# Randomized Factorial Designs

## Regular Two-Level Factorial Designs

The Regular Two-Level Factorial Design Builder offers two-level full factorial and regular fractional factorial designs. You can investigate 2 to 21 factors using 4 to 512 runs. This collection of designs provides an effective means for screening through many factors to find the critical few.

Full two-level factorial designs may be run for up to 9 factors. These designs permit estimation of all main effects and all interaction effects (except those confounded with blocks.)

Stat-Ease offers a wide variety of fractional factorial designs. The software calculates detailed information about the alias structure when the design is built. This evaluation should be inspected to ensure the selected design can cleanly estimate the interactions of interest.

Replicates: The value in this box is the number of times the requested design will be produced. The number of runs in the final design is the number of replicates times the number of runs in the requested base design.

We do not recommend replicating a fractional factorial design.

Blocks: The value in this box is the number of pieces to break the design into to account for known sources of variation. See Blocking for more information.

When the number of blocks equals the number of replicates. The replicated designs will be in their own blocks.

Center points per block: When there are numeric factors, center points can be added to check for curvature. The number entered here will be the number of center points randomized within each block.

Show Generators: Checking this box brings up the advanced generators interface when Next is clicked.

The roman numerals on this screen are the resolution. They are also color coded such that no color is a full factorial, red are Resolution III, yellow are resolution IV, and green are at least resolution V.

Red means Stop and Think: A resolution III design indicates that main effects may be aliased with two factor interactions. Resolution III designs can be misleading when significant two-factor interactions affect the response.

Yellow means Proceed with Caution: A resolution IV design indicates that main effects may be aliased with three-factor interactions. Two-factor interactions may be aliased with other two-factor interactions. Resolution IV designs are a good choice for a screening design because the main effects will be clear of two-factor interactions.

Green means Go Ahead: Resolution V (or higher) designs are just about as good as a full factorial, while saving some runs. There is an assumption that main effects and two-factor interactions can adequately model the response surface.

## Minimum Run, Resolution V Factorial Designs (Characterization)

These are a class of designs containing the minimum number of runs to estimate all main effects and all two-factor interactions (Resolution V) while maintaining treatment balance for each factor. If any of the runs are not completed (resulting in missing data) then the resulting design will be Resolution IV.

Because they are minimum run designs, the power to detect small effects will be limited.

Because this is an irregular fraction, the significance of effects depends on the other terms in the model. As terms are added to or taken away from the model during analysis, the ANOVA and model selection plots will change accordingly.

## Irregular Rev V Fraction Design

These are two-level resolution V designs that use unusual fractions like 3/4, 3/8, etc. of the number of runs that a full factorial would need. These fractional designs can fit a model that includes the linear and two-factor interaction terms for all factors.

Because this is an irregular fraction, the significance of effects depends on the other terms in the model. As terms are added to or taken away from the model during analysis, the ANOVA and model selection plots will change accordingly.

## Minimum Run, Resolution IV Factorial Designs (Screening)

These designs allow all main effects to be estimated, clear of two-factor interactions. The two-factor interactions will be aliased with each other. Like all minimum run designs, they are extremely sensitive to missing data. Even one missing data point will result in aliasing and cause the design to become Resolution III. To protect against this possibility, the default option is to include two extra runs to protect against one botched run and improve the power of the design.

Because they are minimum run designs, the power to detect small effects will be limited.

Because this is an irregular fraction, the significance of effects depends on the other terms in the model. As terms are added to or taken away from the model during analysis, the ANOVA and model selection plots will change accordingly.

## Plackett-Burman Design

This item generates a set of saturated screening designs based on the Plackett-Burman structures. Since these are generally run as Resolution III designs, you must assume the absence of interactions, otherwise you should choose a higher resolution two-level factorial design. Plackett-Burman designs are useful for ruggedness testing (validation) where you hope to find little or no effect on the response due to any of the factors. Plackett-Burman designs are not recommended for Screening when there is the possibility that two-factor interactions exist. These interactions, if present, will bias the main effect estimates and can cause serious analysis problems.

Stat-Ease offers a selection of Plackett-Burman designs. The number of factors allowed is up to one less than the number of runs (for example 11 factors in 12 runs.) Choose the design with the number of factors equal to or just larger than the number you actually have. Fill in the factor names, units, type, and actual low and high levels. Factors can be specified as numerical or categorical. If you have unused factors, leave them lettered, or name them dummy 1, dummy 2, etc. Enter the response names and units.

If a “dummy” factor appears significant on the probability plot of effects, an aliased interaction is likely the culprit.

Note

Plackett-Burman designs have extremely complex alias structures. Use ONLY if your process doesn’t have interactions, or if you are doing ruggedness testing and don’t expect to find significant effects.

Plackett-Burman (PB) designs are built “saturated” in Stat-Ease because an unsaturated PB creates interactions that are non-orthogonal to the main effects. This can cause serious problems in the alias structure, even to the extent that partial aliasing will occur with coefficients possibly greater than 1.0. Don’t delete the dummy columns.

## Taguchi Designs

Taguchi designs are a type of factorial design. Design options are available with differing numbers of factors and levels. Stat-Ease, Inc. encourages the use of standard Factorial, Multilevel Categoric, or optimal (custom) designs, because these may provide you with additional flexibility and a less complex alias structure. We do not recommend the use of Taguchi design given more modern design and analysis techniques.

To use the Taguchi designs, first pick the design you need from the pull-down list. Then click on Next to inspect the alias structure for this design. It is likely to be very complex. On the next screen, enter your response names. Then click on Next for the upfront power and then Finish to create the design layout.

Stat-Ease sets up saturated Taguchi designs. Use Taguchi’s linear graphs (available in Taguchi textbooks) to determine which columns are used and which are eliminated from the analysis.

The analysis of Taguchi designs is done using standard analysis of variance techniques.

## Multilevel Categoric Design

The multilevel categoric (general factorial) design allows you to have factors that each have a different number of levels. It will create an experiment that includes all possible combinations of your factor levels. All factors should be categoric (i.e. batch type, tool type, process method) rather than numeric.

Note

If you have numeric factors, it may be more efficient to set up a response surface design with added categoric factors.