Mixture Model of the Exponential, Gamma and Weibull Distributions to Analyse Heterogeneous Survival Data

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1 Joual of Scetfc Reseach & Repots 5: -9, 05; Atcle o.jsrr ISSN: 0-07 SCIENCEDOMAIN teatoal Mxtue Model of the Expoetal, Gamma ad Webull Dstbutos to Aalyse Heteogeeous Suvval Data Yusuf Abbaka Mohammed,*, Bd Yatm ad Suzlah Ismal Depatmet of Mathematcs ad Statstcs School of Quattatve Sceces, Uvesty Utaa Malaysa, Stok, Malaysa. Depatmet of Mathematcs ad Statstcs, Uvesty of Madugu, Madugu, Ngea. Authos cotbutos Ths wok was caed out collaboato betwee all authos. All authos ead ad appoved the fal mauscpt. Atcle Ifomato DOI: 0.974/JSRR/05/504 Edtos: Jausz Bzdek, Depatmet of Mathematcs, Pedagogcal Uvesty, Polad. Revewes: Aoymous, Abaham Adesaya Polytechc, Ijebu-Igbo, Ngea. Ogutude P. E, Depatmet of Mathematcs, Coveat Uvesty, Ngea. Complete Pee evew Hstoy: Ogal Reseach Atcle Receved st Novembe 04 Accepted 6 th Novembe 04 Publshed 5 th Decembe 04 ABSTRACT Ams: I ths study a suvval mxtue model of thee compoets s cosdeed to aalyse suvval data of heteogeeous atue. The suvval mxtue model s of the Expoetal, Gamma ad Webull dstbutos. Methodology: The poposed model was vestgated ad the Maxmum Lkelhood ML estmatos of the paametes of the model wee evaluated by the applcato of the Expectato Maxmzato Algothm EM. Gaphs, log lkelhood LL ad the Akake Ifomato Cteo AIC wee used to compae the poposed model wth the pue classcal paametc suvval models coespodg to each compoet usg eal suvval data. The model was compaed wth the suvval mxtue models coespodg to each compoet. Results: The gaphs, LL ad AIC values showed that the poposed model fts the eal data bette tha the pue classcal suvval models coespodg to each compoet. Also the poposed model fts the eal data bette tha the suvval mxtue models coespodg to each compoet. Cocluso: The poposed model showed that suvval mxtue models ae flexble ad mata the featues of the pue classcal suvval model ad ae bette opto fo modellg heteogeeous suvval data. *Coespodg autho: Emal: yusufabbakam@yahoo.com;

2 Mohammed et al.; JSRR, 5: -9, 05; Atcle o.jsrr Keywods: EM; expoetal; gamma; mxtue model; thee compoets; webull.. INTRODUCTION Suvval aalyss s coceed wth the vestgato of a patcula evet happeg wth a gve duato of tme. Suvval aalyss s wdely appled may felds such as Medcal studes, bology, socal sceces, ecoomcs ad egeeg to meto a few. The most commoly used methods suvval aalyss ae the opaametc methods. Pue classcal paametc suvval models ae commoly employed suvval aalyss; they ae bette opto whe the chose dstbuto seems to ft the data popely. The Expoetal, Gamma, ad Webull dstbutos ae commoly used the lteatue fo modelg suvval data [-4]. Suvval mxtue models ae most appopate fo modelg suvval data whe the data ae beleved to be heteogeeous atue. Recetly, may eseach woks employed the methods of suvval mxtue models to aalyse suvval data. A two compoet mxtue model of Webull dstbutos was poposed to alaysed suvval data whee the paametes of the model wee estmated by the weghted least squaes method [5]. A two compoet suvval mxtue model of Webull dstbutos was poposed; whee the paametes of the models wee estmated by gaphcal appoach [6]. Also a ew techque fo evaluatg the paametes of a two compoet suvval mxtue model of Webull dstbuto was developed to aalyse suvval data [7]. The Expectato Maxmzato Algothm EM was employed to evaluate the paametes of the Webull-Webull suvval mxtue model of two compoets ad the EM stablty was vestgated [8]. Two compoets suvval mxtue models of Gamma-Gamma, Webul- Webull ad Logomal-logomal wee poposed to aalyse suvval data. Model selecto method was used to select the model whch bette epesets the eal data [9]. A suvval mxtue of mxed dstbuto was employed fo aalyzg heteogeeous suvval data. The mxed dstbuto model s a two compoets suvval model of the Exteded Expoetal-Geometc EEG dstbuto whee the EM was employed to estmate the model paametes [0]. Few eseaches cosdeed suvval mxtue models of dffeet dstbutos. A two compoet paametc suvval mxtue model of dffeet dstbutos of Expoetated Paeto ad Expoetal dstbutos was used to model suvval data []. Two compoets suvval mxtue models of dffeet dstbutos cosstg of a Expoetal-Gamma, a Expoetal-Webull ad a Gamma-Webull models wee poposed fo aalysg heteogeeous suvval data by employg EM []. Thee compoets paametc suvval mxtue models dd ot eceve much atteto. I a stuato of a ope heat sugey study; the sk of death afte sugey was dvded to thee dffeet tme ovelappg phases whch ae bette aalysed by a thee compoet mxtue model [-5]. A paametc suvval mxtue model of the Expoetal, Gamma ad Webull dstbutos was cosdeed to ft heteogeeous suvval data. Smulated data wee used to vestgate the stablty ad cosstecy of the EM [6]. I aothe study, model selecto techque was employed to compae the paametc suvval mxtue model of the Expoetal, Gamma ad Webull dstbutos wth the paametc suvval mxtue model coespodg to each compoet [7]. A thee compoet paametc suvval mxtue model of Webull dstbutos was poposed to model suvval data by applyg Bayesa estmato method [8]. EM was usually employed o data beleved to cosst of some mssg o uobseved obsevatos [9]. The paametes of suvval mxtue models ae commoly evaluated by mplemetg the EM Algothm [0,]. I ths study, eal data wee used to vestgate the flexblty ad appopateess of a thee compoet suvval mxtue of the Expoetal, Gamma ad Webull dstbutos modellg heteogeeous suvval data. The aagemet of ths atcle s as follows; secto two, the suvval aalyss ad some mpotat pobablty fuctos wee hghlghted. The developmet of suvval mxtue model of thee compoets ad the applcato of the EM estmatg the ML paametes of the model wee dscussed. Secto thee was devoted to data applcato to evaluate the paametes of the poposed model ad the dscusso of the esult. Fally secto fou the summay ad cocluso wee peseted.

3 Mohammed et al.; JSRR, 5: -9, 05; Atcle o.jsrr SURVIVAL ANALYSIS AND THREE COMPONENTS MIXTURE MODEL Suvval aalyss deals wth applyg patcula statstcal methods to model ad aalyse suvval data. The focus of teest s the occuece of a patcula evet of teest wth a gve peod of tme. The espose of pmay teest s the adom vaable T whch s o-egatve ad gves the suvval tme of a object o a dvdual. The suvval tme ca be epeseted by thee fuctos whch ae techageable The pobablty desty fucto pdf s deoted by f t, whch s expessed as f t df t dt Whee Ft s the dstbuto fucto of the adom vaable T. The gaphcal epesetato of the pobablty desty fucto s fequetly used the lteatue, the gaph of f t, s commoly efeed to as the desty cuve. The aea betwee the cuve ad the t axs of the oegatve desty fucto f t s equal to. The suvval fucto St s commoly expessed as S t F t Whch estmated the pobablty of a dvdual suvvg beyod a specfed tme t. The suvval fucto St s a cotuous mootoc deceasg fucto wth S 0 lm 0 S t ad S lmt S t 0. The hazad fucto deoted by ht, ad s gve by f t t S t h Whch gves the pobablty of a dvdual wll fal wth a small teval t, t t, povded that the dvdual was alve utl the begg of that teval. Pue classcal paametc suvval models ae poweful methods suvval aalyss. They ae pefeed whe the chose pobablty dstbuto appopately epesets the data. The Expoetal, Gamma ad Webull dstbutos ae amog the most mpotat ad fequetly used dstbutos suvval aalyss [,,,4]. t The pobablty desty fucto f t ad suvval fucto S t of these dstbutos ae hghlghted below. Expoetal Dstbuto f E t t e t, 0 Whee s a scale paamete S Gamma dstbuto 4 t E t e 5 - t e f G t t t,, 0 6 Whee s the shape paamete ad s the scale paamete - Whee t e dt s kow as the x S G complete Gamma fucto. Webull Dstbuto t x t x 0 t,, 0 7 t fw t exp - 8 Whee s the shape paamete ad s the scale paamete t S W t exp - 9 I suvval aalyss, mxtue models ae fequetly used because they ae flexble. They ae the best opto whee pue classcal paametc suvval models do ot ft the data of heteogeeous atue [0,]. Suvval mxtue model of thee compoets s used whe t s beleved that the data cosst of thee subpopulato o subgoups. Equato 0 epesets a suvval mxtue model of thee compoets. f t; f t; f t; f t; X, Y, Q X X Y Y Q Q 0 4

4 Mohammed et al.; JSRR, 5: -9, 05; Atcle o.jsrr Whee the vecto,,,,,, X Y Q epesets the vecto the paametes of the mxtue model. The fuctos f t;, f t; ad f t ; ae the X X Y Y Q Q pobablty desty fucto coespodg to each compoet wth some paametes ad Q espectvely., I ths pape, a suvval mxtue model of thee compoets of dffeet dstbutos s poposed to aalyse suvval data whch s beleved to be heteogeeous. The pape poposed a suvval mxtue model of the Expoetal, Gamma ad Webull dstbutos. The poposed model s defed as f E _ G _ W t;, t; f E t; f G X Y t;, f Whee s epeset the mxg pobabltes of ' the thee subpopulatos wth. The fuctos f E, fg ad W f, as defed 4, 6 ad 8, epeset the pobablty fuctos of the Expoetal, Gamma ad Webull dstbutos espectvely. Oe of the most effcet ad effectve methods commoly used to estmate the ML estmatos of fte mxtue models s the EM Algothm []. Let t,...,, t t be a set of obsevatos of z, z, z be a set of complete data ad mssg obsevatos, whee z k zk t, f the obsevato belogs to the k th compoet ad 0 othewse fo k,, ad,,...,. O the mplemetato of the EM to the mxtue model, the vaables z`s ae cosdeed as mssg values. The EM cossts of two dffeet steps, the fst oe s the Expectato step o the E-step ad the secod oe s the Maxmzato step o the M-step. The z`s vaables ae teated as mssg obsevatos the E-step, the hdde vaable vecto z z, z, z ] ae estmated by the [ W evaluato of the codtoal expectato E z k t. Thus z z z E z E z E z t t f X t ; X f X t ; X fy t ; Y fq t ; Q fq t ; Q f X t ; X fy t ; Y fq t ; Q The fuctos E z t, E z t 4 5 ad E z t calculated the E-step wll be t maxmzed the M-step of the EM ude the codto. The evaluato of the mxg fy t ; Y f X t ; X fy t ; Y fq t ; Q pobabltes ad vecto of paamete [ X, Y, Q ], s by the mplemetato of the Lagage method. The mxg pobabltes wll be obtaed by; z 6 z 7 z 8 The ML estmato of the paamete of the Expoetal dstbuto fo the poposed model ca be obtaed by the equato 9 [,6,7]. z z t 9 The maxmum lkelhood estmatos of the paametes ad of the Gamma dstbuto fo poposed model ae evaluated usg equatos 0 ad espectvely [9,,6,7]. 5

5 Mohammed et al.; JSRR, 5: -9, 05; Atcle o.jsrr l l t z z t l z z ' 0 z t z Ad z t z Whee s the umbe of Newto-Raphso ' teato wth EM Algothm ad. ad. ae a dgamma ad tgamma fuctos espectvely. The shape ad scale paametes ad of the Webull dstbuto the poposed model ae obtaed by solvg the equatos ad espectvely [9,,6,7]. Whee D A C A B B D C B z z l t, B z t z C ad s the umbe of Newto-Raphso teato wth EM. t l t z t l t. REAL DATA APPLICATION AND DISCUSSION The eal data aalysed ths secto ae the Kdey Cathete data whch s cluded as oe of the datasets famous suvval package [] of the R statstcal softwae [4]. The data wee studed ogally [5]. The data gve the ecuece tmes to fecto, at the tme of setg cathetes of kdey patets usg potable dalyss equpmet. It cossts of 76 obsevatos ad 7 vaables. The poposed model was used to aalyse the data ad the t was compaed wth the pue classcal paametc suvval models coespodg to each compoet usg Log-lkelhood LL ad Akake Ifomato Cteo AIC value. Table. shows that the LL value of the poposed model s hghe tha that of the pue classcal suvval models ad also the AIC value of the poposed model s lowe tha that of the pue classcal suvval models whch makes the poposed model sutable fo the eal data used. The poposed model was gaphcally compaed wth pue classcal paametc suvval models coespodg to each compoet of the mxtue model. The pobablty fuctos of poposed model ad the pue classcal paametc suvval models alog wth the hstogam of the Kdey Cathe data wee peseted Fg.. Fg.. shows that the poposed model aalysed the eal data bette tha the dvdual pue paametc suvval models. Table. The paametes, LL ad AIC values fo the Kdey Cathete data Model Estmates LL AIC Expoetal = Webull = 0.89, = Gamma = 0.86, = Mxtue model = = 0.8, = 7.7 = =.9, = 0.5, = 0.9 6

6 Mohammed et al.; JSRR, 5: -9, 05; Atcle o.jsrr The Kdey Cathete data wee used to compae the poposed model wth the suvval mxtue models of the Expoetal, Gamma ad Webull dstbutos, espectvely to select the model that fts the data appopately. Table. dsplays the estmated paametes of each model togethe wth the LL ad AIC values. It s obseved that the poposed model epesets the eal data bette tha the othe models. Also poposed model was compaed gaphcally wth the suvval mxtue models of the Expoetal, Gamma ad Webull dstbutos, espectvely. Fg.. shows the compaso of the desty fucto of the poposed model wth the othe models. It s also obseved that the poposed model epesets the eal data bette tha the othe models. Fg.. The pobablty desty fuctos of poposed model ad the pue classcal dstbuto of the Kdey Cathete data Fg.. The pobablty desty fuctos of the poposed model ad the mxtue models of each compoet of the Kdey Cathete data 7

7 Mohammed et al.; JSRR, 5: -9, 05; Atcle o.jsrr Table. LL ad AIC of the poposed model ad paametc suvval mxtue models of the same dstbuto E_G_W E_E_E G_G_G W_W_W LL -.50 LL LL -7.0 LL -.9 AIC AIC 68.9 AIC AIC CONCLUSION Ths atcle poposed a suvval mxtue model of thee compoets of the Expoetal, Gamma ad Webull dstbutos to aalyse suvval data whch s beleved to be heteogeeous. Real data wee used to estmate the paametes of the model. EM algothm was employed estmatg the ML paametes of the poposed model. The compaso of the poposed model wth the pue classcal suvval models ad the suvval mxtue models coespodg to each dstbuto showed that the poposed model epesets the data bette tha the othe models. The poposed model showed that the suvval mxtue models ae flexble ad mata the featue of pue classcal suvval models ad they ae bette opto to model heteogeeous suvval data. *Note: The R laguage veso was used fo all the calculatos ad gaphs COMPETING INTERESTS Authos have declaed that o competg teests exst. REFERENCES. Ibahm JG, Che MH, Sha D. Bayesa suvval aalyss. New Yok: Spgevelag; 00.. Kalbflesch JD, Petce RL. The statstcal aalyss of falue tme data. d ed. New Jesey: Joh Wley & Sos, Ic. Hoboke; 00.. Lawless JF. Statstcal models ad methods of lfetme data. d ed. New Jesey: Joh Wley ad Sos, Ic. Hoboke; Lee ET, Wag JW. Statstcal methods fo suvval tme data aalyss. d ed. New Jesey: Joh Wley & so; Cheg SW, Fu JC. Estmato of mxed webull paametes lfe testg. Relablty, IEEE Tasactos. 98;4:77-8. Avalable: ledetals.jsp?aumbe=580.09/ TR Jag S, Kececoglu D. Gaphcal epesetato of two mxed-webull dstbutos. IEEE Tasacto o Relablty. 99a; 4:4-47. Avalable: ledetals.jsp?aumbe= / Jag S, Kececoglu D. Maxmum lkelhood estmates, fom cesoed data, fo mxed- Webull dstbutos. IEEE Tasacto o Relablty. 99b;4, DOI:0.09/ Zhag Y. Paametc mxtue models suvval aalyss wth applcato, Doctoal Dssetato UMI Numbe: 0087, Gaduate School, Temple Uvesty; Esoglu U. Esoglu M. Eol H. Mxtue model appoach to the aalyss of heteogeeous suvval tme data. Paksta Joual of Statstcs. 0;8:5-0. 8

8 Mohammed et al.; JSRR, 5: -9, 05; Atcle o.jsrr Esoglu U. Eol H. Modellg heteogeeous suvval data usg mxtue of exteded expoetal-geometc dstbutos. Commucatos Statstcs -Smulato ad Computato. 00;90:99-5. Avalable: Abu -Zadah HH. A study o mxtue of expoetated paeto ad expoetal dstbutos. Joual of Appled Sceces Reseach. 00;64: Esoglu U, Esoglu M, Eol H. A mxtue model of two dffeet dstbutos appoach to the aalyss of heteogeeous suvval data. Iteatoal Joual of Computatoal ad Mathematcal Sceces.0;5. Avalable: cod.ul?ed=-s &pateid=40&md5=90faa 5759d0767b0b676000e789c. Blackstoe EH, Naftel DC, Tue ME. The decomposto of tme-vayg hazad to phases, each copoatg a sepaate steam of cocomtat fomato. Joual of the Ameca Statstcal Assocato. 986;895: Avalable: Ng ASK, McLachla GJ, Yau KKW, Lee AH. Modellg the dstbuto of schaemc stoke-specfc suvval tme usg a EMbased mxtue appoach wth adom effects adjustmet. Statstcs Medce. 004;7: Avalable: cod.ul?ed=-s &pateid=40&md5=9f7 9bd555b68c78faf99e 5. Phllps N, Coldma A, McBde ML. Estmatg cace pevalece usg mxtue models fo cace suvval. Statstcs Medce. 00;9: Avalable: DO /sm.0 6. Mohammed YA, Yatm B, Ismal S. A smulato study of paametc mxtue model of thee dffeet dstbutos to aalyse heteogeeous suvval data. Mode Appled Scece. 0;77:-9. Avalable: 7p 7. Mohammed YA, Yatm B, Ismal S. A paametc mxtue model of thee dffeet dstbutos: A appoach to aalyse heteogeeous suvval data. Poceedgs of the st Natoal Symposum o Mathematcal Sceces SKSM AIP Cof. Poc. 04;605: DOI:0.06/ Maí JM, Rodíguez-Beal MT, Wpe MP. Usg webull mxtue dstbutos to model heteogeeous suvval data. Commucatos Statstcs: Smulato ad Computato. 005;4: Dempste AP, Lad NM, Rub DB. Maxmum lkelhood estmato fom complete data va the EM algothm wth dscusso. Joual of Royal Statstcal Socety. Sees B. 977;9:-8. Avalable: McLachla GJ, Peel D. Fte mxtue models. New Jesey: Joh Wley & Sos, Ic.; McLachla GJ, Ksha T. The EM algothm ad extesos d ed. New Jesey: Joh Wley & Sos, Ic.; Fuhwth-Schatte S. Fte mxtue ad makovs swtchg models. Spge; Theeau T. A package fo suvval aalyss S. R package veso..7-4; 0 Reteved fom Avalable: 4. Team RC. R: A laguage ad evomet fo statstcal computg: ISBN R Foudato fo Statstcal Computg. Vea, Austa; 0. Avalable: R-poject. og 5. Mc Glchst CA, Asbett CW. Regesso wth falty suvval aalyss. Bometcs 99;47: Mohammed et al.; Ths s a Ope Access atcle dstbuted ude the tems of the Ceatve Commos Attbuto Lcese whch pemts uestcted use, dstbuto, ad epoducto ay medum, povded the ogal wok s popely cted. Pee-evew hstoy: The pee evew hstoy fo ths pape ca be accessed hee: 9