Number of runs

Fractional factorial experiments require a number of runs that is a power of 2. In the previous sections, we examined in detail designs for 22 = 4, 23 = 8 and 24 = 16 runs. Fractional factorial experiments can be designed in a similar way for experiments using 32 and 64 runs. (Experiments with higher powers of two are usually too expensive.)

As before, the design is based on a single replicate of a complete factorial design and additional factors are added whose levels are given by interactions between the original factors.

An experiment with 32 runs

An experiment with 25 = 32 runs would be based on a complete factorial experiment with factors A, B, C, D and E. Additional factors would be confounded with interactions between these. For example, two extra factors could be defined by

F  =   ABCD

G  =   BCDE

The table below shows the details of the design.

    Factor  
Run   A     B     C     D    E    F =
ABCD  
 G =
BCDE  
1 -1 -1 -1 -1 -1 +1 +1
2 -1 -1 -1 -1 +1 +1 -1
3 -1 -1 -1 +1 -1 -1 -1
4 -1 -1 -1 +1 +1 -1 +1
5 -1 -1 +1 -1 -1 -1 -1
6 -1 -1 +1 -1 +1 -1 +1
7 -1 -1 +1 +1 -1 +1 +1
8 -1 -1 +1 +1 +1 +1 -1
9 -1 +1 -1 -1 -1 -1 -1
10 -1 +1 -1 -1 +1 -1 +1
11 -1 +1 -1 +1 -1 +1 +1
12 -1 +1 -1 +1 +1 +1 -1
13 -1 +1 +1 -1 -1 +1 +1
14 -1 +1 +1 -1 +1 +1 -1
15 -1 +1 +1 +1 -1 -1 -1
16 -1 +1 +1 +1 +1 -1 +1
17 +1 -1 -1 -1 -1 -1 +1
18 +1 -1 -1 -1 +1 -1 -1
19 +1 -1 -1 +1 -1 +1 -1
20 +1 -1 -1 +1 +1 +1 +1
21 +1 -1 +1 -1 -1 +1 -1
22 +1 -1 +1 -1 +1 +1 +1
23 +1 -1 +1 +1 -1 -1 +1
24 +1 -1 +1 +1 +1 -1 -1
25 +1 +1 -1 -1 -1 +1 -1
26 +1 +1 -1 -1 +1 +1 +1
27 +1 +1 -1 +1 -1 -1 +1
28 +1 +1 -1 +1 +1 -1 -1
29 +1 +1 +1 -1 -1 -1 +1
30 +1 +1 +1 -1 +1 -1 -1
31 +1 +1 +1 +1 -1 +1 -1
32 +1 +1 +1 +1 +1 +1 +1

This is called a 27-2 fractional factorial design.

Resolution

Experimental designs are classified according to which main effects and interactions can be independently estimated and which are aliased with others.

Resolution III
All main effects are orthogonal and can be independently estimated but some main effects are confounded with 2-factor interactions
Resolution IV
The main effects are orthogonal and none are confounded with 2-factor interactions. However some 2-factor interactions are confounded with each other.
Resolution V and higher
All main effects and 2-factor interactions are orthogonal and can be independently estimated.

(The notation extends to resolution VI and higher according to how well 3-factor and higher-order interactions can be estimated.)

Given any number of runs and target resolution, the table below shows how many factors can be used in a fractional factorial experiment.

  Number of factors
 Best resolution  4 runs 8 runs 16 runs 32 runs 64 runs
III 3 5-7 9-15 12-31 12-63
IV - 4 6-8 7-11 9-11
V+ 2 3 4-5 5-6 7-8

The table also allows you to find the number of runs needed, given the number of factors and design resolution.

An experiment with 32 runs

The 27-2 fractional factorial design that is defined by

F  =   ABCD

G  =   BCDE

is of resolution IV since some 2-factor interactions are confounded with each other but no main effects are confounded with 2-factor interactions.

FG   =   ABCD x BCDE   =   AB2C2D2E   =   AE