The successful use of fractional factorial designs is based on three key ideas:

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Fractional Factorial. The successful use of fractional factorial designs is based on three key ideas: The sparsity of effects principle. When there are several variables, the system or process is likely to be driven primarily by some of the main effects an low order interactions.

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Fractional Factorial

• The successful use of fractional factorial designs is based on three key ideas:
• The sparsity of effects principle. When there are several variables, the system or process is likely to be driven primarily by some of the main effects an low order interactions.
• The projection property. Fractional factorial designs can be projected into stronger designs in the subset of significant factors.
• Sequential experimentation.

Fractional Factorial

For a 24 design (factors A, B, C and D) a one-half fraction, 24-1, can be constructed as follows:

Choose an interaction term to completely confound, say ABCD.

Using the defining contrast L = x1 + x2 + x3 + x4 like we did before we get:

Fractional Factorial

Hence, our design with ABCD completely confounded is as follows:

The fractional factorial design

Fractional Factorial

Each calculated sum of squares will be associated with two sources of variation.

Fractional Factorial

Lets clean a bit:

Fractional Factorial

Lets reorganize:

Complete 23 Design

Fractional Factorial

So to analyze a 24-1 fractional factorial design we need to run a complete 23 factorial design (ignoring one of the factors) and analyze the data based on that design and re-interpret it in terms of the 24-1 design.

Fractional Factorial

Resolution:

• Many resolutions the three listed in the book are:
• Resolution III designs: No main effect is aliased with any other main effect, they are aliased with two factor interactions and two factor interactions are aliased with each other. Example 2III3-1 with ABC as the principle fractions.
• Resolution IV designs: No main effect is aliased with any other main effect or any two factor interaction, but two factor interactions are aliased with each other. Example, 2IV4-1 with ABCD as the principle fraction.
• Resolution V designs. No main effect or two-factor interactions is aliased with any other main effect or two-factor interaction, but two-factor interactions are aliased with three factor interactions. Example, 2V5-1 with ABCDE as the principle fraction.

Fractional Factorial

Assuming all factors are fixed, the linear model is as follows:

Fractional Factorial

If we still cant run this design all at once, we can block; that is we can implement a group-interaction confounding step. We can confound the highest level interaction of the 23 design, as we did before.

Group-Interaction Confounded designs

Partial confounding:

Confounding ABC replicate 1:

Group-Interaction Confounded designs

Partial confounding:

Confounding ABC replicate 1:

Fractional Factorial

For a 24 design (factors A, B, C and D) a one-quarter fraction, 24-2, can be constructed as follows:

Choose two interaction terms to confound, say ABD and ACD, these will serve as our principle fractions. The third interaction, called the generalized interaction, that we confounded in the way is: A2BCD2 = BC.

Need two defining contrasts

L1 = x1 + x2 + 0+ x4

and

L2 = x1 + 0 + x3 + x4

Fractional Factorial

Hence, our design with ABCD completely confounded is as follows:

Fractional Factorial

One of the possible one-quarter designs is:

Fractional Factorial

Each calculated sum of squares will be associated with four sources of variation.

Fractional Factorial

Each calculated sum of squares will be associated with four sources of variation.

Fractional Factorial

The above is not quite satisfactory because we are aliasing some of the main effects with other main effects;

i.e. the resolution is not good enough!!!

Fractional Factorial

What happens after analyzing the data:

Can do a confirmatory experiment, complete the block!!

Fractional Factorial

Hence, our design with ABCD completely confounded is as follows:

Fractional Factorial

Each calculated sum of squares will be associated with four sources of variation.

Fractional Factorial

Each calculated sum of squares will be associated with four sources of variation.