Abstract
Distributed lag non-linear models (DLNMs) have gained popularity for modeling nonlinear lagged relationships between exposures and outcomes. When applied to spatially referenced data, these models must account for spatial dependence, a challenge that has yet to be thoroughly explored within the penalized DLNM framework. This gap is mainly due to the complex model structure and high computational demands, particularly when dealing with large spatio-temporal datasets. To address this, we propose a novel Bayesian DLNM-Laplacian-P-splines (DLNM-LPS) approach that incorporates spatial dependence using conditional autoregressive (CAR) priors, a method commonly applied in disease mapping. Our approach offers a flexible framework for capturing nonlinear associations while accounting for spatial dependence. It uses the Laplace approximation to approximate the conditional posterior distribution of the regression parameters, eliminating the need for Markov chain Monte Carlo (MCMC) sampling, often used in Bayesian inference, thus improving computational efficiency. The methodology is evaluated through simulation studies and applied to analyze the relationship between temperature and mortality in London.
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Data Availability
Data and R codes used to generate the results are available on GitHub using the following link: https://github.com/Rutten-Sara/DLNM---Laplace.
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Acknowledgements
The computational resources and services used in this work were provided by the VSC (Flemish Supercomputer Center), funded by the Research Foundation - Flanders (FWO) and the Flemish Government - department EWI.
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TN gratefully acknowledges funding by the Research Foundation - Flanders (grant number G0A3M24N).
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S.R. and B.S. contributed equally to this work. They were responsible for writing the main manuscript text and developing the computational code. O.G., T.N., E.D., N.H. and C.F. provided conceptual guidance, supervised the research, critically reviewed and revised the manuscript, and verified the methodology and results.
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Rutten, S., Sumalinab, B., Gressani, O. et al. Penalized distributed lag non-linear models for small area data using Laplacian-P-splines. Stat Comput 36, 38 (2026). https://doi.org/10.1007/s11222-025-10790-9
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DOI: https://doi.org/10.1007/s11222-025-10790-9