Research Article Received 9 January 2015, Accepted 22 December 2015 Published online 26 January 2016 in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/sim.6873 Age–space–time CAR models in Bayesian disease mapping T. Goicoa, a,b,c M. D. Ugarte, a,b J. Etxeberria a,b,d and A. F. Militino a,b * † Mortality counts are usually aggregated over age groups assuming similar effects of both time and region, yet the spatio-temporal evolution of cancer mortality rates may depend on changing age structures. In this paper, mortality rates are analyzed by region, time period and age group, and models including space–time, space– age, and age–time interactions are considered. The integrated nested Laplace approximation method, known as INLA, is adopted for model ftting and inference in order to reduce computing time in comparison with Markov chain Monte Carlo (McMC) methods. The methodology provides full posterior distributions of the quantities of interest while avoiding complex simulation techniques. The proposed models are used to analyze prostate cancer mortality data in 50 Spanish provinces over the period 1986–2010. The results reveal a decline in mortality since the late 1990s, particularly in the age group [65, 70), probably because of the inclusion of the PSA (prostate- specifc antigen) test and better treatment of early-stage disease. The decline is not clearly observed in the oldest age groups. Copyright © 2016 John Wiley & Sons, Ltd. Keywords: INLA; interaction models; mortality rates 1. Introduction Aggregating counts over all age groups inside a region in a specifc period of time is a common practice to study age-standardized mortality (or incidence) risks or rates. This means that region and temporal effects are supposed to be the same over the age groups, and hence, a single estimate is provided. This is a simple framework of analysis, and it should be regarded as a starting point [1]. If a single estimated rate is provided for each area and time, it is implicitly assumed that age groups are similarly affected by the disease, and this may not be necessarily true as the effects on children or elderly people could be more pronounced than on youngsters. Therefore, if the age groups are not equally affected by the disease, this practice could lead to misleading conclusions. Conditional autoregressive (CAR) models have been and still are one of the most popular approaches to disease mapping since the pioneering research by Besag et al. [2]. The more common assumption is based on the same spatial and spatio-temporal effect on the age groups [3–7]. It can be argued that in certain diseases there is no biological reason to assume a different age effect for each area [8], but in other cases region effects such as pollution, may have different consequences for distinct age groups [9]. Models incorporating age effects usually follow the proportional assumption [10], whereby separate area and age effects multiply to produce area–age rates. In any case, regardless of whether area–age interactions are considered, the necessity of dealing with the potential age effect in these models is clear, because when the evolution of the disease is not the same among the age groups, age-specifc rates within each region should be provided. Embedding the effect of age in mortality rates is not new in spatial or spatio-temporal disease mapping, yet it has been only partially studied. For example, Nandram et al. [11] consider local health areas within larger regions to study mortality rates a Department of Statistics and O. R. Universidad Pública de Navarra, Campus de Arrosadia, 31006 Pamplona, Spain b Institute for Advanced Materials (INAMAT), Universidad Pública de Navarra, Campus de Arrosadia, 31006 Pamplona, Spain c Research Network on Health Services in Chronic Diseases (REDISSEC), Spain d Consortium for Biomedical Research in Epidemiology and Public Health (CIBERESP), Spain * Correspondence to: A. F. Militino, Departamento de Estadística e Investigación Operativa, Universidad Pública de Navarra, Campus de Arrosadía, 31006 Pamplona, Spain. † E-mail: militino@unavarra.es Copyright © 2016 John Wiley & Sons, Ltd. Statist. Med. 2016, 35 2391–2405 2391