Distributed lag models (DLMs) have been widely used in environmental epidemiology to quantify the lagged effects of air pollution on a health outcome of interest such as mortality and morbidity. Most previous DLM approaches only consider one pollutant at a time. In this article, we propose distributed lag interaction model (DLIM) to characterize the joint lagged effect of two pollutants. One natural way to model the interaction surface is by assuming that the underlying basis functions are tensor products of the basis functions that generate the main-effect distributed lag functions. We extend Tukey's one-degree-of-freedom interaction structure to the two-dimensional DLM context. We also consider shrinkage versions of the two to allow departure from the specified Tukey's interaction structure and achieve bias-variance tradeoff. We derive the marginal lag effects of one pollutant when the other pollutant is fixed at certain quantiles. In a simulation study, we show that the shrinkage methods have better average performance in terms of mean squared error (MSE) across different scenarios. We illustrate the proposed methods by using the National Morbidity, Mortality, and Air Pollution Study (NMMAPS) data to model the joint effects of PM10 and O3 on mortality count in Chicago, Illinois, from 1987 to 2000.

中文翻译：

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具有两种污染物的分布式滞后相互作用模型。

分布式滞后模型（DLM）已广泛用于环境流行病学中，以量化空气污染对诸如死亡率和发病率之类的健康结果的滞后效应。以前的大多数DLM方法一次只考虑一种污染物。在本文中，我们提出了分布式滞后相互作用模型（DLIM）来表征两种污染物的联合滞后效应。对交互表面进行建模的一种自然方法是，假设基础基函数是生成主效应分布滞后函数的基函数的张量积。我们将Tukey的一自由度交互结构扩展到二维DLM上下文。我们还考虑了两者的缩小版本，以允许偏离指定的Tukey的交互结构并实现偏差方差折衷。当另一种污染物固定在一定的分位数时，我们推导出一种污染物的边际滞后效应。在仿真研究中，我们显示出收缩方法在不同情况下的均方误差（MSE）方面具有更好的平均性能。我们通过使用国家发病率，死亡率和空气污染研究（NMMAPS）数据来模拟1987年至2000年伊利诺伊州芝加哥市PM10和O3对死亡率计数的联合影响来说明所提出的方法。