WIGGLE WORRY

Abstract

Introduction The principle complexity in time series studies of dependence of mortality on air pollution arises from the need to disentangle the effect of air pollution from the usually larger variations of death counts over time due to other factors. Entering specific variables, such as temperature, humidity, and influenza reports alongside pollution in a Poisson regression of daily death count on air pollution helps, but leaves residual variation that can confound the pollution effect. To reduce this risk, investigators usually include in models terms for slow changes in mortality over time, using a variety of forms including moving averages, stratification, trigonometric functions of time, LOESS smooths, smoothing or natural cubic splines, or by case-crossover matching. For all of these, investigators must choose how wiggly to allow this nuisance term to be, for example choose degrees of freedom (df) in a smoothing spline. Insufficient wiggles (df) can leave residual confounding. Too many wiggles (df) can leave insufficient information from which to estimate the pollution effect coefficient, and exacerbate bias to the null caused by errors in mismeasuring or mismodelling air pollution. (A common form of mismodelling is when single pollution lag is used as proxy for a distributed lag effect.) Extent of wigglyness (df) has been decided by a priori judgement and a variety of data-based methods, including minimising AIC, absolute autocorrelation in residuals, or other test statistics for departure of residuals from white noise. Methods We have examined the properties of such methods by reference to simple principles and by simulation. Results General principles suggest that data-based methods can never determine whether a model (i.e. a specific wiggliness) is sufficient to control confounding bias, though impact of wiggliness on precision can be predicted given exposure series. Simulations confirm that in some realistic situations, application of any of the data-based criteria mentioned above can leave bias in pollution coefficients that is reduced if wiggliness of the time function is increased. Further, in the absence of mismeasurement/mismodelling of air pollution, using large numbers of df does not in general cause bias in the pollution coefficient, even if negative autocorrelation in residuals is induced. However, simulations also confirm negative consequences of increasing wiggles (df): (i) precision can be lost, though often not substantially, and (ii) bias due to mismeasuring or mismodelling pollution can be increased. Conclusions In the absence of mismeasurement/mismodelling of pollution, choice of smoothness in time functions comprises a simple trade-off between reduced vulnerability to confounding bias against reduced precision. In the presence of classical pollution measurement error or an imperfectly modelled pollution effect, however, adding more df to the time function also increases bias to the null of the pollution effect.