Matrix Form of The Bayes Theorem And Diagnostic Tests

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1 IOSR Joural of Maheaic IOSR-JM e-issn: , p-issn: X. Volue 14, Iue 6 Ver. I Nov - Dec 018, PP Marix For of The Baye Theore Ad Diagoic Te María Magdala Pérez-Nio 1 Joé A. Caúñez-Ruiz 1 1 Deparaeo de Ecooía Aplicada I Uiveridad de Sevilla Faculad de Ciecia Ecoóica y Epreariale Avda. Raó y Cajal Sevilla Spai Correpodig Auhor; María Magdala Pérez-Nio Abrac: I order o olve cerai proble i calculaig probabiliie, uch a Markov chai or codiioal pecificaio of dicree diribuio, he ue of arix ad vecor reae of codiioed probabiliie ad of vecor of argial probabiliie i coo. Followig hee idea, he pree udy obai arix for of oe eleeary reul of probabiliy heory, uch a he oal probabiliy ad Baye heore. Thee reul ad ehodology are applied o he arix udy of reul of diagoic e, allowig a iediae geeralizaio o e wih ore ha wo reul. I addiio,we propoe afey ad validiy eaure of a e baed o arix rule, which i oe cae are relaed o he well-kow Youde idex. Key Word: Codiioed Probabiliie, BayeTheore, Marix, Diagoic Te Dae of Subiio: Dae of accepace: I. Iroducio I he Markov chai heory [1];[] a well a i codiioal pecificaio proble for fiie rado variable [3], he arix reae of codiioal probabiliie i coo. Thi lie i followed i he pree udy, ad he ai reul of codiioal probabiliie are obaied, icludig a arix for of he Baye' heore. The reul obaied, are applied o he udy of diagoic e. II. Marix Approach Le u coider wo coplee e of eve, A1, A,..., Ai,..., A ad B1, B,..., B j,..., B, boh wih ozero probabiliie P B1, P B,..., P B j,..., P B. We defie he P A1, P A,..., P Ai,..., P A vecor. 1 PB 1 PA PB α β PB Ad he arice A a a P A B, i 1,,..., ; j 1,,..., ij ij i j P A1 B1 P A1 B... P A1 B P A B1 P A B... P A B A P A B1 P A B... P A B B b b P B A, j 1,,..., ; i 1,,..., ij ij j i DOI: / Page

2 Marix For Of The Baye Theore Ad Diagoic Te P B1 A1 P B A1... P B A1 P B1 A P B A... P B A B P B1 A P B A... P B A Boh arice have dieio. We deoe 1 he colu vecor of dieio wih all i copoe equal o oe, ad by 0 he colu vecor of dieio wih all copoe equal o zero. O he oher had, i hould be reebered ha a arix i aid o be ochaic by row colu if i i o-egaive ad he u of he elee i each row colu i equal o oe. Noe ha 1 A 1 ad B1 I [3] Arold ad Pre obai he heore 1: Theore 1. Toal probabiliy heore, arix for. 1, herefore, A i ochaic by colu ad B i ochaic by row. Aβ α B α β Deoraio: we rely o he oal probabiliy heore i i uual for: Siilarly, i i how ha P A1 B1 P A1 B... P A1 B P B1 P A B1 P A B... P A B P B Aβ P A B P A B... P A B P B 1 P A B P B 1 j j j1 1 P A B j P B j PA j1 α PA P A B j P B j j1 B α β. Theore. The arice AB ad ha value ad, repecively, each arix. Deoraio: cobiig he forula of he previou heore we have: A a reul: BA have eigevalue oe wih α ad β eigevecor aociaed wih Aβ α AB α α B α β B Aβ β AB α α AB - I α 0 de AB - I 0 Theore 3. BAβ β B A-I β 0 de B A-I 0 α Aβ α Deoraio: Ju eed o apply Theore 1. β B α β α Aβ α α α α Bβ β DOI: / Page

3 Marix For Of The Baye Theore Ad Diagoic Te β B α β β β O he oher had, he la equaliy i obaied by rapoig he ecod. Le M be he arix of he probabiliie of he ierecio of he wo coplee e of eve: P A1 B1 P A1 B... P A1 B P A B1 P A B... P A B M P A B1 P A B... P A B Theore 4. No-codiioal oal probabiliy heore. 1Mβ M1 β M1 α Deoraio: Siply operae o he fir eber of each equaio, P A1 B1 P A1 B... P A1 B P A B1 P A B... P A B 1M P A B1 P A B... P A B P A B P A B... P A B β i 1 i i i1 i1 i1 The ecod equaliy i obaied by rapoiio of he fir. The hird equaliy i deoraed laer i eed a he fir oe. Le M be he e of quare arice of dieio. Le u ow defie he applicaio D : M, which aociae o each vecor v he quare arix whoe diagoal i v ad he re of elee are ull. If here i o roo for cofuio we will oi he ubcrip, ha i, D D. I hi way, we obai: PA... 0 Dα PA Theore 5. Deoraio: ADβ M DαB PB PB... 0 Dβ PB P A1 B1 P A1 B... P A1 B P B P A B1 P A B... P A B 0 P B... 0 ADβ P A B P A B... P A B P B 1 DOI: / Page

4 Marix For Of The Baye Theore Ad Diagoic Te P A1 B1 P B1 P A1 B P B... P A1 B P B P A B1 P B1 P A B P B... P A B P B P A B1 P B1 P A B P B... P A B P B P A1 B1 P A1 B... P A1 B P A B1 P A B... P A B M B1 P A B... P A B The deoraio of he oher equaliy i aalogou. Theore 6.Baye' heore, arix for. I proof i a iediae coequece of Theore A DαBDβ B Dα ADβ 3.- Diagoic e. Coider a cliical e for he diagoi of a cerai dieae ee, [4]. Suppoe, a i uual i hi coex, ha each paie ca be ick eve D or healhy ad ha he e ca oly pree poiive reul eve R if i deec he dieae, ad egaive reul oherwie. Defie he vecor: PD PD Dieae prevalece vecor: α. I coai, aog oher elee, he prevalece of he dieae PD PR Te reul vecor: β PR. Coaiig he probabiliie of poiive ad egaive reul, PR ad PR. We alo defie he arice: P D R P D R Te ecuriy arix: A P D R P D R I coai, aog oher elee, he predicive value of he e, P D R poiive ad P D R egaive. P R D P R D Te validiy arix: B. P R D P R D I coai, aog oher elee, P R D eiiviy ad P R D pecificiy. P D R P D R Te Marix: M. P D R P D R Coaiig he rue poiive P D R, rue egaive P D R, fale poiive P D R ad fale egaive P D R. The relaio bewee he e arix ad he prevalece ad reul vecor i he obaied i arix for M1 α ad M1 β : a well a he relaio bewee all he elee of a diagoic e: ADβ M DαB. O he oher had, applyig he for of he oal probabiliy heore ad he Baye' heore, we obai -1 A DB α BDα. Thi i he arix verio of well-kow forula of he ype DOI: / Page

5 TPV Marix For Of The Baye Theore Ad Diagoic Te prevalece x eibiliy prevalece x eibiliy 1 prevalece 1 pecificiy, which correlae prevalece, eiiviy ad pecificiy wih predicive value. Noe ha he ue of arice allow a eay geeralizaio o e wih everal level of diagoi or o dieae wih differe ypologie. I addiio, i allow defiige ecuriy eaure ad idexe i er of diace. The afey of a e uually require he udy of i predicive value. A e i copleely afe if P D R P D R 1.Tha i, boh predicive value are worh oe. For hee e, he afey arix A defied above i he ui. I hi cae, a afey eaure for a e wih he afey arix A, ca be defied a a cerai diace fro he arix A o he ideiy MS A d I, A I A, for oe arix or See, for iace, [5]. For a copleely uafe e hee eaure are worh zero. For a copleely uele e, ha i, P D R P D R 0, he afey arix i 0 1 J 1 0 ad MSJu be axial. I value will deped o he or ued. Noralizig ad raferrig he MS eaure,ecuriy idexe -SI- ca be obaied, o ha if he axiu value of MS i he SI 1 MS,akig value bewee 0 ad 1. The e i copleely afe if i ake he value 1 ad copleely uele whe i ake he value 0. Le' look a oe pecific cae. i.-row or. MS 1 P D R P D R A P D R 1 P D R áx 1 P D R P D R, P D R 1 P D R MS A P D R P D R 0 MS A By oralizig, we obai he aociaed afey idex, ii.-colu or. P D R P D R SI A 1 1 P D R P D R MS1 A 1 P D R 1 P D R áx 1 P D R P D R, P D R 1 P D R 1 1 MS A áx P D R, P D R 0 MS A By oralizig we obai he aociaed afey idex, SI A 1 iii.-euclidea or pecral or. 1 ax P D R, P D R 1 P D R P D R MS A I - A P D R 1 P D R DOI: / Page

6 Marix For Of The Baye Theore Ad Diagoic Te which i he quare roo of he greae igular value. The igular value of ha arix are 0 ad P D R P D R hu: MS P D R P D R A I A 0 MS A The aociaed afey idex i SI 1 P D R P D R A. The ae reaoig ay be ued wih he validiy arix Bad defie e validiy eaure ad idexe. I u be ake io accou ha for eiiviy ad pecificiy equal o oe, we have ha B = I, MV B d I, B I B ad he axiu validiy i foud for zero MV, while he axiu validiy i give i he cae of B = J. Nor MV Idex Row P R D P R D Colu, P R D P R D 1 áx P R D P R D 1 áxp R D, P R D Specral P R D P R D MV i bouded bewee 0 ad i he hree cae udied. Noe ha i he cae of he row or, he validiy idex 1 1 Y VI P R D P R D,where Y i he well-kow Youde idex [6]. I addiio, hi way of dealig wih codiioal probabiliie ca be ued i eachig a a exu bewee ubjec coaiig eleeary heory of probabiliy ad hoe coaiig elee of liear algebra, which are uually eparaed. Referece. [1]. Bera A. Pleo R.J Noegaive Marice i he Maheaical Sciece. SIAM. Piladelphia []. Iaaco D.L. y Made R.W Markov Chai: Theory ad aplicaio. Joh Wiley ad So. New York. [3]. Arold B.C. Pre S. J Copaible codiioal diribuio. Joural of he Aerica Saiical Aociaio. 84, [4]. ] Gordi, L. 014 Epideiology. 5ª edició. Elevier Sauder Philadelphia [5]. Scho J.R.1997 Marix Aalyi for Saiic Joh Wiley ad o New York. [6]. Youde, W.J Idex for raig diagoic e. Cacer 3: María Magdala Pérez-Nio. " Marix For Of The Baye Theore Ad Diagoic Te." IOSR Joural of Maheaic IOSR-JM : DOI: / Page