Comparative analysis of bayesian control chart estimation and conventional multivariate control chart

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America Jural f Theretical ad Applied Statistics 3; ( : 7- ublished lie Jauary, 3 (http://www.sciecepublishiggrup.cm//atas di:.648/.atas.3. Cmparative aalysis f bayesia ctrl chart estimati ad cvetial multivariate ctrl chart Adewara, J.

1 America Jural f Theretical ad Applied Statistics 3; ( : 7- ublished lie Jauary, 3 ( di:.648/.atas.3. Cmparative aalysis f bayesia ctrl chart estimati ad cvetial multivariate ctrl chart Adewara, J. Ademla, Ogudei K. Rtimi Distace Learig Istitute, Uiversity f Lags, Lags, Nigeria Departmet f Mathematics, Faculty f Sciece, Uiversity f Lags, Lags, Nigeria address: adewaraa@yah.cm (Adewara, adewara@uilag.edu.g (J. Ademla, rk3@yah.cm (O. K. Rrimi T cite this article: Adewara, J. Ademla, Ogudei K. Rrimi. Cmparative Aalysis f Bayesia Ctrl Chart Estimati ad Cvetial Multivariate Ctrl Chart, America Jural f Theretical ad Applied Statistics. Vl., N., 3, pp. 7-. di:.648/.atas.3. Abstract: Bayesia mdel r Beta-bimial cugate usig Bayesia sequetial estimati methd t estimate the prprti f differet age grups is cmpared with the cvetial multivariate ctrl chart methd. The parameters fr the techiques were derived ad applied. The result shws that the patiets betwee the ages f 5-44 i 9 ad ad 64 ad abve i are ut f ctrl. This implies the Bayesia sequetial estimati methd is very efficiet t tice ay small shift that ccurs amg patiets that make use f the hspital. Als the bracket metied abve was very high amg the peple that used the hspital cmpared t thers. The result f shws that there was a high shift i the ages f the patiets that atteded the hspital fr the ages betwee ad 64 ad abve respectively. Keywrds: Beta-Bimial, Sequetial Estimati, Hyperparameters, Cugates Beta-Bimial, Shrikage Factr Ad Multivariate Radm Variables. Itrducti Statistical prcess ctrl (SC chart is a imprtat tl i the ctrl chart. It ca be used t detect chages i prducti prcesses, assess prcess stability, ad idetify chages that idicate either imprvemet r deterirati i quality ad als t measure icrease i perfrmaces f a particular sectr. The mmetum is chagig as result f the adpti f these SC techiques i the healthcare system t aid i prcess uderstadig ad t measure the delivery f the services redered t the public. Hspitals, i particular, are a part f the health care service idustry that rutiely cllect data but d t use it t the best advatage. Cases treated at times i hspital are bth uivariate ad multivariate cases. The Uivariate has ly e variable r sickess at time ad the multivariate aalysis ivlves variables that have mre tha e quality characteristic r sickesses []. These quality characteristics are clearly crrelated ad ctrl chart fr mitrig the idividual quality characteristic may t be adequate fr detectig chages i the verall quality f the prduct. The system experieces sme challeges i the applicati f SC t mitr perfrmace systems which icludig idetificati f the best statistical mdel fr the cmm cause variability, grupig f data, selecti f type f ctrl chart, the cst f false alarms ad lack f sigals, ad difficulty i idetifyig the special causes whe a chage is sigaled [] [3] [4]. Nevertheless, carefully cstructed ctrl charts are pwerful methds t mitr perfrmace systems i a hspital. Ctrl charts were itrduced by Dr. Shewhart i9 s ad ivlve tw phases. I phase I, a set f histrical data is aalyzed t assess stability ad idetify special causes. If special causes are preset, the i-ctrl prcess parameters are estimated ad ctrl limits are established. I phase II, the data are sequetially cllected ver time t assess whether the perfrmace has chaged frm the estimated value [5] [6] [7]. The bective f this paper is t used a Bayesia sequetial estimati ctrl chart t determie a small shift that ca easily shw a ut f ctrl sigals. Als the phase II apprach which ivlves sequetial cllecti f data ver a perid f time is adpted i this research usig Natial Orthpaedic data ad the result is cmpared with the cvetial Htelligs T.. The Bayesia Sequetial Methdlgy If a set f bservatis x, x, x3,..., x geerates a psterir distributi ad, i a similar situati, additial

2 8 Adewara et al.: Cmparative aalysis f bayesia ctrl chart estimati ad cvetial multivariate ctrl chart data are cllected beyd these bservatis, the the psterir distributi fud with earlier bservatis becmes the ew prir distributi ad the additial bservatis give a ew psterir distributi ad iferece ca be made frm the secd psterir distributi. This prcedure ca ctiue with ewer ad mre bservatis. That is, the secd psterir becmes the ew prir, ad the ext set f bservatis give the ext psterir frm which the iferece ca be made [8]. This is the priciple f Bayesia sequetial methdlgy that we prpse t estimate the prprti f cuts data btaied frm the hspital. Based the Bayesia apprach described abve, data were cllected mthly ad cllated yearly fr three years (9, ad frm the hspital recrds. The ppulati prprti f patiets admitted fr rthpaedic surgery is deted by while the prprti f patiets admitted fr rthpaedic surgery i age grup is (,,..., 5. X represets a radm utcme f patiet i examied i age grup. ik i i i if ith a tie t is a d m itte d f r rth p a e d ic s u rg e ry i a g e g r u p a d i year k O th e rw is e the ttal umber f patiets admitted fr rthpaedic surgery i age grup i year k. the ttal umber f patiets admitted fr treatmets (bth rthpaedic ad -rthpaedic surgeries i the hspital i age grup i year k. the prprti f patiets admitted fr rthpaedic surgery i age grup ad year k. Fr each year i each age grup, we cmputed sample prprtis as fllws:. I 9 ad age grup : I ad age grup : I ad age grup : Estimatrs f sample prprtis: ( Var (... The Beta-Bimial Mdel ˆ ad The mdel t be applied is a cugate beta-bimial mdel where the bimial distributi represets the likelihd f the bserved data likelihd ad the beta distributi serves as the prir distributi f the bimial parameter. The psterir mea is ~ η ( f (, d A key cmpet f this itegral is f (, η, the psterir distributi f which is. Uder the geeral Bayesia framewrk ad usig the beta cugate prir plus the bimial likelihd, the psterir distributi f is: f (, η d B ( r, s r s ( ( B ( r, s r s ( ( η ( r, s There is eed t estimate the hyperparameters r ad s f the beta distributi i rder t cmpletely specify the prir. This ca be achieved easily thrugh re-parameterizati f f ( η, ad usig mmet estimati [9]. Lettig r r + ; s ; M r + s ad usig the prir distributi f rs ( E ( ad Var( ( r + s + ( r + s M + These are kw as prir mea ad variace respectively. Csequetly, f ( i, ˆ, µ Μ ˆ Where, (. α β ( B ( α, β ˆ α + Μˆ ˆ ; ˆ β + Μˆ (, Ad where S With Μ ad i ˆ ( S Μ ( S N N ( N ɶ ( estimated, the; i E ( /,, Μ ˆ α α + β. (3 + Μˆ Μˆ + ˆ ˆ ˆ + Μ + Μ + Μ (4 αβ. (5 ( α + β + ( α + β Μˆ ˆ Csequetly, λ + Μˆ ad it ca be readily see

3 America Jural f Theretical ad Applied Statistics 3, ( : 7-9 that where Μˆ (the scale factr is large relative t, λ is large ad receives a larger weight tha. But large Μˆ implies small prir variace. Thus, the estimate which is assciated with smaller variace receives larger weight i determiig the psterir mea. O the ther had, if Μˆ ~ is small relative t, the sample mea receives mre weight. We te that the psterir desity fr the verall age grup prprti is btaied by replacig ad i equati (3 with ad N, respective- ly. Uder cugacy, the estimatr f a prprti is a weighted mea f tw estimatrs, the mea f the prir desity ad the sample prprti estimatr. Thus, ~ ~ λ + ( λ (6 is the empirical Bayes Estimatrs with λ as the shrikage factr. λ is a fucti f the prir ad sample estimatr variace such that, if variace f sample estimatr is large, the weight f (i.e. λ will be large ad ~ will shrik twards. Tw cmpets f the abve mdel λ ad []. are derived frm the prcess,.. Multivariate Htellig s T Ctrl Chart Htellig s T is a very versatile multivariate ctrl chart statistic. It ca be used t ly t idetify utliers i the histrical data set but als t detect prcess shift usig ew icmig bservati. I the uivariate test f meas, the test statistic emplyed is Studet t give by t X µ s / where Χ Χ ad s ( Χ Χ This test statistic has a Studet t distributi with degrees f freedm. Whe the bserved t exceeds a specified percetage pit f the t distributi with degrees f freedm, H is reected. The multivariate aalgue f the square f t was prpsed by Htellig s i 93, it was prpsed fr the - sample case as; [] t ( X µ s / (( X µ ( s ( X µ Reectig H whe abslute value f t ( t is large is equivalet t reectig H: if t, the squared distace frm sample mea x t the test value µ, is large. Whe t is geeralized t p multivariate radm variables, it becmes S T T ( p p 3. Results ( Χ µ ' S ( Χ µ ( S ( Χ µ ( Χ µ ' Χ ( Χ Χ, Χ ( Χ Χ ' The results f the applicati f Beta-Bimial mdel ad Bayesia sequetial methds t the data f differet age grup patiets fr the three years (9, ad are preseted i Table ad belw. The hyperparameters µ ad M are estimated usig sample ifrmati. These are subsequetly used t determie the parameters f the psterir distributis α ad β, thereby cmpletely specifyig them. I ur aalyses, we btai the yearly results fr Bayesia Sequetial (see Table.The result is pltted as shw i Figure. Cmparig the yearly basis estimated sample prprtis ad prprtis as well as variaces f estimated sample prprtis ad prprtis. Table. Cmparative Aalysis f Estimated Sample rprtis ad rprtis. ear: 9 Age grup < yr yrs yrs yrs > 64yrs Overall

4 Adewara et al.: Cmparative aalysis f bayesia ctrl chart estimati ad cvetial multivariate ctrl chart Figure.The ctrl chart fr Bayesia Sequetial f prprti. The result shws that the chart fr patiets betwee the ages f 5-44years (.389 i 9 is ut f ctrl. This implies that amg the peple that make use f the hspital the age bracket 5 44 recrds very high figure cmpared t thers. I the result shws that there is a shift i the ages f the patiets that atteded the hspital frm 5-44 years t ad 64 ad abve respectively (see Table ad Figure abve respectively. The result shws that this ew apprach is able t idetify a small r slight shift that may ccur amg thse that atteded the hspital. Table is the estimated values btaied fr the cvariace s ad Table 3 is the variace ad cvariace values btaied frm cmputati f Htelligs. Cmparig the results f figures ad, figure cat idetify ay slight chage that ccur while the result f the sequetial Bayesia aalysis des. Als the values f the variaces btaied frm the sequetial Bayesia aalysis are better tha that f Htelligs. Table. Cmparative Aalysis f Variaces f Estimated Sample rprtis ad rprtis. ear: 9 Age grup Var( Var( Var( Age grup Var( Var( < yr yrs yrs yrs > 64yrs Overall Table 3. Cmputati f variace f Htelligs T Square. uder yr - 4RS 5-44RS 45-64RS 65RS & ABOVE uder yr RS RS RS RS & ABOVE

5 America Jural f Theretical ad Applied Statistics 3, ( : 7- Figure. The ctrl chart fr Multivariate HtelligsT4. Cclusi. This paper has bee able t used a Bayesia sequetial estimati ctrl chart t determie a small shift that ca easily shw a ut f ctrl sigals. Als the phase II apprach which ivlves sequetial cllecti f data ver a perid f time is adpted i this research usig Natial Orthpaedic data ad the result is cmpared with the cvetial Htelligs T. Bayesia sequetial estimati f prprti is suitable t idetify ad slight chage that ccurs tha the usual r cvetial techique. Similarly, the verall variaces f the prprtis ted mre t zer ver the three years uder review tha that f the variace f Htelligs T square. Thus, the results shw that the estimatrs are better estimatrs the basis f efficiecy ad csistecy prperties f gd estimatrs. Refereces [] Resul Oduk ( Ctrl Charts fr Serially Depedet Multivariate Data Thesis submitted t the Departmet f Ifrmatics ad Mathematical Mdelig at Techical Uiversity f Demark i partial fulfillmet f the requiremets fr the degree f Master f Sciece i Mathematical Mdelig ad Cmputati Techical Uiversity Of Demark. [] Shewart W.A (95 The Applicati f Statistics as a aid i maitaiig quality f a maufactured prduct Jural f America Statistical Assciati: [3] Shewhart WA, (986, Statistical Methd frm the Viewpit f Quality Ctrl Geeral ublishig Cmpay, ISBN [4] Shewhart WA, Ecmic ctrl f quality f maufactured prduct (93, ricet, NJ:Reihld C. [5] Wdall, W. H. (. Ctrversies ad Ctradictis i Statistical rcess Ctrl (with discussi. Jural f Quality Techlgy 3, pp (available at ww.asq.org/pub/qt. [6] Wdall, W.H. Review f Imprvig Healthcare with Ctrl Charts by Raymd G. Carey, Jural f Quality Techlgy, 36, (4. 3. [7] Wdall, W.H. Use f ctrl charts i health-care ad public-health surveillace (with discussi, Jural f Quality Techlgy, 38, 89-4 (6. [8] Lee,. (4, Bayesia Statistics A Itrducti, Hdder Arld, New rk. [9] Bradel, J. (4, Empirical Bayes Methds fr missig data aalysis, Departmet f Mathematics Uppsala Uiversity, rect Reprt. [] Carli, B.. ad Luis, T. A. (b, Bayes ad Empirical Bayes Methds fr Data Aalysis, Bca Rat, Flrida: Chapma ad Hall/CRC ress. [] Richard A.J. ad Dea W.W.(988 Secd Editi Applied Multivariate Statistical Aalysis retice, Hall Iteratial, Ic. 67.

ENGI 4421 Central Limit Theorem Page Central Limit Theorem [Navidi, section 4.11; Devore sections ]

ENGI 4421 Central Limit Theorem Page Central Limit Theorem [Navidi, section 4.11; Devore sections ] ENGI 441 Cetral Limit Therem Page 11-01 Cetral Limit Therem [Navidi, secti 4.11; Devre sectis 5.3-5.4] If X i is t rmally distributed, but E X i, V X i ad is large (apprximately 30 r mre), the, t a gd