Design of Experiments – Optimal Designs

When designing an experiment it is not always possible to generate a regular, balanced design such as a full or fractional factorial design plan. There are usually restrictions of the total number of experiments that can be undertaken or constraints on the factor settings both individually or in combination with each other.

In these scenarios computer generated designs, the optimal designs of a given size, can be identified from a list of candidate factor combinations. The library AlgDesign in R has facilities for optimal design searches based on the Federov exchange algorithm. An optimality criterion has to be selected by the investigator, currently D, A or I, and this criterion is minimise by searching for an optimal subset of a given size from the candidate design list.

Given the total number of treatment runs for an experiment and a specified model, the computer algorithm chooses the optimal set of design runs from a candidate set of possible design treatment runs. This candidate set of treatment runs usually consists of all possible combinations of various factor levels that one wishes to use in the experiment.

First stage, as always, is to make the package available for use:

library(AlgDesign)

For illustrative purposes consider a four factor experiment, where the factors have 4, 3, 2, and 2 levels each respectively. Using the expand.grid function we can create a data frame of all possible combinations of the factor settings:

cand.list = expand.grid(Factor1 = c("A", "B", "C", "D"),
  Factor2 = c("I", "II", "III"),
  Factor3 = c("Low", "High"),
  Factor4 = c("Yes", "No"))

The random number seed is set so that the algorithm can run:

set.seed(69)

The function optFederov calculates an exact or approximate algorithmic design for one of three criteria, using Federov’s exchange algorithm. The first argument to the function is a formula for the intended model for the data and the data argument specifies the list of candidate points:

optFederov( ~ ., data = cand.list, nTrials = 13)

In this example all of the factors in the candidate list appear in the model with a linear term. Quadratic or cubic terms can be included in this formula. The argument nTrials specifies the number of design points to select from the candidate list. The output from this function is:

$D
[1] 0.226687
 
$A
[1] 7.022811
 
$Ge
[1] 0.718
 
$Dea
[1] 0.676
 
$design
   Factor1 Factor2 Factor3 Factor4
3        C       I     Low     Yes
6        B      II     Low     Yes
12       D     III     Low     Yes
16       D       I    High     Yes
19       C      II    High     Yes
21       A     III    High     Yes
25       A       I     Low      No
26       B       I     Low      No
29       A      II     Low      No
35       C     III     Low      No
39       C       I    High      No
44       D      II    High      No
46       B     III    High      No
 
$rows
 [1]  3  6 12 16 19 21 25 26 29 35 39 44 46

This provides details of the values of the optimality criteria for the design points selected from the candidate list, the row numbers and the levels for the factors for the chosen design points.