Projection, Search, and Optimality in Fractional Factorial Experiments.

摘要

We propose a general Up-Down method to search for efficient 2 m fractional factorial designs in fitting a class of models when the number of factors is m, and the number of runs is n. The orthogonal array designs exist for some specific values of n. The orthogonal array designs are optimal under the resolution assumptions. The proposed Up-Down method searches for efficient designs having the number of runs in between two values of n for orthogonal array designs satisfying a resolution assumption. We present the efficient resolution III designs obtained by the Up-Down method for 3 ≤ m ≤ 10 and a range of practical values of n. While many of these designs are found to be the global optimal resolution III designs by exhaustive computer search, the other designs are near global optimal designs. For m=4 and 5, we compare our designs with the optimal resolution III+k (k=0,1,2,...) designs in Ghosh and Tian (2006). Moreover, we utilize the method to obtain unbalanced Up-Res V designs performing slightly better than the balanced optimal fractional factorial designs (BOFFD) given in Srivastava and Chopra (1971) with respect to A- and D-optimality criteria. For a given n, all our designs are isomorphic having same optimality properties. For general m and n, the conditions are derived for obtaining such isomorphic designs with respect to Trace and Determinant.;Several interesting projection properties are known in the literature for orthogonal arrays and in particular for the Plackett-Burman (PB) designs. In this dissertation, the projection properties are investigated for both orthogonal and non-orthogonal array designs under different model assumptions. The structure of the variance-covariance matrix for the estimates of the model parameters is characterized. The optimality properties of these designs are also investigated. For m=5, we consider seven 12-run designs di,i=1,...,7 and a collection of classes of models. The designs di,i = 1,...,5 are balanced arrays of full strength, d6 and d 7 are orthogonal arrays of strength 2. The designs d 6 and d7 are two non-isomorphic designs obtained from the PB design by projecting 11 factors onto 5 factors. Overall, our designs d1 and d3 are at the top of their performances. By projection, all possible t (≤ m) factors out of m factors are considered. As t increases from 2, to 3 and 4, the design d1 becomes better and better compared to the design d3. When t=5, the design d 3 is optimal under resolution III model. For fitting resolution III plus k (k=1,2,3) models, the design d1 again becomes better and better compared to the design d3 as k increases.