How to create a fractional factorial design with Python

Introduction

Fractional factorial designs offer a resource-efficient alternative to full factorial designs by focusing on a subset of the possible experimental runs. This approach is especially valuable when dealing with a large number of factors, where a full factorial design would be impractical due to the sheer number of required experiments. In this blog post, we'll explore how to create a fractional factorial design using Python and the pyDOE2 package.

What is a Fractional Factorial Design?

A fractional factorial design tests only a fraction of the possible combinations of levels for each factor, reducing the total number of experiments needed. This reduction is achieved by strategically selecting subsets of the experimental runs. While this approach saves time and resources, it introduces aliasing, where certain effects are confounded with each other.

Aliasing isn't always a problem. For example, in initial screening experiments or cases involving many factors, higher-order interactions (e.g., three-factor or four-factor interactions) are often considered negligible compared to main effects and two-factor interactions. As a result, aliasing higher-order interactions with main effects or two-factor interactions can be a reasonable trade-off.

Why Use Python and pyDOE2?

The pyDOE2 package simplifies the creation of experimental designs, including fractional factorial designs. Using Python and pyDOE2, we can efficiently plan, execute, and analyze experiments, even with limited resources.

Setting Up the Environment

First, ensure you have Python and the necessary packages installed. If you haven't already, install pyDOE2 using pip:

pip install pyDOE2

*note, if you are on Python 3.12 or higher, you need to install pyDOE3

Creating the Fractional Factorial Design

The pyDOE2 package provides a function called fracfact to generate fractional designs. This function takes a generator string as an input variable and provides the design matrix as output.

Here's the code to generate a half fractional design for an experiment with four factors ($2^{4-1}$):

# Import necessary packages
from pyDOE2 import fracfact
import pandas as pd

# Define the generator for the fractional factorial design
generator = 'a b c abc'

# Generate the fractional factorial design matrix
design_matrix = fracfact(generator)

# Convert the design matrix to a DataFrame for better readability
factor_names = ['T', 'P', 'C', 'RPM']
df = pd.DataFrame(design_matrix, columns=factor_names)

# Display the design matrix
print("Design Matrix for the Fractional Factorial Design:")
print(df)

The resulting design matrix:

|    | T    | P    | C    | RPM  |
|----|------|------|------|------|
| 0  | -1.0 | -1.0 | -1.0 | -1.0 |
| 1  |  1.0 | -1.0 | -1.0 |  1.0 |
| 2  | -1.0 |  1.0 | -1.0 |  1.0 |
| 3  |  1.0 |  1.0 | -1.0 | -1.0 |
| 4  | -1.0 | -1.0 |  1.0 |  1.0 |
| 5  |  1.0 | -1.0 |  1.0 | -1.0 |
| 6  | -1.0 |  1.0 |  1.0 | -1.0 |
| 7  |  1.0 |  1.0 |  1.0 |  1.0 |

This design is known as a half fractional design because it contains 8 experimental runs, which is half of the 16 runs that would be required for a full factorial design with 4 factors.

Understanding the Generator

In fractional factorial designs, the generator is crucial. It defines which fraction of the full factorial design is used and determines the alias structure, specifying how some factors are confounded with interactions among other factors.

The generator string 'a b c abc' used in the example indicates that the fourth factor (d) is confounded with the interaction (abc). This means that the main effect of d cannot be distinguished from the three-way interaction abc.

An important concept in fractional designs is the defining relation or identity, often denoted by I. The defining relation summarizes the entire alias structure of our design, allowing us to understand what effects are confounded with each other. To find the defining relation for this design, we take the generator equation: 

• d = abc

and multiply both sides by d:

• d*d = abcd

In fractional design, multiplying a factor by itself yields the identity I, this simplifies to:

• I = abcd

Any interaction that is equal to the identity cannot be studied at all. The remaining alias pattern can then be determined through multiplying by each factor.

So, the alias structure for the design 'a b c abc' is:

• a = bcd
• b = acd
• c = abd
• d = abc
• ab = cd
• ac = bd
• bc = ad

That means that the main effects are aliased with three-way interactions and the two-way interactions are aliased with each other. As mentioned earlier, the four way interaction abcd cannot be studied at all.

Generating Strings for More Factors

For a half-fractional design with more factors, the process of constructing the generator string follows similar principles. Here are examples for half-fractional designs with 5 factors:

Five Factors

With 5 factors (a, b, c, d, e):

• Generator string: a b c d abcd

Explanation:

• e = abcd
• I = abcde
• Aliases:
  • e = abcd
  • a = bcde
  • b = acde
  • c = abde
  • d = abce
  • ae = bcd
  • be = acd
  • ce = abd
  • de = abc

From this example, it becomes clear why fractional designs are especially helpful when a high number of factors are present. The main effects and important two-way interactions are only aliased with higher-order interactions like three-way and four-way interactions.

One-quarter fractional designs

There are other fractional designs, for example, the one-quarter fractional design that only has a quarter of the experimental runs compared to a corresponding full factorial design. An example would be the $2^{5-2}$ design.

Here we could use the generator a b c d=ab e=ac

In this case we have three identities:

• I = abd
• I = ace
• I = bcde

Thus, the three-way interactions abd and ace as well as the four-way interaction bcde cannot be identified.

The alias structure for the remaining design can be obtained similarly as earlier. For example the main effect of a is aliased with:

• a = bd
• a = ce
• a = abcde

Thus, the main effect a is confounded with the two-way interactions bd, ce as well as the five-way interaction abcde.

Conclusion

Fractional factorial designs are not as straight forward as full factorial designs. But it’s worth understanding them since they can literally be the difference between performing 100 experiments 50 experiments or 25 experiments. So try it.