We present a novel approach for analysing multivariate case-control georeferenced data in a Bayesian disease mapping context using stochastic partial differential equations (SPDEs) and the integrated nested Laplace approximation (INLA) for model fitting. In particular, we propose smooth terms based on SPDE models to estimate the underlying spatial variation as well as risk associated to pollution sources. Log-Gaussian Cox processes are used to estimate the intensity of the cases and controls, to account for risk factors and include a term to measure spatial residual variation. Each intensity is modelled on a baseline spatial effect (estimated from both controls and cases), a disease-specific spatial term and the effects of some covariates. By fitting these models, the residual spatial terms can be easily compared to detect high-risk areas not explained by the covariates. Three different types of effects to model exposure to pollution sources are considered on the distance to the source: a fixed effect, a smooth term to model non-linear effects by means of a discrete random walk of order one and a Gaussian process in one dimension with a Matérn covariance function. Spatial terms are modelled using a Gaussian process in two dimensions with a Matérn covariance function and are approximated using an approach based on solving an SPDE through INLA. Finally, this new framework is applied to a dataset of three different types of cancer and a set of controls from Alcalá de Henares (Madrid, Spain). Covariates available include the distance to several polluting industries and socioeconomic indicators. Our findings point to a possible risk increase due to the proximity to some of these industries.

中文翻译：

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疾病映射中多元点模式分析的近似贝叶斯推理

我们提出了一种使用随机偏微分方程 (SPDE) 和用于模型拟合的集成嵌套拉普拉斯近似 (INLA) 在贝叶斯疾病映射上下文中分析多变量病例控制地理参考数据的新方法。特别是，我们提出了基于 SPDE 模型的平滑项来估计潜在的空间变化以及与污染源相关的风险。Log-Gaussian Cox 过程用于估计案例和控制的强度，考虑风险因素并包括测量空间残差变化的术语。每个强度都基于基线空间效应（从对照和病例估计）、特定疾病的空间项和一些协变量的影响进行建模。通过拟合这些模型，残差空间项可以很容易地进行比较，以检测协变量无法解释的高风险区域。考虑到污染源的距离对污染源暴露建模的三种不同类型的影响：固定效应、通过一阶离散随机游走对非线性效应建模的平滑项和一维高斯过程具有 Matérn 协方差函数。空间项使用具有 Matérn 协方差函数的二维高斯过程建模，并使用基于通过 INLA 求解 SPDE 的方法进行近似。最后，这个新框架应用于三种不同类型癌症的数据集和一组来自 Alcalá de Henares（西班牙马德里）的对照。可用的协变量包括与几个污染行业的距离和社会经济指标。