### Experimental Design

#### Motivation for experimental design

• Minimize number of experiments
• Maximize useful information
• Note: categorical factors assumed throughout, and mostly we assume a linear model.

• Response
• Factors
• Levels
• Replication
• Design

#### Families of experimental designs

• ``Simple design:'' base case plus vary one factor at a time. Bad idea.
• Full factorial design: all combinations of factor/levels, 1 or more replication.
• Fractional factorial design: subset of full factorial, carefully chosen.

#### factorial design (chap. 17)

• Simple case of full factorial design: each of k factors has only two levels.
• Model (for k=2)

where

and similarly for .

• unknowns (effects) and experiments. So a direct method (with tabular tricks) yields values for the effects and for allocation of variation (via sums-of-squares arguments).

#### factorial design with replication (chap. 18)

• Idea: replicate experiments to yield error estimates and rigorous confidence intervals (t-tests) and ANOVA (F-tests).
• Model (for k=2)

• Tabular tricks still work, with sample means replacing single experiment values.

• Given variance estimates, can compute confidence intervals for effects and for predicted future responses.

• And F-tests can be used to determine if a significant amount of variation is attributable to a particular factor.

#### Important visual tests to verify assumptions

• Are errors independent? Plot residuals vs. or residuals vs. experiment number.
• Are errors distributed normally? Normal quantile-quantile plot.

#### fractional factorial designs (chap. 19)

• Idea: reduce cost of full design by only doing experiments, carefully chosen to give information about 1st order effects at expense of 2nd, 3rd, 4th, ...

#### One-factor designs (chap. 20)

• Another simple case of full factorial design -- this time only have 1 factor, but > 2 levels.

#### General full factorial designs (chap. 23)

CS 5014, C. J. Ribbens, 10/17/2001