Bayesian random effects model for disease mapping of relative risks

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1 Avalable onlne at Scholars Research Lbrary Annals of Bologcal Research, 014, 5 (1):3-31 ( ISS CODE (USA): ABRBW Bayesan random effects model for dsease mappng of relatve rsks Srnvasan R. and Venkatesan P.* atonal Insttute for Research n Tuberculoss, ICMR, Chenna ABSTRACT The use of random effects assumes that every regon has some common rsk rates and allows estmates to combne nformaton from several regons. Ths random effect s common n Bayesan approach dsease mappng usng lognormal model. Bayesan posteror dstrbutons are obtaned va Markov Chan Monte Carlo (MCMC) computatons. HIV Data was obtaned from atonal Insttute for Research n Tuberculoss. The result of the study reveals that the random effects model, gves the smoother values of relatve rsk than the Posson gamma model. Spatal analyss s proved to be more useful for studyng spread of HIV analyss. Keywords: Dsease mappng, Posson gamma, Log-normal and random effects model. ITRODUCTIO The applcatons of Bayesan methods for dsease mappng, rsk assessment and predcton wthn spatal research are numerous. There are two domnant approaches called, emprcal and fully Bayesan method. In the Emprcal Bayesan (EB) method, parameters of pror dstrbutons are estmated usng observed margnal dstrbutons, but n the fully Bayesan approach, the pror and posteror dstrbutons are obtaned va Markov Chan Monte Carlo (MCMC) computatons. The dsease mappng s useful to fnd the geographcal dstrbuton of dsease burden and dseases ncdence n whch dsease s shaded accordng to the hgh and low rsk areas n Chenna ward. Bayesan methods for dsease mappng of dsease n Chenna ward usng Posson-Gamma model and Log-normal model are explored. The am s to compare whether Bayesan estmates of random effects model of log normal model are more stable than the Posson gamma model estmates. The dea of Emprcal Bayesan and Fully Bayesan methods for dsease mappng was revewed and explored n dfferent tmes pont [1,9,16]. Many author dscussed about mportance of pror and errors n covarates model n dsease mappng [,3]. The comparsons of varous models for dsease mappng provdes a comprehensve revew of the recent developments n Bayesan dsease mappng[7, 14, 15]. In the Full Bayesan, Inference s made by usng MCMC[11,1] technques that provde an estmate of the posteror dstrbuton of the parameters of the model. Fully Bayesan dsease mappng was explored n dfferent stuaton [4,5,6,10] and Spato-temporal model was used to fnd the spato temporal changes between years and place[0]. The most recent Herarchcal Bayesan models that are used for dsease mappng usng full Bayes estmaton was revewed by [8]. The several model for nfectous dsease lke HIV and Tuberculoss was done by [18,19]. Geographcal analyss of heart dseases for tamlnadu was by studed [17]. Spatal mappng of cholera for Chenna were done by [13]. Scholars Research Lbrary 3

2 Srnvasan R. and Venkatesan P. Annals of Bologcal Research, 014, 5 (1):3-31 Posson-Gamma Model The Posson model holds the assumpton that mean and varance are same, but t s n the spatal context, the data are over dspersed and varance s hgher than mean. A smple way to allow for a hgher varance s to use negatve bnomal dstrbuton nstead of Posson. The negatve bnomal dstrbuton can also be regarded as a mxed model n whch random effect follow a Gamma dstrbuton for each area. Ths combnaton s known as Posson Gamma model. A sample model s; Y n ~ Po( Eθ ), = 1,..., I and θ ~ Gamma( a, b), and Varance σ a = b whch denotes gamma dstrbuton wth mean a µ = b The crude SMR rates and RR are often subject to large chance varaton when rare dseases are nvestgated n small areas. Lterature on Bayesan dsease mappng presents mxed effects Posson models that are characterzed as spatal smoothng. The methods assume spatally varyng or randomly varyng RRs and the assocated jont pror probablty for poolng data and borrowng strength. In ths secton, these consderatons are motvated and explored through fully Bayesan mappng models whch requres a pror dstrbuton on parameters. Y θ, β θ 0 ~ Posson(E e ~ Gamma( α, α) here pror for β 0 α and ~ orm(m,v) α ~ Gamma(a,b) β 0 s β0 θ ) (1) () two ssues s normally arsng, when expected count of E s small, a small varaton n O can produce dramatc changes n the value of SMR, secondly, n Posson model, nformaton s borrowed from all the areas n order to construct the posteror estmates gven that a and b are the same for every regon. Log-ormal Model Clayton and Kaldor (1987) proposed another rsk estmator based on assumpton that the logarthm of the relatve β = follows multvarate normal dstrbuton wth mean µ and varance σ. The estmate of the rsks ( log( θ ) log relatve rsk s log ( O + 1/ ) / E nstead of log ( O )/ E zero., because n the Posson model s not defned f O s In the Posson gamma model whch s easy to ft but we cannot nclude the non-spatal random effect and zero case s not dfferentated, so we have to see alternatve technque lke Posson log-normal model. A Posson log-normal non-spatal random effects model s gven by Y β, V V ~ d ~ d or(0, σ ) Posson( E µ e v V ) (3) Where V are area-specfc random effects that capture the resdual or unexplaned (log) relatve rsk of dsease n area, = 1... n. Here we have V θ = e ~ Lognormal(0, σ ). (4) Scholars Research Lbrary 4

3 Srnvasan R. and Venkatesan P. Annals of Bologcal Research, 014, 5 (1):3-31 MATERIALS AD METHODS The lattce or areal data of Chenna wards for HIV/AIDS dsease mappng are consdered for ths work. Chenna cty conssts of 155 wards. Let Y (=1,,,155) s the observed count of HIV/AIDS dsease and E (=1,,,155) s the expected count n the th regon. If the model follows the Posson dstrbuton, θ (=1, n) s the relatve rsk n the th regon. It measures of how much of a rsk factor n regons leads to dfference n the varance of regonal estmates. Here Y s a random varable, and E s a fxed and known functon of θ where θ s a number of persons at rsk for the dsease n regon. We assume that E = y r θ θ θ, (5) where r s the overall dsease rates n the entre study regon and here we assume that ths dsease rates constant n all the regons. Ths method of calculatng expected count s called nternal standardzaton or drect standardzaton. The HIV patents were regstered durng 004 to 006 at the atonal Insttute for Research n Tuberculoss was consdered for ths work. The observed cases were aggregated for each ward level n Chenna and expected numbers of cases were calculated usng ndrect standardzaton method from ward populaton. The total number of cases recorded durng 004 and 006 were collected and also the populaton of each ward was collected from census β and gamma (1,1) pror for α and run the teraton 001. In the Posson gamma model, specfed the flat pror for 0 for after dscardng ntal 3000 as burn-n. The convergence of the model was checked by Gelman Rubn convergence test. RESULTS The results are presented n Table 1. The posteror relatve rsk estmate ranges from 0.51 to.74 and hghest relatve rsk shrunk toward the mean. Bayesan Posson gamma model smoothed towards the global mean and the varatons n the map dsappeared. Table 1 Posteror Means of RR under Bayesan Posson Gamma and Lognormal Model Ward Posson Gamma Model Lognormal Model Mean SD Credble Interval Mean SD Credble Interval RR[1] , , 1.1 RR[] , , RR[3] , , RR[4] , , RR[5] , , 1.89 RR[6] , , 1.57 RR[7] , , RR[8] , , RR[9] , , RR[10] , , RR[11] , , RR[1] , , 1.61 RR[13] , ,.071 RR[14] , , RR[15] , , RR[16] , , RR[17] , , 1.39 RR[18] , , 1.76 RR[19] , , 1.58 RR[0] , , RR[1] , , RR[] , , 1.57 RR[3] , , 1.75 RR[4] , , RR[5] , , RR[6] , , RR[7] , , RR[8] , RR[9] , , 1.78 RR[30] , , 1.71 Scholars Research Lbrary 5

4 Srnvasan R. and Venkatesan P. Annals of Bologcal Research, 014, 5 (1):3-31 RR[31] , , RR[3] , , RR[33] , , RR[34] , , RR[35] , ,.014 RR[36] , , RR[37] , , RR[38] , , RR[39] , , 1.40 RR[40] , , RR[41] , , RR[4] , , RR[43] , , RR[44] , , RR[45] , RR[46] , , RR[47] , , RR[48] , , 1.58 RR[49] , , RR[50] , , RR[51] , , 1.54 RR[5] , , 1.69 RR[53] , , RR[54] , , RR[55] , , 1.93 RR[56] , , RR[57] , ,.13 RR[58] , , RR[59] , , RR[60] , , RR[61] , , RR[6] , , RR[63] , , 1.36 RR[64] , , 1.15 RR[65] , , RR[66] , , 1.19 RR[67] , , 1.45 RR[68] , , RR[69] , , RR[70] , , RR[71] , , RR[7] , , 1.53 RR[73] , , RR[74] , , RR[75] , , 1.8 RR[76] , , RR[77] , , RR[78] , , RR[79] , , RR[80] , , RR[81] , , RR[8] , , RR[83] , , RR[84] , , 1.70 RR[85] , , RR[86] , , RR[87] , , RR[88] , , RR[89] , , RR[90] , , RR[91] , , RR[9] , , RR[93] , , RR[94] , ,.096 RR[95] , , RR[96] , ,.007 RR[97] , , 1.64 RR[98] , , RR[99] , , RR[100] , , RR[101] , , Scholars Research Lbrary 6

5 Srnvasan R. and Venkatesan P. Annals of Bologcal Research, 014, 5 (1):3-31 RR[10] , , 1.58 RR[103] , , 1.78 RR[104] , , RR[105] , , 1.94 RR[106] , , RR[107] , , RR[108] , ,.069 RR[109] , , RR[110] , , RR[111] , , RR[11] , ,.014 RR[113] , , 1.64 RR[114] , , RR[115] , , RR[116] , , RR[117] , , RR[118] , , 1.58 RR[119] , , RR[10] , , RR[11] , , 1.57 RR[1] , , RR[13] , ,. RR[14] , , 1.70 RR[15] , , 1.44 RR[16] , , RR[17] , , RR[18] , , 1.39 RR[19] , , 1.30 RR[130] , , RR[131] , , RR[13] , , RR[133] , , RR[134] , RR[135] , , RR[136] , , RR[137] , ,. RR[138] , , RR[139] , , RR[140] , , RR[141] , , RR[14] , RR[143] , , RR[144] , , RR[145] , ,.047 RR[146] , ,.198 RR[147] , , RR[148] , , 1.9 RR[149] , RR[150] , , RR[151] , , 1.66 RR[15] , , RR[153] , , 1.35 RR[154] , , RR[155] , , The posteror RR for Posson gamma model ranges between comparng to lognormal model whch mples that log normal model gves better smoothng for ths HIV data due to addng random effect n Chenna ward. After ncorporatng the random effects n the log normal model, the extreme values dsappear and relatve rsk extremely shrunk toward the global mean. Fgure () dsplays the relatve rsk estmates for all the wards n Chenna after convergence was acheved. The spatal pattern n relatve rsk s very smlar to the one obtaned usng the Posson-gamma model but more smoother map wth less extremes n the relatve rsk estmates. The relatve rsk ranges from 0.5 to Scholars Research Lbrary 7

6 Srnvasan R. and Venkatesan P. Annals of Bologcal Research, 014, 5 (1):3-31 The map of the Posson model revealed that the extreme values shrunk towards the mean and there s no ward comes under RR > 1. (samples)means f or RR (0) < 0.5 medan f or RR () < 0.5 (65) (9) >= 1.0 (79) (76) >= 1.0 (samples)means f or theta () < 0.5 medan f or theta (6) < 0.5 (9) (99) (63) >= 1.0 (5) >= 1.0 Fgure 1 Posteror Expected Relatve (Mean and Medan) and Posteror Probablty of Theta (Mean and Medan) under Posson Gamma Model Table Posteror summares for Posson gamma model and Log normal model Posson Gamma model Log-normal model Parameter Mean SD MC error Credble Interval Mean SD MC error Credble Interval α , 4.68 β , σ , 0.68 Scholars Research Lbrary 8

7 Srnvasan R. and Venkatesan P. Annals of Bologcal Research, 014, 5 (1):3-31 (samples)means f or RR (0) < 0.5 medan f or RR () < 0.5 (157) (155) (0) >= 1.0 (0) >= 1.0 medan f or h (140) < 0.5 (samples)means f or h (140) < 0.5 (16) ( 16) (1) >= 1.0 (1) >= 1.0 Fgure Posteror Expected Relatve Rsk (Mean and Medan) and Resdual (Mean and Medan) under Lognormal Model The posteror estmates of the parameter for Bayesan Posson gamma and Log-normal model are gven n Table. The posteror mean for α s 3.07, compared to 3.17 under log-normal, and the posteror mean for β s 0.17, compared to The MC error for Posson gamma model s hgh compared wth random effect log-normal model. Also, Credble Interval for log normal model s very narrow. Scholars Research Lbrary 9

8 Srnvasan R. and Venkatesan P. Annals of Bologcal Research, 014, 5 (1):3-31 Table 3 Devance values for Posson gamma and Log normal models Model D Dˆ D p DIC Posson-gamma Log-normal The Devance Informaton Crteron (DIC) for log-normal model s less (584.35) compared wth Posson-gamma model whch mples that log-normal model has the advantage of spatal random effect and better ft for ths HIV dsease mappng model. box plot: RR box plot: h [137] [57] [13] [146] [94] [13] [108] [14] [1] [35] [96] [11] [145] [7] [7] [4] [8] [60] [81] [33] [86] [89] [105] [107] [16] [148] [14] [9] [37] [49] [18] [0] [3] [30] [41] [44] [5] [61] [71] [84] [] [4] [6][9] [1] [16] [5] [43] [46] [47] [73] [76] [77] [8] [85] [87] [90] [93] [103] [95] [99] [100] [101] [104] [106] [109] [8] [15] [19] [] [38] [40] [11] [6][3] [4] [45] [48] [51] [59] [70] [53] [7] [56] [17] [3] [31] [34] [36] [39] [54] [58] [67] [68] [69] [78] [80] [9] [97] [110] [113] [117] [14] [136] [143] [1] [134] [144] [147] [151] [98] [10] [83] [88] [91] [118] [11] [15] [131] [111] [116] [139] [150] [156] [157] [74] [79] [115] [119] [10] [15] [17] [13] [133] [5] [55] [114] [18] [130] [135] [138] [155] [140] [141] [154] [1] [63] [10] [50] [64] [66] [75] [19] [149] [153] [6] [65] [86] [96] [4] [57] [85] [89] [13] [13] [11] [137] [146] [14] [7] [94] [108] [145] [] [7] [18] [1] [35] [44] [46] [60] [8] [9] [33] [49] [81] [10] [105] [61] [107] [16] [148] [14] [143] [37] [3] [30] [40] [47] [51] [95] [131] [38] [84] [87] [8] [16] [41] [5] [] [1] [3] [4] [6][9] [11] [15] [0] [5] [3] [34] [36] [54] [56] [63] [70] [58] [67] [68] [7] [73] [74] [76] [77] [8] [93][99] [103] [117] [136] [151] [155] [80] [97][104] [106] [109] [113] [1] [90] [5] [39] [4][48] [53][59] [6] [64] [65] [69] [75] [101] [110][118] [11] [14] [144] [134] [147] [153] [1] [17] [19] [43] [66] [71] [78] [6] [10] [31] [45] [50] [55] [79] [83] [88] [91] [9][98] [100] [114] [116] [18] [130] [13] [138] [139] [15] [150] [156] [135] [141] [154] [157] [10] [15] [19] [111] [115] [119][17] [133] [140] [149] Fgure 3 Box plot for Relatve rsk and resdual usng Lognormal Model COCLUSIO Bayesan dsease mappng technques n whch the random effect model method gves smoother relatve rsk, especally when rare dseases are nvestgated n an area wth a small populaton. The result reveals that there are 8 wards havng no rsk HIV/AIDS ward and 75 wards havng hgher rsk whch was scattered throughout Chenna. Bayesan model wth random effect gves better shrnkage and smaller DIC than Posson gamma model. Ths approach would observe the unobserved and unexplaned spatal varaton of nterest. Bayesan random effect dsease mappng of RR shrunk extreme towards the global mean and lowest RR pulls upwards and hghest RR pulls downwards. The MC error and credble Interval for Bayesan random effect model s very narrow and posteror medan also close wth posteror mean whch mples that random effect model s better for dsease mappng of HIV/AIDS data. REFERECES [1] Banerjee S, Carln B and Gelfand A E (004): Herarchcal Modelng and Analyss for Spatal Data. Boca Raton: Chapman & Hall. [] Bernardnell L, Clayton D and Montomol C (1995): Statstcs n Medcne, 14: [3] Bernardnell L, Pascutto C, Best G and Glks W R (1997): Statstcs n Medcne, 16: [4] Besag J (1974): Journal of Royal Statstcal Socety, Seres-B, 36: [5] Besag J and Green P J (1993): Journal of Royal Statstcal Socety, Seres-B, 55: [6] Besag J, York J C and Molle A (1991): Annals of the Insttute of Statstcal Mathematcs, 43: [7] Best, Waller L, Thomas A, Conlon E and Arnold R (1999): Bayesan models for spatally correlated dseases and exposure data. In Bayesan Statstcs 6, eds. J.M. Bernardo et al. Oxford: Oxford Unversty Press, pp [8] Best, Rchardson S and Thomson A (005): Statstcal Methods n Medcal Research. 14: [9] Carln B P and Lous T A (1996): Bayes and Emprcal Bayes Methods for Data Analyss. London: Chapman & Hall. [10] Clayton D and Kaldor J (1987): Bometrcs 43: [11] Ellott P, Wakefeld J C, Best G and Brggs D J (000): Spatal Epdemology: Methods and Applcatons. Oxford: Oxford Unversty Press. [1] GeoBUGS User Manual (004): GeoBUGS User Manual. Verson 1.. [13] Jayakumar K and Malarvannan S (013): Archves of Appled Scence Research, 5 (3):93-99 [14] Lawson A (000): Statstcs n Medcne, 19: [15] Lawson A (009): Bayesan dsease mappng: Herarchcal modelng n spatal epdemology, Chapman & Hall/ Scholars Research Lbrary 30

9 Srnvasan R. and Venkatesan P. Annals of Bologcal Research, 014, 5 (1):3-31 CRC. [16] Marshall, R (1991): Appled Statstcs, 40: [17] Tamlenth1 S, Arul P, Punthavath P and Manonman I.K(011), Archves of Appled Scence Research, 3 ():63-74 [18] Venkatesan P and Srnvasan R (008): Appled Bayesan Statstcal Analyss : [19] Venkatesan P, Srnvasan R and Bose M S C (007): Varahmr Journal of Mathematcal Scences, 7 : [0] Waller L A and Gotway C A (004): Appled Spatal Statstcs for Publc Health Data. ew York: Wley. Scholars Research Lbrary 31