A distributed lag approach to fitting non-linear dose–response models in particulate matter air pollution time series investigations

The majority of studies that have investigated the relationship between particulate matter (PM) air pollution and mortality have assumed a linear dose–response relationship and have used either a single-day's PM or a 2- or 3-day moving average of PM as the measure of PM exposure. Both of these modeling choices have come under scrutiny in the literature, the linear assumption because it does not allow for non-linearities in the dose–response relationship, and the use of the single- or multi-day moving average PM measure because it does not allow for differential PM–mortality effects spread over time. These two problems have been dealt with on a piecemeal basis with non-linear dose–response models used in some studies and distributed lag models (DLMs) used in others. In this paper, we propose a method for investigating the shape of the PM–mortality dose–response relationship that combines a non-linear dose–response model with a DLM. This combined model will be shown to produce satisfactory estimates of the PM–mortality dose–response relationship in situations where non-linear dose response models and DLMs alone do not; that is, the combined model did not systemically underestimate or overestimate the effect of PM on mortality. The combined model is applied to ten cities in the US and a pooled dose–response model formed. When fitted with a change-point value of 60 μg/m 3, the pooled model provides evidence for a positive association between PM and mortality. The combined model produced larger estimates for the effect of PM on mortality than when using a non-linear dose–response model or a DLM in isolation. For the combined model, the estimated percentage increase in mortality for PM concentrations of 25 and 75 μg/m 3 were 3.3% and 5.4%, respectively. In contrast, the corresponding values from a DLM used in isolation were 1.2% and 3.5%, respectively.