Star design

Two-level factorial designs are useful for estimating first-order factor effects and interaction effects between factors, but they cannot be used to estimate additional second-order curvature effects such as those represented by the terms P, and P22 -in the model 

A different class of experimental designs, the star designs, provide information that can be used to fit models of the general form described by Equation 12.44. Models of this class contain 2kparameters, where k is the number of factors included in the model. 

Because 2k + 1 parameters were estimated from data obtained at + 1 factor combinations, there are no degrees of freedom for lack of fit (/ - p = 5 - 5 = 0) again, there was no replication in this example, so the estimated model should give a perfect fit to the data. This is verified by estimating the responses using the fitted model parameters. 

This reproduces the original data exactly, as expected. 

Other models can be fit to data from star designs. For example, the model 

These limitations can be seen by comparing the experimental procedure for a three factor, three level star design, shown in Table 5.10, with the experimental procedure for a reflected saturated fractional design, which also tests three factors at three levels, shown in Table 5.11. 

In Tables 5.10 and 5.11 the method conditions are represented by 0 and the extreme conditions by 1 and -1. The star design requires one measurement at each extreme value whereas the reflected design has two measurements. This difference becomes more marked for larger k values. Star designs can not test interaction effects. 

Central composite designs are constructed by a juxtaposition of a two level full factorial and a star design. They can be used to determine the effects of changing from the method value to the extremes and the effects of changing from extreme to extreme. 

---

As an example of the use of star designs, let us fit the model of Equation 12.44 to data obtained at locations specified by the star design in Figure 12.10. 

Inspection of the coded experimental design matrix shows that the first four experiments belong to the two-level two-factor factorial part of the design, the next four experiments are the extreme points of the star design, and the last four experiments are replicates of the center point. The corresponding matrix for the six-parameter model of Equation 12.54 is... 

Can the following design be used to fit the model expressed by Equation 12.44 Can it be used to fit the model expressed by Equation 12.15 How is this design related to a star design To a comer design ... 

A small star design within a larger factorial design... 

Figure 13.14 shows a star design that can be used to fit a two-factor FSOP model. The experimental design matrix is... 

Figure 13.14 A star design. (The replicates are Mizar and Alcor in Ursa Major.) DF = 1, DF...

Add one design point to a two-factor star design to generate a design that is sufficient to fit a full second-order polynomial model ( y, = Po + PiJCj,- + P Tj, + Pn- ii + P22 i + Pi2 iA2i + "ii)- Hint see Figure 13.11. 

Figure 5.7 A star design for three factors tested at three levels...

However, a simple UV spectroscopic method may require the testing of five factors and thus an eleven experiment star design would be sufficient. 

Full two-level factorial design. Star design. Center point. 

Scheme 3.1 Design strategies for making labeled RNA constructs. Numeration corresponds to the outline numbering in the text. Brown lines designate RNA, blue lines designate DNA oligos, stars designate fluorescent dyes.

---