# od.crit: Optimality criterion In OptimalDesign: Algorithms for D-, A-, and IV-Optimal Designs

## Description

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

## Usage

 `1` ``` 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.

## References

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

`od.infmat, od.print, od.plot`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18``` ```# 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") ```