Spatial Frailty Survival Model for Infection of Tuberculosis among HIV Infected Individuals

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1 SSRG Internatonal Journal of Medcal Scence (SSRG-IJMS) Volume 4 Issue 2 February 217 Spatal Fralty Survval Model for Infecton of uberculoss among HIV Infected Indvduals Srnvasan R 1, Ponnuraja C 1, Rajendran S 1 and Venkatesan P 2 1 Dept. of Statstcs, Natonal Insttute for Research n uberculoss, ICMR, Chenna, Inda. 2 Sr Ramachandra Unversty, Chenna, Inda. Abstract Capturng spatal varaton s mportant whle studyng dsease survval n dfferent geographcal regons where regons has ts own rsk factor assocated wth dsease. o account ths rsk factor, fralty effect was ntroduced n ths model that captures correlaton and varaton between neghborng locatons usng condtonally autoregressve (CAR) pror n Bayesan parametrc survval model for studyng dual nfecton of tuberculoss and HIV. Natonal Insttute for Research n uberculoss data on tuberculoss and HIV were used n ths study. Monte Carlo Markov Chan (MCMC) technque was used to estmate the parameter. WnBUGS software was used for Bayesan Survval model estmaton. he result of the study revealed that the spatal fralty model accounts hgher heterogenety along wth weght at baselne was one of the sgnfcant factors assocated wth death and conclude that there were unmeasured covarates and rsk factors nfluencng death n the Chenna regons. Keywords : Bayesan spatal, Survval, CAR, Webull, random effects Background Survval analyss used when outcome varable of nterest s tme untl an event occurs for ndvduals. he tme ndcates any unt of tme for ther tme to end pont. he event of nterest n ths study s death n an area along wth covarates wthn model, random effects relatng to ndvdual heterogenety and spatal correlaton effect between regons 1. Capturng spatal correlaton and varaton s mportant whle studyng dsease survval n dfferent geographcal regons where regons has ts own rsk factor assocated wth dsease. o account ths rsk factor, fralty or random effect was ntroduced n ths model that captures spatal correlaton and varaton between geographcal regons 2-4. Normally, n conventonal analyss regons are assumed to be ndependent, but random effects correspondng to regon that are spatally arranged and suspect that regons n closer proxmty to each other mght also be smlar n magntude4 5. It wll affect the robustness of the estmates of the parameter but n Bayesan fralty model ncorporates random effects were ntroduced at neghborng locatons that are allowed to exhbt spatal correlaton n ths model by specfyng a condtonally autoregressve (CAR) pror developed by Besag 6 for applcaton n effects n tme-to-event data across neghborng unts, wth the neghbors defned va an adjacency matrx. uberculoss and HIV have been closely lnked snce the emergence of AIDS. B s the most common opportunstc nfecton affectng HIV ndvduals and t reman the most common cause of death n patents wth AIDS. HIV nfecton has contrbuted to a sgnfcant ncrease n the worldwde ncdence of B. B and HIV co-nfecton each dsease speeds up the progress of the other. In addton to HIV nfecton speedng up the progresson from latent to actve B, B bactera also accelerate the progress of HIV nfecton 7. he rsk of developng tuberculoss (B) s estmated to be between 26 and 31 tmes greater n people lvng wth HIV (PLHIV) than among those wthout HIV nfecton. hs work may accounts all ths unmeasured rsk factors consdered n ths model along wth spatal varaton. In ths work, Parametrc model of Webull s explored for survval analyss due to ts versatlty. It contans shape parameter β, the scale parameter η, and locaton parameter γ. Also, parametrc model wth a Webull formulaton for the baselne hazards, placng a unvarate Condtonal autoregressve (CAR) structure on the fralty ntercept terms was used to account the correlaton between regons. hs CAR model ndcates the exstence of spatal dependence on the composton of covarance where the CAR parameter dstrbuton statng precson or varance nverse of ts random effect dstrbuton. CAR dstrbuton captures correlaton across both geographc regons and the random effects for a gven regon. hs model mplemented usng MCMC computatonal technques n a herarchcal Bayesan approach that permts borrowng of strength across regons. ISSN: Page 5

2 SSRG Internatonal Journal of Medcal Scence (SSRG-IJMS) Volume 4 Issue 2 February 217 In the lterature, Parametrc proportonal hazards model wth a Webull formulaton for the baselne hazards, placng a unvarate CAR structure on the fralty ntercept terms was studed for nfant mortalty data and hghlghted the mportance of spatal model 7. Bayesan spatal survval models for poltcal event process were extensvely studed n spatal, non-spatal model wth n parametrc and semparametrc approach and proved that spatal dependence n the random effects also produces changes n the effects of covarates 8. Spato-temporal model usng CAR pror for tuberculoss dsease was extensvely studed for other dseases. BAYESIAN SPAIAL MODEL SPECIFICAION Webull model was estmated usng WnBUGS software to model the hazard rate of death or relapse and sgnfcance of factors assocated wth the same. Cox proportonal hazard model s of the form where s the tme to death or censorng for ndvdual n the ward,, s a vector of ndvdual-specfc covarates and s a vector of parameters, s the baselne hazard. Extendng ths model to nclude the spatal dependency, we propose the model he lkelhood for the Cox model wth spatal and non-spatal fraltes s; L k n j j ', W ; t, h t j ; xj exp H t j exp xj W V j1 1 Where W s the spatal fraltes and V s non-spatal fraltes for the same regon. he event of nterest s death durng treatment as well as follow up perod. he event s coded as 1 and s coded for censorng. here are other mportant covarates are ncluded n the analyss such as age, se treatment group, weght at baselne. W represents CAR model dstrbuton supportng the possblty of correlated random effects at the wards level ~ N(,1 ) and V represents non-fralty random effect and ths exchangeable pror V ~ N(,1 ). Webull models wth spatal fraltes he jont posteror dstrbuton for the Bayesan parametrc Webull model s p(, W,, t, ) L(, W, ; t, ) p( W ) p( ) p( ) p( ) (1) Where ρ s the shape parameter for the baselne hazard n the Webull model. he lkelhood for the Bayesan Webull model wth spatal ndvdual fraltes s; I L(, W, ; t, ) t exp( x W ) exp t exp( 1 1 x W ) he ndvdual Webull fraltes are completed by assgnng sutable pror for the parameter. he lkelhood for the Bayesan Webull model wth spatal ndvdual fraltes s; 1a) 1b) 2a) 2b) 3a) W (2) ISSN: Page 6

3 SSRG Internatonal Journal of Medcal Scence (SSRG-IJMS) Volume 4 Issue 2 February 217 I L(, W, ; t, ) t exp( x W ) exp t exp( x 1 1 Where V represents the non-spatal fraltes wth ~ N(,1 ) V W V ) (3) Materal and methods he data have been collected from Natonal Insttute for Research n uberculoss, Chenna. he data conssts of 15 HIV nfected uberculoss patents admtted n clncal tral who were treated wth three types of treatments of 6 months to 8 months duraton. he covarates consdered for analyss are age (n years), sex (Male-1 and Female-), treatment group, weght at baselne (n kg). he event of nterest s death durng treatment and follow up perod. he event of nterest s coded as1 and censorng s coded as. Webull dstrbuton assumed for the tme dstrbuton and a number of covarate s consdered for the ndvdual level and at a hgher spatal level the wards wthn whch the ndvdual was dstrbuted s consdered. Webull tme to endpont model s defned as Where f ( t ) t 1 exp( p t ) W V s modeled as log( ) 1x1 2 x2... W V (5) where X 1, X 2 Se X 3 reatment group, X 4 Weght at baselne, W CAR model dstrbuton supportng the possblty of correlated random effects at the wards level and V represents non-fralty random effect and ths exchangeable pror V ~ N(,1 ). he WnBUGS software used for Webull model analyss for whch observed tme, censorng tme wth other covarate mentoned above and adjancy matrx for Chenna wards were ncluded for analyss. he pror for ths model s hyper pror.e., Gamma pror whch s dstrbuted wth a small precson, thus takng a larger neghborhood structure nto account. he software used for Cox model analyss s WnBUGS 14.. he pror for ths model s hyper pror.e., Gamma pror whch s dstrbuted wth a small precson, thus takng a larger neghborhood structure nto account. he non-nformatve prors were consdered where ~ N(,.1). Usng a burn n of 1 samples and addtonal of 1, from 1 to 3 Gbbs samples were drawn, posteror estmates of s gven n the table. he comparsons of the posteror estmates ndcate that the convergence has acheved n 3 teratons. RESULS he able 1 shows the non-fralty and fralty Webull model for the HIV nfected B data consderng the covarates age, treatment (cat), sex and wt were gven n the table. he analyss was carred over by SAA software for these models. able 1 Summary Statstcs for Webull model Cov. Webull model wthout fralty Webull model wth fralty Beta SE 95 % CI Beta SE 95 % CI , , , , , , wtm -.49* , * , -.29 Cons , , *P<.5 (4) ISSN: Page 7

4 SSRG Internatonal Journal of Medcal Scence (SSRG-IJMS) Volume 4 Issue 2 February 217 he above table shows the Webull models for fralty and non-fralty n whch weght s one factors assocated wth death for both model, but SE for fralty model s substantally reduced for all the covarates. he heterogenety accounts between areas are 2.37 n ths fralty model. Webull regresson Survval analyss tme sex=male sex=female Fgure 1 Survval curve for Webull model for gender Fgure 1 shows the survval curve for male and female n whch male are havng better survval pattern than female. able 2 Posteror summares for spatal Webull fralty model Spatal Webull fralty Model Parameter Mean MC error Credble Interval [] ,.198 [] ,.17 [] , 1.21 [] , -.15 [Const] , , , , ISSN: Page 8

5 SSRG Internatonal Journal of Medcal Scence (SSRG-IJMS) Volume 4 Issue 2 February 217 he table 2 shows the Bayesan spatal Webull fralty model for the HIV-B data. he Bayesan spatal Webull fralty model s MC error s less for all the covarates. he credble Interval for spatal Webull fralty model s also. he precson for s to the Webull model. able 3 Goodness of ft for Webull models Bayesan Webull fralty No. Spatal Non-Spatal * Model * W V * W V * W V * W V I * * W V * * * * * * * * pd DIC pd DIC he Devance Informaton Crteron(DIC) was used to assess the model ft for ths data n whch frst model contans full model wth correlated effect were captured through (W) and uncorrelated effects were captured through (V). he model wth lowest DIC value was consdered to be better model. From the above sx models, spatal fralty models fts better than the other non-spatal fralty model and the mnmum value of DIC 71.5 n model(1) of Webull model better than other spatal model. he model(5) conssts of age, sex and weght had the next lowest DIC(853.8) after gnorng reatment regmen, ndcates that the effect of reatment regmen was very less mpact on death. he model(4) and model(3) are almost havng same DIC value vz, and respectvely. he model(2) contans treatment regmen, age and sex were havng hgh value DIC(864.4) and gnorng weght n ths model ndcates that weght at baselne s mportant factor for death n our analyss. he last model(6) s a fxed effect model, wth hghest DIC value of 973.7, ndcates that random effects wth n regon and spatal autocorrelaton between ward were mportant factors for estmatng any parameter n spatal analyss. Concluson In the standard approach on fralty modelng, the random effects are treated as ndependent. But n the Bayesan spatal survval modelng approach, spatal dependence between neghborng effects are modeled wth a spatal pror. he condtonally autoregressve pror allows to ncorporate ths spatal dependence n ther survval models. he random effect model consders the other unmeasured factors nfluencng death n the regon. Also, t was found ISSN: Page 9

6 SSRG Internatonal Journal of Medcal Scence (SSRG-IJMS) Volume 4 Issue 2 February 217 that weght at baselne, was one of the factor assocated wth death n our study and hazard rate of male havng two fold hgh as comparng to female. he spatal correlaton was dverse n Chenna ward and north-eastern wards n Chenna havng spatal hgh spatal dependence. Spatal random effect model accounts hgher heterogenety (2.3) n our model whch ndcates that the regonal varaton and other envronmental factors nfluencng survval pattern of dsease n ths study. Webull spatal fralty model fts better other than the non-spatal Webull fralty model. Spatal fralty accounts hgher heterogenety between regons. It s clear that a non-spatal and non-fralty model understates the unexplaned heterogenety n the data. Hence, spatal fralty model captures the unexplaned spatal heterogenety and t draws accurate nferences about other spatal covarates of nterest. REFERENCES 1) Banerjee S, Carln B and Gelfand A E (24): Herarchcal Modelng and Analyss for Spatal Data. Boca Raton: Chapman & Hall. 2) Banerjee S, Wall M M and Carln B P (23): Fralty modelng for spatally-correlated survval data wth applcaton to nfant mortalty n Mnnesota. Bostatstcs, 4: ) Besag J (1974): Spatal nteracton and the statstcal analyss of lattce systems. Journal of Royal Statstcal Socety, Seres-B, 36: ) Besag J and Kooperberg C (1995): On condtonal and ntrnsc autoregressons. Bometrka, 82: ) Besag J, York J C and Molle A (1991): Bayesan mage restoraton, wth two applcatons n spatal statstcs. Annals of the Insttute of Statstcal Mathematcs, 43: ) Carln B P and Banerjee S (23): Herarchcal multvarate CAR models for spato-temporally correlated survval data, Bayesan Statstcs 7. Oxford: Oxford Unversty Press. 7) Carln B P and Hodges J S (1999): Herarchcal proportonal hazards regresson models for hghly stratfed data. Bometrcs, 55: ) Cox D and Oakes D (1984): Analyss of Survval Data. London: Chapman and Hall. 9) Gelfand A Dggle P and Guttorp P (21): Handbook of spatal statstcs, Chapman & Hall / CRC. 1) GeoBUGS User Manual (24): GeoBUGS User Manual. Verson ) Lawson A, Browne W and Vdal Rodero C (23): Dsease mappng wth WnBUGS and MLwN. John Wley & Sons Ltd. 12) L Y and Ryan L (22): Modelng spatal survval data usng semparametrc fralty models. Bometrcs, 58, ) Spegelhalter D, homas A, Best N and Lunn D (22b): WnBugs User Manual Verson 1.4. Cambrdge, England: MRC Bostatstcs Unt. 14) Venkatesan P and Srnvasan R (28): Bayesan model of HIV/AIDS n Inda: A spatal Analyss. Appled Bayesan Statstcal Analyss : ) Venkatesan P and Srnvasan R (211): Bayesan ntrnsc condtonal autoregressve random effect model of tuberculoss : An applcaton, Recent trends n Statstcs and computer applcaton, Journal from Manonmanan Sundaranar Unversty : ) Venkatesan P, Srnvasan R and Dharuman C (212): Bayesan condtonal autoregressve model for mappng tuberculoss prevalence n Inda, Internatonal Journal of Pharmaceutcal Studes and Research, Vol.III, onlne journal. 17) Xa H and Carln B (25): Multvarate parametrc spatotemporal models for country level breast cancer survval data, Lfe tme Data analyss, 11: ISSN: Page 1