and V is a p p positive definite matrix. A normal-inverse-gamma distribution.

Transcription

1 OSR Journal of athematcs (OSR-J) e-ssn: , p-ssn: X. Volume 3, ssue 3 Ver. V (ay - June 07), PP Comparng The Performance of Bayesan And Frequentst Analyss ethods of rregular Fractonal Factorals Usng Desgn Based optmalty And Effcency Crtera * Al H. U., Lass K. E., Nwaosu S. C. * Department of athematcs, Unversty of Jos, Ngera Department of athematcal Scences, Abubakar Tafawa Balewa Unversty, P.. B. 048, Bauch, Ngera. Department of athematcs/statstcs,unversty of Agrculture, akurd, Ngera Abstract: rregular fractonal factoral desgns are wdely used n screenng experments. Several analyss methods were proposed for dentfyng man effects and ther nteractons n the past twenty years. The goal of ths research was to evaluate the performance of two Bayesan and two Frequentst approaches usng some real data. The two Bayesan approaches are Bayes screenng method and Emprcal Bayes method. The two Frequentst approaches are tradtonal stepwse varable selecton method and least angle regresson method. The comparson of analyss methods s demonstrated usng A & D optmalty and A & D effcency, the results obtaned showed that the Frequentst approach s the best. We recommend the use of Frequentst approach n analyzng rregular fractonal factoral desgns at two levels because t works wth current data and takes nto consderaton that varablty comes from our samplng. Keywords: Bayesan, Frequentst, rregular, and Screenng.. ntroducton Screenng s the process of dscoverng, through statstcal desgn of experments and modelng, those controllable factors or nput varables that have a substantve mpact on the response or output whch s ether calculated from a numercal model or observed from a physcal process. [] n many scentfc nvestgatons, the man nterest s n the study of effects of many factors smultaneously. Factoral desgns, especally twolevel or three-level factoral desgn, are the most commonly used expermental plans for ths type of nvestgaton. A full factoral experment allows all factoral effects to be estmated ndependently. However, t s often too costly to perform a full factoral experment, so a fractonal factoral desgn, whch s a subset or fracton of a full factoral desgn, s preferred snce t s cost-effectve. [] rregular two-level fractonal factoral desgns such as Plackett-Burman desgns are becomng ncreasngly popular choces n many felds of scentfc nvestgaton due to ther run sze economy and flexblty. The run sze of rregular two-level factoral desgn s a multple of 4. They fll the gaps left by the regular two-level fractonal factoral desgns whose run sze s always a power of.e. 4,8,6, n rregular factoral desgns each man effect s partally confounded wth all the two-factor nteractons not nvolvng tself. Because of ths complex alasng structure, rregular factoral desgns had not receved suffcent attenton untl recently. [3]. aterals And ethods The analyss was carred out usng Optex procedure of the SAS software package and R software package. The Emprcal Bayes ethod: The emprcal work of [] has demonstrated that emprcal Bayes method makes use of lkelhood functons and conjugate prors for estmatng the varances at hgh and low levels respectvely. A conjugate pror s gven by f d d p d V exp m V m, () Wth >0, d>0, m and V s a p p postve defnte matrx. A normal-nverse-gamma dstrbuton. The Bayes Screenng ethod: The Bayes screenng method proposed by Box and eyer (986) [] use the margnal posteror probablty to dentfy actve factors and effects. n the screenng approach, models wth dfferent combnatons of man effects and nteractons are created. We consder a set of models DO: / Page

2 Comparng The Performance Of Bayesan And Frequentst Analyss ethods Of rregular 0,..., m. Each model has a parameter Y, Y PrY, Pr d Pr. The pror probablty of model Pr. The predctve densty of Y s denoted as such that the probablty densty of response Y s Pr and the pror probablty densty of s s Pr for all. () The posteror probablty of model N gven Y s then calculated by the followng equaton: Pr PrY Pr Y (3) Pr Pr Y allj j j The Tradtonal Frequentst ethod: Ths method [4] makes use of the half-normal probablty plot to dentfy sgnfcant man effects based on the assumptons of effect sparsty and effect heredty. Effect sparsty s the Pareto Prncple n expermental desgn states that only a small number of effects n the experment are relatvely mportant. oreover, effect heredty prncple descrbes that an nteracton s actve only f at least one of ts component man effects s consdered as mportant. The Least Angle Regresson (LARS) ethod: Ths approach [5] consders three levels of heredty prncples: no heredty, weak heredty and strong heredty. For the purpose of ths work, the strong heredty wll be consdered snce here both of ts component factors have to be sgnfcant. Under the strong heredty prncple, f an nteracton term s selected, the correspondng parent effects must be ncluded. One can calculate the average predctablty of all varables that must be ncluded.. Results And Dscusson Table (): A -run Plackett-Burman Desgn generated from ntap Software RUN A B C D E F G H J K RESPONSE A-EFFCENCY FOR THE FREQUENTST APPROACH A Effcency 00 N Dtrace X X p D-EFFCENCY FOR THE FREQUENTST APPROACH D Effcency 00 N X X D 00 p (4) (5) DO: / Page

3 Comparng The Performance Of Bayesan And Frequentst Analyss ethods Of rregular A optmalty trace D optmalty X X X X 7.430e A-OPTALTY FOR THE FREQUENTST APPROACH X X A optmalty trace (6) D-OPTALTY FOR THE FREQUENTST APPROACH D optmalty X X 7.430e (7) D-EFFCENCY FOR THE BAYESAN APPROACH The Optex procedure of the SAS software was used to obtan the effcences and optmaltes and results presented thus; Bayesan D-effcency, X X, A-EFFCENCY FOR THE BAYESAN APPROACH trx, true X X Bayesan A-effcency, X true D opt, true (8) Aopt, true (9) D-OPTALTY FOR THE BAYESAN APPROACH Bayesan D-optmalty, (0) X detx X R e 0 A-OPTALTY FOR THE BAYESAN APPROACH Bayesan A-optmalty, X tr X X R.04 where R s a p x p known matrx () Table (): The Optex Procedure results for Bayesan Effcences Standard Error Desgn Number D-effcency A-effcency * * DO: / Page

4 Comparng The Performance Of Bayesan And Frequentst Analyss ethods Of rregular Table (3): Results Of Analyss Summary of results ethod A-effcency D-effcency A-optmalty D-optmalty Frequentst e+ Bayesan e+0 V. Dscusson From table (3) above, the values of A-effcency and D-effcency for the Frequentst approach are hgher compared wth the Bayesan approach, ths shows that the Frequentst s more effcent. From table (3) above, the values of A-optmalty and D-optmalty for the Frequentst approach are hgher when compared wth the Bayesan approach, and ths showed that the Frequentst approach s a better method of analyzng rregular fractonal factoral desgns. Ths work has successfully compared the Bayesan and Frequentst methods of analyzng rregular fractonal factoral desgns usng the desgned based optmalty crtera at two levels. Accordng to the numercal results obtaned usng SAS, ntap and SPSS statstcal software, we nfer that: The Frequentst approach s more A-optmal than the Bayesan approach n the analyss of rregular fractonal factoral desgns at two levels as shown above. The Frequentst approach s more D-optmal than the Bayesan approach n the analyss of rregular fractonal factoral desgns at two levels as shown above. The Frequentst approach s more A-effcent than the Bayesan approach n the analyss of rregular fractonal factoral desgns at two levels as shown above. The Frequentst approach s more D-effcent than the Bayesan approach n the analyss of rregular fractonal factoral desgns at two levels. V. Concluson n ths work, we have successfully dscussed the two Bayesan methods: Bayes screenng and Emprcal Bayes methods, and appled them to a two level Plackett-Burman desgn of -runs. We also dscussed the two Frequentst methods we consdered n ths work: Least Angle Regresson and the Tradtonal Frequentst approaches and appled them to a two level Plackett-Burman desgn of -runs. Bayesan approaches are always straghtforward but complex and dffcult to understand. Furthermore, they are not flexble. The Emprcal Bayes method s more flexble and effcent than the Bayes screenng method as the emprcal Bayes method consders addtonal prors and the model space n more detal. t was used to obtan the estmates of the responses. The LARS approach s a fast and effcent strategy to dentfy mportant man effects. Ths approach s a good choce to deal wth data sets wth a large number of factors. Based on the analyss we performed, the results of A & D optmalty as well as A & D effcency showed that the Frequentst approach s a better method of analyzng a -run rregular fractonal factoral desgns at two levels. The Frequentst approach s more practcal and logcal snce t seems reasonable that these events have a specfc probablty and that the varaton s n our samplng. ore so, most data analyss from studes are usually done usng the Frequentst approach (.e. confdence nterval, hypothess testng wth p value) snce t s easly understandable. Hence, when one s usng a Frequentst method the expermenter s makng predctons on underlyng truths of the experment usng only data from the current experment. We recommend the use of Frequentst approach n analyzng rregular fractonal factoral desgns at two levels because t beleves n workng wth current data and takes nto consderaton that varablty comes from our samplng. From fgure below, we observed that seres of lambda were set from 0 to wth the correspondng crossvaldated mean squares error, t was observed that lambda L-norm between 0.6 to 0.8 proved to be the best wth the mnmum mean squares error. The lasso algorthm automatcally chose the lambda L-norm parameters to shrnk the parameters towards zero due to the presence of multcollnearty n the covarates A to K. and fgure also confrm the result of fgure. n fgure 3, t was observed that factor G has the least probablty whereas factor D has the hghest probablty, ther contrbutons are n order of the probablty. References []. Box, G.E.P. & eyer, R.D. (993): Fndng the Actve Factors n Fractonated Screenng Experments; Journal of Qualty Technology. []. Shek, Y. W. (0): Comparson of Analyss ethods for Non-regular Fractonal Factoral Desgns; Unversty of Calforna, Los Angeles. [3]. Woods, D. C. & Lows S.. (05): odel Selecton va Bayesan nformaton capacty desgns for generalzed lnear models; Technometrcs. [4]. Jaynes, J., Dng, X., Xu, H., Wong, W.K., & Ho, C.. (03). Applcaton of Fractonal Factoral Desgns to Study Drug Combnatons. Statstcs n edcne, 3, DO: / Page

5 Comparng The Performance Of Bayesan And Frequentst Analyss ethods Of rregular [5]. Hongquan, Xu, Frederck K. H. Phoa & Weng Kee Wong (009): Recent Developments n Nonregular Fractonal Factoral Desgns; Statstcs Surveys. Unversty of Calforna, Los Angeles. [6]. Hamada, & Wu, C.F.J. (000): Analyss of Desgned Experments Wth Complex Alasng; Journal of Qualty Technology. [7]. Ke, Yao. (008): Selecton of Non-Regular Fractonal Factoral Desgns When Some Two-Factor nteractons are mportant; South Dakota State Unversty. [8]. Phoa, F. K. H., Xu, H. & Wong, W. K. (009). The use of nonregular fractonal factoral desgns n combnaton toxcty studes. Food and Chemcal Toxcology, 47, [9]. Yng Zhang. (006): Bayesan D-Optmal Desgn for Generalzed Lnear odels; Blacksburg, Vrgna. V. Fgures Fgure : The Least Angle Regresson plot showng ean Square Error. Fgure : The Bayesan plot showng the ean Square Error. Fgure 3: The screenng posteror probablty plot wth all man effects Al H. U. "Comparng The Performance of Bayesan And Frequentst Analyss ethods of rregular Fractonal Factorals Usng Desgn Based optmalty And Effcency Crtera." OSR Journal of athematcs (OSR-J) 3.3 (07): DO: / Page