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Distributed lag non-linear models (DLNMs) are a modelling tool for describing potentially non-linear and delayed dependencies. Here, we illustrate an extension of the DLNM framework through the use of penalized splines within generalized additive models (GAM). This extension offers built-in model selection procedures and the possibility of accommodating assumptions on the shape of the lag structure through specific penalties. In addition, this framework includes, as special cases, simpler models previously proposed for linear relationships (DLMs). Alternative versions of penalized DLNMs are compared with each other and with the standard unpenalized version in a simulation study. Results show that this penalized extension to the DLNM class provides greater flexibility and improved inferential properties. The framework exploits recent theoretical developments of GAMs and is implemented using efficient routines within freely available software. Real-data applications are illustrated through two reproducible examples in time series and survival analysis.

Many studies have examined the impact of air pollution on cardiovascular hospitalization (CVH), but few have looked at the delayed effects of air pollution on CVH. Additionally, there has been no research on the spatial and temporal differences in how environmental pollutants affect CVH. This study seeks to identify spatial heteroscedasticity in the relation between PM2.5 and CVH by developing a Generalized Additive Distributed Lag (GADL) model. Data on hospitalization due to cardiovascular disease were collected from the Hospital Information System (HIS) of Mashhad University of Medical Science from 2017 to 2020. Air pollution data from 22 air quality monitoring (AQM) stations were obtained from the Environmental Pollution Monitoring Center of Mashhad administrates. Markov Random Field (MRF) smoother was utilized in the GADL model to account for spatial heteroscedasticity in the observations. This developed model is a Spatial Heteroscedastic Generalized Additive Distributed Lag (SHGADL) model. Our use of GADL allowed us to discover a significant relationship between PM2.5 exposures and the risk of CVH at lags 0 and 1 in all districts. Our results reveal heteroscedasticity in the Relative Risks (RR) of PM2.5 on CVH across different districts. After accounting for this spatial heteroscedasticity, we found that the RR of PM2.5 on CVH at lags 0 and 1 were 1.0102 (95% CI: 1.0034, 1.0170) and 1.0043 (95% CI: 1.0009, 1.0078) respectively. The central and southeastern districts showed higher RR for CVH. The developed SHGADL model provides evidence of a significant lagged effect of PM2.5 exposures on CVH, and identifies low- and high-risk districts for CVH in Mashhad. This finding can assist decision-makers in allocating resources and planning strategically, with a focus on local interventions to manage ambient air pollution and providing emergency care for CVH.

The results of studies on intrauterine fetal death (IUFD) caused by exposure to fine particulate matter (PM 2.5 ) during pregnancy are inconsistent. Further exploration of the dose–response relationship and exposure window is required. We aimed to provide a reference for policy formulation by estimating the exposure-lag relationship of PM 2.5 on IUFD and looking for sensitive exposure windows. IUFD data was obtained from China Children Under 5 Death Surveillance Network in Linxia Hui Autonomous Prefecture from 2016 to 2020. Air pollution data and temperature data were obtained from ambient air monitoring stations and China Meteorological Data Network, respectively. The moving average is used to describe the trend and seasonality of PM 2.5 exposure; the distributed lag non-linear model (DLNM) is used to estimate the exposure-lag effect; the sandwich estimators are used to correct the variance–covariance matrix; and the model selected by Akaike’s Information Criterion (AIC) finally adjusts gender, temperature, and district. About 180,622 infants were enrolled in the study, including 952 IUFDs (5.27‰). The median of PM 2.5 exposure is 34.08 μg/m 3 . There is an exposure-lag effect of PM 2.5 on IUFD approximate to a wavy shape; the concentration with effect is 40–90 μg/m 3 ; and the sensitive lag time is 1, 2, 3, 8, 9, and 10 months. The maximum RR value of the exposure-lag effect of PM 2.5 on IUFD is 1.61 [95% CI 1.19, 2.19], in which the concentration of PM 2.5 is 62 μg/m 3 , and the lag month is 9 months. In the case of less than 6 months lag, the maximum RR value of the exposure-lag effect of PM 2.5 on IUFD is 1.43 [95% CI 1.24, 1.67], in which the concentration of PM 2.5 is 73 μg/m 3 , and the lag month is 3 months. Exposure to PM 2.5 concentrations above 40 μg/m 3 may increase the risk of IUFD, especially in the first and third trimesters.

Exposure–lag–response associations shed light on the duration of pathogenesis for radiation-induced diseases. To investigate such relations for lung cancer mortality in the German uranium miners of the Wismut company, we apply distributed lag non-linear models (DLNMs) which offer a flexible description of the lagged risk response to protracted radon exposure. Exposure–lag functions are implemented with B-Splines in Cox models of proportional hazards. The DLNM approach yielded good agreement of exposure–lag–response surfaces for the German cohort and for the previously studied cohort of American Colorado miners. For both cohorts, a minimum lag of about 2 year for the onset of risk after first exposure explained the data well, but possibly with large uncertainty. Risk estimates from DLNMs were directly compared with estimates from both standard radio-epidemiological models and biologically based mechanistic models. For age > 45 year, all models predict decreasing estimates of the Excess Relative Risk (ERR). However, at younger age, marked differences appear as DLNMs exhibit ERR peaks, which are not detected by the other models. After comparing exposure–responses for biological processes in mechanistic risk models with exposure–responses for hazard ratios in DLNMs, we propose a typical period of 15 year for radon-related lung carcinogenesis. The period covers the onset of radiation-induced inflammation of lung tissue until cancer death. The DLNM framework provides a view on age-risk patterns supplemental to the standard radio-epidemiological approach and to biologically based modeling.