# Optimality criterion

### 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 |

`w` |
The non-negative vector of length |

`crit` |
The optimality criterion. Possible values are |

`R` |
The region of summation for the IV-optimality criterion. The argument |

`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

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")
``` |