Skip to content

MANE 6313

Week 16, Module E

Student Learning Outcome

  • Select an appropriate experimental design with one or more factors,
  • Select an appropriate model with one or more factors,
  • Evaluate statistical analyses of experimental designs,
  • Assess the model adequacy of any experimental design, and
  • Interpret model results.

Module Learning Outcome

Relate computer-generated designs.

Computer-Generated Designs

There are two broad categories of computer-generated designs:

  • Optimal Designs
  • Space-filling Designs

Optimal Designs for Response Surfaces

  • Presented in 11.4.4

  • Situations where an optimal design may be appropriate:

    • An irregular experimental region
    • A non-standard model is being used
    • Unusual sample size requirements

  • Different types of optimal designs:

    • D-optimal
    • A-optimal
    • G-optimal
    • V-optimal
    • I-optimal

D-optimality Criterion

A design is D-optimal if

\[ \left|\left(X^\prime X)^{-1}\right)\right| \]

is minimized.

  • D-optimal designs minimize the volume of the joint confidence region on the vector of regressoin coefficients

A-optimality Criterion

  • A design is A-optimal if it minimizes the sum of the main diagonal elements of \(\left(X^\prime X\right)\)

  • A-optimal designs minimize the sum of the variances of the regression coefficients

G-optimality Criterion

A design is G-optimal if it minimizes the maximum scaled prediction variance over the design region; that is, if the maximum value of

\[ \frac{NV\left[\hat{y}\left(x\right)\right]}{\sigma^2} \]

Over the design region is a minimum, where N is the number of points in the design

V-optimality Criterion

The V-criterion considers the prediction variance at a set of ponits of interest in the design region, \(x_1,x_2,\ldots,x_m\). The set of poitns could be candidate set from which the design was selected, or it could be some other collection of points that have a specific meaning to the experimenter. A design that minimizes the average prediction variance over this set of m points is a V-optimal design.

I-optimality Criterion

Computes an average or integrated variance over the design space, say

\[ I=\frac{1}{A}\int_RV\left[\hat{y}\left(x\right)\right]\,dx \]

where R is the design region and A is the volume of the region.

R Support of Optimal Designs

  • Package OptimalDesign
    • Supports D-, A-, I- and c-optimal designs
    • Some functions require the Gurobi software and its accompanying R package

Space-Filling Designs

  • Relatively new approach

  • Well suited to "deterministic computer models"

  • Goal is to spread the design points out nearly evenly or uniformly

    • Desirable if the form of the model is not known
    • Do not contain replicates (which are redundant in deterministic systems)

Latin Hypercube Design

Sphere-packing Design

Goal is that the minimum distance between pairs of points is maximized. Also called maximin designs

Uniform Design

Goal is place the design points so that they are uniformly scattered through the regions as would a sample from a uniform distribution.

Maximum Entropy Design

R Support

  • DiceDesign. Space-filling designs and space-filing criteria (distance-based and uniformity-based), with emphasis to computer experiments