A bootstrap method to avoid the effect of concurvity in generalised additive models in time series studies of air pollution

Background: In recent years a great number of studies have applied generalised additive models (GAMs) to time series data to estimate the short term health effects of air pollution. Lately, however, it has been found that concurvity-the non-parametric analogue of multicollinearity-might lead to underestimation of standard errors of the effects of independent variables. Underestimation of standard errors means that for concurvity levels commonly present in the data, the risk of committing type I error rises by over threefold. Methods: This study developed a conditional bootstrap methology that consists of assuming that the outcome in any observation is conditional upon the values of the set of independent variables used. It then tested this procedure by means of a simulation study using a Poisson additive model. The response variable of this model is a function of an unobserved confounding variable (that introduces trend and seasonality), real black smoke data, and temperature. Scenarios were created with different coefficients and degrees of concurvity. Results: Conditional bootstrap provides confidence intervals with coverages close to nominal (95%), irrespective of the degree of concurvity, number of variables in the model or magnitude of the coefficient to be estimated (for example, for a concurvity of 0.85, bootstrap confidence interval coverage is 95% compared with 71% in the case of the asymptotic interval obtained directly with S-plus gam function). Conclusions: The bootstrap method avoids the problem of concurvity in time series studies of air pollution, and is easily generalised to non-linear dose-risk effects. All bootstrap calculations described in this paper can be performed using S-Plus gam.boot software.