Lesson 8: 2-level Fractional Factorial Designs

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Learning Objectives & Outcomes 

The learning objectives for this lesson include:

  • Understanding the application of Fractional Factorial designs, one of the most important designs for screening
  • Becoming familiar with the terms “design generator”, “alias structure” and “design resolution”
  • Knowing how to analyze fractional factorial designs in which there aren’t normally enough degrees of freedom for error
  • Becoming familiar with the concept of “foldover” either on all factors or on a single factor and application of each case
  • Being introduced to “Plackett-Burman Designs” as another class of major screening designs

Introduction to Fractional Factorial Designs

What we did in the last chapter is consider just one replicate of a full factorial design and run it in blocks. The treatment combinations in each block of a full factorial can be thought of as a fraction of the full factorial.

In setting up the blocks within the experiment we have been picking the effects we know would be confounded and then using these to determine the layout of the blocks.

We begin with a simple example.

In an example where we have k = 3 treatments factors with 23 = 8 runs, we select 2p = 2 blocks, and use the 3-way interaction ABC to confound with blocks and to generate the following design.

trt
A
B
C
AB
AC
BC
ABC
I
(1)
-
-
-
     
-
 
a
+
-
-
     
+
 
b
-
+
-
     
+
 
ab
+
+
-
     
-
 
c
-
-
+
     
+
 
ac
+
-
+
     
-
 
bc
-
+
+
     
-
 
abc
+
+
+
     
+
 

Here are the two blocks that result using the ABC as the generator:

Block
1
2
ABC
-
+
 
(1)
a
 
ab
b
 
ac
c
 
bc
abc

A fractional factorial design is useful when we can't afford even one full replicate of the full factorial design. In a typical situation our total number of runs is N = 2k-p, which is a fraction of the total number of treatments.

Using our example above, where k = 3, p = 1, therefore, N = 22 = 4

So, in this case, either one of these blocks above is a one half fraction of a 23 design. Just as in the block designs where we had AB confounded with blocks - where we were not able to say anything about AB. Now, where ABC is confounded in the fractional factorial we can not say anything about the ABC interaction.

Let's take a look at the first block which is a half fraction of the full design. ABC is the generator of the 1/2 fraction of the 23 design. Now, take just the fraction of the full design where ABC = -1 and we  place it within its own table:

trt
A
B
C
AB
AC
BC
ABC
I
(1)
-
-
-
+
+
+
-
+
ab
+
+
-
+
-
-
-
+
ac
+
-
+
-
+
-
-
+
bc
-
+
+
-
-
+
-
+

Notice the contrast defining the main effects (similar colors) - there is an aliasing of these effects.  Notice that columns with the same color are just -1 times one another.  

In this half fraction of the design we have 4 observations, therefore we have 3 degrees of freedom to estimate. The degrees of freedom estimate the following effects: A - BC, B - AC, and C - AB. Thus, this design is only useful if the 2-way interactions are not important, since the effects we can estimate are the combined effect of main effects and 2-way interactions.

This is referred to as a Resolution III Design. It is called a Resolution III Design because the generator ABC has three letters, but the properties of this design and all Resolution III designs are such that the main effects are confounded with 2-way interactions.

Let's take a look at how this is handled in Minitab:

This design is only useful if you can be assured that the 2-way interactions are not important. If this is the case then you will find Resolution III designs to be very useful and efficient. When runs are expensive and factors are plentiful these are popular designs.