Optimality criterion

Description

Computes the value of the criterion of D-, A-, or IV-optimality of a given design.

Usage

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  od.crit(F, w, crit="D", R=NULL, tol=1e-12)

Arguments

F

The n times m matrix of real numbers. Rows of F represent the m-dimensional regressors corresponding to n design points.

w

The non-negative vector of length n representing the design.

crit

The optimality criterion. Possible values are "D", "A","IV".

R

The region of summation for the IV-optimality criterion. The argument R must be a subvector of 1:n, or NULL. If R=NULL, the procedure uses R=1:n. Argument R is ignored if crit="D", or if crit="A".

tol

A small positive number to determine singularity of the information matrix.

Details

Let w be a design with information matrix M, let n be the number of design points and let m be the number of parameters of the model.

For w, the value of the criterion of D-optimality is computed as (det(M))^(1/m) and the value of the criterion of A-optimality is computed as m/trace(M.inv), where M.inv is the inverse of M.

The IV-optimal design, sometimes called I-optimal or V-optimal, minimizes the integral of the variances of the BLUEs of the response surface over a region R, or the sum of the variances over R, if R is finite; see Section 10.6 in Atkinson et al. Let the matrix L be the integral (or the sum) of F[x,]%*%t(F[x,]) over x in R. If the criterion of IV-optimality is selected, the region R should be chosen such that the associated matrix L is non-singular. Then, let L=t(C)%*%C be the Cholesky decomposition of L. The design w is IV-optimal in the model given by F, if and only if w is A-optimal for the model with the regressors matrix F%*%C.inv, where C.inv is the inverse of C.

For the purpose of this package, the value of the IV-criterion for w is m/trace(N.inv), where N.inv is the inverse of the information matrix of w in the model given by regressors matrix F%*%C.inv, and every computational problem of IV-optimality is converted to the corresponding problem of A-optimality. The argument R is assumed to be a subset of 1:n. If the application requires that R is not a subset of the set of design points, the user should compute the matrix C, transform the model as described above, and use the procedures for A-optimality.

If the information matrix is singular, the value of all three criteria is zero. An information matrix is considered to be singular, if its minimal eigenvalue is smaller than m*tol.

Value

The value of the concave, positive homogeneous version of the selected real-valued criterion applied to the information matrix of the design w in the linear regression model with m-dimensional regressors F[1,],...,F[n,] corresponding to n design points.

Author(s)

Radoslav Harman, Lenka Filova

References

Atkinson AC, Donev AN, Tobias RD (2007): Optimum Experimental Designs, with SAS, Oxford University Press, Oxford

See Also

od.infmat, od.print, od.plot

Examples

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# The matrix of regressors for the spring balance weighing model with 
# 6 weighed items.
F.sbw <- F.cube(~x1 + x2 + x3 + x4 + x5 + x6 - 1, rep(0, 6), 
                rep(1, 6), rep(2, 6))

# The value of all 3 optimality criteria for the design of size 15 
# that weighs each pair of items exactly once.
w2 <- rep(0, 64); w2[apply(F.sbw, 1, sum)==2] <- 1
od.crit(F.sbw, w2, "D")
od.crit(F.sbw, w2, "A")
od.crit(F.sbw, w2, "IV")

# The value of all 3 optimality criteria for the design of size 15 that 
# weighs each quadruple of items exactly once.
w4 <- rep(0, 64); w4[apply(F.sbw, 1, sum)==4] <- 1
od.crit(F.sbw, w4, "D")
od.crit(F.sbw, w4, "A")
od.crit(F.sbw, w4, "IV")