
mu0 <- c(5.8, 3.0, 4.3, 1.3)
# carry out the test
test1 <- meanTest.single(X, mu0)
test2 <- meanTest.single(X, mu0, Sigma0 = diag(1, 4))
test3 <- meanTest.single(X, mu0, verbose = FALSE)
# get the elements
test1$Stat
test1$SampMean
test1$SampA
test1$Df

Two Mean Vectors Hypothesis Testing

Description

Test whether the mean vectors of two multivariate normal populations are equal when the covariance matrices are equal or unequal. The null hypothesis is that "H0: mu1 = mu2".

Usage

meanTest.two(
  data1,
  data2,
  alpha = 0.05,
  equal = TRUE,
  method = c("None", "Coupled", "Transformed"),
  verbose = TRUE
)
meanTest.two(
  data1,
  data2,
  alpha = 0.05,
  equal = TRUE,
  method = c("None", "Coupled", "Transformed"),
  verbose = TRUE
)

Arguments

`data1`	A matrix or data frame of group 1.
`data2`	A matrix or data frame of group 2.
`alpha`	The significance level. Default is 0.05.
`equal`	A boolean value. Default is TRUE. If TRUE, the covariance matrix is equal. If FALSE, the covariance matrix is not equal.
`method`	A string value. Default is "None". When equal is FALSE, you must choose a method in "Coupled" or "Transformed". Choose "Coupled" when the sample size of two groups is equal. Choose "Transformed" when the sample size of two groups is not equal.
`verbose`	A boolean value. Default is TRUE. If TRUE, the null hypothesis will be displayed. If FALSE, the test will be carried out silently.

Value

An object of class "testResult", which is a list with the following elements: Return when the param equal is TRUE.

`Conclusion`	The conclusion of the test.
`Stat`	A data frame containing the statistics, p value and critical value.
`SampMean1`	The sample mean of group 1.
`SampMean2`	The sample mean of group 2.
`SampA1`	The sample deviation of group 1.
`SampA2`	The sample deviation of group 2.
`MixSampA`	The mixed sample deviation.
`Df`	The degree of freedom.

Return when the param equal is FALSE and method is "Coupled".

`Conclusion`	The conclusion of the test.
`Stat`	A data frame containing the statistics, p value and critical value.
`SampMeanC`	The sample mean of coupled data.
`SampAC`	The sample deviation of coupled data.
`Df`	The degree of freedom.
`dataC`	The coupled data.

Return when the param equal is FALSE and method is "Transformed".

`Conclusion`	The conclusion of the test.
`Stat`	A data frame containing the statistics, p value and critical value.
`SampMeanT`	The sample mean of transformed data.
`SampAT`	The sample deviation of transformed data.
`Df`	The degree of freedom.
`dataT`	The transformed data. Return when the param equal is FALSE and method is "Transformed".

Author(s)

Xifeng Zhang

References

Huixuan, Gao. Applied Multivariate Statistical Analysis. Peking University Press, 2005: pp.76-80.

Examples

data(iris)
X <- iris[1:50, 1:4]
Y <- iris[51:100, 1:4]
# carry out the test
test1 <- meanTest.two(X, Y)
test2 <- meanTest.two(X, Y, verbose = TRUE)
test3 <- meanTest.two(X, Y, equal = FALSE, method = "Coupled")
test4 <- meanTest.two(X, Y, equal = FALSE, method = "Transformed")
# get the elements
test1$Stat
test1$SampMean1
test3$SampMeanC
test4$dataT
data(iris)
X <- iris[1:50, 1:4]
Y <- iris[51:100, 1:4]
# carry out the test
test1 <- meanTest.two(X, Y)
test2 <- meanTest.two(X, Y, verbose = TRUE)
test3 <- meanTest.two(X, Y, equal = FALSE, method = "Coupled")
test4 <- meanTest.two(X, Y, equal = FALSE, method = "Transformed")
# get the elements
test1$Stat
test1$SampMean1
test3$SampMeanC
test4$dataT

Package 'MNormTest'

Help Index

Multiple Covariance Matrix Hypothesis Testing

Description

Usage

Arguments

Value

Author(s)

References

Examples

Single Covariance Matrix Hypothesis Testing

Description

Usage

Arguments

Value

Author(s)

References

Examples

Multivariate Normal Independence Test

Description

Usage

Arguments

Value

Author(s)

References

Examples

Mean and Covariance Matrix Hypothesis Testing (Simultaneously)

Description

Usage

Arguments

Value

Author(s)

References

Examples

Multiple Mean Vectors Hypothesis Testing

Description

Usage

Arguments

Value

Author(s)

References

Examples

Single Mean Vector Hypothesis Testing

Description

Usage

Arguments

Value

Author(s)

References

Examples

Two Mean Vectors Hypothesis Testing

Description

Usage

Arguments

Value

Author(s)

References

Examples