Traditionally, when normalized designs are discussed, the normalization factor is equal to the number of experiments N; see (7.3). In this section we will consider cost-normalized designs. Each measurement at point Xi is assumed to be associated with a cost c(Xi) and a restriction on the total cost is given by
In this case it is quite natural to normalize the information matrix by the total cost C and introduce
Once the cost function c(X) is defined, we can introduce a cost-based design ξC = {wi,Xi and use exactly the same techniques of constructing continuous optimal designs for various optimality criteria as described in the previous sections,
As usual, to obtain counts ni, values ñi = wi C/c(Xi) have to be rounded to the nearest integers ni subject to ∑i nic(Xi) ≤ C.
We believe that the introduction of cost constraints and the alternative normalization (7.23) makes the construction of optimal designs for models with multiple responses more sound. It allows for a meaningful comparison of "points" X with distinct number of responses.
The example considered in this section is the example discussed in Section 7.8.3 with cost functions added. Program 7.10 computes a D-optimal design by invoking the same SAS macros and SAS/IML modules as Program 7.9. The computational tools for calculating the cost-normalized information matrices are built into the macros and modules, we need only to specify the cost function and set the COST variable to 2 (to indicate that a user-defined cost function will be provided).
The cost function must be entered by the user into the SAS/IML module MCOST. In Program 7.10, we select a linear cost function with two components, the CV variable is associated with an overall cost of the study (i.e., the cost of a patient visit) and CS is the cost of obtaining/analyzing a single sample:
* Cost module; start mcost(t); * t is vector sampling times; Cv = 1; Cs = 0.3; C = Cv + sum(Cs[1:nrow(t[loc(t>0)])]); return(C); finish(mcost);
In general, the cost function can have almost any form.
The design region is defined as follows
that is, we allow any combination of r sampling times for each patient from the original sequence χ, 3 ≤ r ≤ 5. The values of r are specified in the KS data set. Note that the a total of 6,748 information matrices (47,236 rows and 7 columns of data) are calculated and stored in this example.
* Design parameters; %let h=0.001; * Delta for finite difference derivative approximation; %let paran=7; * Number of parameters in the model; %let nf=2; * Number of fixed effect parameters; %let cost=2; * Cost function (1, no cost function, 2, user-specified function); * Algorithm parameters; %let convc=1e-9; %let maximit=1000; %let const1=2; %let const2=1; %let cmerge=5; * PK parameters; data para; input CL V vCL vV covCLV m s; datalines; 0.211 5.50 0.0365 0.0949 0.0443 0.0213 8060 ; * All candidate points; data cand; input x @@; datalines; 0.083 0.25 0.5 0.75 1 2 3 4 5 6 12 24 36 48 72 144 ; * Number of time points in the final design; data ks; input r @@; datalines; 3 4 5 ; * Initial design; data sample; input x1 x2 x3 x4 x5 w @@; datalines; 0.083 0.5 4 24 144 1.0 ; run; |
Determinant of the covariance matrix D (initial design) IDED 0.231426 Determinant of the covariance matrix D (optimal design) DETD 0.0209139 Optimal design Obs COL1 COL2 COL3 COL4 COL5 1 0.083 0.25 48 72 144 2 0.083 0.25 72 144 Optimal weights Obs W 1 0.59125 2 0.40875 |
Output 7.10 displays the D-optimal design based on the linear cost funtion defined in the MCOST module. The D-optimal design is a collection of two sampling sequences,
with weights w1 = 0.59 and w2 = 0.41, respectively. This example shows that once costs are taken into account, sampling sequences with a smaller number of samples may become optimal.