Abstract
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. © 2013 The Authors. Statistics in Medicine published by John Wiley & Sons, Ltd.
Highlights
In biomedical research, it is commonly appreciated that an exposure event produces effects lasting well beyond the exposure period, with an increase in risk occurring from few hours to many years later, depending on the physiological processes linking the exposure and the health outcome
Model 1 is specified by a constant function, producing a lag-basis identical to the traditional index of unweighted cumulative exposure; model 2 is an example of a distributed lag models (DLMs) with a piecewise constant function; the best-fitting B-spline models with and without intercept, specified by a single knot at 13.3 lags are reported as models 3 and 4, respectively
The true simulated exposure–lag response associations are displayed in the top panels, while the other panels offer a comparison of the true lag–response and exposure–response curves at specific values with the average of the estimates from AIC and BIC-selected models, together with a sample of 25 individual curves
Summary
It is commonly appreciated that an exposure event produces effects lasting well beyond the exposure period, with an increase in risk occurring from few hours to many years later, depending on the physiological processes linking the exposure and the health outcome. Equivalent approaches were previously established in time series analysis, on the basis of distributed lag models (DLMs), a methodology originally formulated in econometrics [21], applied in epidemiological research [22] These models involve the definition of a distributed lag function, analogous to the weighting function described before. I aim to establish a general conceptual and statistical framework for modeling exposure– lag–response associations, built upon the paradigm of DLMs and DLNMs. In this paper, I aim to establish a general conceptual and statistical framework for modeling exposure– lag–response associations, built upon the paradigm of DLMs and DLNMs This modeling class, extended beyond time series analysis, provides a unified methodology applicable in different study designs, data structures, and regression models, including most of the previous methods as specific cases. The R code and data are included in the supporting information together with additional details, making the results of the illustrative example and of the simulation study entirely reproducible
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