A New Aliasing Pattern And Its Applications

For analyzing fractional factorial2m-p designs,Cheng and Tang(2005)intro-duced a ANOVA model Y=γ0I+W1γ1+W2γ2+…+WJγJ+ε,where γ1is the set of effects we want to estimate.We Suppose that we can divide the remaining effects into J-1groups based on some prior knowledge,denoted by γ2,…,γJ.It is also Supposed that the effects in γj are more important than those in7j+1. Let W=(A3,...,A%)be Wordlength Pattern(WLP)of design,based on the WLP,they introduced a new pattern vector(N2,...,NJ)which has some relationships with W=(A3,...,Ak).It was proved that in some special situations, sequentially minimizing the vector is equivalent to sequentially minimizing the WLP.In order to overcome the limitations of existing optimality criteria and explore a more wide and new optimality theory with applications,Zhang,Li,Zhao and Ai(2008)introduced a new aliasing pattern for classifying regular designs,called Aliased Effect Number Pattern(AENP),denoted by{#iC(k),i,j=0,…,m,k=0,...,Kj).They proved that all the existing criteria can be expressed by functions of AENP,which indicates that all the existing optimality criteria can be unified by the pattern AENP.Especially,based on the AENP,they proposed a new optimal-ity criterion,called General Minimum Lower Order Confounding(GMC)criterion and established a kind of optimal designs,called GMC designs,which filled the gaps there were no optimal designs for the situation that experimenter has a prior about importance order of factors.In recent year the GMC theory has got a rapid development,including many aspects,such as construction of GMC designs,ex-tensions of GMC theory to multi-level design,blocked design,robust parameter design,split-plot design,non-regular design and so on. In this paper, by using AENP, based on the work of Cheng and Tang (2005) we further analyze the relationships between the vector pattern (N2,...,NJ) and WLP, and propose a new vector pattern, denoted by (N02,...,Nk22,..., NOJ,..., NkJJ). This vector pattern is a further expression of the original vector pattern (N2,…, NJ). In terms of the AENP we obtain some meaningful result relating to the relation of the new vector pattern with WLP and (N2,...,NJ). Moreover, we propose a new method to select deigns by sequentially maximizing the components of (N02,…, Nk22,..., N0J,..., NkJJ). This approach is more convenient for experimenter to find MA-type optimal designs. Also, we give an algorithm for computing the new vector pattern for several cases of{γi,1≤i≤J} taking different sets. At the last, we apply the new pattern into the case of blocked fractional factorial designs.