Although distributed lag non-linear models (DLNMs) are commonly used to quantify delayed and non-linear exposure-response relationships, most existing applications assume that these relationships are constant across space. However, in many geographical and environmental studies, local characteristics vary substantially across areas, making a spatially varying effect more realistic. Extending DLNMs to allow for spatial heterogeneity remains challenging, and only a limited number of modelling strategies have been proposed in literature. The most popular extension is a two-stage meta-analysis approach, which requires sufficiently large sample sizes at each location. Therefore, its usefulness is limited when working with sparse count data in small area data analyses. Although a number of alternative one-stage approaches have been introduced, their computational burden restricts their applicability in real-life data applications. In this paper, we introduce a computationally efficient Bayesian one-stage spatially-varying DLNM for count data. We define four model variants, differing in the assumed spatial dependence structure and the flexibility of the DLNM spline specification. To address the computational burden typically associated with these flexible models, we use Laplace approximations, offering an efficient alternative to classically used Markov Chain Monte Carlo (MCMC) approaches. Model comparison criteria are provided to facilitate the selection of a suitable model in a real-life data application. The proposed methods are evaluated through simulation studies, and their practical usefulness is illustrated through a real-life data application, investigating the temperature-mortality relationship in every municipality of Sicily, Italy.

翻译：尽管分布滞后非线性模型（DLNMs）被广泛用于量化延迟且非线性的暴露-反应关系，但现有大多数应用均假设这些关系在空间上恒定不变。然而在许多地理与环境研究中，局部特征在不同区域间存在显著差异，这使得空间变化效应更具现实意义。将DLNM扩展以允许空间异质性仍具挑战性，目前文献中仅有少数建模策略被提出。最流行的扩展是两阶段元分析方法，这要求每个位置具有足够大的样本量。因此，在处理小区域数据分析中的稀疏计数数据时，其实用性受到限制。尽管已引入若干替代性单阶段方法，但它们的计算负担限制了其在实际数据应用中的可行性。本文针对计数数据提出一种计算高效的贝叶斯单阶段空间变化DLNM。我们定义了四种模型变体，其在空间依赖结构假设和DLNM样条规范灵活性方面存在差异。为解决这些灵活模型通常伴随的计算负担，我们采用拉普拉斯近似，为经典的马尔可夫链蒙特卡洛（MCMC）方法提供了高效替代方案。提供模型比较准则以协助在实际数据应用中选择合适模型。通过模拟研究评估所提方法，并通过实际数据应用（研究意大利西西里岛各市镇的温度-死亡率关系）展示其实用价值。