stats (version 3.4.3)

mauchly.test: Mauchly's Test of Sphericity

Description

Tests whether a Wishart-distributed covariance matrix (or transformation thereof) is proportional to a given matrix.

Usage

mauchly.test(object, ...)
# S3 method for mlm
mauchly.test(object, ...)
# S3 method for SSD
mauchly.test(object, Sigma = diag(nrow = p),
   T = Thin.row(proj(M) - proj(X)), M = diag(nrow = p), X = ~0,
   idata = data.frame(index = seq_len(p)), ...)

Arguments

object

object of class SSD or mlm.

Sigma

matrix to be proportional to.

T

transformation matrix. By default computed from M and X.

M

formula or matrix describing the outer projection (see below).

X

formula or matrix describing the inner projection (see below).

idata

data frame describing intra-block design.

…

arguments to be passed to or from other methods.

Value

An object of class "htest"

Details

Mauchly's test test for whether a covariance matrix can be assumed to be proportional to a given matrix.

This is a generic function with methods for classes "mlm" and "SSD".

The basic method is for objects of class SSD the method for mlm objects just extracts the SSD matrix and invokes the corresponding method with the same options and arguments.

The T argument is used to transform the observations prior to testing. This typically involves transformation to intra-block differences, but more complicated within-block designs can be encountered, making more elaborate transformations necessary. A matrix T can be given directly or specified as the difference between two projections onto the spaces spanned by M and X, which in turn can be given as matrices or as model formulas with respect to idata (the tests will be invariant to parametrization of the quotient space M/X).

The common use of this test is in repeated measurements designs, with X = ~1. This is almost, but not quite the same as testing for compound symmetry in the untransformed covariance matrix.

Notice that the defaults involve p, which is calculated internally as the dimension of the SSD matrix, and a couple of hidden functions in the stats namespace, namely proj which calculates projection matrices from design matrices or model formulas and Thin.row which removes linearly dependent rows from a matrix until it has full row rank.

References

T. W. Anderson (1958). An Introduction to Multivariate Statistical Analysis. Wiley.

See Also

SSD, anova.mlm, rWishart

Examples

# NOT RUN {
utils::example(SSD) # Brings in the mlmfit and reacttime objects

### traditional test of intrasubj. contrasts
mauchly.test(mlmfit, X = ~1)

### tests using intra-subject 3x2 design
idata <- data.frame(deg = gl(3, 1, 6, labels = c(0,4,8)),
                    noise = gl(2, 3, 6, labels = c("A","P")))
mauchly.test(mlmfit, X = ~ deg + noise, idata = idata)
mauchly.test(mlmfit, M = ~ deg + noise, X = ~ noise, idata = idata)
# }