Comparison on Modelling the Relative. Risk Estimation: Bayesian Study

Transcription

1 Appled Mathematcal Scences, Vol. 4, 00, no. 54, Comparson on Modellng the Relatve Rsk Estmaton: Bayesan Study Rafda M. Elobad and Noor Akma I. Laboratory of Appled & Computatonal Statstcs Insttute for Mathematcal Research Unversty Putra Malaysa, Serdang, Selangor, Malaysa Abstract The estmaton of the dsease ncdents was prevously analyzed usng a classcal approach. However, ths approach features large outlyng relatve rsks and consdered as msleadng due to several major problems. Some approaches such as the herarchcal Bayesan method have been adopted n the lterature n order to overcome these problems. The purpose of ths study s to compare between herarchcal Bayesan models that mprove the relatve rsk estmaton. The focus les on examnng the performance of dfferent sets of denstes va montorng the hstory graphs, estmatng the potental scale reducton factors and conductng senstvty analyss for dfferent choce of pror nformaton. The best model ft s accomplshed by conductng a goodness of ft test. The study s appled on Scotland lp cancer data set. The results show that for models wth large number of parameters, more teraton s needed to acheve the convergence. The study also shows that dagnostc test and senstvty analyss are mportant to decde about the stablty and the the nfluence of the choce of the pror denstes. The DIC results were n lne wth the prevous results and provde a good method of comparson. Mathematcs Subject Classfcaton: 6-07, 6P0

2 664 R. M. Elobad and N. Akma I. Keywords: herarchcal Bayesan models, relatve rsk, potental scale reducton, senstvty analyss, convergence, pror denstes, DIC. Introducton The Bayesan approach to statstcal analyss provdes a cohesve framework for combnng complex data models and external knowledge. The advantages n usng the Bayesan method justfy the ncreased efforts n computatons and pror determnaton [0]. The herarchcal Bayesan approach has become a standard n the epdemology and publc health lterature, see [, 8, 7,, 8]. Ths approach s called herarchcal because t uses multple level of analyss n an teratve way. Unlke the conventonal statstcal nference, whch derves the average estmates of the parameters, the herarchcal Bayesan modellng produces parameter estmates for each ndvdual analyss unt by borrowng nformaton from all analyss unts,.e. the customary Bayesan borrow of strength effect. As many herarchcal Bayesan models used n the lterature to nvestgate the relatve rsk estmaton, we lmt ths study to nvestgate three types of herarchcal Bayesan models. These models were classfed as follow. The nterregonal varablty model (IVM) whch ncludes the covarate effect and the nterregonal (local or non spatal) varablty effect, the correlated varablty model (CVM) that has the capablty of ncorporatng the covarate effect and the effect of the rsk n the adjacent countes (the spatal varablty) and the global spatal model (GSM) that accommodates the covarate, nterregonal and the spatal effects. The three models were ntroduced n such away to produce stable, accurate and smooth relatve rsk estmaton. In ths study, the relatve rsk n each county assumed to depend on the adjacent neghbours [9, 3, 5]. The MCMC smulaton was used to estmate the dfferent sources that caused the relatve rsk varablty n each county usng the three Bayesan selected models. However, the selected models have dfferent nfluence n the relatve rsk estmaton. Ths usually depends on the choce of the pror denstes, the number of teraton used n the smulaton, the number of parameter n each model...etc. Ths study ntroduces dfferent methods to compare between the selected herarchcal Bayesan models.. Method. Estmatng relatve rsks usng the IVM, CVM and GSM Several factors affectng the relatve rsks among the countes wthn the lattce grd lead to relatve rsk heterogenety. In ths study, two man sources were dentfed as

3 Comparson on modellng the relatve rsk estmaton 665 the causes of ths heterogenety, the covarate effect whch nfluences the occurrences of the dsease n the dfferent countes and the effect of the over-dsperson [5,, 0] between the countes. The over-dsperson was separated nto two man effects, the nterregonal varablty due to dfferent envronmental (local, natural, or bult) factors, and the spatal varablty whch s due to the spatal autocorrelaton among the dfferent countes [4]. The dsease relatve rsk was estmated usng the standardzed mortalty rates (SMR). In the presence of the prevous factors, three herarchcal models IVM, CVM and GSM were used. Two stages were consdered to buld all the three models. In the frst stage, and snce rare events were dealt wth n buldng the herarchcal Bayesan models, the Posson lkelhood was specfed for the observed dsease ncdents, gven the vector of the dsease relatve rsk. In the second stage, a set of pror denstes for each model was selected over the space of the possble relatve rsk parameters and hyper-parameters to get a set of posteror means for the relatve rsks, gven the observed dsease ncdences. The constructon method for each model s elaborated as follow. For the IVM, CVM and GSM the SMR s defned as n the followng equatons respectvely, ˆ X ˆ SMR e β + ψ β + = = θ, () ˆ X ˆ SMR e β + ψ β + = =, () ψˆ = SMR ˆ X e β + β + θ + =, (3) where ψ s the relatve rsk parameter n regon whch estmated usng the standardzed mortalty rate ( SMR ˆ ), β s the ntercept term, β s the coeffcent for the covarate X, θ s the random effect representng the nterregonal varablty, and s the spatal varablty random effect n countes =,,.,N. Pror denstes were specfed for each parameter n each model. As ponted out by Kelsall and Wakefeld [9], care s needed when choosng the pror denstes for the parameters of nterest because the posteror denstes of the parameters can be senstve. In the IVM, the nterregonal varablty θ was assumed as an ndependent gamma pror, condtonal on the hyper-parameters α and α. Wthout the constant term, the gamma pror densty was proportonal to the followng: f(, ) α θ α α θ e α θ, (4) where α α s the pror expectaton for θ, and α α s the pror varance. Wthout pror expectatons about the drecton and magntude of the covarate effects

4 666 R. M. Elobad and N. Akma I. and condtonal on known hyper-parameters, the ntercept β and the covarate coeffcent β are assumed to have vague but nformatve normal pror, ( a) b f( β a, b) e β, (5) f( β r, d) e β ( r) d. (6) where abr,, and d are fxed known values. The hyper-parameters α and α of the gamma densty are also gven prors p( α Y ) and p( α Y ). The prors selected was the exponental dstrbuton wth constant second order hyper-parameters c and m for each α and α, respectvely, that s: c f ( α c) e α, (7) m f( α m) e α. (8) For the CVM, condtonal on the hyper-parameter τ whch controls the varablty of, the spatal varablty was modelled as ntrnsc condtonal autoregressve (ICAR) model [7]. The most common densty of ths pror [] has the jont dstrbuton proportonal to: τ n / τ / τ exp ( ) τ exp ω ( ), (9) = where τ = σ s the precson term, = ω denotes the average of the neghbourng that s adjacent to, and ω s the number of the adjacent countes. The vague flat pror dstrbuton was used for the ntercept term β to gve the desred propertes of the mproper ICAR pror [6]. The covarate coeffcent β was assumed to have a normal pror dstrbuton, as n Equaton (6) wth fxed hyper-parameters r and d. The vague Gamma dstrbuton was chosen as a pror densty p( τ Y ) for the hyper-parameter τ of the spatal varablty, wth mean l l and varance l l. Wth no pror estmaton for the precsons of the random effects, small values of l, l were chosen to assume the large varance. The pror for precson hyper-parameter τ s proportonal to the followng densty, l f ( τ l, l ) τ e l τ. (0)

5 Comparson on modellng the relatve rsk estmaton 667 For the thrd model (the GSM), condtonal on the precson hyper-parameter τ θ, the nterregonal varablty θ s assumed as a conjugate normal ndependent pror, that s: τθ τθ f ( θ τθ ) = exp( θ ), () π where τθ = σθ s the precson term, whch s the nverse of the nterregonal varance. Condtonal on the hyper-parameter τ, the spatal varablty s modelled as the ntrnsc condtonal autoregressve (ICAR) model, where has a normal dstrbuton as gven n Equaton (9). The pror denstes for the fx parameters β and β were represented as n the CVM. Prors p( τ θ Y ) and p( τ Y ) were chosen for the hyper-parameters, τ θ and τ of the nterregonal and spatal varablty. The vague Gamma dstrbuton wth mean k k and varance k k was chosen for τ θ as follow, k kτθ f( τθ k, k) τθ e. () The gamma pror defned n Equaton (0) was chosen for the spatal precson term τ. Small values for k, k, l and l were chosen to assume large varances, as there were no pror nformaton for precsons of the random effects.. The smulaton study The IVM smulaton: no pror expectaton about the ntercept β and the covarate effects β were avalable, hence a large value was chosen for the fxed hyperparameters whch represented the varances b and d of the pror normal denstes of β and β, respectvely ( b= d = 00 ). For the pror exponental densty of the hyperparameters α and α, small values for the second order hyper-parameters, c= m= 0.0, were chosen to assume large varances, c and α, m for α and respectvely. Two thousand teratons, wth the frst 500 teratons dscarded from each chan as pre-convergence burn n, were used to start the smulaton. The CVM smulaton: to carry out the posteror analyss, fxed values were selected n such a way that produces vague denstes whch can be used to obtan the posteror denstes for the parameters. A large value was chosen for the varance (d=000) of the pror normal dstrbuton of the covarate term β. Wth no pror estmaton for the precson of the random effect termτ, the fx values of 0.5 and were chosen for the second order hyper-parameters of Gamma pror densty l, and l, respectvely.

6 668 R. M. Elobad and N. Akma I. Usng the MCMC method, more teratons were expected to be needed n order to start the analyss, due to the spatal structure whch has been ncluded n the model. Hence, the smulaton started wth three thousand smulatons from the posteror dstrbuton wth the frst 500 teratons dscarded from the results. The GSM smulaton: large value for the varance (d = 0000) was selected for the pror densty of β to produce vague densty, whle vague but proper pror dstrbutons Gamma (0.5, 0.05) and Gamma (0.5, 0.0) were adopted for the precson of the random effects τ θ andτ, respectvely. The analyss was started wth three thousand smulatons from the posteror dstrbuton..3 Models evaluaton To nvestgate the performance of the three models, the analyss was appled to the lp cancer data n 56 countes n Scotland over the perod 973 to 980. The data ncluded the name and the number of 56 dstrcts n Scotland, the observed number of male lp cancer cases n each county Y, the number of male populaton n, a covarate X measurng the percentage of the county s populaton engaged n agrculture, forestry, and fshng (AFF), and the poston of each county expressed as a lst of adjacent countes. Ths data had been orgnally analyzed by Clayton and Kaldor [4] and Breslow and Clayton [6]. Snce the lp cancer dsease s a rare and non-contagous, the number of the observed ncdents s assumed to be mutually ndependent, and to follow Posson dstrbuton. The sets of pror denstes, n addton to the Posson lkelhood are used to obtan the jont posteror densty and consequently the condtonal posteror densty for each parameter n each model. The convergence of the samplng process used to estmate the parameters was assessed by montorng the trace plots of the parameters. To acheve the convergence two parallel chans were used for each model to ensure a complete coverage of the sample space. Sutable number of teraton was also used after takng nto consderaton the number and the expresson of the parameters, the sample sze and the conjugacy of the pror nformaton for each model. Convergence dagnostc was also appled by estmatng the potental scale reducton factor R ) for each parameter of nterest []. The unty value of R ) ndcates that the convergence s reached; otherwse the smulaton can be contnuously run by ncreasng the number of teraton per chan. The precson of the pror nformaton was assessed usng senstvty analyss for each model. The senstvty analyss was conducted n order to frst, nvestgate the nfluence of the choce of the pror denstes n estmatng the relatve rsk, and second, to nvestgate whether the results n the analyss remaned essentally unchanged n the presence of dfferent pror nformaton. Once the models performance was assessed, the study made use of the development of the MCMC methods, whch had made t possble to ft and compare

7 Comparson on modellng the relatve rsk estmaton 669 ncreasngly large classes of models [3]. We used the devance nformaton crteron (DIC), whch was ntroduced by Spegelhater et al. [3] to compare between the herarchcal Bayesan models. The DIC s represented by the followng equaton; DIC = D + p = D D( Θ ), (3) D where D s the posteror expectaton of the devance, pd s the effectve number of parameters and D( Θ ) s the devance evaluated at the posteror expectaton. As wth all penalzed lkelhood crtera, the DIC conssts of two terms; one represents the goodness of ft, whle the other, a penalty for ncreasng the complexty of the model. One of the man reasons, whch encouraged the use of ths method n makng a comparson between the proposed Bayesan models, s that, besdes ts generalty, t may readly be calculated n the MCMC process. 3. Results and Dscusson The convergence of the MCMC smulaton to the posteror dstrbutons for the selected parameters was observed usng the WnBUGS software. Two parallel samplng chans wth two dfferent sets of ntal values for each model were used. Fgures (), () and (3) llustrate the hstory graphs for selected posteror means of the SMR and selected parameters for the IVM, CVM and GSM, respectvely.

8 670 R. M. Elobad and N. Akma I. (a) (b) Fgure. Hstory graphs for selected a) posteror means of the SMR and b) posteror parameters usng the IVM.

9 Comparson on modellng the relatve rsk estmaton 67 (a) (b) Fgure. Hstory graphs for selected a) posteror means of the SMR and b) posteror parameters usng the CVM. The hstory graphs n Fgure () show the output from the two parallel Gbbs samplng chans usng the IVM. Fgure (a) ndcates that the posteror means of the estmated SMR have converged well at 000 and 000 teratons, respectvely; whle Fgure (b) shows that the posteror means for some parameters such as α are not acheved at 000 teratons. By montorng the convergence n the trace plots of the CVM, the prmary results observed from the hstory graphs n Fgures (a) and (b) showed that the convergence had been reached for all the parameters. However, the trace plots of the parameters n the GSM show that the convergence has not been reached for some

10 67 R. M. Elobad and N. Akma I. parameters wth less than sx thousand teratons. Ths s due to the large number of parameters n the model. Thus, three thousand more teratons have been updated. The hstory graphs for the selected posteror means of the SMR and selected parameters are dsplayed n Fgures (3a) and (3b), respectvely. (a) (b) Fgure 3. Hstory graphs for selected a) posteror means of the SMR and b) posteror parameters usng the GSM. To confrm that the convergence of all the parameters had been acheved, the potental scale reducton factor R ) was estmated for each parameter of nterest. In the IVM, Ths factor showed that the convergence had not been acheved for some parameters by usng only 000 teratons,.e. the value of R ) s not equal to one. Ths was depcted n Fgures (4a) and (4b) whch show the overlap of the two chans, R )

11 Comparson on modellng the relatve rsk estmaton 673 values and the posteror nferences for each estmator usng the IVM. Fgures (4a) and (4b) depct the output from 000 and 000 MCMC smulatons, respectvely. The results n these two fgures show obvous results of non-convergence for some parameters than that obtaned by the hstory graphs (Fgure ). Another 000 teratons were carred out n order to acheve the convergence. Usng a total of 3000 teratons ndcated that the potental scale reducton factor was between and. for all the parameters as dsplayed n Fgure (4c). From the fgure, the R ) values confrm that the convergence has been acheved for all the parameters. (a) (b)

12 674 R. M. Elobad and N. Akma I. (c) Fgure 4. Inference for the IVM ncludng the R ) Estmaton from a) 000 b) 000 and c) 3000 MCMC smulatons. Fgure 5. Inference for the CVM ncludng the R ) estmaton from 3000 MCMC smulatons.

13 Comparson on modellng the relatve rsk estmaton 675 Fgure 6. Inference for the GSM ncludng the R ) estmaton from 6000 MCMC smulaton. The potental scale reducton factor R ) was re-estmated for the parameters usng 3000 MCMC smulaton for CVM and 6000 MCMC smulatons for GSM. It was found that the convergence had been acheved for all the parameters. Fgures (5) and (6) show that R ) values are equal to the unty value, suggestng that the models converge well after several thousand teratons. Ths result ndcated a good agreement wth the hstory graphs results (Fgures and 3 for CVM and GSM, respectvely) and show that no more teratons are needed. The senstvty analyss was carred out by conductng several trals to dfferent choces of fxed values for the hyper-parameters abr,, and d, and the second order hyper-parameters c and m n the IVM. The MCMC smulaton was carred out for each tral, and the samples of the posterors were summarzed. Later, these samples were checked to fnd out whether they had mposed a practcal mpact on the nterpretatons or decsons made. All the trals gave almost dentcal results to the results shown n Fgure (7). The fgure dsplays the relatve rsk estmaton for the 56 countes of Scotland. Overall, the mean of the relatve rsk s.45, whle the standard devaton s 0.9. Another senstvty analyss was conducted usng dfferent pror nformaton n CVM. The posteror quanttes of nterest were recomputed by specfyng vague but proper pror dstrbutons, Normal (0, 00) and Gamma (0.5, 0.05) for the covarate and the precson parameters, respectvely; followed by other trals wth dfferent sets

14 [7] [8] [9] [0][] [][3] 676 R. M. Elobad and N. Akma I. of fxed values for both prors. The output n all the tral remaned almost the same for all the parameters, ndcatng the robustness of the results wth dfferent choces of pror nformaton. The overall mean of the relatve rsk usng the CVM s.43, as shown n the box plot of the estmated SMR n Fgure (8). The standard devaton has also been reduced to.06, as compared to the results derved from the IVM. countes vs SMRhat 0.0 [] [] [3] [4] [5] 5.0 [6] [4] [5][6] [7] [8] [9] [0] [] [3] [][3] [4] [5][6] [7] [8][9][30] [3] [33] [39] [34][35][36][37][38] [4] [43] [55] [56] [40] [46][47] [5][5] [4] [44] [48][49] [53][54] [45] [50] 0.0 Countes Fgure 7. Box plots of the IVM posteror estmaton of the SMR for Scotland countes. countes vs SMRhat 0.0 [] [] [3] [] 5.0 [4] [5] [7] [0] [3] [6] [9] [4] [5] [6][7] [9] [0] [3] [8] [] [8] [][] [3] [5] [9] [6][7][8] [39] [43] [33] [4] [30][3] [34] [35][36][37] [55] [56] [4] [38] [40] [46] [4] [44] [47][48][49] [50][5][5] [45] [53][54] 0.0 countes Fgure 8. Box plots of the CVM posteror estmaton of SMR for Scotland countes.

15 Comparson on modellng the relatve rsk estmaton 677 countes vs SMRhat 0.0 [] [] [3] [] 5.0 [4] [5] [6] [7] [0] [9] [] [8] [3] [4] [7] [6] [5] [9] [3] [8] [0] [][][3] [4] [5][6][7][8][9] [30][3] [33] [39] [43] [34][35][36] [37] [55] [56] [38] [40] [4] [4] [46] [44] [47][48] [45] [49] [50][5][5] [53][54] 0.0 countes Fgure 9. Box plots of the GSM posteror estmaton of the SMR for Scotland countes. In the GSM the senstvty analyss ndcates that the pror dstrbutons gave almost dentcal results, suggestng that the results are robust to changes n pror nformaton. The GSM was found to produce shrnkage n the estmaton of the posteror relatve rsk towards the overall mean (.456), as shown n Fgure (9). The standard devaton of the dfferent trals of the relatve rsk estmaton showed the smallest value (0.89) among the Bayesan models. Ths confrms that the pror used for ths model are more robust and accurate comparng wth the other two models. The DIC was obtaned to compare between the models. The selecton was done based on the DIC values computed for each model, usng the MCMC method. Model wth a smaller DIC ndcates a better fttng model. Table () lsts the devance summares for each model, the posteror mean of the devance, D, the devance evaluated at the posteror mean of the parameters, D( Θ ), the effectve number of the parameters p D and the DIC. The smaller values of D and D( Θ ) showed that models wth spatal random effect (CVM and GSM) have a better average ft to data than the model wth the nterregonal varablty (IVM). The results also show that the best fttng s acheved when the full model (GSM) s used due to ts small D and D( Θ ) values. Ths partcular model also has a smaller DIC value (306.69) and extra effectve parameter, p D = As a result, t s suggested that ths s the most robust and approprate model to be used.

16 678 R. M. Elobad and N. Akma I. Table. Devance summares for the herarchcal Bayesan models. Model D D( Θ ) p D DIC IVM CVM GSM Concluson The work n ths paper had prmarly been concerned wth producng stable estmaton for the relatve rsk assocated wth the spread of the dsease n spatally arranged regons. The purpose s to provde dfferent methods to assess the performance of the herarchcal Bayesan models IVM, CVM and GSM. These models nvestgate the covarate and over-dsperson effects that mght affect the relatve rsk estmaton. The study was carred out to choose the most approprate model whch could ft the data well. The Posson lkelhood and dfferent sets of the pror denstes were used to obtan the jont posteror densty, followed by the condtonal posteror densty for each parameter. The study dscussed the results of fttng these models to the Scotland lp cancer data. An MCMC smulaton was conducted usng two chans wth selected ntal values and specfc number of teratons for each model. Burn-n perods were also used to confrm a statonary convergence whch s ndependent of the ntal parameter values for each model. The convergences were assessed for each model by montorng the trace plots whch summarzed n hstory graphs for all the parameters of nterest. Meanwhle, the results from the potental reducton factor R ) show more obvous results of nonconvergence for some parameters than that obtaned by the hstory graphs n IVM. Ths ndcates the mportant of applyng dagnostc test to examne the convergence. Increasng the number of teraton n the IVM and n the subsequent models showed that the R ) values vary between and. ndcatng that the convergence was acheved. A senstvty analyss was also conducted n order to study the nfluence of the choce of the pror denstes. Ths analyss nvestgated the adequacy of the dfferent pror nformaton used n each herarchcal model. The results show that the GSM provde more adequate pror denstes compared to the IVM and CVM. Ths was also confrmed when adoptng the DIC to compare between the nested models. The DIC results showed that the GSM had smaller value and extra effectve parameters. Ths

17 Comparson on modellng the relatve rsk estmaton 679 suggested that the GSM could perform well as compared to the other herarchcal models. Ths result also ndcates a good agreement wth the results obtaned by Banerjee et al. []. The present study show that wthn the Bayesan approach the dfferent methods used to demonstrate the models provde good compromse between satsfactory and approprateness for the data of nterest. The model wth suffcent characterstcs wll mprove the relatve rsk estmaton and help the researcher to plan and evaluate strateges used to prevent llness. REFERENCES [] A. Gelman, J.B. Carlne, H.S. Stern and D.B Rubn, Bayesan Data Analyss, st ed. Chapman and Hall Text, In Statstcal Scence Seres, Chapman & Hall, London, 995. [] B.P. Carln and T.A. Lous, Bayes and Emprcal Bayes Methods for Data Analyss, Chapman and Hall/CRC, New York, 000. [3] D. Spegelhalter, N. Best, B.P. Carln and A. van dn Lnde, Bayesan measures of model complexty and ft (wth dscusson), J. Roy. Statst. Soc. Ser. B, 64 (00), pp [4] D.G. Clayton and J. Kaldor, Emprcal Bayes estmates of age-standardzed relatve rsks for use n dsease mappng, Bometrcs, 43(987), pp [5] D.G. Clayton, A Monte Carlo method for Bayesan nference n fralty models, Bometrcs, 47 (99), pp [6] J. Besag and C. Kooperberg, On condtonal and ntrnsc auto regressons. Bomrtrka, 8(4) (995), pp [7] J. Besag, J. York and A. Molle, Bayesan mage restoraton, wth two applcatons n spatal statstc, Ann. Inst. Stat. Math., 43 (99), pp. -0. [8] J. Gll, Bayesan Methods for the Socal and Behavoural Scences, Chapman & Hall, New York, 00. [9] J.E. Kelsall and J.C. Wakefeld, Dscusson of Bayesan models for spatally correlated dsease and exposure data, by Best et al., In Bayesan Statstcs 6, ed., J.M. Bernardo, J.O. Berger, A.P. Davd, A.F.M Smth, Oxford Unversty Press, Oxford, 999, pp. 5.

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19 Comparson on modellng the relatve rsk estmaton 68 [3] W.R. Glks, S. Rchardson and D.J. Spegelhalter, Markov Chan Monte Carlo n Practce, Chapman and Hall, New York, 996. Receved: November, 009