Factorial Designs Overview
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Factorial designs allow for the simultaneous study of the effects that several factors may have on a process. When performing an experiment, varying the levels of the factors simultaneously rather than one at a time is efficient in terms of time and cost, and also allows for the study of interactions between the factors. Interactions are the driving force in many processes. Without the use of factorial experiments, important interactions may remain undetected.

Screening designs

In many process development and manufacturing applications, the number of potential input variables (factors) is large. Screening (process characterization) is used to reduce the number of input variables by identifying the key input variables or process conditions that affect product quality. This reduction allows you to focus process improvement efforts on the few really important variables, or the "vital few." Screening may also suggest the "best" or optimal settings for these factors, and indicate whether or not curvature exists in the responses. Optimization experiments can then be done to determine the best settings and define the nature of the curvature.

In industry, two-level full and fractional factorial designs, and Plackett-Burman designs are often used to "screen" for the really important factors that influence process output measures or product quality. These designs are useful for fitting first-order models (which detect linear effects), and can provide information on the existence of second-order effects (curvature) when the design includes center points.

Full factorial designs

In a full factorial experiment, responses are measured at all combinations of the experimental factor levels. The combinations of factor levels represent the conditions at which responses will be measured. Each experimental condition is called a "run" and the response measurement an observation. The entire set of runs is the "design."

The following diagrams show two and three factor designs. The points represent unique combinations of factor levels. For example, in the two-factor design, the point on the lower left corner represents the experimental run when Factor A is set at its low level and Factor B is also set at its low level.

Two-level full factorial designs

In a two-level full factorial design, each experimental factor has only two levels. The experimental runs include all combinations of these factor levels. Although two-level factorial designs are unable to explore fully a wide region in the factor space, they provide useful information for relatively few runs per factor. Because two-level factorials can indicate major trends, you can use them to provide direction for further experimentation. For example, when you need to further explore a region where you believe optimal settings may exist, you can augment a factorial design to form a central composite design.

Two-level split-plot designs

In a two-level split-plot design, you can specify hard-to-change factors in full or fractional 2-level designs. The levels of the hard-to-change factors are held constant for several runs, which are collectively treated as a whole plot, while easy-to-change factors are varied over these runs, each of which is a subplot.

General full factorial designs

In a general full factorial design, the experimental factors can have any number of levels. For example, Factor A may have two levels, Factor B may have three levels, and Factor C may have five levels. The experimental runs include all combinations of these factor levels. General full factorial designs may be used in optimization experiments.

Fractional factorial designs

In a full factorial experiment, responses are measured at all combinations of the factor levels, which may result in a prohibitive number of runs. For example, a two-level full factorial design with 6 factors requires 64 runs; a design with 9 factors requires 512 runs. To minimize time and cost, you can use designs that exclude some of the factor level combinations. Factorial designs in which one or more level combinations are excluded are called fractional factorial designs. Minitab generates two-level fractional factorial designs for up to 15 factors.

Fractional factorial designs are useful in factor screening because they reduce the number of runs to a manageable size. The runs that are performed are a selected subset or fraction of the full factorial design. When you do not run all factor level combinations, some of the effects will be confounded. Confounded effects cannot be estimated separately and are said to be aliased. Minitab displays an alias table which specifies the confounding patterns. Because some effects are confounded and cannot be separated from other effects, the fraction must be carefully chosen to achieve meaningful results. Choosing the "best fraction" often requires specialized knowledge of the product or process under investigation.

Plackett-Burman designs

Plackett-Burman designs are a class of resolution III, two-level fractional factorial designs that are often used to study main effects. In a resolution III design, main effects are aliased with two-way interactions.

Minitab generates designs for up to 47 factors. Designs can have 12 to 48 runs; the number of possible runs is always a multiple of 4. The number of factors must be less than the number of runs.