Distributed lag non-linear models (DLNMs) are a popular approach to flexibly model the effect of time-delayed exposures. Classical DLNMs specify a common exposure-lag-response relationship across geographical areas. However, this relationship might be altered by an effect modifier that differs between spatial units. Although some methods have been proposed to account for effect modification, their applicability is context-dependent. For example, a meta-analysis can account for heterogeneity between groups, but this technique requires sufficiently large study groups. This limitation is particularly relevant when working with count data, where small numbers of events are often encountered. In this paper, we review existing methods that allow for spatial effect modification for count-based outcomes and propose a Bayesian DLNM alternative method that accounts for the modifier through flexible interaction effects. Through the use of Laplacian P-splines, we provide a computationally fast estimation procedure by avoiding the use of classical Markov Chain Monte Carlo (MCMC) approaches. The performance of the different methods is evaluated through simulation studies. Moreover, the practical applicability of our proposed method is showcased through a data application, containing daily temperature and mortality count data in 87 Italian cities.

翻译：分布式滞后非线性模型（DLNMs）是一种灵活建模时间延迟暴露效应的常用方法。经典DLNMs假设各地理区域存在共同的暴露-滞后-响应关系。然而，这种关系可能因空间单元间不同的效应修正因子而发生改变。尽管已有部分方法可考虑效应修正，但其适用性依赖于特定情境。例如，荟萃分析可处理组间异质性，但该方法要求研究组规模足够大——这一限制在小样本计数数据情境下尤为突出。本文回顾了现有允许对计数型结局进行空间效应修正的方法，并提出了一种贝叶斯DLNM替代方法，通过灵活交互效应纳入修正因子。通过采用拉普拉斯P样条技术，我们避免了传统马尔可夫链蒙特卡洛（MCMC）方法的使用，实现了计算高效的估计流程。通过模拟研究评估了不同方法的性能，并利用包含87个意大利城市每日温度与死亡率计数数据的实际应用案例展示了所提方法的实践适用性。