Computes the value of the criterion of D-, A-, or IV-optimality of a given design.
The non-negative vector of length
The optimality criterion. Possible values are
The region of summation for the IV-optimality criterion. The argument
A small positive number to determine singularity of the information matrix.
w be a design with information matrix
n be the number of design points and let
m be the number of
parameters of the model.
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.inv is the inverse of
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 is finite;
see Section 10.6 in Atkinson et al. Let the matrix
L be the integral (or the sum) of
If the criterion of IV-optimality is selected, the region
R should be chosen such that the associated matrix
L is non-singular.
L=t(C)%*%C be the Cholesky decomposition of
L. The design
w is IV-optimal in the model given by
if and only if
w is A-optimal for the model with the regressors matrix
C.inv is the inverse of
For the purpose of this package, the value of the IV-criterion for
N.inv is the inverse of the information matrix of
w in the model given by regressors matrix
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
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
F[1,],...,F[n,] corresponding to
n design points.
Radoslav Harman, Lenka Filova
Atkinson AC, Donev AN, Tobias RD (2007): Optimum Experimental Designs, with SAS, Oxford University Press, Oxford
od.infmat, od.print, od.plot
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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")