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Mortality data provide valuable information for the study of the spatial distribution of mortality risk, in disciplines such as spatial epidemiology, medical demography, and public health. However, they are often available in an aggregated form over irregular geographical units, hindering the visualization of the underlying mortality risk and the detection of meaningful patterns. Also, it could be of interest to obtain mortality risk estimates on a finer spatial resolution, such that they can be linked with potential risk factors — in a posterior correlation analysis — that are usually measured in a different spatial resolution than mortality data. In this paper, we propose the use of the penalized composite link model and its representation as a mixed model to deal with these issues. This model takes into account the nature of mortality rates by incorporating the population size at the finest resolution, and allows the creation of mortality maps at a desirable scale, reducing the visual bias resulting from the spatial aggregation within original units. We illustrate our proposal with the analysis of several datasets related with deaths by respiratory diseases, cardiovascular diseases, and lung cancer. ; Acknowledgements: The first and second authors acknowledge financial support from the Spanish Ministry of Economy and Competitiveness grants MTM2011-28285-C02-02 and MTM2014-52184. The third author acknowledges financial support from the Basque Government through the BERC 2014-2017 program and by the Spanish Ministry of Economy and Competitiveness MINECO: BCAM Severo Ochoa excellence accreditation SEV-2013-0323.

There are two main approaches to carrying out prediction in the context of penalized regression: with low-rank basis and penalties or through the smooth mixed models. In this article, we give further insight in the case of P-splines showing the influence of the penalty on the prediction. In the context of mixed models, we can connect the new predicted values to the observed values through a joint normal distribution, which allows us to compute prediction intervals. In this work, we propose an alternative approach, called the extended mixed model approach, that allows us to fit and predict data simultaneously. The methodology is illustrated with two real datasets, one of them on aboveground biomass and the other on monthly sulphur dioxide (SO2) levels in a selection of monitoring sites in Europe. ; The first and second authors acknowledge financial support from the Spanish Ministry of Economy and Competitiveness MTM2014-52184-P. The fourth author acknowledges the financial support by the Basque Government through the BERC 2018-2021 programme and by Spanish Ministry of Economy and Competitiveness MINECO through BCAM Severo Ochoa excellence accreditation SEV-2013-0323 and through project MTM2017-82379-R funded by (AEI/FEDER, UE) and acronym AFTERAM.

Mortality data provide valuable information for the study of the spatial distribution of mortality risk, in disciplines such as spatial epidemiology and public health. However, they are frequently available in an aggregated form over irregular geographical units, hindering the visualization of the underlying mortality risk. Also, it can be of interest to obtain mortality risk estimates on a finer spatial resolution, such that they can be linked to potential risk factors that are usually measured in a different spatial resolution. In this paper, we propose the use of the penalized composite link model and its mixed model representation. This model considers the nature of mortality rates by incorporating the population size at the finest resolution, and allows the creation of mortality maps at a finer scale, thus reducing the visual bias resulting from the spatial aggregation within original units. We also extend the model by considering individual random effects at the aggregated scale, in order to take into account the overdispersion. We illustrate our novel proposal using two datasets: female deaths by lung cancer in Indiana, USA, and male lip cancer incidence in Scotland counties. We also compare the performance of our proposal with the area-to-point Poisson kriging approach. ; We would like to thank two reviewers and an associate editor for their constructive comments and suggestions on the original manuscript. We also thank Dr. Pierre Goovaerts, who provided the high resolution population estimates described in Section 3.1. This research was supported by the Spanish Ministry of Economy and Competitiveness grants MTM2011-28285-C02-02 and MTM2014-52184-P. The research of Dae-Jin Lee was also supported by the Basque Government through the BERC 2014-2017 and ELKARTEK programs and by the Spanish Ministry of Economy and Competitiveness MINECO: BCAM Severo Ochoa excellence accreditation SEV-2013-0323. The research of Paul H. C. Eilers was also supported by the Universidad Carlos III de Madrid-Banco Santander Chair of Excellence program.