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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

IntechOpen
Biostatistics Edited by Cruz Vargas-De-León

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Biostatistics [Working Title]

Prof. Cruz Vargas-De-León

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Abstract

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.

Keywords

  • 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.

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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:

EWs=0,E1
VarWs=σ2,E2

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

CorrWsWs=ρh,E3

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

WN0σ2ρhE4

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

WNM0σ2R,E5

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
Exponentialρhϕ=exphϕ
Gaussianρhϕ=exph2ϕ2
Sphericalρhϕ=11.5hϕ+0.5hϕ3
Circularρhϕ=12πa1a2+sin1a
Cubicρhϕ=17hϕ2354hϕ3+72hϕ534hϕ7
Waveρhϕ=ϕhsinhϕ
Matérnρhϕκ=12κ1ΓκhϕκKκhϕ
Powered exponentialρhϕκ=exphϕκ
Cauchyρhϕκ=1+hϕ2κ
Stableρhϕκ=exphϕ
Besselρhϕκ=2ϕhκΓκ+1Jκhϕ

Table 1.

Models for the spatial correlation structure of a spatial process.

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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

Ys=μ+Ws+Zs,sA,E6

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

Ysi=μ+Wsi+Zsi,E7

where

  • 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,

YsiWNμ+Wsiτ2.E8

The joint distribution of Ysis normal multivariate given by

YsNMμ1σ2Rϕ+τ2I,E9

where

  • 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

Rϕij=ρYhijϕ,E10

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

Ys=Xsβ+Ws+Zs,sA,E11

with

YsNMXsβσ2Rϕ+τ2I,E12

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.

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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

YiWipμi.E13

where

WNMXβσ2RE14

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.

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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

hti=h0tiexpxiβ,E15

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

htijxij=h0tijexpxijβ+Wi,E16

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.

WiN0σ2.E17

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

WNM0σ2Rϕ.E18

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

htijxij=h0tijexijβ+Wi.E19

The corresponding survival function and the density are given by

Stijxij=S0tijexijβ+Wi,ftijxij=exijβ+WiS0tijexijβ+Wi1f0tij,E20

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:

htijxij=h0exijβ+Witijexijβ+Wi.E21

The survival function and density are given by

Stijxij=S0exijβ+Witij,ftijxij=exijβ+Wif0exijβ+Witij,E22

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

htijxij=h011+exijβWi1S0tij.E23

The survival function and density are given by

Stijxij=S0tijexijβWi1+exijβWi1S0tij,ftijxij=f0tijexijβWi1+exijβWi1S0tij2,E24

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.

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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

PMnbn/anzGz,E25

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

Gyητξ=exp1+ξyητ1ξ,E26

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

ηs=fηsβη+Wηsσηϕηκη,τs=fτsβτ+Wτsστϕτκτ,ξs=fξsβξ+Wξsσξϕξκξ,E27

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.

WηN(0,ση2ρhϕηκη,WτN(0,στ2ρhϕτκτ,WξN(0,σξ2ρhϕξκξ.E28

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

Ysiηsj,τsj,ξsjGEVηsjτsjξsjE29

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

Davison et al.in 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.

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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.

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Conflict of interest

The authors declare no conflict of interest.

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Abbreviations

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.

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Written By

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

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