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Barnett, Adrian*; Williams, Gail*; Neller, Anne†; Best, Trudi†; Simpson, Rod†
*The University of Queensland; †University of the Sunshine Coast
Polynomial Distributed Lag (PDL) models are a useful method for studying the long-term effects of air pollution exposure on morbidity and mortality. The model assumes that the past effects of air pollution (in days) vary smoothly according to a parametric polynomial shape. The model's key parameters are the order of the polynomial and the number of past days (P); both of which are ideally chosen to give an optimal fit to the data. In making this optimal selection two problems occur: 1) increasing the number of past days (P) does not add extra terms to the Akaike Information Criteria (AIC), and so it cannot be used to assess the optimal value of P; 2) the polynomial assumption means that very non-linear patterns require a high order model. In this paper, we tackled these problems by fitting a non-parametric window to a set of unconstrained lagged covariates, and used the Deviance Information Criteria (DIC) to select the optimal value of P.
We smoothed a set unconstrained lagged covariates β to β[P], using a moving average (MA): β[i]*=(β[i−1]+ β[i]+β[i+1])/3, i=2,. . .P−1; β*=β; β[P]*=β[P]. The order of the PDL model was chosen by testing the orthogonal polynomial estimates. We compared the performance of the methods using a simulation study of a non-linear time-varying effect.
In the simulation study a PDL model of order six captured the non-linear pattern well (Figure 1); the area under the curve was estimated as 18.7 (95% Confidence Interval: 17.3, 20.0) compared to the true value of 17.5. The DIC performed well as an indicator of the correct value of P, whereas the AIC monotonically decreased with P (Figure 2.)
In our simulation study the DIC proved a useful statistic for choosing the crucial parameter of the number of past days in a distributed lag model. The PDL model gave estimates that were closer to the true effect compared those from a MA model which were comparatively noisy. We recommend combining the methods to give optimal estimates of long-term pollution effects.
© 2004 Lippincott Williams & Wilkins, Inc.
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