Box's M-test

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.

Usage

boxM(Y, ...)

# Default S3 method
boxM(Y, group, ...)

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

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

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

Arguments

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
...: Arguments passed down to methods.
group: a factor defining groups, or a vector of length n doing the same.
data: a numeric data.frame or matrix containing n observations of p variables; it is expected that n > p.
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

Value

A list with class c("htest", "boxM") containing the following components:

statistic: an approximated value of the chi-square distribution.
parameter: the degrees of freedom related of the test statistic in this case that it follows a Chi-square distribution.
p.value: the p-value of the test.
cov: a list containing the within covariance matrix for each level of grouping.
pooled: the pooled covariance matrix.
logDet: a vector containing the natural logarithm of each matrix in cov, followed by the value for the pooled covariance matrix
means: a matrix of the means for all groups, followed by the grand means
df: a vector of the degrees of freedom for all groups, followed by that for the pooled covariance matrix
data.name: a character string giving the names of the data.
method: the character string "Box's M-test for Homogeneity of Covariance Matrices".

Details

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, \(\lambda_1\)

References

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.

Author

The default method was taken from the biotools package, Anderson Rodrigo da Silva anderson.agro@hotmail.com

Generalized by Michael Friendly and John Fox

Examples


data(iris)

# default method
res <- boxM(iris[, 1:4], iris[, "Species"])
res
#> 
#> 	Box's M-test for Homogeneity of Covariance Matrices
#> 
#> data:  iris[, 1:4]
#> Chi-Sq (approx.) = 140.94, df = 20, p-value < 2.2e-16
#> 

summary(res)
#> Summary for Box's M-test of Equality of Covariance Matrices
#> 
#> Chi-Sq:	 140.943 
#> df:	 20 
#> p-value: < 2.2e-16 
#> 
#> log of Covariance determinants:
#>     setosa versicolor  virginica     pooled 
#> -13.067360 -10.874325  -8.927058  -9.958539 
#> 
#> Eigenvalues:
#>        setosa  versicolor  virginica     pooled
#> 1 0.236455690 0.487873944 0.69525484 0.44356592
#> 2 0.036918732 0.072384096 0.10655123 0.08618331
#> 3 0.026796399 0.054776085 0.05229543 0.05535235
#> 4 0.009033261 0.009790365 0.03426585 0.02236372
#> 
#> Statistics based on eigenvalues:
#>                 setosa   versicolor    virginica       pooled
#> product   2.113088e-06 1.893828e-05 0.0001327479 4.732183e-05
#> sum       3.092041e-01 6.248245e-01 0.8883673469 6.074653e-01
#> precision 5.576122e-03 7.338788e-03 0.0169121236 1.304819e-02
#> max       2.364557e-01 4.878739e-01 0.6952548382 4.435659e-01

# 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)
#> 
#> 	Box's M-test for Homogeneity of Covariance Matrices
#> 
#> data:  Y
#> Chi-Sq (approx.) = 140.94, df = 20, p-value < 2.2e-16
#> 

### Skulls dat
data(Skulls)
# lm method
skulls.mod <- lm(cbind(mb, bh, bl, nh) ~ epoch, data=Skulls)
boxM(skulls.mod)
#> 
#> 	Box's M-test for Homogeneity of Covariance Matrices
#> 
#> data:  Y
#> Chi-Sq (approx.) = 45.667, df = 40, p-value = 0.2483
#>