Fractional vs. Full Factorial Design
The benefit of fractional designs
Switching from a full factorial design to a fractional design can save a lot of resources—time, materials, and effort. Why? Because fractional designs only use a carefully selected subset of experiments from a full factorial design. This smart reduction means you’re still gathering valuable data without running every possible combination. However, there’s a trade-off.
By reducing the number of experiments, some effects start to overlap and you’re only able to understand their combined effect on the response rather than their individual contributions. This phenomenon is called aliasing.
What is aliasing?
Aliasing is an unavoidable trade-off when you choose to run fewer experiments. Think of it like reading a summary of a novel instead of the whole book—you get the main storyline, but you might miss out on some of the finer details. The length of the summary that you read is determined by the design resolution.
Design resolution
The design resolution refers to the degree of aliasing in a fractional design. Higher-resolution designs reduce aliasing, making it easier to distinguish between effects, but they come at the cost of more experimental runs.
Common design resolutions are:
- Resolution III: Main effects are aliased with two-way interactions. Suitable for a rough screening.
- Resolution IV: Main effects are not aliased with two-way interactions, but two-way interactions are aliased with each other.
- Resolution V: Main effects and two-way interactions are not aliased with each other. Higher-order interactions (three-way and beyond) are aliased with those. These designs come very close to full factorial designs and the additional information gain from a full factorial design might be negligible.
Choosing the right resolution depends on the goals of your experiment and the expected complexity of the system you’re studying. Resolution IV designs often offer a good balance between number of experimental runs and information gained.
Full factorial vs. fractional design
Let’s take a practical example to see the differences between a full factorial and a half fractional design—and to understand how aliasing works and why it is not always a problem.
Imagine we’re trying to improve the filtration rate of a chemical product, which currently filters at 75 gallons per hour. To optimize this, we’re testing four factors: temperature (T), pressure (P), concentration of formaldehyde (CoF), and stirring rate (RPM).
In a full factorial design, we would need 16 experimental runs to test every possible combination of these four factors at two levels.
By choosing a half fractional design with the resolution of IV, we reduce the effort to just 8 runs. But at the tradeoff that two of the two-way interactions are always aliased or "confused" with each other, making it impossible to distinguish them. Additionally, the main effects are aliased with the three-way interactions.
Main effects
If we visually compare the main effects for the full factorial design and the fractional design then we notice that the absolute values slightly differ. This deviation is due to the main effects being aliased with the three-way interactions and it is actually the sum of main effect and three-way interaction that we are seeing here. Despite this, the overall impact of the three-way interactions on the main effects is minimal. In fact, the contribution of the three-way interactions is only about 10%, making their influence negligible in this example.
Two-way interactions
When comparing two-way interactions between the full factorial design and the fractional design, we notice a mirroring effect in the fractional design due to aliasing. This happens because certain interactions are “confused” or aliased with each other. For example, the interaction between temperature (T) and stirring rate (RPM) is aliased with the interaction between pressure (P) and concentration of formaldehyde (CoF). As a result, we only see their combined impact rather than their individual contributions.
It is not easy to interpret these results. It might be that one of the interactions has a large impact on the filtration rate and the other one only a small one (similar to what we saw with the main effects). But it might as well be possible that both interactions affect the filtration similarly so that they are equally important.
The biggest cue here of which of those two assumptions might be correct are the main effects. Since the main effect of pressure (P) is relatively low, interactions involving the pressure are also likely to be small. Comparing this to the fully resolved full factorial design confirms that this assumption is correct. In addition to that, you can leverage your domain-specific knowledge to predict which interactions are more likely to be significant. After all, DOE is just a tool and does not replace a good engineer but makes them better.
ANOVA
When running ANOVA on a fractional design, you can’t estimate the aliased terms independently. Instead, you estimate their combined effect and assess its significance. However, as explained earlier, by applying your knowledge and insights into which factors are likely to have the most significant impact, you can make informed decisions about which aliased terms to interpret. By doing this, you can produce results from a fractional design that closely match those of a full factorial design.
It's not always as obvious as here
This example was fairly straightforward, and running the additional 8 experiments to complete the full factorial design wouldn’t have provided much additional value. In practice, though, it’s not always this simple. If you’re unsure about your conclusions or encounter ambiguity in your results, running the remaining experiments to complete the full factorial design can resolve any uncertainties.