`boxM`

performs the Box's (1949) M-test for homogeneity of covariance matrices
obtained from multivariate normal data according to one or more classification
factors. The test compares the product of the log determinants of the
separate covariance matrices to the log determinant of the pooled
covariance matrix, analogous to a likelihood ratio test.
The test statistic uses a chi-square approximation.

`boxM(Y, ...)`# S3 method for formula
boxM(Y, data, ...)

# S3 method for lm
boxM(Y, ...)

# S3 method for default
boxM(Y, group, ...)

# S3 method for boxM
summary(object,
digits = getOption("digits"),
cov=FALSE, quiet=FALSE, ...)

Y

The response variable matrix for the default method, or a `"mlm"`

or `"formula"`

object
for a multivariate linear model.
If `Y`

is a linear-model object or a formula, the variables on the right-hand-side of the model must all be factors
and must be completely crossed, e.g., `A:B`

data

a numeric data.frame or matrix containing *n* observations of *p* variables;
it is expected that *n > p*.

group

a factor defining groups, or a vector of length *n* doing the same.

object

a `"boxM"`

object for the `summary`

method

digits

number of digits to print for the `summary`

method

cov

logical; if `TRUE`

the covariance matrices are printed.

quiet

logical; if `TRUE`

printing from the `summary`

is suppressed

...

Arguments passed down to methods.

A list with class `c("htest", "boxM")`

containing the following components:

an approximated value of the chi-square distribution.

the degrees of freedom related of the test statistic in this case that it follows a Chi-square distribution.

the p-value of the test.

a list containing the within covariance matrix for each level of `grouping`

.

the pooled covariance matrix.

a vector containing the natural logarithm of each matrix in `cov`

, followed by the value for the pooled
covariance matrix

a matrix of the means for all groups, followed by the grand means

a vector of the degrees of freedom for all groups, followed by that for the pooled covariance matrix

a character string giving the names of the data.

the character string "Box's M-test for Homogeneity of Covariance Matrices".

As an object of class `"htest"`

, the statistical test is printed normally by default.
As an object of class `"boxM"`

, a few methods are available.

There is no general provision as yet for handling missing data. Missing data are simply removed, with a warning.

As well, the computation assumes that the covariance matrix for each group is non-singular, so that \(log det(S_i)\) can be calculated for each group. At the minimum, this requires that \(n > p\) for each group.

Box's M test for a multivariate linear model highly sensitive to departures from multivariate normality, just as the analogous univariate test. It is also affected adversely by unbalanced designs. Some people recommend to ignore the result unless it is very highly significant, e.g., p < .0001 or worse.

The `summary`

method prints a variety of additional statistics based on the eigenvalues of
the covariance matrices. These are returned invisibly, as a list containing the following
components:

`logDet`

- log determinants`eigs`

- eigenvalues of the covariance matrices`eigstats`

- statistics computed on the eigenvalues for each covariance matrix:`product`

: the product of eigenvalues, \(\prod{\lambda_i}\);`sum`

: the sum of eigenvalues, \(\sum{\lambda_i}\);`precision`

: the average precision of eigenvalues, \(1/\sum(1/\lambda_i)\);`max`

: the maximum eigenvalue, \(\lambda_1\)

Box, G. E. P. (1949). A general distribution theory for a class of likelihood criteria. *Biometrika*, 36, 317-346.

Morrison, D.F. (1976) *Multivariate Statistical Methods*.

`leveneTest`

carries out homogeneity of variance tests for
univariate models with better statistical properties.

`plot.boxM`

, a simple plot of the log determinants

`covEllipses`

plots covariance ellipses in variable space for several groups.

# NOT RUN { data(iris) # default method res <- boxM(iris[, 1:4], iris[, "Species"]) res summary(res) # visualize (what is done in the plot method) dets <- res$logDet ng <- length(res$logDet)-1 dotchart(dets, xlab = "log determinant") points(dets , 1:4, cex=c(rep(1.5, ng), 2.5), pch=c(rep(16, ng), 15), col= c(rep("blue", ng), "red")) # formula method boxM( cbind(Sepal.Length, Sepal.Width, Petal.Length, Petal.Width) ~ Species, data=iris) ### Skulls dat data(Skulls) # lm method skulls.mod <- lm(cbind(mb, bh, bl, nh) ~ epoch, data=Skulls) boxM(skulls.mod) # }