tau2.coef {subselect} | R Documentation |
Computes the Tau squared coefficient for a multivariate linear hypothesis
Description
Computes the Tau squared index of "effect magnitude". The maximization of this criterion is equivalent to the minimization of Wilk's lambda statistic.
Usage
tau2.coef(mat, H, r, indices,
tolval=10*.Machine$double.eps, tolsym=1000*.Machine$double.eps)
Arguments
mat |
the Variance or Total sums of squares and products matrix for the full data set. |
H |
the Effect description sums of squares and products matrix (defined in the same way as the |
r |
the Expected rank of the H matrix. See the |
indices |
a numerical vector, matrix or 3-d array of integers giving the indices of the variables in the subset. If a matrix is specified, each row is taken to represent a different k-variable subset. If a 3-d array is given, it is assumed that the third dimension corresponds to different cardinalities. |
tolval |
the tolerance level to be used in checks for
ill-conditioning and positive-definiteness of the 'total' and
'effects' (H) matrices. Values smaller than |
tolsym |
the tolerance level for symmetry of the
covariance/correlation/total matrix and for the effects ( |
Details
Different kinds of statistical methodologies are considered within the framework, of a multivariate linear model:
X = A \Psi + U
where X
is the (nxp) data
matrix of original variables, A
is a known (nxp) design matrix,
\Psi
an (qxp) matrix of unknown parameters and U
an
(nxp) matrix of residual vectors.
The \tau^2
index is related to the traditional test statistic
(Wilk's lambda statistic) and
measures the contribution of each subset to an Effect characterized by
the violation of a linear hypothesis of the form
C \Psi = 0
, where C
is a known cofficient matrix
of rank r. The Wilk's lambda statistic (\lambda
) is given by:
\Lambda=\frac{det(E)}{det(T)}
where
E
is the Error matrix and T
is the Total matrix.
The index \tau^2
is related to the Wilk's lambda statistic
(\Lambda
) by:
\tau^2 = 1 - \lambda^{(1/r)}
where r
is the
rank of H
the Effect matrix.
The fact that indices
can be a matrix or 3-d array allows for
the computation of the \tau^2
values of subsets produced by the search
functions anneal
, genetic
, improve
and
eleaps
(whose output option $subsets
are
matrices or 3-d arrays), using a different criterion (see the example
below).
Value
The value of the \tau^2
coefficient.
Examples
## ---------------------------------------------------------------
## 1) A Linear Discriminant Analysis example with a very small data set.
## We considered the Iris data and three groups,
## defined by species (setosa, versicolor and virginica).
data(iris)
irisHmat <- ldaHmat(iris[1:4],iris$Species)
tau2.coef(irisHmat$mat,H=irisHmat$H,r=2,c(1,3))
## [1] 0.8003044
## ---------------------------------------------------------------
## 2) An example computing the value of the tau_2 criterion for two
## subsets produced when the anneal function attempted to optimize
## the xi_2 criterion (using an absurdly small number of iterations).
xiresults<-anneal(irisHmat$mat,2,nsol=2,niter=2,criterion="xi2",
H=irisHmat$H,r=2)
tau2.coef(irisHmat$mat,H=irisHmat$H,r=2,xiresults$subsets)
## Card.2
##Solution 1 0.8079476
##Solution 2 0.7907710
## ---------------------------------------------------------------