Nonregular designs

When a regular design is used, any two factorial effects (main effects or interactions) are either orthogonal or are completely aliased. Partial aliasing of factorial effects makes the alias structures of nonregular designs much more complex (see Chapter 8 for information on the difficulties in the analysis when complex aliasing is present), but it also leads to some superior projection properties. 

The tests should be repeated thereafter at regular or nonregular intervals. Therefore, any test case should be designed for reuse using sets of data inputs and known outputs. If the program is used infrequently, it is good practice to run the tests each time the program is used. Spreadsheet validation is discussed in detail in Chapter 18 and in Ref. [11]. 

Projection properties of designs with more than two levels have been relatively unexplored. Cheng and Wu (2001) found a nonregular OA(27, 38, 2) such that the quadratic model can be estimated in all four-factor projections. For/factors, each with three quantitative levels, the quadratic model specifies the mean response at factor level combination (xi,..., Xf) to be... 

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. 

Cheng, C. S., Deng, L. Y., and Tang, B. (2002). Generalized minimum aberration and design efficiency for nonregular fractional factorial designs. Statistica Sinica, 12, 991-1000. 

The nonregular nature of the fractional factorial design makes it possible to consider interaction effects as well as main effects see also Chapter 7. An interaction between two factors, each with three levels, has four degrees of freedom which can be decomposed into linear x linear, linear x quadratic, quadratic x quadratic, and quadratic x linear effects. The contrast coefficients for these effects are formed by multiplying the coefficients of the corresponding main effect contrasts. 

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