# Full Factorial Designs

Full factorial experiments are used if the KPIVs have two or more discrete levels. These designs use linear transformation equations, which means that, although a low level and a high level of a KPIV are run in the model (this is a type of regression model), we cannot interpolate between the discrete factor levels. Figure 9.35 provides an example of a full factorial experimental design that optimizes monthly sales. Price level is the most significant variable for predicting monthly sales. This is confirmed by its probability value of -0.00 in the ANOVA table. Recall that a variable is statistically significant if its probability value (of not making a Type I error) is less than 0.05 (or 5%). The other variables all have probability values greater than 0.05 and are not statistically significant. The factor plots show large discrete changes in monthly sales as the price level changes. In contrast, the other factors show no change as price level increases.

FIGURE 9.35

Full factorial designs.

price level will remain an important predictor of monthly sales. This design shows the impact of each KPIV on the KPOV independently as well as in combination with other KPIVs (these are interactions between the independent variables). Interactions exist if combinations of KPIV levels cause unusual changes to the level of a KPOV. In fact, in some models, the KPIVs by themselves may not be significant, but combinations of the KPIVS may be statistically significant. In this example, the

FIGURE 9.36

2k Design.

interaction between Industry and Price Level is statistically significant with a probability value of 0.031.

# Fractional Factorial Designs

Fractional factorial designs are a special type of 2k experimental designs in which not all combinations of the KPIVs or factors are evaluated in an experiment. Fractionation can be very useful in situations with many KPIVs because some of their higher-order interaction information is usually not practically useful. This is shown in Figure 9.37, in which a 23 experimental design containing three KPIVs was fractionated into two parts, or a one-half fraction, for experimentation. Fractionation saves experimental resources, and it reduces the information obtained from an experiment. As an example, in Figure 9.37 we saw that, prior to fractionation, each of

FIGURE 9.37

Fractional designs.

the KPIVs (i.e., A, B, and C) and their interactions AB, AC, BC, and ABC had different patterns of +1 and -1 levels. After fractionation, factors A and BC, factors В and AC, and factors C and AB had similar patterns. In other words, we cannot know if the change in the KPOV will be caused by level changes in either factor in these factor pairs. This fractionalization is also called aliasing because some variables are indistinguishable relative to their effect on the KPOV. It is also called confounding because there is confusion related to which factors are affecting the KPOV (i.e., the main factor or its interaction). The interpretation of fractional factorial designs is like full factorial interpretation, except not all KPIV and interaction terms are required. Screening designs are created when the fractionalization is continued to a point that all KPIVs are aliased with lower level interaction information. In these highly fractioned experimental designs statistically insignificant variables and their aliases can be eliminated from the analysis to save experimental resources. Then the statistically significant variables and their aliases can be studied as full factorial designs. The number of overall experiments is reduced with the reduction in KPIVs studied.

# Response Surface Designs

The previously discussed experimental designs were linear relative to relationships between the KPIVs and the KPOV. In contrast, response surface models explain curvilinear relationships between one or more of the KPIVs and the KPOV. Figure 9.38 shows a relationship between adhesion, which is the KPOV, and the KPIVs of temperature and pressure. The KPIV terms are of the form X2 and X. There are also several versions of these models, with the central composite and Box-Behnken designs being the two most popular. Figure 9.38 shows that a central composite design can be built from a 2k level design using axial points if a curvilinear relationship is found by running an intermediate center point in the 2k level design.