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Distributed lag non-linear models

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      Abstract

      Environmental stressors often show effects that are delayed in time, requiring the use of statistical models that are flexible enough to describe the additional time dimension of the exposure–response relationship. Here we develop the family of distributed lag non-linear models (DLNM), a modelling framework that can simultaneously represent non-linear exposure–response dependencies and delayed effects. This methodology is based on the definition of a ‘cross-basis’, a bi-dimensional space of functions that describes simultaneously the shape of the relationship along both the space of the predictor and the lag dimension of its occurrence. In this way the approach provides a unified framework for a range of models that have previously been used in this setting, and new more flexible variants. This family of models is implemented in the package dlnm within the statistical environment R. To illustrate the methodology we use examples of DLNMs to represent the relationship between temperature and mortality, using data from the National Morbidity, Mortality, and Air Pollution Study (NMMAPS) for New York during the period 1987–2000. Copyright © 2010 John Wiley & Sons, Ltd.

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      Most cited references 44

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      Computational Many-Particle Physics

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        Weather-related mortality: how heat, cold, and heat waves affect mortality in the United States.

        Many studies have linked weather to mortality; however, role of such critical factors as regional variation, susceptible populations, and acclimatization remain unresolved. We applied time-series models to 107 US communities allowing a nonlinear relationship between temperature and mortality by using a 14-year dataset. Second-stage analysis was used to relate cold, heat, and heat wave effect estimates to community-specific variables. We considered exposure timeframe, susceptibility, age, cause of death, and confounding from pollutants. Heat waves were modeled with varying intensity and duration. Heat-related mortality was most associated with a shorter lag (average of same day and previous day), with an overall increase of 3.0% (95% posterior interval: 2.4%-3.6%) in mortality risk comparing the 99th and 90th percentile temperatures for the community. Cold-related mortality was most associated with a longer lag (average of current day up to 25 days previous), with a 4.2% (3.2%-5.3%) increase in risk comparing the first and 10th percentile temperatures for the community. Mortality risk increased with the intensity or duration of heat waves. Spatial heterogeneity in effects indicates that weather-mortality relationships from 1 community may not be applicable in another. Larger spatial heterogeneity for absolute temperature estimates (comparing risk at specific temperatures) than for relative temperature estimates (comparing risk at community-specific temperature percentiles) provides evidence for acclimatization. We identified susceptibility based on age, socioeconomic conditions, urbanicity, and central air conditioning. Acclimatization, individual susceptibility, and community characteristics all affect heat-related effects on mortality.
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          Models for the relationship between ambient temperature and daily mortality.

          Ambient temperature is an important determinant of daily mortality that is of interest both in its own right and as a confounder of other determinants investigated using time-series regressions, in particular, air pollution. The temperature-mortality relationship is often found to be substantially nonlinear and to persist (but change shape) with increasing lag. We review and extend models for such nonlinear multilag forms. Popular models for mortality by temperature at given lag include polynomial and natural cubic spline curves, and the simple but more easily interpreted linear thresholds model, comprising linear relationships for temperatures below and above thresholds and a flat middle section. Most published analyses that have allowed the relationship to persist over multiple lags have done so by assuming that spline or threshold models apply to mean temperature in several lag strata (e.g., lags 0-1, 2-6, and 7-13). However, more flexible models are possible, and a modeling framework using products of basis functions ("cross-basis" functions) suggests a wide range, some used previously and some new. These allow for stepped or smooth changes in the model coefficients as lags increase. Applying a range of models to data from London suggest evidence for relationships up to at least 2 weeks' lag, with smooth models fitting best but lag-stratified threshold models allowing the most direct interpretation. A wide range of multilag nonlinear temperature-mortality relationships can be modeled. More awareness of options should improve investigation of these relationships and help control for confounding by them.
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            Author and article information

            Affiliations
            [a ]simplePublic Health and Policy Department, London School of Hygiene and Tropical Medicine Keppel Street, London W1C 7HT, U.K.
            [b ]simpleEpidemiology and Population Health Department, London School of Hygiene and Tropical Medicine London, U.K.
            Author notes
            * Correspondence to: A. Gasparrini, Public Health and Policy Department, London School of Hygiene and Tropical Medicine, Keppel Street, London W1C 7HT, U.K.
            Journal
            Stat Med
            sim
            Statistics in Medicine
            John Wiley & Sons, Ltd.
            0277-6715
            1097-0258
            20 September 2010
            07 May 2010
            : 29
            : 21
            : 2224-2234
            2998707
            20812303
            10.1002/sim.3940
            Copyright © 2010 John Wiley & Sons, Ltd.

            Re-use of this article is permitted in accordance with the Creative Commons Deed, Attribution 2.5, which does not permit commercial exploitation.

            Categories
            Research Article

            Biostatistics

            smoothing, distributed lag, delayed effects, time series

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