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Chapter 3   Fractional Designs

3.1   Introduction

3.1.1   Screening experiments

In many studies with a high number of potential factors, an initial experiment is conducted to identify which factors are most important.

3.1.2   Cost of complete factorial design

The number of runs for a complete factorial design increases exponentially with the number of factors. Even a single replicate can be prohibitively expensive if there are many factors.

3.1.3   Designs with fewer runs

There are experimental designs that for a subset of the runs needed for a complete factorial experiment. Such a design should ensure that all factors are orthogonal.

3.1.4   Problems

If there are fewer runs than required for a complete factorial, it is impossible to independently estimate all possible interactions between factors -- some interactions get mixed up (confounded) with main effects and other interactions.

3.2   Designs for 4 and 8 runs

3.2.1   Two factors

One replicate of a complete factorial experiment for 2 factors requires 4 runs. The two factors and their interaction are orthogonal, so all effects can be independently estimated.

3.2.2   Three factors

If the levels for C are defined by the AB interaction term in a complete factorial experiment for A and B, all three factors are orthogonal. However the main effects are confounded with 2-factor interactions.

3.2.3   Assessing factor importance

Since main effects are confounded with 2-factor interactions in a fractional factorial experiment for 3 factors in 4 runs, the interactions must be assumed to be negligible before the main effects can be used to rank the importance of the factors.

3.2.4   Two or three factors

Two replicates of a complete factorial experiment with 2 factors or one replicate of a complete factorial experiment with 3 factors can be conducted in 8 runs. All main effects and interactions can be estimated.

3.2.5   Four factors

A fractional factorial experiment for 4 factors in 8 runs can be designed by confounding D with the ABC interaction. Main effects are not confounded with 2-factor interactions but 2-factor interactions are confounded with each other.

3.2.6   Five to seven factors

A design with up to seven orthogonal factors can be defined by confounding D, E, F and G with the three 2-factor interactions and the 3-factor interaction in a complete factorial experiment for A, B and C. Main effects are confounded with 2-factor interactions.

3.2.7   Ranking factor importance

Importance of factors is determined by their main effects. However this ranking assumes that 2-factor interactions are negligible if there are 5 or more factors.

3.3   Designs with 16 runs

3.3.1   Four factors

A complete factorial experiment for 4 factors can be designed with 16 runs. All main effects and interactions can be estimated.

3.3.2   Five factors

If factor E is confounded with the 4-factor interaction between A, B, C and D in a complete factorial experiment for these 4 factors, all main effects and 2-factor interactions can be independently estimated.

3.3.3   Six to eight factors

The additional factors E, F, G and H should be confounded with 3-factor interactions between A, B, C and D. The factors are orthogonal and are not confounded with 2-factor interactions, but some 2-factor interactions are confounded with others.

3.3.4   Nine to fifteen factors

The factors E, F, ... can be confounded with any selection of the interactions between A, B, C and D. When there are 9 or more factors, it is impossible to avoid confounding main effects with 2-factor interactions.

3.4   Designs with other numbers of runs

3.4.1   Other fractional factorial designs

The number of runs in a fractional factorial design must be a power of 2. A table shows the resolution possible for designs with up to 64 runs.

3.4.2   Plackett-Burman designs for 12 runs

For 8-11 factors, a Plackett-Burman design uses only 12 runs, as opposed to the 16 runs needed for a fractional factorial experiment. The factors are still orthogonal.

3.4.3   Plackett-Burman designs for 24+ runs

A Plackett-Burman design for 16-19 factors in 20 runs is much less expensive than the 32 runs needed for a fractional factorial experiment. Plackett-Burman designs exist for any multiple of 4 runs.

3.4.4   Foldover designs

Repeating the runs in a resolution III design with the high and low levels for all factors swapped, results in a resolution IV design. Foldover designs can be used sequentially to augment a previous resolution III design. Some foldover Plackett-Burman designs are resolution IV designs with fewer runs than any fractional factorial design.