# A Spatially Discrete Approximation to Log-Gaussian Cox Processes for Modelling Aggregated Disease Count Data

#### Olatunji Johnson

#### 2021-03-16

# Introduction

This article presents a simple tutorial code from SDALGCP package to make inference on spatially aggregated disease count data when one assume that the disease risk is spatially continious. There are two main functions provided by the package, for parameter estimation and for prediction.

## Model

Our goal is to analyse of diease count data, more specifically when disease cases are aggregated over a partition, say \((\mathcal{R}_{1}, \ldots, \mathcal{R}_{n})\), of the area of interest, \(A\), which can be written mathematically as \[\begin{eqnarray} \label{eq:data} \mathcal{D} = \left\{(y_{i}, d_{i}, \mathcal{R}_{i}): i=1,\ldots,n\right\} \end{eqnarray}\] where \(y_{i}\) and \(d_{i}\) are the number of reported cases and a vector of explanatory variables associated with \(i\)-th region \(\mathcal{R}_{i}\), respectively. Hence, we model \(y_{i}\) conditional on the stochastic process \(S(X)\) as poission distribution with mean \(\lambda_i= m_{i} \exp\{d_{i}\beta^* + S_{i}^*\}\). Then we assume that \(S^* \sim MVN(0, \Sigma)\), where \[\Sigma_{ij} = \sigma^2 \int_{\mathcal{R}_{i}} \int_{\mathcal{R}_{j}} w_i(x) w_j(x') \: \rho(\|x-x'\|; \phi) \: dx \: dx'\], where \(w(x)\) is population density weight. There are two classes of models in this package; one is when we approximate \[S_i^* = \int_{\mathcal{R}_{i}} w_i(x) S^*(x) \: dx \] and the other is \[S_i^* = \frac{1}{\mathcal{R}_{i}} \int_{\mathcal{R}_{i}} S^*(x) \: dx. \] ## Inference We used Monte Carlo Maximum Likelihood for inference. The likelihood function for this class of model is usually intractible, hence we approximate the likelihood function as \[\frac{1}{N}~ \sum_{j=1}^N~\frac{f(\eta_{(j)}; \psi)}{f(\eta_{(j)}; \psi_0)}.\], where \(\psi\) is the vector of the parameters.

This vignette walk you through how to analyse spatial and spatio-temporal dataset using package. Two illustrative examples were provided; application to primary biliary cirrhosis in Newcastle-upon-tyne, UK (static spatial case) and Lung cancer mortality in Ohio, USA (spatio-temporal case).

# Tutorial

This part illustrates how to fit an SDALGCP model to spatially aggregated data. We used the example dataset that is supplied in the package.

load the package

    require(SDALGCP)

load the data

    data("PBCshp")

extract the dataframe containing data from the object loaded

    data <- as.data.frame(PBCshp@data)

load the population density raster

    data("pop_den")

set any population density that is NA to zero

    pop_den[is.na(pop_den[])] <- 0

write a formula of the model you want to fit

    FORM <- X ~ propmale + Income + Employment + Education + Barriers + Crime +
      Environment +  offset(log(pop))

Now to proceed to fitting the model, note that there two types of model that can be fitted. One is when approximate the intensity of LGCP by taking the population weighted average and the other is by taking the simple average. We shall consider both cases in this tutorial, starting with population weighted since we have population density on a raster grid of 300m by 300m.

## SDALGCP I (population weighted): application to primary biliary cirrhosis in Newcastle-upon-tyne, UK

Here we estimate the parameters of the model

Discretise the value of scale parameter \(\phi\)

    phi <- seq(500, 1700, length.out = 20)

estimate the parameter using MCML

    my_est <- SDALGCPMCML(data=data, formula=FORM, my_shp=PBCshp, delta=200, phi=phi, method=1, pop_shp=pop_den,
                          weighted=TRUE, par0=NULL, control.mcmc=NULL)

To print the summary of the parameter estimates as well as the confidence interval, use;

    summary(my_est)
    #and for confidence interval use
    confint(my_est)

We create a function to compute the confidence interval of the scale parameter using the deviance method. It also provides the deviance plot.

    phiCI(my_est, coverage = 0.95, plot = TRUE)

Having estimated the parameters of the model, one might be interested in area-level inference or spatially continuous inference.

1. If interested in STRICTLY area-level inference use the code below. This can either give either region-specific covariate-adjusted relative risk or region-specific incidence. This is achieved by simply setting in the function.

    Dis_pred <- SDALGCPPred(para_est=my_est,  continuous=FALSE)

From this discrete inference one can map either the region-specific incidence or the covariate adjusted relative risk.

    #to map the incidence
    plot(Dis_pred, type="incidence", continuous = FALSE)
    #and its standard error
    plot(Dis_pred, type="SEincidence", continuous = FALSE)
    #to map the covariate adjusted relative risk
    plot(Dis_pred, type="CovAdjRelRisk", continuous = FALSE)
    #and its standard error
    plot(Dis_pred, type="SECovAdjRelRisk", continuous = FALSE)
    #to map the exceedance probability that the covariate-adjusted relative risk is greter than a particular threshold
    plot(Dis_pred, type="CovAdjRelRisk", continuous = FALSE, thresholds=3.0)

2. If interested in spatially continuous prediction of the covariate adjusted relative risk. This is achieved by simply setting in the function.

    Con_pred <- SDALGCPPred(para_est=my_est, cellsize = 300, continuous=TRUE)

Then we map the spatially continuous covariate adjusted relative risk.

    #to map the covariate adjusted relative risk
    plot(Con_pred, type="relrisk")
    #and its standard error
    plot(Con_pred, type="SErelrisk")
    #to map the exceedance probability that the relative risk is greter than a particular threshold
    plot(Con_pred, type="relrisk", thresholds=1.5)

## SDALGCP II (Unweighted)

As for the unweighted which is typically by taking the simple average of the intensity an LGCP model, the entire code in the weighted can be used by just setting in the line below.

    my_est <- SDALGCPMCML(data=data, formula=FORM, my_shp=PBCshp, delta=200, phi=phi, method=1,
                          weighted=FALSE,  plot=FALSE, par0=NULL, control.mcmc=NULL, messages = TRUE, plot_profile = TRUE)

# Spatio-temporal SDALGCP: application to Lung cancer mortality in Ohio.

Download the dataset

    require(rgdal)
    require(sp)
    require(spacetime)
    ohiorespMort <- read.csv("https://raw.githubusercontent.com/olatunjijohnson/dataset/master/OhioRespMort.csv")
    download.file("https://github.com/olatunjijohnson/dataset/raw/master/ohio_shapefile.zip", "ohio_shapefile.zip")
    unzip("ohio_shapefile.zip")
    ohio_shp <- rgdal::readOGR("ohio_shapefile/","tl_2010_39_county00")
    ohio_shp <- sp::spTransform(ohio_shp, sp::CRS("+init=epsg:32617"))

create a spacetime object as an input of the spatio-temporal SDALGCP model

    m <- length(ohio_shp)
    TT <- 21
    Y <- ohiorespMort$y
    X <- ohiorespMort$year
    pop <- ohiorespMort$n
    E <- ohiorespMort$E
    data <- data.frame(Y=Y, X=X, pop=pop, E=E)
    formula <- Y ~  X + offset(log(E))
    phi <- seq(10, 300, length.out = 10)
    control.mcmc <- controlmcmcSDA(n.sim=10000, burnin=2000, thin=80, h=1.65/((m*TT)^(1/6)), c1.h=0.01, c2.h=0.0001)
    time <- as.POSIXct(paste(1968:1988, "-01-01", sep = ""), tz = "")
    st_data <- spacetime::STFDF(sp = ohio_shp, time = time, data = data)

Plot the spatio-temporal count data

    spacetime::stplot(st_data[,,"Y"])

Parameter estimation

    model.fit <- SDALGCPMCML_ST(formula=formula, st_data = st_data,  delta=800,
                                phi=phi, method=2, pop_shp=NULL,  kappa=0.5,
                                weighted=FALSE, par0=NULL, control.mcmc=control.mcmc,
                                plot=TRUE, plot_profile=TRUE, rho=NULL,
                                giveup=50, messages=TRUE)
    summary(model.fit)

Area-level of the spatio-temporal prediction

    dis_pred <- SDALGCPPred_ST(para_est = model.fit, continuous = FALSE)

Ploting the area-level incidence and the covariate adjusted relative risk

    plot(dis_pred, type="CovAdjRelRisk", main="Relative Risk", continuous=FALSE)
    plot(dis_pred,  type="incidence", main="Incidence", continuous=FALSE)

Spatially continuous prediction of the covariate adjusted relative risk

    con_pred <- SDALGCPPred_ST(para_est = model.fit, cellsize = 2500, continuous=TRUE, n.window = 1)

Ploting the spatially continuous covariate-adjusted relative risk

    plot(con_pred, type="relrisk", continuous=TRUE)

#Discussion Using SDALGCP package for analysis of spatially aggregated data provides two main advantages. One, it allows the user to make spatially continous inference irrespective of the level of aggregation of the data. Second, it is more computationally efficient than the lgcp model for aggregated data that was implemented in package.