Orthogonal Bases for Different Levels

In general, if an /-level factorial design is of interest, then the orthogonal basis for this factorial design will be formed by the set of polynomials from 0 to / - 1, evaluated at each of the treatment levels. The zeroth-order polynomial, Lq, will be used for the zero-order interaction (/ o)- The first-order polynomial, Lj, will be used for the factors whose powers are 1. Similarly, the second-order polynomials, L2, will be used for the factors whose powers are 2. The results can be obtained by solving Eqs. (4.35) and (4.36). In the following sections, the results for 1 = 2, 3, and 4 will be provided, as well as generalised orthonormal polynomials for 1 = 2 and 3. Examples will be provided as appropriate. 

Consider first the situation of determining the orthogonal basis for the simplest factorial design with two levels. Since the two treatment points are —1 and 1, and the basis function has the form 

Note that, by condition (4.35), Yn+Yi2 = 0. This gives that Eq. (4.41) can be written as 

consider the case of determining the orthogonal basis for the factorial design with three levels. The three equispaced coded treatment points are —1, 0, and 8i= ). Ignoring the zero-power basis function, there will be two additional basis functions of general form  

For the factors raised to the first power, the treatment coefficients can be written as 

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