Adjustment for confounding factors is a common goal in the analysis of both observational and controlled studies. The choice of which confounding factors should be included in the model used to estimate an effect of interest is both critical and uncertain. For this reason it is important to develop methods that estimate an effect, while accounting not only for confounders, but also for the uncertainty about which confounders should be included. In a recent article, Crainiceanu et al. (2008) have identified limitations and potential biases of Bayesian Model Averaging (BMA) (Raftery et al., 1997; Hoeting et al., 1999)when applied to adjustment uncertainty, that arise because BMA weights models by their ability to make predictions and this may not reflect the models' ability to correctly adjust for confounding.
An important remaining question is whether it is possible to design approaches that account for adjustment uncertainty by treating the selection of variables as an unknown parameter, as BMA does, but do not suffer from the same limitations. In this paper, we propose a novel Bayesian formulation, called "Bayesian Confounding Adjustment" (BCA)to account for adjustment uncertainty in effect estimation from a Bayesian perspective. BCA uses a different weighting mechanism than BMA, wherein effect estimation is obtained by weighting effect estimates from models, all of which attempt to be fully adjusted for confounding. In simulation studies we show that BCA provides estimates of the exposure effect that have lower mean squared error than BMA and correct coverage. We then compare BCA, the approach of Crainiceanu et al. (2008), and tra- ditional BMA in a time series data set of hospital admissions, air pollution levels and weather variables in Nassau, NY for the period 1999-2005. Using each approach, we estimated the short-term effects of PM2.5 on emergency admissions for cardiovascular diseases, accounting for confounding. This application illustrates the potentially significant pitfalls of misusing variable selection methods in the con- text of adjustment uncertainty.
Available at: http://works.bepress.com/ciprian_crainiceanu/22/