Some Studies Of The General Minimum Aberration Theory

The regular fractional factorial designs at two levels are well known and widely used in experimental designs.Such designs have a simple aliasing structure in that any two effects are either orthogonal or fully aliased.The effect hierarchy principle is one of the most important principles in experimental design(Wu and Hamada(2000)).It states that in a design,lower-order effects are more likely to be important than higher-order ones and effects with the same order are equally like to be important.Therefore,in order to choose good designs,we should minimize the confounding between the lower-order effects.Towards this aim,many criteria have been advised in the literature,of which the minimum aberration(MA) criterion proposed by Fries and Hunter(1980) and the clear effects(CE) criterion proposed by Wu and Chen(1992) received most attention in the past two and half decades. For more details,we refer to Chen,Sun and Wu(1993) and Mukerjee and Wu(2006).Recently,through introducing an aliased effect-number pattern(AENP),Zhang, Li,Zhao and Ai(2008) proposed a general minimum lower-order confounding(GMC) criterion for selecting two-level regular designs,which can really reach the aim in a very elaborate and explicit manner.The optimal design under the GMC criterion is called a GMC design.Since the AENP reveals the confounding information of designs,all kinds of investigations based on the AENP,together with GMC itself,are called GMC theory.They proved that all the existing optimal criteria are functions of the AENP. Thus,the AENP can manage all kinds of optimal criteria.For example,the word-length pattern(WLP),as the core of MA criterion,is a function of AENP.The same also holds for the clear effects(CE) criterion.They also proved that,the GMC design must be the best one among the optimal designs under the CE criterion,and the GMC criterion works for any case,in spite of a design containing clear effects or not.Furthermore,the construction of GMC designs becomes very easy,for more details,we refer to Zhang and Mukerjee(2009),Li,Zhao and Zhang(2008),Zhang and Cheng(2008) and Cheng and Zhang(2008)). The main purpose of this dissertation is to investigate some further properties and applications of GMC theory.First,we prove that a GMC design minimizes A₃,the leading term of word-length pattern,in all the designs with same parameter.This strictly reveals that minimizing A₃ is a necessary condition for the purpose of selecting optimal designs under the effect hierarchy principle.Furthermore,we find out the uniquely optimal confounding structure form between main effects and 2fi's,possessed by resolutionâ…¢designs with and only with minimum A₃.The form reveals the essential of the minimum A₃ and is shared by both GMC and minimum aberration designs.Based on GMC theory and the information matrices for estimating main effects and two-factor interactions(2fi's),we propose the definition of minimal sufficient information for estimating main effects and 2fi's.The newly proposed concept shows that GMC is the optimal criterion in selecting regular two-level fractional factorial designs under the condition that three and higher order interactions are negligible.Particularly, this concept has many important applications.For the ease of computation,the(M,S) procedure proposed by Eccleston and Hedayat (1974) has been widely used and advocated in optimal design literature,of which many examples of theory and applications of(M,S)-optimality were discussed by Shah and Sinha(1989).Cheng,Deng and Tang(2002) and Mandal and Mukerjee(2005) also studied(M,S)-optimality in factorial designs.Recently,(M,S)-optimality was proposed by Qu,Kushler and Ogunyemi(2008) in selecting two-level factorial designs. Both Jacroux(2004) and Qu,Kushler and Ogunyemi(2008) considered the connection between the(M,S) and MA criteria for two-level regular designs of resolutionâ…¢or higher.They showed that for designs of resolutionâ…£or higher,MA designs must be (M,S)-optimal.Furthermore,Qu,Kushler and Ogunyemi(2008) showed that all designs of resolutionâ…¢up to 64 runs are also(M,S)-optimal.However,for designs of resolutionâ…¢with N(＞64) runs,whether an MA design is(M,S)-optimal is still under consideration.As an important application of the minimal sufficient information for estimating main effects and 2fi's,we show that minimum aberration(MA) designs must be(M,S)- optimal designs.In addition,by the minimal sufficient information for estimating main effects and 2fi's,we show that sequentially minimizing M_(1,2)1,M_(2,2)2 and M_(2,2)1,as the core of the minimum M-aberration criterion proposed by Zhu and Zeng(2005),is equivalent to sequentially minimizing A₃ and A₄.One model robustness criterion considered in Cheng,Steinberg and Sun(1999) is estimation capacity.Cheng,Steinberg and Sun(1999) showed that minimum aberration is a good surrogate for maximum estimation capacity(MEC) and that the two criteria often produce quite consistent results.But,they did not discuss the further connection between the MEC and MA criteria in theory.By the minimal sufficient information for estimating main effects and 2fi's,we show that sequentially minimizing A₃ and A₄ is equivalent to sequentially maximizing the first two components of the maximum estimation capacity,i.e.,E₁(d) and E₂(d), defined in Cheng,Steinberg and Sun(1999).Thus,we have almost completely explained why the MEC and MA criteria often produce quite consistent results since most of the MA designs are uniquely determined by sequentially minimizing A₃,A₄.Chen and Cheng(2006) discussed the method of doubling for constructing two-level regular designs of resolutionâ…£.By the minimal sufficient information for estimating main effects and 2fi's,we give a more easily verified necessary and sufficient condition for the maximal designs of resolutionâ…£or higher,and consider some further properties of doubling,which is a generalization of a key result,Theorem 3.3,of Chen and Cheng(2006).The theorem there only discussed the maximal designs with resolutionâ…£or higher,but our result here has no such limitation.