mauchly.test              package:stats              R Documentation

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, 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.

Details:

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

     It is common 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 transformation 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.

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

Value:

     An object of class '"htest"'

Note:

     The p-value differs slightly from that of SAS because a second
     order term is included in the asymptotic approximation in R.

References:

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

See Also:

     'SSD', 'anova.mlm'

Examples:

     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)

