Factorial design projection

Projection Properties of Factorial Designs for Factor Screening... 

Table 1 displays an orthogonal array of strength three because it contains each of the 23 = 8 level combinations of any set of three factors exactly twice. In other words, its projection onto any set of three factors consists of two replicates of the complete 23 factorial design. 

Bulutoglu, D. A. and Cheng, C. S. (2003). Hidden projection properties of some nonregular fractional factorial designs and their applications. Annals of Statistics, 31, 1012— 1026. 

Chen, H. (1998). Some projective properties of fractional factorial designs. Statistics and Probability Letters, 40, 185-188. 

We can make a projection of the original design down to the space spanned by the remaining variables. The principles are Illustrated in Fig. 6.12, in which a 2 fractional factorial design is projected down to the planes spanned by the remaining variables. 

Fig.6.14 A fractional factorial design can be projected down to a lower-dimensional variable space if it should turn out that some variables are insignificant.

The factorial design 2 may be represented by a cube, the 8 experiments being situated at each corner (figure 3.5a). To study the interaction Pj, between the variables and we project the corners of the cube on the (Xj, Xj) plane to give a square, and calculate the mean value of the responses at each comer of the square, as shown in figure 3.5b. 

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