Areal data: disease mapping and small area estimation

This thesis focuses on the analysis of areal data, where an outcome is observed across different areas of a region. Areal data commonly arise in disease mapping or small area estimation (SAE). We aim to provide flexible spatial and spatio-temporal models in the analysis of disease mapping data. Furthermore, we investigate different methods for SAE, including machine learning (ML) approaches. When the number of cases of a disease is recorded across different areas within a region, disease mapping is useful to estimate the areal relative risk. The number of cases in an area is often assumed to follow a Poisson distribution whose log risk may be written as the sum of fixed and random effects. The BYM2 model decomposes each latent effect into a weighted sum of independent and spatial effects. In the first manuscript, we extend the BYM2 model to allow for heavy-tailed latent effects and accommodate potentially outlying risks, after accounting for the fixed effects. We assume a scale mixture wherein the variance of the latent process changes across areas and allows for outlier identification. We explore two prior specifications of the scaling parameters and compare the proposed model to another proposal in the literature, in simulation studies and in the analysis of Zika cases from the 2015-2016 epidemic in Rio de Janeiro.Further, disease counts are increasingly recorded over time and across areas, and spatio-temporal disease mapping models help understand the spread of the disease over time. Commonly, the areal number of cases is assumed to follow a Poisson distribution, where the log risk varies with space and time. Models have been proposed to account for a spatio-temporal trend in the latent effects. In the second manuscript, we extend a spatio-temporal model to allow for heavy-tailed effects to accommodate and identify outliers. At each time point, we assume the latent effects to be spatially structured and include scaling parameters in the precision matrix to allow for heavy tails. We investigate the performance of the proposed model through simulation studies and analyse the weekly evolution of COVID-19 cases across Montreal and France during the second wave.When an outcome is measured across a fraction of the areas of a region through a survey that samples few units per area, SAE methods are useful to obtain reliable estimates at the areal level. In the third manuscript, we propose a comparison of different approaches for model-based small area prediction when there are abundant auxiliary data for the sampled and non-sampled areas. Random forest (RF) and LASSO approaches are compared with a frequentist forward selection procedure and a Bayesian shrinkage method. To provide uncertainty quantification of estimates obtained from RF and LASSO methods, we propose a modification of the split conformal (SC) procedure that relaxes the assumption of exchangeable data. Through simulation studies, we assess the performance of the proposed SC procedure and compare the four modelling approaches. Further, we estimate the areal mean household log consumption in the Greater Accra Metropolitan Area using data available from the sixth Ghanaian Living Standard Survey (GLSS) and the 2010 Population and Housing Census. The dependent variable is measured only in the GLSS for 3% of all the areas, and 174 covariates are available from both datasets. For this analysis, a cross-validation study showed that the Bayesian shrinkage method yielded smaller bias and MSE. The methods proposed in the three manuscripts of this thesis contribute to the literature on disease mapping, SAE, and ML. The first two add to the disease mapping literature by accommodating and identifying outlying areas in spatial and spatio-temporal models. The third manuscript contributes to the SAE literature by studying model-based approaches in a high-dimensional setting, and to the ML literature by proposing a procedure to provide uncertainty quantification of ML estimates

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