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% File src/library/stats/man/anova.mlm.Rd

% Part of the R package, https://www.R-project.org

% Copyright 1995-2014 R Core Team

% Distributed under GPL 2 or later

\name{anova.mlm}

\alias{anova.mlm}

%\alias{anova.mlmlist}

\title{Comparisons between Multivariate Linear Models}

\description{

  Compute a (generalized) analysis of variance table for one or more

  multivariate linear models.

}

\usage{

\method{anova}{mlm}(object, \dots,

      test = c("Pillai", "Wilks", "Hotelling-Lawley", "Roy",

               "Spherical"),

      Sigma = diag(nrow = p), T = Thin.row(proj(M) - proj(X)),

      M = diag(nrow = p), X = ~0,

      idata = data.frame(index = seq_len(p)), tol = 1e-7)

}

\arguments{

  \item{object}{an object of class \code{"mlm"}.}

  \item{\dots}{further objects of class \code{"mlm"}.}

  \item{test}{choice of test statistic (see below).  Can be abbreviated.}

  \item{Sigma}{(only relevant if  \code{test == "Spherical"}).  Covariance

    matrix assumed proportional to \code{Sigma}.}

  \item{T}{transformation matrix.  By default computed from \code{M} and

    \code{X}.}

  \item{M}{formula or matrix describing the outer projection (see below).}

  \item{X}{formula or matrix describing the inner projection (see below).}

  \item{idata}{data frame describing intra-block design.}

  % avoid Matrix's grab of qr.

  \item{tol}{tolerance to be used in deciding if the residuals are

    rank-deficient: see \code{\link{qr}}.}

}

\details{

  The \code{anova.mlm} method uses either a multivariate test statistic for

  the summary table, or a test based on sphericity assumptions (i.e.

  that the covariance is proportional to a given matrix).

  For the multivariate test, Wilks' statistic is most popular in the

  literature, but the default Pillai--Bartlett statistic is

  recommended by Hand and Taylor (1987).  See

  \code{\link{summary.manova}} for further details.

  For the \code{"Spherical"} test, proportionality is usually with the

  identity matrix but a different matrix can be specified using \code{Sigma}.

  Corrections for asphericity known as the Greenhouse--Geisser,

  respectively Huynh--Feldt, epsilons are given and adjusted \eqn{F} tests are

  performed.

  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 \code{T} can be given directly or specified as

  the difference between two projections onto the spaces spanned by

  \code{M} and \code{X}, which in turn can be given as matrices or as

  model formulas with respect to \code{idata} (the tests will be

  invariant to parametrization of the quotient space \code{M/X}).

  As with \code{anova.lm}, all test statistics use the SSD matrix from

  the largest model considered as the (generalized) denominator.

  Contrary to other \code{anova} methods, the intercept is not excluded

  from the display in the single-model case.  When contrast

  transformations are involved, it often makes good sense to test for a

  zero intercept.

}

\value{

   An object of class \code{"anova"} inheriting from class \code{"data.frame"}

}

\note{

  The Huynh--Feldt epsilon differs from that calculated by SAS (as of

  v.\sspace{}8.2) except when the DF is equal to the number of observations

  minus one.  This is believed to be a bug in SAS, not in \R.

}

\references{

  Hand, D. J. and Taylor, C. C.  (1987)

  \emph{Multivariate Analysis of Variance and Repeated Measures.}

  Chapman and Hall.

}

%% Probably use example from Baron/Li

\seealso{

  \code{\link{summary.manova}}

}

\examples{

require(graphics)

utils::example(SSD) # Brings in the mlmfit and reacttime objects

mlmfit0 <- update(mlmfit, ~0)

### Traditional tests of intrasubj. contrasts

## Using MANOVA techniques on contrasts:

anova(mlmfit, mlmfit0, X = ~1)

## Assuming sphericity

anova(mlmfit, mlmfit0, X = ~1, test = "Spherical")

### 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")))

anova(mlmfit, mlmfit0, X = ~ deg + noise,

      idata = idata, test = "Spherical")

anova(mlmfit, mlmfit0, M = ~ deg + noise, X = ~ noise,

      idata = idata, test = "Spherical" )

anova(mlmfit, mlmfit0, M = ~ deg + noise, X = ~ deg,

      idata = idata, test = "Spherical" )

f <- factor(rep(1:2, 5)) # bogus, just for illustration

mlmfit2 <- update(mlmfit, ~f)

anova(mlmfit2, mlmfit, mlmfit0, X = ~1, test = "Spherical")

anova(mlmfit2, X = ~1, test = "Spherical")

# one-model form, eqiv. to previous

### There seems to be a strong interaction in these data

plot(colMeans(reacttime))

}

\keyword{regression}

\keyword{models}

\keyword{multivariate}