A Study On The Robustness Of The Model In Fractional Factorial Design

Experiment is one of the most common activities of human beings.From the experiment of crop planting in the primitive society to the experiment of new materi-al,new medicine,and biotechnology today,experiment has always been an important way for humans to explore nature and understand the world.In order to get precise data and decrease the cost at the same time,scientific experimental design is neces-sary.The experimental design is a subject which studies how to arrange experiments,obtain data,and make reasonable analysis scientifically.As one of important branch-es of statistics,experimental design plays a vital role in comparison of processing,screening of variable,exploration of response surface,and quality control.Especially for the analysis of the problems under the multi-factor and multi-level comprehensive effects in the objective world,scientific and rational design is more important.In an experiment,each impact factor,or we say experimental factor is usually set at multiple levels,and a combination of different levels among the factors is called a treatmen-t.The collection of a number of treatments forms the experiment.The experiment with all the treatment combinations is called a complete experiment.Since the scale of the complete experiment will increase sharply with the increasing of the number of factors and levels,only a part of the treatment is selected in practice which is called fractional factorial design.The fractional factorial design has always been a focused content in the research of experimental design.Since Fisher established the subject of experi-ment design at the beginning of the twentieth century,many optimal design criteria for selecting fractional designs have been raised.The typical design criteria with deep ap-plication and research include the maximum resolution criterion(MR),the minimum aberration criterion(MA),and the maximum estimated capacity criterion(MEC),clear effect criterion(CE),general minimum lower order confounding criterion(GMC),etc.These studies are all based on the principle of effect hierarchy,giving priority to the low-order effects especially the estimability of the main effect and the second-order interaction effect.In the case the model is unknown,a good design could provide the experimenter with more optional robust models.The robustness of the model here refers to the lighter confounding between the main effects and the second-order interaction effect-s and the second-order interaction effects each other in the model.The maximum estimated capacity criterion maximizes the number of estimable models from the per-spective of fully ensuring the diversity of the model;The other typical optimal design criteria get the aliased features of the designs themselves,extracting the related infor-mation first,and establish the Word Length Pattern(abbreviated as WLP)and the Aliased Effect Number Pattern(abbreviated as AENP),and then choose the optimal design accordingly.While the AENP extracts the aliased information contained in the design in a more detailed manner,which allows the experimenter to arrange the ex-periment in a more reasonably way according to some existed prior information.One focus of this paper is to propose the concept of model quality by means of AENP un-der the effect hierarchy principle,and divide models into three classes.According to the quality of the model,the maximum quantity that can be estimated for each class of model is found from high order to low order.By comparing the robustness of the design,we found that GMC design have the best robustnessAnother focus of this paper is model discrimination of design.For the previous optimal design criteria mainly focus on the maximum number of estimable models without considering the difference between the models,from the perspective of model space,the problem of the model discrimination about regular fractional factorial design with two levels is studiedThe main innovations of the research include the following aspects:(1)For two level fractional factorial design,the second-order interaction effects with different aliased degree will have different influence on the estimation and predic-tion of the model.The concept of model quality is proposed to describe the robustness of the design.By means of the aliased information between various order effects given in AENP,combined with the number of second-order interaction effects contained in the model,all of models are classified.The upper bounds of the estimatable models contained in each class are given.(2)Based on the classification of models,a new statistical pattern,the robust mod-el number pattern(RMNP)is established based on the number of models with different classes provided by a fractional factorial design.Based on this pattern,according to the principle of effect hierarchy,a new optimal design criterion-the best model robust-ness design criterion is proposed,and the fractional factorial design that achieves the best robustness is named as the best robust design.(3)Based on the research of Zhang et al.[85],the key parameters {|Ci|,i=0.1,…,l-1} in the calculation formula of best robustness model number are given in the GMC and MEC designs with 16-run and 32-run.The number of estimatable second-order interactions with different aliased degree in GMC and MA designs of 16-run,32-run and 64-run are calculated and compared.(4)A method of computer simulation is established,which directly demonstrates that the second-order interaction with high-order confounding results in a larger es-timation bias than lower ones,which also shows necessity of the proposal on model quality.(5)Paying attention to maximizing the number of estimatable model,the problem of the difference between these estimatable models is studied,that is,the problem of model discrimination of design.Starting from the measurement of the space difference between the two model arrays,six optimal criteria of designs are established which can be used to evaluate the identification ability of design to models,and apply them into the two-level regular fractional factorial design,getting some conclusions.(6)Calculate all values of non-isomorphic two-level regular fractional factorial designs with 16-run,32-run and 64-run by MATLAB under six optimal model dis-crimination criteria in case of including one and two second-order interaction effects in the model,find the corresponding optimal design,and compare with GMC and MA design.At the end of the thesis,based on the summary of the full text,the next research direction and content are prospected.