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Stat Med. 2014 Feb 28;33(5):881-99. doi: 10.1002/sim.5963. Epub 2013 Sep 12.

Modeling exposure-lag-response associations with distributed lag non-linear models.

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Medical Statistics Department, London School of Hygiene and Tropical Medicine, London, U.K.

Erratum in

  • Stat Med. 2014 Feb 28;33(5):900.


In biomedical research, a health effect is frequently associated with protracted exposures of varying intensity sustained in the past. The main complexity of modeling and interpreting such phenomena lies in the additional temporal dimension needed to express the association, as the risk depends on both intensity and timing of past exposures. This type of dependency is defined here as exposure-lag-response association. In this contribution, I illustrate a general statistical framework for such associations, established through the extension of distributed lag non-linear models, originally developed in time series analysis. This modeling class is based on the definition of a cross-basis, obtained by the combination of two functions to flexibly model linear or nonlinear exposure-responses and the lag structure of the relationship, respectively. The methodology is illustrated with an example application to cohort data and validated through a simulation study. This modeling framework generalizes to various study designs and regression models, and can be applied to study the health effects of protracted exposures to environmental factors, drugs or carcinogenic agents, among others.


delayed effects; distributed lag models; exposure-lag-response; latency; splines

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