Open access peer-reviewed chapter - ONLINE FIRST

Spatial Modeling in Epidemiology

Written By

María Guzmán Martínez, Eduardo Pérez-Castro, Ramón Reyes-Carreto and Rocio Acosta-Pech

Reviewed: March 25th, 2022 Published: May 10th, 2022

DOI: 10.5772/intechopen.104693

Biostatistics Edited by Cruz Vargas-De-León

From the Edited Volume

Biostatistics [Working Title]

Prof. Cruz Vargas-De-León

Chapter metrics overview

6 Chapter Downloads

View Full Metrics


The objective of this chapter is to present the methodology of some of the models used in the area of epidemiology, which are used to study, understand, model and predict diseases (infectious and non-infectious) occurring in a given region. These models, which belong to the area of geostatistics, are usually composed of a fixed part and a random part. The fixed part includes the explanatory variables of the model and the random part includes, in addition to the error term, a random term that generally has a multivariate Gaussian distribution. Based on the random effect, the spatial correlation (or covariance) structure of the data will be explained. In this way, the spatial variability of the data in the region of interest is accounted for, thus avoiding that this information is added to the model error term. The chapter begins by introducing Gaussian processes, and then looks at their inclusion in generalized spatial linear models, spatial survival analysis and finally in the generalized extreme value distribution for spatial data. The review also mentions some of the main packages that exist in the R statistical software and that help with the implementation of the mentioned spatial models.


  • Geostatistic
  • gaussian process
  • spatial GLM
  • spatial survival analysis
  • spatial extremes

1. Introduction

The term spatial statistics is used to describe a wide range of statistical models and methods for the analysis of geo-referenced data [1]. Its rapid use has been increasing in various fields of science, such as biology, image processing, environmental and earth sciences, ecology, epidemiology, agronomy, forestry, among others [2]. In epidemiology, spatial statistics are used to study the occurrence of health-disease events or deaths in a region of interest. It is now known that several public health problems tend to exhibit spatial dependence (spatial autocorrelation, spatial variability), and that sometimes these problems are related to climatic factors that are generally of a spatially continuous nature or with factors specific to the study region. The use of classical statistical techniques to model spatial data generally leads to an overestimation of model parameters [1]; and although they may eventually help, these models, lacking adequate structure, will not be able to model the spatial variability of the data; valuable information that will be sent to model error and cannot be used to explain the nature of the phenomenon under study.

Recent studies have shown that spatial models can help identify spatial patterns in infectious and non-infectious diseases. These models also help determine the factors that favor them, such as sociodemographic, environmental, etc.; as well as generate maps to visualize the distribution of morbidity or mortality of infectious and non-infectious diseases, and identify critical points in the spatial distribution [3, 4].

Generalized linear spatial models (GLSM), which are a particular class of multilevel or hierarchical models, have been used for the study of certain diseases (infectious and non-infectious). The estimation of GLSM parameters can be done under the frequentist or Bayesian approach [1], some examples are given below. A spatial Poisson regression model, where parameter estimation was performed under the frequentist approach, was used to study esophageal cancer incidence rates [5] and the sociodemographic risk factors for diabetes [6]. Under the Bayesian approach, these models have been used to study the relationship between Visceral Leishmaniasis incidence rates and climatological variables [7], as well as to identify risk factors associated with nontuberculous mycobacterial infections [8]. Spatial Binomial regression models, under the Bayesian approach, have been used to describe patterns of occurrence of dengue and chikungunya [9], and filariasis [10]. Under the classical approach, spatial binomial regression models have been used to investigate environmental and sociodemographic factors associated with leptoserosis disease [11]; are also used to study risk factors associated with HIV infection among drug users [12].

On the other hand, survival analysis under the spatial approach has also received great attention in recent years, because geographic location can play a relevant role in predicting disease survival [13]. Fragility models (spatial survival models) can be an option to analyze the heterogeneity of the data when it cannot be explained by the covariates in a classical survival model. In spatial survival models, in addition to covariates, a random effect known as frailtyis added, which modifies the hazard function of an individual, or of spatially correlated individuals [14]. Generally, the random factor, which is assigned a multivariate normal distribution, plays an important role in modeling survival times; since in this term the differences that exist in the socioeconomic level, access to medical care, population density, weather conditions, among others, can be taken into account. It is worth mentioning that spatial survival models have been applied in studies such as: recovery time in patients with COVID-19 [15], hospitalization time in dengue patients [16], HIV/AIDS survival [17] and breast cancer [18] to name a few. In all these works, the estimation of the model parameters was under the Bayesian approach.

Extreme events in public health (for example, the saturation of hospitals) are generally analyzed through measures of central tendency or time series, however, these approaches are not the most appropriate to understand extreme events (unusual events); that when they occur they strongly impact the health care network, thus often collapsing the system [19]. The extreme value theory (EVT) aims to study the probability of occurrence of extreme events (values) of a phenomenon of interest over time, generally these values only occur when they exceed a threshold. Although the applications of EVT in public health are scarce, if they exist at all; an application was presented when predicting extreme events of annual seasonal influenza mortality and the number of emergency department visits in a network of hospitals [20], another application was presented when modeling elevated cholesterol levels using the spikes-over-threshold model [21]. In both cases, the parameters were estimated under the frequentist approach. Given the advantages they have with the application of a spatial model, it would be convenient to study the extreme events of the health sector in space, for which there is already a methodology known as spatial modeling of extreme values [22].

The objective of this work is to provide a general review of the theoretical framework of spatial statistical models developed in the area of geostatistics, which have been used in the area of epidemiology to analyze, model and predict the phenomena of interest. Some of the packages that exist in the statistical software R [23] to carry out said spatial analyzes are also mentioned.


2. Gaussian processes

A stochastic process Wt,tTis a collection of random variables. That is, for each tT, Wtis a random variable [24]; if the stochastic process is indexed by a coordinate space sARd, then the stochastic process is called a random field [25]. A realization of the random field, Ws,sA, is given by Ws1=ys1Wsn=ysn. Generally from the sample ys1,,ysnone tries to know the characteristics of the process Win si, i=1,,n; and with this information to make inference of the process Wson all ARd, a convex set where svaries continuously. To the geo-referenced data ys1,,ysnis often referred to as geocoded, geostatistical data or point-referenced data. The study of this type of data is known as geostatistics, which is a part of spatial statistics that studies phenomena with continuous variation in space, a convex region denoted A[26].

A process Wis second order stationary if it has finite variance, constant mean and its covariance function depends only on distance. Having second-order stationarity in a stochastic process implies having intrinsic stationarity, i.e., second-order stationarity is stronger than intrinsic stationarity. On the other hand, weak stationarity and second-order stationarity are equivalent in the space [27]. The following defines what is known as a Gaussian process (field).

Definition 1.A stochastic process Ws:sAR2, where svaries continuously on a fixed subset Acontent in R2, is a Gaussian process if for any collection of locations s1,,snwith siA, the joint distribution of Ws1Wsnis multivariate Gaussian [1].

What is known as a stationary Gaussian process is defined below.

Definition 2.A Gaussian process Ws:sAR2, is stationary if sA:


and its correlation function depends only on the distance, i.e.


where h=ssis the Euclidean distance that exists between sand s.

That is, the mean and variance of Wsare constant and its correlation function only depends on the distance, so that


Given W=W1Wn, where Wi=Wsi, the distribution of Wis normal multivariate NM,i.e.


where the ijelement of Ris given by Rij=CorrWsiWsj=ρhij, hij=sisjis the Euclidean distance between siand sj. Note that the covariance of the Gaussian process is given by CovW=σ2R.

In this way, the correlation structure of a stationary Gaussian process can be studied through the ρhfunction. Several parametric expressions for this function are shown in the Table 1. In these correlation functions, ϕ>0is a range parameter controlling the spatial decay over distance; h=ssis the Euclidean distance between sand sand h0; Γdenotes the gamma function. κ>0, in theory of spatial extremes Jκand Kκare the Bessel and modified Bessel function of the third kind of order κ[28], while in the spatial survival analysis and generalized linear models Kκis the modified Bessel function of the second kind of order κ[29]; κis a shape parameter that determines the analytic smoothness of the underlying process W[1]. In the powered exponential correlation function 0<κ2and in the Bessel correlation function κ0.

FamilyCorrelation function
Powered exponentialρhϕκ=exphϕκ

Table 1.

Models for the spatial correlation structure of a spatial process.


3. Gaussian spatial model

Generally from process Ws:sAR2, there is a noisy version, i.e., a set of observation ys1,,ysnof the random variables Ys1,,Ysn, siA. In this way Ysis a measurement process of Ws, sA[1, 26].

The Gaussian geostatistical model, in the absence of independent variables, is given by


where μis a constant mean effect, Wsis a stationary Gaussian process (1) and Zsis the error term in the model with ZsN0τ2; τ2is the nugget effect variance. Zsis known as measurement error, micro-scale variation or a non-identifiable combination of the two [22, 26].

Thus for a realization of a stationary Gaussian spatial process, Ys=Ys1Ysn, siAand i=1,,nwith



  • WsiN0σ2.

  • Zsiare mutually independent and identically distributed, ZsiN0τ2,i=1,,n.

  • Zsiare independent of the process W[26].

  • Conditional on W, random variables Ysi, i=1,,n, are mutually independent with normal distribution,


The joint distribution of Ysis normal multivariate given by



  • μis the mean of the Gaussian process Wand 1is a vector of dimension n×1.

  • σ2is the variance of the process W.

  • Rϕis a matrix of correlations of dimension n×n, whose elements given by


where hij=sisjis the euclidean distance that exists between siand sjthat are in A, and ϕis a spatial scale parameter.

  • τ2is the variance of Zand Iis the identity matrix of dimension n×n.

  • Note that the covariance of the Ysis given by CovYs=σ2Rϕ+τ2I.

When Yscan be explained by a set of covariates that also depend on the location, Xs=X1s.Xps, then the model is given by




where β=β0βpis a vector of unknown regression parameters; in this case also CovYs=σ2Rϕ+τ2I. The unknown parameters in this model are β, σ2, τ2and ϕ. The parameters of the Models (4) y (5) can be estimated under the classical approach (maximum likelihood or maximum restricted likelihood) and under the Bayesian statistical approach [1, 30]. Among the most important points in geostatistics is the modeling of the covariance structure of the spatial process and the identification of the interpolation method that will be used to perform the prediction of the process in the non sampled points in A. Regarding the last point, [31] made a compilation of the most used criteria for assessing the performance of the spatial interpolation method.

The geoRpackage contains the likfitfunction that allows to estimate, under Maximum likelihood (ML) or restricted maximum likelihood (REML), the parameters of a Gaussian process [32]. The function likfitestimates the coefficients of the models (4) y (5).

The function krige.covof the same package helps to perform the spatial prediction of a Gaussian process using simple kriging (SK), ordinary kriging (OK), external trend kriging (KTE) and universal kriging (UK) [33]. The package glmmfieldsallows to fit Gaussian models [34] under the Bayesian approach.

On the other hand, with the function glmmfieldsof the package glmmfields, the coefficients of the models (4) and (5) can be estimated under the Bayesian approach. The function glmmfieldsreports the posterior median of the parameters with their respective 95%credible intervals; this function, also reports the values of the Gelman and Rubin statistic [35], where values less than 1.20would indicate convergence of the chain.


4. Generalized linear spatial models

Generalized linear models (GLM) [36, 37] are very useful when the response variable does not follow a normal distribution. The assumptions of GLMs are

  1. 1. Yi, i=1,,nare mutually independent with expectations μi.

  2. The μiare specified by gμi=ηi, where gis a known link function.

  3. The linear predictor is given by ηi=xiβ, where xiis a vector of explanatory variables associated with the response Yi, and βis a vector of unknown parameters.

The Yifollow a common distributional family, indexed by their expectations μi, and possibly by additional parameters common to all nresponses.

An important extension of this basic class of models is the generalized linear mixed model (GLMM) [38], in which Y1,,Ynare mutually independent conditional on the realized values of a set latent random variables (random effects) U1,,Unand the conditional expectations are given by gμi=Ui+xi'β. A generalized linear spatial model is a GLMM in which the U1,,Unare derived from spatial process. Diggle and Ribeiro in 2007[1], refers to these models as generalized linear geostatistical model (GLGM). In accordance with Diggle et al. (1998) [39], the assumptions of the generalized linear spatial models are as follows

  1. Wis a stationary Gaussian process, WN0σ2ρh, (Eq. (1)).

  2. Conditionally an W, the random variables Yi, i=1,,nare mutually independent, with distributions fiyWsi=fyMi, specified by the values of the conditional expectations Mi=EYiWsi.

  3. gMi=xiβ+Wsifor some known link function gand explanatory variable xi=xsi.

Then Mi=g1xiβ+Wsi, where the linear predictor would be given by ηi=xiβ+Wsi.

Taking Diggle and Tawn as a precedent (1998) [39]; Jing and De Oliveira in 2015[40] state the GLSM as follows




Ris of the same form as the Gaussian process (2)

  • Ysi:i=1,,nare conditionally independent given Wwith pdfs or pmfs pμi.

  • EYiWi=μiand gis a known one-to-one link function.

  • X=1x1xpis a known n×p+1design matrix assumed of full-rank, with 1a vector of n×1of ones and xj=xjs1.xjsn', where xjsiis the value of the j-th covariate of the i-th sampling location, and β=β0β1βpis the vector of unknown regression parameters.

Since gis the link function then gμi=ηiand μi=g1ηi, i=1,,n, where the linear predictor is given by ηi=Wi, then μi=g1Wi. The unknown parameters in GLSM are β, σ2and ϕ.

The two most widely used GLSM for spatial count data are the Poisson and Binomial spatial models [39, 41].

The geoCount[40] package implements the GLSM; the function runMCMCis used to generate posterior samples of the Gaussian process and the GLSM parameters, with which the parameter estimates and their credibility intervals can be obtained.

In the package geoRglm[42, 43], the functions glsm.krige, pois.krigeand binom.krigeimplement the GLSMs, in this case, parameter estimation is performed under the frequentist approach. While the functions krige.bayes, pois.krige.bayesand binom.krige.bayes, which also implement the GLSMs, estimate the parameters under the Bayesian approach. These functions report estimates of β, σ2and ϕ.

The glmmfieldspackage implements the Gamma, Poisson, Negative Binomial, Binomial and Lognormal models using the function glmmfields[34], parameter estimation is performed under the Bayesian approach. The function glmmfieldsreports the parameter estimates using the posterior median with their respective 95%percentile credible intervals; it also reports the Gelman and Rubin diagnostic values.


5. Spatial survival models

Generally, survival analysis models are specified through their hazard function, ht, whose intuitive interpretation is that htδtis the conditional probability that a patient will die in the interval tt+δt, given tat they have survived until time t. The most widely used approach to modeling ht, at least in medical applications, is to use a semi-parametric formulation [44]. In this approach, the hazard for the i-th patient is modeled as


where xiis a vector of explanatory variables for patient iand h0tis an unspecified baseline hazard function. This is known as a proportional hazards (PH) model, because for any two patients iand j, hti/htjdoes not change over time [1].

Another key idea in survival analysis is frailty, this corresponds to the random effects term used; time-to-event data will be group into strata, such as clinical sites, geographic regions, etc. This gives rise to mixed models, which include a random effect (the frailty) that correspond to a stratum’s overall health status [30]. To illustrate, let tijbe the time to death or censoring for subject jin stratum i, j=1,,ni, i=1,,m. Let xijbe a vector of individual specific covariates, then


where Wies the stratum-specific frailty term, designed to capture differences among strata; strata are typically denoted by si, i=1,,m, so sidenotes the location of the i-th patient and Wi=Wsi. It can be assumed that the Wiare independent identical distribution (iid), i.e.


But it can also be assumed that Wiarises from a Gaussian process, i.e. if W=W1Wm, then


This way, suppose subjects are observed at mdistinct spatial locations s1,,smA. Let tijbe a random event time associated with the j-th subject in si, assume the survival time tijlies in the interval aijbij, i=1,,m, j=1,,ni; and xijbe a related p-dimensional vector of covariates, then are defined proportional hazard (PH) frailty models, accelerated failure time (AFT) frailty models and proportional odds (PO) frailty models.

PH frailty models are the extensions of the population hazards model which is best known as the Cox model [44] a widely pursued model in survival analysis. PH frailty models extends the Cox model such that the hazard of an individual depends in addition on an unobserved random variable W, then introducing an additive frailty term Wifor each individual in the exponent of the hazard function as follows


The corresponding survival function and the density are given by


where S0, f0and h0are the baseline survival function, baseline density and baseline hazard function assumed to be unique for all individual in the study population respectively.

Accelerated failure time frailty model extends the AFT model such that the hazard of an individual depends in addition on an unobserved random variable W[45, 46, 47]. Introducing an additive frailty term Wifor each individual in the exponent of the hazard function it becomes:


The survival function and density are given by


where S0, f0and h0are the baseline survival function, baseline density and baseline hazard function assumed to be unique for all individual in the study population respectively.

Finally, proportional odds frailty model is given by


The survival function and density are given by


where S0, f0and h0are the baseline survival function, baseline density and baseline hazard function assumed to be unique for all individual in the study population respectively.

In the frailty models, it is possible to deal with left, right and interval censoring of the data. Among the packages that exist in the R statistical software to perform spatial survival analysis is the spBayesSurvpackage [48]; the function survregbayesestimates the parameters of the PH, AFT and PO spatial models under the classical and Bayesian approach; also reports the posterior mean and median of the regression coefficients and of the parameters of the covariance function of the Gaussian process, σ2and ϕ, with their 95%credible intervals. The spBayesSurvpackage uses the powered exponential function (Table 0) to model the spatial correlation of the data.

Also in R, there is the spatsurvpackage [49], which implements the function survspatthat fits parametric PH spatial survival models. This function reports the estimates and posterior median of the parameters β, σ2and ϕwith the respective credibility intervals.


6. Spatial generalized extreme value model

According to Coles (2001) [50], given Y1,,Yna sequence of independent random variables with a common distribution function Fwith Mn=maxY1Yn, if there a sequences of constants an>0and bnsuch that


when n, for a non-degenerative distribution function G, then Gis a member of the generalized extreme value (GEV) distribution family


defined on z:1+ξzη/τ>0, where <η<, τ>0and <ξ<.

Davison et al. in 2012[51], describe spatial GVE as follows. For each sin R2, suppose that Ysis GEV distributed whose parameters μs, σsand ξsvary smoothly for sin R2according to a stochastic process Ws. We assume that the processes for each GEV parameters are mutually independent Gaussian processes [52]. Then


where fη, fτand fxiare deterministic functions depending on a regression parameters βη, βτand βξrespectively. While Wη, Wτand Wξare a zero mean stationary Gaussian process with correlation function ρhϕηκη, ρhϕτκτand ρhϕξκξrespectively, i.e.


Then conditional on the values of the tree Gaussian process at the sites s1sk, the maxima are assumed to follow GEV distributions


z independently for each location s1,,sk, j=1,,kand i=1,,n.

Davison et 2012[51], proposed the construction of Bayesian hierarchical models for spatial extremes.

The SpatialExtremespackage [53] allows modeling spatial extremes, through max-stable processes with the function fitmaxstab, which reports the values of the parameter estimates with their respective standard errors.

To implement hierarchical Bayesian models, the function latentis used, this reports the posterior median of the scale, shape and location parameters with their respective credible intervals.

Another package in the literature to model spatial extremes is glmmfields[34], with the function glmmfields, parameter estimation is performed under the Bayesian approach. The function glmmfieldsalso allows modeling spatial extreme events incorporating temporally, that is, time, these models are known as spatio-temporal models.


7. Conclusions

The main characteristic of spatial data is that observations close in space tend to be correlated, and in spatial modeling this correlation is used to understand the behavior of the phenomenon under study in a region of interest.

Omitting the spatial dependence of the data can generate a bias of the information and, consequently, lead to an incorrect inference. Therefore, adequately describing the spatial pattern of an event can provide sufficient elements to elaborate possible hypotheses of its cause. As we have seen, the spatial variability of georeferenced data can be studied with the spatial models developed in geostatistics. The usefulness of these models has been demonstrated in several applications related to the identification of social structures, disease patterns, occupational patterns, as well as in the identification of populations (or subgroups) that are at greater or lesser risk of an event. As we have seen, in statistics, all correctly processed information helps in correct decision making. In this sense, this paper aims to introduce the reader to the use of spatial models in geostatistics.

If the response or variable of interest is the cases (counts) of sick people in a given region, or the new cases of a disease in a given period of time (incidence), then Poisson GLSMs can be useful to know the spread of the disease in the population of interest, predict new cases, and identify the variables that influence the occurrence of the disease. On the other hand, when the response variable is a binary or ratio variable, such as mortality rates or infection rates, then binomial GLSMs can be helpful. These models have been used to study the prevalence of dengue and to identify the variables associated with the event.


Conflict of interest

The authors declare no conflict of interest.



AFTAccelerated failure time
EVTExtreme value theory.
GEVgeneralized extreme value.
GLSMGeneralized Linear Spatial Models.
GLMGeneralized linear models.
GLMMgeneralized linear mixed model
GLGMgeneralized linear geostatistical model
iidIndependent identical distribution
NMNormal multivariate.
PHProportional hazards.
POProportional odds.


  1. 1. Diggle PJ, Ribeiro PJ. Model-Based Geostatistics. New York: Springer; 2007. p. 228
  2. 2. Gaetan C, Guyon X. Spatial Statistics and Modeling. New York, NY: Springer Science+Business Media, LLC; 2010
  3. 3. Rezaeian M, Dunn G, St. Leger S, Appleby L. Geographical epidemiology, spatial analysis and geographical information systems: A multidisciplinary glossary. Journal of Epidemiology and Community Health. 2007;61(2):98-102. DOI: 10.1136/jech.2005.043117
  4. 4. Chowell G, Rothenberg R. Spatial infectious disease epidemiology: On the cusp. BMC Medicine. 2018;16(1):1-5. DOI: 10.1186/s12916-018-1184-6
  5. 5. Mohebbi M, Wolfe R, Jolley D. A poisson regression approach for modelling spatial autocorrelation between geographically referenced observations. BMC Medical Research Methodology. 2011;11(1):1-11. DOI: 10.1186/1471-2288-11-133
  6. 6. Kauhl B, Schweikart J, Krafft T, Keste A, Moskwyn M. Do the risk factors for type 2 diabetes mellitus vary by location? A spatial analysis of health insurance claims in Northeastern Germany using kernel density estimation and geographically weighted regression. International Journal of Health Geographics. 2016;15(1):1-12. DOI: 10.1186/s12942-016-0068-2
  7. 7. Ben-Ahmed K, Aoun K, Jeddi F, Ghrab J, El-Aroui MA, Bouratbine A. Visceral leishmaniasis in Tunisia: Spatial distribution and association with climatic factors. The American Journal of Tropical Medicine and Hygiene. 2009;81(1):40
  8. 8. Lipner EM, Knox D, French J, Rudman J, Strong M, Crooks JL. A geospatial epidemiologic analysis of nontuberculous mycobacterial infection: An ecological study in Colorado. Annals of the American Thoracic Society. 2017;14(10):1523-1532
  9. 9. Hira FS, Asad A, Farrah Z, Basit RS, Mehreen F, Muhammad K. Patterns of occurrence of dengue and chikungunya, and spatial distribution of mosquito vector Aedes albopictus in Swabi district, Pakistan. Trop Med Int Heal. 2018;23(9):1002-1013. DOI: 10.1111/tmi.13125
  10. 10. Slater H, Michael E. Mapping, Bayesian geostatistical analysis and spatial prediction of lymphatic filariasis prevalence in Africa. PLoS One. 2013;8(8):28-32. DOI: 10.1371/journal.pone.0071574
  11. 11. Mayfield HJ, Lowry JH, Watson CH, Kama M, Nilles EJ, Lau CL. Use of geographically weighted logistic regression to quantify spatial variation in the environmental and sociodemographic drivers of leptospirosis in Fiji: A modelling study. Lancet Planet Heal. 2018;2(5):223-232. DOI: 10.1016/S2542-5196(18)30066-4
  12. 12. Zhou YB, Wang QX, Liang S, Gong YH, Yang MX, Chen Y, et al. Geographical variations in risk factors associated with HIV infection among drug users in a prefecture in Southwest China. Infectious Diseases of Poverty. 2015;4(1):1-10. DOI: 10.1186/s40249-015-0073-x
  13. 13. Zhou H, Hanson T, Zhang J. SpBayesSurv: Fitting bayesian spatial survival models using R. Journal of Statistical Software. 2020;92(9):1-33. DOI: 10.18637/jss.v092.i09
  14. 14. Banerjee S, Wall MM, Carlin BP. Frailty modeling for spatially correlated survival data, with application to infant mortality in Minnesota. Biostatistics. 2003;4(1):123-142. DOI: 10.1093/biostatistics/4.1.123
  15. 15. Mahanta KK, Hazarika J, Barman MP, Rahman T. An application of spatial frailty models to recovery times of COVID-19 patients in India under Bayesian approach. Journal of Scientific Research. 2021;65(03):150-155. DOI: 10.37398/JSR.2021.650318
  16. 16. Aswi A, Cramb S, Duncan E, Hu W, White G, Mengersen K. Bayesian spatial survival models for hospitalisation of dengue: A case study of Wahidin hospital in Makassar, Indonesia. International Journal of Environmental Research and Public Health. 2020;17(3):1-12. DOI: 10.3390/ijerph17030878
  17. 17. Martins R, Silva GL, Andreozzi V. Bayesian joint modeling of longitudinal and spatial survival AIDS data. Statistics in Medicine. 2016;35(19):3368-3384. DOI: 10.1002/sim.6937
  18. 18. Zhou H, Hanson T, Jara A, Zhang J. Modeling county level breast cancer survival data using a covariate-adjusted frailty proportional hazards model. The Annals of Applied Statistics. 2015;9(1):43-68. DOI: 10.1214/14-AOAS793
  19. 19. Chiu Y, Chebana F, Abdous B, Bélanger D, Gosselin P. Mortality and morbidity peaks modeling: An extreme value theory approach. Statistical Methods in Medical Research. 2018;27(5):1498-1512. DOI: 10.1177/0962280216662494
  20. 20. Thomas M, Lemaitre M, Wilson ML, Viboud C, Yordanov Y, Wackernagel H, et al. Applications of extreme value theory in public health. PLoS One. 2016;11(7):3-9. DOI: 10.1371/journal.pone.0159312
  21. 21. De Zea BP, Mendes Z. Extreme value theory in medical sciences: Modeling total high cholesterol levels. J Stat Theory Pract. 2012;6(3):468-491. DOI: 10.1080/15598608.2012.695673
  22. 22. Gelfand AE, Schliep EM. Spatial statistics and gaussian processes: A beautiful marriage. Spatial Statistics. 2016;18:86-104. DOI: 10.1016/j.spasta.2016.03.006
  23. 23. R Core Team. R: A Language and Environment for Statistical Computing. Vienna, Austria: R Foundation for Statistical Computing; 2021. URL
  24. 24. Ross SM et al. Stochastic Processes. Vol. 2. New York: Wiley; 1996
  25. 25. Grigoriu M. Stochastic processes. In: Calculus S, editor. Birkhäuser. Boston: MA; 2002. pp. 103-204
  26. 26. Diggle PJ, Ribeiro PJ, Christensen OF. An introduction to model-based geostatistics. In: Møller J, editor. Spatial Statistics and Computational Methods. New York: Springer Verlag; 2003. pp. 43-86
  27. 27. Cressie N, Wikle CK. Statistics for Spatio-Temporal Data. John Wiley & Sons; 2015
  28. 28. Ribatet M. A user’s Guide to the SpatialExtremes Package. Lausanne, Switzerland: EPFL; 2009
  29. 29. Chilès JP, Delfiner P. Geostatistics: Modeling Spatial Uncertainty. New York: Wiley Series In Probability and Statistics; Vol. 497; 2009
  30. 30. Banerjee S, Carlin BP, Gelfand AE. Hierarchical Modeling and Analysis for Spatial Data. 2nd ed. United States of America: Chapman and Hall/CRC; 2004. p. 472
  31. 31. Li J, Heap A. A review of comparative studies of spatial interpolation methods in environmental sciences: Performance and impact factors. Ecol Inform. 2011;6(3–4):228-241. DOI: 10.1016/j.ecoinf.2010.12.003
  32. 32. Ribeiro PJ, Diggle PJ. geoR: A package for geostatistical analysis. R-NEWS. 2001;1:15-18
  33. 33. Goovaerts P. Geostatistics for Natural Resources Evaluation. New York: Oxford University Press; 1997
  34. 34. Anderson SC, Ward EJ. Black swans in space: Modeling spatiotemporal processes with extremes. Ecology. 2019;100(1):1-23. DOI: 10.1002/ecy.2403
  35. 35. Gelman A, Rubin DB. Inference from iterative simulation using multiple sequences. Statistical Science. 1992;7(4):457-511. DOI: 10.1214/ss/1177011136
  36. 36. Nelder J, Wedderburn R. Generalized linear models. Journal of the Royal Statistical Society Series A (General). 1972;135(3):370-384. DOI: 10.2307/2344614
  37. 37. McCullagh P, Nelder JA. Generalized Linear Models. 2nd ed. London: Chapman and Hall; 1989
  38. 38. Breslow NE, Clayton DG. Approximate inference in generalized linear mixed models. Journal of the American Statistical Association. 1993;88(421):9-25
  39. 39. Diggle PJ, Tawn JA, Moyeed RA. Model-based geostatistics. Journal of the Royal Statistical Society: Series C (Applied Statistics). 1998;47(3):299-350
  40. 40. Jing L, De Oliveira V. Geocount: An R package for the analysis of geostatistical count data. Journal of Statistical Software. 2015;63:1-33. DOI: 10.18637/jss.v063.i11
  41. 41. Christensen OF, Waagepetersen R. Bayesian prediction of spatial count data using generalized linear mixed models. Biometrics. 2002;58(2):280-286. DOI: 10.1111/j.0006-341x.2002.00280.x
  42. 42. Christensen OF, Ribeiro PJ Jr. geoRglm-a package for generalised linear spatial models. R News. 2002;2(2):26-28
  43. 43. Ribeiro PJ Jr, Christensen OF, Diggle PJ. geoR and geoRglm: Software for model-based geostatistics. Hornik K, Leisch F, Zeileis A, editors. Vienna: 3rd International Workshop on Distributed Statistical Computing (DSC 2003): 20-22 March 2003; p. 2
  44. 44. Cox DR. Regression models and life-tables. J R Stat Soc [B]. 1972;34(2):187-202. DOI: 10.1111/j.2517-6161.1972.tb00899.x
  45. 45. Buckley J, James I. Linear regression with censored data. Biometrika. 1979;66(3):429-436. DOI: 10.2307/2335161
  46. 46. Wei L. The accelerated failure time model: A useful alternative to the cox regression model in survival analysis. Statistics in Medicine. 1992;11(14–15):1871-1879. DOI: 10.1002/sim.4780111409
  47. 47. Zhang J, Lawson AB. Bayesian parametric accelerated failure time spatial model and its application to prostate cancer. Journal of Applied Statistics. 2011;38(3):591-603. DOI: 10.1080/02664760903521476
  48. 48. Zhou H, Hanson T, Zhang J. spBayesSurv: Fitting Bayesian Spatial Survival Models Using R. 2017. arXiv preprint arXiv:1705.04584
  49. 49. Taylor BM, Rowlingson BS. Spatsurv: An R package for bayesian inference with spatial survival models. Journal of Statistical Software. 2017;77(4):1-32. DOI: 10.18637/jss.v077.i04
  50. 50. Coles S, Bawa J, Trenner L, Dorazio P. An Introduction to Statistical Modeling of Extreme Values. Vol. 208. London: Springer; 2001. p. 208
  51. 51. Davison AC, Padoan SA, Ribatet M. Statistical modeling of spatial extremes. Statistical Science. 2012;27(2):161-186. DOI: 10.1214/11-STS376
  52. 52. Casson E, Coles S. Spatial regression models for extremes. Extremes. 1999;1(4):449-468. DOI: 10.1023/A:1009931222386
  53. 53. Ribatet M. SpatialExtremes: An R Package for Modelling Spatial Extremes. R Package Version 2.1-0. 2020. Available from:

Written By

María Guzmán Martínez, Eduardo Pérez-Castro, Ramón Reyes-Carreto and Rocio Acosta-Pech

Reviewed: March 25th, 2022 Published: May 10th, 2022