Three-Stage Method to Estimate Distributed Lag Effects for a Time-Series Analysis of Noncontinuous Exposure Data

Distributed lag models (DLMs) are essential for estimating the delayed health effects of environmental exposures in a time-series analysis. However, their application is often limited by the requirement for complete continuous exposure data. Simpler lag-specific models with a single exposure can use more data but suffer from an omitted variable bias (OVB). This study develops and validates a three-stage method to accurately estimate distributed lag effects from noncontinuous exposure data by correcting for OVB. We propose a three-stage approach, fitting a series of lag-specific models using the noncontinuous data to obtain biased estimates in the first stage, correcting the bias by the autocorrelation structure of the exposure time series in the second stage, and smoothing the calibrated estimates in the third stage. The method's validity was confirmed by statistical simulations. We then applied it to a case study analyzing the association between PM_2.5, its chemical constituents, and daily all-cause mortality in Beijing from 2013 to 2015. Simulations showed that our three-stage method robustly and accurately recovered the true lag-response functions and cumulative effects, even when only one valid daily record is available for each week's data. In the Beijing case study, the traditional DLM failed to find significant associations due to a substantially restricted sample size, due to the missingness. In contrast, our method, by utilizing nearly 90% more observations, found significant health effects. For every 10 μg/m³ increase in PM_2.5, the cumulative effect (lag 0-13 days) on all-cause mortality was a 0.37% increase (95% CI: 0.23, 0.50%). Significant associations were also found for the PM_2.5 constituents. The proposed three-stage method is an effective and robust statistical tool for overcoming challenges posed by noncontinuous exposure data in time-series analysis. By correcting for omitted variable bias, it maximizes the use of available sparse data and significantly enhances statistical power, offering a practical and reliable solution for environmental epidemiology facing similar data limitations.