Fractional Factorial Designs

Suppose we have seven factors. If we choose two levels per factor, we must perform experiments.

How much information could we get from many fewer experiments - perhaps just eight?

A design with just eight experiments is called a design.

In general, a design is called a half-replicate of a design. A design is a quarter-replicate.

Example of sign table...

Recall the sign table...

IF the interactions AB, AC, AD, ABCD, ..., are negligible, we can relabel the rightmost four columns by factors D,E,F,G to obtain the table:

The model is

The percent variation suggests more experiments with factors A and C. The percent variation figures apply to D, E, F, and G only if the interactions of two or more effects is negligible

Preparing the sign table for a design...

The sign table is not always unique. Consider a design:

The following assignment of factor D is recommended if we believe the ABC interaction is negligible:

However, if we believe the AB interaction is the smallest interaction, the following assignment of D is recommended:

Confounding...

The drawback of fractional factorial designs is that the experiments only yields the combined influence of two or more effects. This is called confounding.

In first design, example D was confounded with ABC:

In the bottom, D was confounded with AB:

In fact, in a design, every column represents a sum of 2 effects. For example, in the first design, for experiment i

In general, in a design, effects are confounded together:

• design confounds effects
• design confounds effects

In the previous example (top), the complete list of confoundings is:

Algebra of confounding...

Recall the complete set of confoundings for the top design:

Formula I=ABCD is called the generator polynomial.

Design is called I=ABCD.

To list all other confoundings given just one confounding, multiply both sides of confounding by different terms, using two rules:

1. For any term x, xI=x.
2. For any term x, .

Given generator polynomial I=ABCD:

• Multiplying both sides by A yields:

  

• Multiplying both sides by AB yields:

  

  and so on.

In fact, two designs are identified by their generator. I=ABCD is the generator for

and I=ABD is the generator for

Design Resolution

The order of an effect is the number of factors included in it:

If an ith order effect is confounded with a jth order term, the confounding is said to be of order i+j:

• The confounding of I and A is of order 1
• The confounding of AB and CD is of order 4

Design resolution is minimum of orders of all confoundings.

For first design:

All confoundings are of order 4. Hence the resolution is 4.

A design with resolution 4 is denoted .

Second example: The design on the bottom contained:

The set of all confoundings is:

The set of all confounding orders is . Thus the design has resolution 3 (denoted ).

Note: Look only at generator polynomial to find resolution:

• I=ABCD is
• I=ABD is

Confounding More Than Two Effects

A generator polynomial can confound more than two terms.

In general, design confounds effects.

Example: Reconsider design:

This has only eight columns. But design needs 128 columns. Therefore 16 sets of eight columns are mapped to the above eight columns to ``collapse'' design to .

Thus effects are confounded:

The above equation is the generator polynomial. It arises because effect D is assigned to the column for AB, and so on:

This yields the generator

Finally, the generator polynomial is obtained by reducing all pairs of terms in the last equation so that I is on one side. For example,

yields I = BCDE, and so on.

Example: design (Wang VS system)...

Case Study in [Jain, 19.2]:

Objective: decide what type of schedule should be used with three different types of workloads on a computer: word processing, interactive processing, and batch processing.

Results and Interpretation

Results of experiments shown in Tables 19.10, 19.11 of [Jain].

• Ideal setting of parameters is different for the three workloads.
  • Effects that impact word processor:

    A (preemption), B (time slice), AB

  • Effects that impact interactive jobs:

    E (fairness), A (preemption), BE, B (time slice)

  • Effects that impact background jobs:

    A (preemption), AB, B (time slice), E (fairness)

• Factors C (queue assignment) and D (requeuing) do not impact any throughput.
• Factor A (preemption) impacts all three workloads significantly.
• Factor B (time slice) has less impact than factor A (preemption).
• Factor E (fairness) is important for interactive jobs and slightly important for background jobs.

Therefore, run two more experiments with:

• Just factors A, B
• Just factors B, E