Mathematical and Computer Modelling 50 (2009) 921–928

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Mathematical and Computer Modelling

journal homepage: www.elsevier.com/locate/mcm

Geographical variation in pharmacological prescriptionI

Carmen Armero ∗, Anabel Forte, Antonio López-QuílezGEeitEma Research Group, Departament d’Estadística i I.O., Universitat de València, Doctor Moliner 50, 46100 Burjassot, València, Spain

a r t i c l e i n f o

Article history:Received 26 September 2008Accepted 12 December 2008

Keywords:Bayesian hierarchical modelsGeneralized linear mixed modelsMarkov Chain Monte Carlo methodsSpatial effects

a b s t r a c t

Promoting rational drug administration in treatments is one of the most important issuesin Public Health. Bayesian hierarchical models are a very useful tool for incorporatinggeographical information into the analysis of pharmacological prescription data. Theyallow the mapping of spatial components which express the trend of geographicalvariation. In addition, these models are able to deal with uncertainty in a sequential waythrough prior distributions on parameters and hyperparameters. Bayes’ theorem combinesall types of information and provides the posterior distributionwhich is computed throughMarkov Chain Monte Carlo (MCMC) simulation methods.Simulated data for pharmacological prescription corresponding to people with a

diagnosis of degenerative osteoarthritis have been analyzed. Specifically, the number ofprescriptions and pharmaceutical costs per patient have been evaluated as well as itsrelationship with gender and age. Geographical variation between different administrativeunits is also introduced and discussed.

© 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Pharmacoepidemiology is devoted to the study of the use and effects of drugs in populations in order to promote rationaldrug administration in treatments. This a recent discipline, strongly tied to epidemiology, which takes advantage of itsmethodologies, applications and connections with other subjects such as mathematics and statistics [1].A very important issue in epidemiology is disease mapping which displays, in a very intuitive way, the geographical

variation of a disease [2,3]. Diseasemaps provide useful information about the spatial pattern of a disease, indicating regionswith high levels of prevalence/incidence, clustered areas, etc. Data can be presented in different ways (aggregate to a regionor municipality, individual cases) and allow the inclusion of potential risk factors, usually in terms of covariates.Spatial statistics is a very fruitful area of research and applications are capable of analyzing data with spatial elements.

They are a fundamental tool in disease mapping because they provide the appropriate statistical methodology for assessingthe estimations and predictions which will be shaded on the maps. Bayesian Hierarchical models, based on Bayesianmethodology, are possibly themost extensively statistical approach in this area [4–6]. They are specially useful formanagingthe statisticalmodeling and analysis of hierarchically structured data (also calledmultilevel data) because they can introduceinto the analysis new parameters which are not directly linked to the data and which can explain or be responsible for aconsiderable part of its variability.All this statistical and epidemiological machinery can be transferred in a very natural way to the pharmacoepidemiology

arena. Pharmacological maps displaying the geographical distribution of the relevant outcomes such as the prescription

I Supported in part by the Ministerio de Educación y Ciencia grant MTM2007-61554, Conselleria d’Empresa, Universitat i Ciència de la GeneralitatValenciana grant GV/2007/079 and a collaboration agreement with Conselleria de Sanitat de la Generalitat Valenciana.∗ Corresponding address: Univ. de Valencia, Dept. Statistics & Operations Research C/ Dr. Moliner 50, 46100 Burjassot, Spain Tel.: +34 963544309; fax:+34 963543238.E-mail addresses: carmen.armero@uv.es (C. Armero), anabel.forte@uv.es (A. Forte), antonio.lopez@uv.es (A. López-Quílez).

0895-7177/$ – see front matter© 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.mcm.2009.05.020

922 C. Armero et al. / Mathematical and Computer Modelling 50 (2009) 921–928

and cost of a given drug, adherence to a therapy in patients receiving a particular treatment, antibiotic consumption linkedwith antimicrobial resistance phenomena, side effects, etc turn out to be a very profitable and effective instrument to betterunderstand the polyhedric nature of the subject.Ourmain purpose in this paper goes in that direction.Wewill use Bayesian hierarchicalmodels to assess the geographical

variation in the prescription and cost of the specific drugs for people suffering from osteoarthritis with regard to genderand age. In particular, we are interested in examining spatial patterns among the municipalities of Castelló, which is oneof the three provinces in the Comunitat Valenciana, an autonomous region in the east of Spain. Osteoarthritis has beenselected because it is a serious disease whose study is of great interest in Public Health research: high prevalence, highpharmacological and consultation costs and directly responsible for the loss of millions of working days. Data in thisstudy are not real but realistic: they are simulated according to real epidemiological information on the prevalence ofosteoarthritis in this province and its demographic structure. They are part of a general project focused on the spatial analysisof pharmacological outcomes in the Comunitat Valenciana in which the research team of this paper is currently involved.The rest of the paper is as follows. Section 2 is a very general, methodological section that contains a brief outline of the

main statistical tools and procedures on which this paper is based. Section 3 discusses in detail the statistical modeling re-lated to the study of the pharmacoutilization and cost of people suffering from osteoarthritis. Section 4 presents the data andreports themain results of the statistical analysis. The paper ends with a short section devoted to some concluding remarks.

2. Statistical methodology

2.1. Generalized, linear and mixed models

One of the more relevant problems in statistics deals with the study of the relationship between a response variable ofinterest and a set of predictor variables. In this scenario, Linear Models (LM) have been the most paradigmatic statisticalmodels for many decades. They establish the normal distribution

[Yi | xi] ∼ N(µi, σ 2), i = 1, . . . , n, (1)where [A | B] denotes the conditional distribution of A given B, Yi is the response variable corresponding to observation i, xiis a k× 1 vector containing the values of predictors for Yi, µi = βtxi and σ 2 are, respectively, the mean and variance of thenormal model and β is a k× 1 vector corresponding to the coefficients associated to fixed effects.Generalized Linear Models (GLM) are a generalization of LM which allows the conditional distribution of Y to be not

necessarily normal. In fact, it can be any one belonging to the exponential family of distributions which includes the mostusual ones in statistics such as the Bernoulli, binomial, exponential, gamma, negative binomial, normal, Poisson andWeibull.GLMmodel a function of the conditional expectation of Yi as a linear combination of the corresponding values of the predictorvariables:

E(Yi | xi) = µi, g(µi) = βtxi. (2)The function g(.) is usually called the link function and provides the relationship between the linear predictors xi andµi.

GLM provide a unified framework for classical statistical methods such as linear regression, logistic and probit regression,Poisson regression and log-linear models among others [7,8].Generalized Linear Mixed Models (GLMM) are the natural extension of GLM when introducing random effects into the

model. In particular

E(Yi | xi) = µi, g(µi) = βtx1i + Z tx2i, (3)where g(.) and β are the same as in (2), x1i are now the data corresponding to predictor variables (denoted as xi in (2)), Z isa random effect with p levels and x2i is a p× 1 design vector whose structure will depend on the particular problem beingconsidered.Random effects have a very important role in epidemiological and pharmacoepidemiological research because they can

assess the influence of latent variables in the variability of the data. They are usually associated with the municipality orregion where each patient lives [9], the hospital where he/she is sent in the case of disease [10], etc. In some cases, randomeffects are assumed to be independent and identically normal distributed variables,

[Zj] ∼ N(0, σ 2Z ), j = 1, . . . , p, (4)

with a common and unknown variance,σ 2Z . Nevertheless, this is not a sensible assumption in situationswhere there is strongevidence of dependence among them. This topic is discussed in more detail in the next subsection.

2.2. Spatially structured random effects

Conditionally autoregressive models (CAR) are the most popular distributions to spatially model correlated randomeffects. In particular, each Zj in (Z1, . . . , Zp) is normally distributed given Z−j = (Z1, . . . , Zj−1, Zj+1, . . . , Zp) as:

[Zj | Z−j] ∼ N

(∑m6=j

wjmZmwj+

,σ 2Z

wj+

), (5)

C. Armero et al. / Mathematical and Computer Modelling 50 (2009) 921–928 923

where wjm are the elements of a proximity matrix W which gives a measure of the relationship among regions, wj+ =∑mwjm and σ

2Z is a variance parameter. Thus each spatial effect Zj, given the rest, is normally distributed with a mean that

depends on the neighbors whose influence is weighted in the sense of how close they are to element j. Neighbors for amunicipality or region are usually chosen by adjacency, and consequently it is considered that two regions are neighbors ifthey have a common border.CARmodels were introduced by Besag (1974) [11], but their use has spread extensively in the last 15 years. This is mainly

due to the increased relevance of Bayesian methods, which have become very popular thanks to the power and extensionof the Markov Chain Monte Carlo (MCMC) methods. They are introduced in more detail below.

2.3. Bayesian approach

Broadly speaking, traditional statistical methods make use of the data (basically) through the likelihood function, whichdepends on the selected probabilistic model and is in charge of connecting the data and the unknown parameters. Thisfunction provides estimates of the parameters whose quality is assessed by means of the sampling distribution of itsrespective estimator.Bayesian statistics is a more complex methodology that allows us to incorporate into the inferential process not only the

data but also all the available prior knowledge about the unknown parameters. This information needs to be expressed inprobabilistic language in the so-called prior distribution. Bayes’ theorem combines both types of information and providesthe posterior distribution, which contains all the relevant knowledge about the parameters of interest.Posterior distributions are not always easy to compute and, moreover, cannot always be done in an analytical way.

Bayesian methods solve these difficulties by using Monte Carlo sampling and Markov Chain Monte Carlo (MCMC) methodssuch as Gibbs sampling and the Metropolis-Hastings algorithm [12]. These useful simulation procedures are based onMarkov processes and result in an approximate sample of the posterior distribution from which we can extract ourinferences (posterior means and medians, credible regions, quantiles, etc) [13]. MCMC methods can be implemented inmany statistical packages. In this paper we will use them through WinBUGS [14], a statistical software which provides asimple implementation of a great number of complex statistical models.Bayesian Hierarchical models are a powerful tool for constructing models for complex scenarios. In particular, prior

distributions for random effects usually depend on unknown parameters. In addition, the knowledge of these parameterscan be expressed in turn by means of prior distributions, and so on following a hierarchical structure [5,15]. They are theso-called hyperparameters and hyperpriors, respectively.Model selection is one of themost attractive and stimulating problems in statistical applications. Among themost popular

criteria, we can quote the Akaike information criterion (AIC) [16], Bayesian information criterion (BIC) [17] and the Devianceinformation criterion (DIC) [18]. Although all of them weight up the goodness-of-fit and the complexity of the selectedmodel, the more useful criterion when comparing models whose posterior distribution has been approximated by MCMC isDIC. This is the case in our paper, and consequently our selected criterion. The smaller the DIC, the better the fit.

3. Modeling pharmacological prescription

3.1. Number of drug prescriptions

Weassume that the randomvariable describing the annual number,Npre, of drug prescriptions per patient suffering fromOsteoarthritis is a discrete variable following a Poisson distribution (very usual formodeling counts as here) with parameterλ. As usual in observable epidemiological studies, our simulated data setwill only contain information for those patientswhohave usedmedical resources directly and consequently with only positive values of the variable of interest. Technically, thisstatistical design provides a known sample with positive observations of the number of drug prescriptions and an unknownnumber N of missing data corresponding to people with osteoarthritis who have not consumed pharmaceutical resources.This missing mechanism produces truncated Poisson data with associated probabilities:

P(Npre = y | y > 0, λ) =λy e−λ

y! (1− e−λ), y > 0. (6)

In this case, a general Bayesian analysis also includes the unknown parameter N in the inferential process whoseprocessing results, with an appropriate flat distribution for the conditional prior distribution of N given λ, in a conventionalBayesian analysis with the truncated Poisson providing the likelihood of the observed data [19]. Obviously, the stability ofthe recording data is a natural issue in our inferential process and consequently the conditional independence between theparameters and hyperparameters defining λ and the truncation rule is assumed.Our interest now is to model the behavior of the expected annual number of drug prescriptions for patient i in terms of

the covariates describing his/her gender and age and also the municipality where he/she is receiving medical treatment forosteoarthritis:

log(λi) = β0 + β1I{male}(i)+ β2I{age<40}(i)+ β3I{age 40-55}(i)+ β4I{age 55–70}(i)+ Ui +Wmuni(i), (7)

924 C. Armero et al. / Mathematical and Computer Modelling 50 (2009) 921–928

where I{A}(i) stands for the indicator function defined as 1 if patient i has the characteristic A and zero otherwise, Ui isa random effect which describes the specific characteristics of the patient i, and Wmuni(i) represents a spatially correlatedrandom effect for the m municipalities which describes a certain level of similarity between contiguous areas. Randomeffects Ui also allow us to deal with the usual phenomenon of overdispersion [20] which is attached to data analyzed withPoisson models.Observe that the notation in (7) is not the same (less compact but more intuitive) as the general one used in the previous

section. But the equivalence between them is easy when considering: the vector of the predictor variables for patient ias x1i = (1, I{male}(i), I{age<40}(i), I{age 40-55}(i), I{age 55-70}(i))t ; the design vector x2i corresponding to an individual randomeffect as a (n × 1) vector with all the elements equal to zero except that one corresponding to individual i which is 1; andthe accompanying design vector for the spatial effect as a (m × 1) vector where all the elements are zero except that onecorresponding to the municipality where patient i is located. It is also worth noting that we have discretized the age ofthe patients into four categories (<40, [40, 55), [55, 70) and>70 years). This division is based on medical information (seefor example [21]) about the different stages and evolution of osteoarthritis. Note also that in expression (7) the interceptparameter β0 assumes the effect of the women older than 70, who have been chosen as the reference category in all thepaper.Specific random effects Ui for patients are assumed to be normally i.i.d. random variables with a common and unknown

population variance σ 2U . Furthermore, a CAR model (as defined in (5)) with unknown variance σ2W controlling the intensity

of the local dependence is considered here to describe the conditional distribution [Wj | W−j].From a Bayesian reasoning the model is completely specified when a prior distribution of the parameters of the model is

elicited. This implies the construction of a joint distribution for the β coefficients in (7) and the variance of each one of therandom effects, σ 2U and σ

2W .

Many pharmacoepidemiological analyses usually need to present their findings with a great deal of objectivity to avoidconflicts and suspicion (a typical case is the development of a new drug in the pharmacological industry). In this paper,and although the data in this paper are simulated, we are interested in getting our analysis as close to reality as possible.Consequently,we are going to use a prior distribution for the parameters that contain little information and leaves the lowestpossible trace on the posterior. Also, in order to avoid any chance of obtaining improper posteriors, we have always workedwith proper densities. To that effect, we consider the coefficients β0, β1, β2, β3 and β4 associated to the fixed effects as priorindependent and normally distributed with a great variance, as for instance N(0, 10 000). The magnitude of this variance isclear when we remember that we are working in the logarithmic scale.The elicitation of hyperprior distributions for the unknown variances in random effects is a very challenging issue in

Bayesian statistics that has produced, and is still generating, very fruitful discussions and results. In this scenario, InverseGamma for variance parameters and Uniform models for standard deviations are possibly the most popular proposals forscale parameters: the first one because of conditional conjugacy and the second one because it seems to play amore neutralrole in the analysis. In this paper, and for both hyperparameters, σU and σW , we opt for the Uniform distribution Un(0, 3)which is sufficiently flat and appropriate to our analysis.

3.2. Cost of drug prescriptions

Data from the randomvariable describing the cost of drug prescriptions per year for a patient receivingmedical treatmentare always positive and usually skewed right. Consequently, a reasonable model to describe the behavior of this variable isa lognormal distribution,

[log(Cost)] ∼ N(µ, σ 2c ), (8)and we are interested in assessing the expected cost of patient i in terms of his/her gender, age and the municipality

where he/she is receiving medical treatment:µi = β0 + β1I{male}(i)+ β2I{age<40}(i)+ β3I{age 40–55}(i)+ β4I{age 55–70}(i)+Wmuni(i), (9)

I{A}(i) and the rest of the elements in (9) being analogous to those discussed in (7). Note that random individual effects havenot been included in the model because overdispersion is not a problem with data modeled with normal distributions. Forthe rest of this model, we follow the same discussion and particular choice, distribution for the spatially random effect, priordistribution for the β ’s parameters and hyperpriors, which are presented to model the number of drug prescriptions in theprevious subsection. The only difference lies in the variance σ 2C of the normal model which is also included in this model,σC also being modeled in terms of the Uniform distribution Un(0, 3).

4. Statistical analysis and results

4.1. The data

Data for both outcomes, the number of prescriptions and the cost of drug prescription, are independently simulatedfollowing the main characteristics of the prevalence of osteoarthritis and the particular demographic structure in Castelló.This province has a population of 543432 inhabitants and 135 municipalities: Castell de Cabres is the least populated (only19 inhabitants) and the city of Castelló is the most populated with 167500 inhabitants.

C. Armero et al. / Mathematical and Computer Modelling 50 (2009) 921–928 925

Table 1Description of theMCMC sample from the posterior distribution of parameters and hyperparameters in themodelling of the number of drug prescriptions.

Node Mean sd 2.5% Median 97.5%

β0 2.958 0.024 2.908 2.961 2.998β1 −0.191 0.019 −0.226 −0.192 −0.150β2 −1.272 0.047 −1.359 −1.275 −1.176β3 −0.658 0.023 −0.704 −0.656 −0.614β4 −0.190 0.018 −0.223 −0.191 −0.152ν 1.184 0.169 0.880 1.170 1.545ψ 1.306 0.012 1.282 1.305 1.330

We have considered a sample of 29617 patients whose municipality has been assigned according to demographiccharacteristics in themunicipalities. Thenwe have simulated the age for each patient by considering this variable as a factorwith the four levels identified previously: under 40, between 40 and 55, between 55 and 70, and older than 70. Because theproportion of men and women suffering from osteoarthritis is different by age group we have also generated the gender foreach patient. Finally, a spatial term according to the neighborhood relationship of the municipalities in Castelló has beenintroduced and immediately afterwards a value from the variable of interest is also simulated.

4.2. Number of drug prescriptions

We consider for this outcome the model presented in (7) with gender and age as covariates, a random effect per patientand a random spatial effect. We have also analyzed two nested and simpler models in order to evaluate the possibilitythat some of the elements in (7) do not provide substantial information about the expected number of drug prescriptions.Specifically, the first one just contains age and gender whereas the second one includes, in addition to these covariates, aspecific random effect for patients.These models have been implemented with WinBUGS and an approximate random sample from the posterior

distribution of parameters and hyperparameters is computed. We have considered 100000 iterations plus 50000corresponding to a burning period to achieve good convergence of the Markovian process.All models have been evaluated through DIC criterion with a total value of 187592.70 for the complete model in (7).

This is the lowest quantity in comparison with the values, 1 156471.00 and 188034.40, corresponding to the two nestedmodels previously quoted, respectively, in order of increasing complexity. Notice that the enormous value provided by themodel containing only age and gender is a clear illustration of the usefulness of including individual random effects foroverdispersed data.Table 1 presents the mean, standard deviation, median and 2.5th and 97.5th percentile of the MCMC sample obtained

from theposterior distribution of the parameters andhyperparameters of themodel. Notice also that the current informationabout the variances σ 2U and σ

2W are here expressed in terms of their precision, ν = 1/σ

2U and ψ = 1/σ

2W , respectively.

Posterior means of the parameters indicate that women older than 70 have the greatest number of drug prescriptions.And for women, this quantity increases directly with age. Men have, in general, the same behavior as women but in the fourage groups they have always lower quantities of drug prescriptions.Fig. 1 displays the posterior expectation of the spatial effect in all themunicipalities in Castelló.We can distinguish a clear

spatial pattern, with two defined regions in the south of the province, one with a high level of prescriptions and the otherquite the opposite. The spatial pattern in the north and central areas is not so clear, but we can appreciate some clusteredmunicipalities.From an epidemiological point of view, spatial information on the evolution of drug prescription with regard to the

different categories defined by the covariates is a useful tool to better understand the etiology of the disease. To that effect,Fig. 2 displays an estimation of the posterior expected number of drugs for women in each group by age and municipality.Obviously, these maps present very clear and intuitive information about the evolution in the number of prescriptions byage in the group of women suffering from osteoarthritis and the geographical variation in Castelló.

4.3. Cost of drug prescription

The analysis of the model discussed in (9) to evaluate the cost of drug prescription per patient and year through theirgender, age and municipality is similar to that presented for the number of prescriptions in (7): MCMC with WinBUGS,burning period, 100000 iterations and DIC criterion. In this case, the obtained value for DIC is 95325.10, which is muchbetter than the one produced with the nested model without the spatial effect (108905.00).In Table 2 we show the mean, standard deviation, median and 2.5th and 97.5th percentile of the MCMC sample obtained

from the posterior distribution of the parameters and hyperparameters of the model. Here also the current information onthe variances σ 2U and σ

2C are expressed in terms of their precision, ν = 1/σ

2U and γ = 1/σ

2W , respectively.

And finally, in Fig. 3 we present a map with an estimation of the posterior expected number of drugs for women in eachgroup by age and municipality. Here also, we can detect the strong presence of the spatial pattern and always a positiveincrease in cost with age.

926 C. Armero et al. / Mathematical and Computer Modelling 50 (2009) 921–928

< –0.4–0.4, –0.08–0.08, 0.080.08, 0.4>= 0.4

Fig. 1. Map of Castelló showing an estimation of the posterior expectation of the spatial random effect associated to the number of drug prescriptions.

<40

< 55, 1010, 1515, 20>= 20

40–55

55–70 >70

Fig. 2. Maps showing an estimation of the number of drug prescriptions for women according to municipality and age group.

C. Armero et al. / Mathematical and Computer Modelling 50 (2009) 921–928 927

Table 2Description of the MCMC sample from the posterior distribution of parameters and hyperparameters in the modeling of the cost of drug prescription.

Node Mean sd 2.5% Median 97.5%

β0 4.004 0.083 3.858 3.996 4.185β1 −0.210 0.015 −0.240 −0.210 −0.178β2 −1.744 0.052 −1.843 −1.744 −1.641β3 −0.927 0.023 −0.973 −0.927 −0.881β4 −0.256 0.015 −0.285 −0.256 −0.226ν 0.195 0.026 0.147 0.194 0.703γ 0.692 0.005 0.681 0.692 0.703

<40

< 1010, 5050, 100100, 150>= 200

40–55

55–70 >70

Fig. 3. Maps showing an estimation of the cost of drug prescription for women according to municipality and age group.

5. Conclusions

Promoting rational drug administration in treatments is one of the most important issues in Public Health. Bayesianhierarchical models are a very useful tool for incorporating geographical information into the analysis of pharmacologicaloutcomes and related factors. They allow us to map spatial components which express the intensity of that geographicalvariation and provide quantitative information about the different uncertainties affecting them.

References

[1] B.L. Strom, Pharmacoepidemiology, Fourth ed., Wiley, 2001.[2] P. Elliott, V.C. Wakefield, N.G. Best, D.J. Briggs, Spatial Epidemiology, Oxford University Press, 2000.[3] A.B. Lawson, Statistical Methods in Spatial Epidemiology, Wiley, 2001.[4] T.C. Bailey, Spatial statistical methods in health, Cadernas Saúde Pública 17 (5) (2001) 1083–1098.

928 C. Armero et al. / Mathematical and Computer Modelling 50 (2009) 921–928

[5] S. Banerjee, B.P. Carlin, A.E. Gelfand, Hierarchical Modeling and Analysis for Spatial Data, Chapman and Hall/CRC, 2004.[6] N. Best, S. Richardson, A. Thomson, A comparison of Bayesian spatial models for disease mapping, Statistical Methods in Medical Research 14 (1)(2005) 35–59.

[7] Y. Lee, J.A. Nelder, Hierarchical generalized linear models, Journal of the Royal Statistical Society. Series B 58 (4) (1996) 619–678.[8] C.E.McCulloch, Generalized linearmodels, in: A.E. Raftery, et al. (Eds.), Statistics in the 21st Century, CRCPress LLC andAmerican Statistical Association,Boca Raton, Alexandria, 2002, pp. 387–396.

[9] M.A. Martínez Beneito, A. López-Quílez, P. Botella-Rocamora, An autoregressive approach to spatio-temporal disease mapping, Statistics in Medicine27 (2008) 2874–2889.

[10] C. Armero, A. López-Quílez, R. López-Sánchez, Bayesian assessment of times to diagnosis in breast cancer screening, Journal of Applied Statistics 35(9) (2008) 997–1009.

[11] J. Besag, Spatial interaction and the statistical analysis of lattice systems (with discussion), Journal of the Royal Statistical Society. Series B 36 (1974)192–236.

[12] P.J. Green, A primer on Markov chain Monte Carlo, in: Complex Stochastic Systems, Chapman and Hall/CRC, 2001, pp. 1–62.[13] D. Gammerman, H.F. Lopes, Monte Carlo Markov Chain: Stochastic Simulation for Bayesian Inference, 2nd ed., Chapman and Hall, 2006.[14] D.J. Spiegelhalter, A. Thomas, N. Best, D. Lunn, WinBUGS version (1.4), 2003. http://www.mrc-bsu.cam.ac.uk/bugs/winbugs/manual14.pdf.[15] Y. Zhao, J. Staudenmayer, B.A. Coull, M.P. Wand, General design Bayesian generalized linear mixed models, Statistical Science 21 (1) (2006) 35–51.[16] H. Akaike, A new look of the statistical model identification, IEEE Transactions on Automatic Control 19 (6) (1974) 716–723.[17] G. Schwarz, Estimating the dimension of a model, Annals of Statistics 6 (2) (1978) 461–464.[18] D.J. Spiegelhalter, N. Best, B.P. Carlin, A. Van der Linde, Bayesianmeasures ofmodel complexity and fit (with discussion), Journal of the Royal Statistical

Society. Series B 64 (4) (2002) 583–639.[19] A. Gelman, J.B. Carlin, H.S. Stern, D.B. Rubin, Bayesian Data Analysis, Chapman and Hall/CRC, 2004.[20] J. Liu, D.K. Dey, Hierarchical overdispersed Poisson model with macrolevel autocorrelation, Statistical Methodology 4 (3) (2007) 354–370.[21] T.B. Kirkwood, What is the relationship between Osteoarthritis and ageing? Baillieres Clininical Rheumatology 11 (4) (1997) 683–694.