Factorial designs notation

A "matrix design" is a suitable way of obtaining all the treatment combinations implicated in a 2 factorial design, but it is not a handy system to notate... 

The first designs for mixture experiments were described by Scheffe [3] in the form of a grid or lattice of points uniformly distributed on the simplex. They are called q, i j simplex-lattice designs. The notation q, v implies a simplex lattice for q components used to construct a mixture polynomial of degree v. The term mixture polynomial is introduced to distinguish it from the polynomials applicable for mutually independent or process variables, which are described later in our discussion of factorial designs (section 8.4). In this way, we distinguish mixture polynomials from classical polynomials. 

One of the most useful symmetrical designs is based on the 4 factorial design (table 2.17), described (using the previous notation) as 4V/4 5 factors at 4 levels are screened in 4 = 16 experiments. We again emphasize that the numbers 0, 1,2, and 3 identify qualitative levels of each variable and have no quantitative significance whatsoever. 

This notation 2 for the half factorial design for 4 variables at 2 levels indicates the fractional nature of this design. 

The columns X, Xi and X2 can be seen to be identical for the factorial experiments. Following the same reasoning as in chapter 3, we conclude that the estimator bo for the constant term, obtained from the factorial points, is biased by any quadratic effects that exist. On the other hand the estimate b o obtained only from the centre point experiments is unbiased. Whatever the polynomial model, the values at the centre of the domain are direct measurements of Pq. So the difference between the estimates, bo - b o, (which we can write as U+22 using the same notation as in chaptw 3) is a measure of the curvature P, + P22- The standard deviation o o is s/v 8 (as it is the mean of 8 data of the factorial design) and that of b o is s/ /2. (being the mean of 2 centre points). We define a function t as ... 

Further reduction of the number of observations can be achieved by employing the fractional factorial design (FFD). The notation 2 is used to denote a 2 fraction of a 2 fractional design. See Table 8 for the one-half fraction of the 2 FD of the weight-watch experiment. 

This notation indicates that the design has four factors, each at two levels, but we perform only eight runs. The presence of the -1 superscript means the full factorial was divided by 2. If it had been divided into four parts, the exponent would be 4—2. 

It will be assumed that a 2 -factorial experiment has been designed with rij full replicates. Furthermore, it will be assumed that all the factors have been coded so that —1 and 1 represent the upper and lower levels in the experiment. The same notation as presented in Chap. 4 will be used. Thus, instead of calculating inverses and transposes, the following simplifications work for a 2 -factorial experiment ... 

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