Fractional Factorial Design

A fractional factorial design is an experimental design (DoE) used to examine the effects of multiple factors (variables) on a response variable. This approach reduces the effort and cost of experiments by analyzing only a subset of the possible combinations of factor levels, instead of testing all combinations as in a full factorial design.

In a full factorial design, all possible combinations of factor levels are tested comprehensively. In contrast, a fractional factorial design deliberately omits specific combinations. This leads to a reduction in the number of required trials, thereby lowering effort and costs, but also results in a reduced amount of information gathered.

Fractional Factorial vs Full Factorial Designs

In a full factorial design, all possible combinations of factor levels are tested comprehensively. In contrast, a fractional factorial design strategically omits certain combinations. This approach reduces the required number of experiments, thus lowering effort and cost, but also results in a reduced amount of information.

Suppose we want to determine which factors influence the frictional torque of a bearing. The frictional torque is our response variable. Potential factors could be lubrication, temperature, and bearing material. Lubrication might have levels such as oil and grease. Temperature could be categorized as low and high, and bearing material as steel and ceramic.

In a full factorial design, we would test all possible combinations of factor levels. This allows us not only to determine whether individual factors affect the frictional torque but also to identify any interaction effects between the factors.

As the number of factors increases, the number of required experiments in full factorial designs grows exponentially, significantly increasing the effort involved. With more than 4-6 factors, conducting a full factorial design often becomes impractical due to the high number of experiments needed. In such cases, screening approaches like fractional factorial designs offer an efficient alternative. These designs focus on identifying the most important main effects and interactions while significantly reducing the number of required experiments.

Resolution

Reducing the number of experiments in fractional factorial designs naturally comes with some information loss. The so-called resolution of these designs is lower, meaning not all interactions can be examined.

What does this mean in practice? In a full factorial design, all interactions, i.e., all possible combinations of factors, are considered. However, with a large number of factors, the number of interactions quickly becomes substantial. For instance, with five factors (A, B, C, D, and E), interactions can occur between two factors, three factors, four factors, and even all five factors.

The key question, therefore, is: Do we need to investigate all interactions, or can the resolution be reduced to focus on the most essential effects? A fractional factorial design provides this simplification by examining a selection of the most relevant effects, thereby reducing the number of experiments.

The extent to which the number of experiments can be reduced in favor of a lower resolution can be seen in this table. The full factorial designs are represented along the diagonal.

For example, with six factors, a full factorial design would require at least 64 experiments.

However, if we choose a fractional factorial design with a resolution of 6, the required number of experiments is reduced to 32.

At a resolution of 4, only 16 experiments are needed, and with a resolution of 3, the number of experiments decreases to just 8.

What Do Resolutions Mean?

In fractional factorial designs, interactions are combined or confounded with other interactions or factors. An example is a design with the so-called generator I = ABCD. This generator indicates which effects are confounded with one another.

Without delving into detailed calculation rules, factors can be freely moved to the left or right side of the equals sign. From the generator I = ABCD, seven combinations can be derived.

In the first case, we can see that the main effect of factor A is confounded with the interaction of factors B, C, and D. 'Confounded' means that after conducting the experiments, it is not possible to distinguish whether the measured effect comes from factor A or from the interaction of factors B, C, and D. Similarly, each other main effect is confounded with an interaction of the remaining three factors. Additionally, interactions between pairs of factors are confounded with interactions between other pairs of factors. For example, the interaction of factors A and B is confounded with the interaction of factors C and D.

Resolution 3

With a resolution of 3, main effects may be confounded with two-factor interactions, such as the main effect of factor A with the interaction between factors B and C.

Experimental designs with a resolution of 3 should therefore be viewed as critical, as it becomes difficult to separate main effects from interactions. They should only be used when the two-factor interactions are significantly smaller than the main effects of the individual factors. It is not enough simply to ignore the interactions; otherwise, two-factor interactions could significantly distort the results of a main effect.

Resolution 4

Experiments with a resolution of 4 are considerably less critical. In this case, two-factor interactions do not distort the main effect. Two-factor interactions are only confounded with other two-factor interactions, or a main effect is only confounded with a three-factor interaction.

Resolution 5

Designs with a resolution of 5 are considered non-critical. Main effects are not confounded with three-factor or lower interactions. Similarly, two-factor interactions are not confounded with other two-factor interactions. Instead, two-factor interactions are only confounded with three-factor interactions, and main effects are only confounded with four-factor interactions, both of which are generally negligible.