Fractional factorial split-plot design has been widely used in many fields due to its advantage of saving experimental cost. The general minimum lower order confounding criterion is usually used as one of the attracti...Fractional factorial split-plot design has been widely used in many fields due to its advantage of saving experimental cost. The general minimum lower order confounding criterion is usually used as one of the attractive design criterion for selecting fractional factorial split-plot design. In this paper, we are interested in the theoretical construction methods of the optimal fractional factorial split-plot designs under the general minimum lower order confounding criterion. We present the theoretical construction methods of optimal fractional factorial split-plot designs under general minimum lower order confounding criterion under several conditions.展开更多
It is very powerful for constructing nearly saturated factorial designs to characterize fractional factorial (FF) designs through their consulting designs when the consulting designs are small. Mukerjee and Fang emplo...It is very powerful for constructing nearly saturated factorial designs to characterize fractional factorial (FF) designs through their consulting designs when the consulting designs are small. Mukerjee and Fang employed the projective geometry theory to find the secondary wordlength pattern of a regular symmetrical fractional factorial split-plot (FFSP) design in terms of its complementary subset, but not in a unified form. In this paper, based on the connection between factorial design theory and coding theory, we obtain some general and unified combinatorial identities that relate the secondary wordlength pattern of a regular symmetrical or mixed-level FFSP design to that of its consulting design. According to these identities, we further establish some general and unified rules for identifying minimum secondary aberration, symmetrical or mixed-level, FFSP designs through their consulting designs.展开更多
Fractional factorial split-plot (FFSP) designs have an important value of investigation for their special structures.There are two types of factors in an FFSP design: the whole-plot (WP) factors and sub-plot (SP) fact...Fractional factorial split-plot (FFSP) designs have an important value of investigation for their special structures.There are two types of factors in an FFSP design: the whole-plot (WP) factors and sub-plot (SP) factors,which can form three types of two-factor interactions:WP2fi,WS2fi and SP2fi.This paper considers FFSP designs with resolution Ⅲ or Ⅳ under the clear effects criterion.It derives the upper and lower bounds on the maximum numbers of clear WP2fis and WS2fis for FFSP designs,and gives some methods for constructing the desired FFSP designs.It further examines the performance of the construction methods.展开更多
The issue of optimal blocking for fractional factorial split-plot (FFSP) designs is considered under the two criteria of minimum aberration and maximum estimation capacity. The criteria of minimum secondary aberration...The issue of optimal blocking for fractional factorial split-plot (FFSP) designs is considered under the two criteria of minimum aberration and maximum estimation capacity. The criteria of minimum secondary aberration (MSA) and maximum secondary estimation capacity (MSEC) are developed for discriminating among rival nonisomorphic blcoked FFSP designs. A general rule for identifying MSA or MSEC blocked FFSP designs through their blocked consulting designs is established.展开更多
Split-plot designs have been widely used in industrial experiments.Up to now,most methods for choosing this kind of designs are based on the minimum aberration (MA) criterion.Recently,by introducing a new pattern,call...Split-plot designs have been widely used in industrial experiments.Up to now,most methods for choosing this kind of designs are based on the minimum aberration (MA) criterion.Recently,by introducing a new pattern,called aliased effect-number pattern (AENP),Zhang et al.proposed a general minimum lowerorder confounding (denoted by GMC for short) criterion and established a general minimum confounding (also denoted by GMC for saving notations) theory.It is proved that,the GMC criterion selects optimal designs in a more elaborate manner than the existing ones,and when an experimenter has a prior about the importance ordering of factors in experiments the GMC designs are better than other optimal designs.In this paper we extend the GMC criterion to the split-plot design case and give a GMC-FFSP criterion for ranking split-plot designs.Some comparisons of the new criterion with the MA-MSA-FFSP criterion are given,and the optimal 32-run split-plot designs up to 14 factors under the two criteria are tabulated for comparison and application.展开更多
This paper discussed Bayesian variable selection methods for models from split-plot mixture designs using samples from Metropolis-Hastings within the Gibbs sampling algorithm. Bayesian variable selection is easy to im...This paper discussed Bayesian variable selection methods for models from split-plot mixture designs using samples from Metropolis-Hastings within the Gibbs sampling algorithm. Bayesian variable selection is easy to implement due to the improvement in computing via MCMC sampling. We described the Bayesian methodology by introducing the Bayesian framework, and explaining Markov Chain Monte Carlo (MCMC) sampling. The Metropolis-Hastings within Gibbs sampling was used to draw dependent samples from the full conditional distributions which were explained. In mixture experiments with process variables, the response depends not only on the proportions of the mixture components but also on the effects of the process variables. In many such mixture-process variable experiments, constraints such as time or cost prohibit the selection of treatments completely at random. In these situations, restrictions on the randomisation force the level combinations of one group of factors to be fixed and the combinations of the other group of factors are run. Then a new level of the first-factor group is set and combinations of the other factors are run. We discussed the computational algorithm for the Stochastic Search Variable Selection (SSVS) in linear mixed models. We extended the computational algorithm of SSVS to fit models from split-plot mixture design by introducing the algorithm of the Stochastic Search Variable Selection for Split-plot Design (SSVS-SPD). The motivation of this extension is that we have two different levels of the experimental units, one for the whole plots and the other for subplots in the split-plot mixture design.展开更多
Fractional factorial split-plot （FFSP） designs are useful in practical experiments. When the num- bers of levels of the factors are not all equal in an experiment, mixed-level design is selected. This paper investig...Fractional factorial split-plot （FFSP） designs are useful in practical experiments. When the num- bers of levels of the factors are not all equal in an experiment, mixed-level design is selected. This paper investigates the conditions of a resolution III or IV FFSP design with both two-level and eight-level factors to have various clear effects, including two types of main effects and three types of two-factor interaction compo- nents.展开更多
文摘Fractional factorial split-plot design has been widely used in many fields due to its advantage of saving experimental cost. The general minimum lower order confounding criterion is usually used as one of the attractive design criterion for selecting fractional factorial split-plot design. In this paper, we are interested in the theoretical construction methods of the optimal fractional factorial split-plot designs under the general minimum lower order confounding criterion. We present the theoretical construction methods of optimal fractional factorial split-plot designs under general minimum lower order confounding criterion under several conditions.
基金supported by the National Natural Science Foundation of China(Grant Nos.10231030&10571093)Specialized Research Fund for the Doctoral Program of Higher Education(Grant No.20050055038).
文摘It is very powerful for constructing nearly saturated factorial designs to characterize fractional factorial (FF) designs through their consulting designs when the consulting designs are small. Mukerjee and Fang employed the projective geometry theory to find the secondary wordlength pattern of a regular symmetrical fractional factorial split-plot (FFSP) design in terms of its complementary subset, but not in a unified form. In this paper, based on the connection between factorial design theory and coding theory, we obtain some general and unified combinatorial identities that relate the secondary wordlength pattern of a regular symmetrical or mixed-level FFSP design to that of its consulting design. According to these identities, we further establish some general and unified rules for identifying minimum secondary aberration, symmetrical or mixed-level, FFSP designs through their consulting designs.
基金This work was partially supported by the National Natural Science Foundation of China(Grant Nos.10301015 and 10571093)the Specialized Research Fund for the Doctoral Program of Higher Education of China(Grant No.20050055038).
文摘Fractional factorial split-plot (FFSP) designs have an important value of investigation for their special structures.There are two types of factors in an FFSP design: the whole-plot (WP) factors and sub-plot (SP) factors,which can form three types of two-factor interactions:WP2fi,WS2fi and SP2fi.This paper considers FFSP designs with resolution Ⅲ or Ⅳ under the clear effects criterion.It derives the upper and lower bounds on the maximum numbers of clear WP2fis and WS2fis for FFSP designs,and gives some methods for constructing the desired FFSP designs.It further examines the performance of the construction methods.
基金This work was partially supported by National Natural Science Foundation of China(Grant No.10231030)Chinese Postdoctoral Science Foundation(Grant No.20040350240).
文摘The issue of optimal blocking for fractional factorial split-plot (FFSP) designs is considered under the two criteria of minimum aberration and maximum estimation capacity. The criteria of minimum secondary aberration (MSA) and maximum secondary estimation capacity (MSEC) are developed for discriminating among rival nonisomorphic blcoked FFSP designs. A general rule for identifying MSA or MSEC blocked FFSP designs through their blocked consulting designs is established.
基金supported by National Natural Science Foundation of China (Grant No.10871104,10971107)the Specialized Research Fund for the Doctoral Program of Higher Education of China (Grant No.20050055038)
文摘Split-plot designs have been widely used in industrial experiments.Up to now,most methods for choosing this kind of designs are based on the minimum aberration (MA) criterion.Recently,by introducing a new pattern,called aliased effect-number pattern (AENP),Zhang et al.proposed a general minimum lowerorder confounding (denoted by GMC for short) criterion and established a general minimum confounding (also denoted by GMC for saving notations) theory.It is proved that,the GMC criterion selects optimal designs in a more elaborate manner than the existing ones,and when an experimenter has a prior about the importance ordering of factors in experiments the GMC designs are better than other optimal designs.In this paper we extend the GMC criterion to the split-plot design case and give a GMC-FFSP criterion for ranking split-plot designs.Some comparisons of the new criterion with the MA-MSA-FFSP criterion are given,and the optimal 32-run split-plot designs up to 14 factors under the two criteria are tabulated for comparison and application.
文摘This paper discussed Bayesian variable selection methods for models from split-plot mixture designs using samples from Metropolis-Hastings within the Gibbs sampling algorithm. Bayesian variable selection is easy to implement due to the improvement in computing via MCMC sampling. We described the Bayesian methodology by introducing the Bayesian framework, and explaining Markov Chain Monte Carlo (MCMC) sampling. The Metropolis-Hastings within Gibbs sampling was used to draw dependent samples from the full conditional distributions which were explained. In mixture experiments with process variables, the response depends not only on the proportions of the mixture components but also on the effects of the process variables. In many such mixture-process variable experiments, constraints such as time or cost prohibit the selection of treatments completely at random. In these situations, restrictions on the randomisation force the level combinations of one group of factors to be fixed and the combinations of the other group of factors are run. Then a new level of the first-factor group is set and combinations of the other factors are run. We discussed the computational algorithm for the Stochastic Search Variable Selection (SSVS) in linear mixed models. We extended the computational algorithm of SSVS to fit models from split-plot mixture design by introducing the algorithm of the Stochastic Search Variable Selection for Split-plot Design (SSVS-SPD). The motivation of this extension is that we have two different levels of the experimental units, one for the whole plots and the other for subplots in the split-plot mixture design.
基金Supported by the National Natural Science Foundation of China(Nos.10901092,11171165,11171188)Shandong Provincial Scientific Research Reward Foundation for Excellent Young and Middle-aged Scientists(BS2011SF006)Program for Scientific Research Innovation Team in Colleges and Universities of Shandong Province
文摘Fractional factorial split-plot （FFSP） designs are useful in practical experiments. When the num- bers of levels of the factors are not all equal in an experiment, mixed-level design is selected. This paper investigates the conditions of a resolution III or IV FFSP design with both two-level and eight-level factors to have various clear effects, including two types of main effects and three types of two-factor interaction compo- nents.
