Parametric and semi-parametric approaches in the analysis of short-term effects of air pollution on health

Since mid-1990s, Generalised Additive Models (GAM) became very popular for the analysis of short-term effects of air pollution on health. Such approach involves specification of non parametric functions to adjust for confounding effect of unobserved variables with a systematic temporal behaviour and to model weather variables and influenza epidemics. Recently critical points in using commercial statistical software for fitting GAMs were stressed (Dominici et al., 2002; Ramsey et al., 2003) and some reanalyses of time series data on air pollution and health were performed. This new attention to semi-parametric models has led researchers to consider alternative estimation methods for GAMs and to wonder whether simpler parametric models can be a better choice than GAMs (Lumley and Sheppard, 2003). The purpose of this work is to show by simulation analyses some of the problems which we could find using GAMs, and to discuss real advantages of semi-parametric approach with respect to a fully parametric alternative, based on specification of Generalized Linear Models with natural cubic splines (GLM + NS). Here we considered the situation in which only the smooth function for time trend is included in the model. Generalized Additive Models were fitted by the direct methods implemented in R software (Wood, 2000). Different simulation analyses were performed, varying the "true" number of degrees of freedom for the smooth function, the concurvity amount in data and the "true" size of air pollutant effect. Our simulations show that GAM provide biased estimates of air pollutant effect, the bias being not negligible for moderate concurvity amount and small effect size. We found also that using semi-parametric approach a certain amount of undersmoothing is needed to obtain appropriated estimation of risk. Good performance was obtained selecting the smoothing parameter by Generalized Cross Validation. On the contrary analysis of partial autocorrelation of residuals from GAM brings to inappropriate model selection. GLM+NS is a good alternative to semi-parametric approach, resulting robust to misspecification of degrees of freedom for the spline. However the applicability of such approach should be considered carefully in presence of particular local variations of seasonality or in presence of outliers, because results could be sensitive to knots placement. Moreover the choice of knots positions could be a very important problem in smoothing other covariates like temperature.