subselect (version 0.15.5)

tau2.coef: 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)

Value

The value of the \(\tau^2\) coefficient.

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 mat matrix).

r

the Expected rank of the H matrix. See the Details below.

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 tolval are considered equivalent to zero.

tolsym

the tolerance level for symmetry of the covariance/correlation/total matrix and for the effects (H) matrix. If corresponding matrix entries differ by more than this value, the input matrices will be considered asymmetric and execution will be aborted. If corresponding entries are different, but by less than this value, the input matrix will be replaced by its symmetric part, i.e., input matrix A becomes (A+t(A))/2.

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).

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

## ---------------------------------------------------------------