Quantification of Uncertainty and Variability for Censored Data Sets in Air Toxics

Two-dimensional Monte Carlo simulation can be used to estimate the variability and uncertainty of emissions of urban air toxics for use in human exposure and risk analysis. The key steps in the twodimensional approach include fitting a parametric distribution to data representing variability in emissions, and to use a method such as bootstrap simulation to estimate uncertainty in average emissions and regarding other statistics of the distribution. Many emission factors for urban air toxics contain several values reported only as below detection limit. Such data sets are referred to as "censored." To analyze the variability and uncertainty of censored data sets, empirical bootstrap simulation was used first to get bootstrap samples from the censored data sets. In the bootstrap simulation, randomly samples values from the original data set that were below the detection limit are treated as non-detected values in the bootstrap samples. A parametric probability distribution was fit to each bootstrap sample. The parameters of the distribution were estimated using Maximum Likelihood Estimation (MLE), which is asymptotically unbiased when applied to censored data sets. The use of MLE allows for a distribution to be fit to the entire domain of the emission factor, including estimation of the portion of the distribution that is below the detection limit. Typically, 500 bootstrap samples and distributions were generated. From the 500 alternative distributions, sampling distributions for uncertainty in any statistic can be calculated, such as for the mean. The method is illustrated with two case studies. In one case study, the detection limit was varied and the sensitivity of the results was evaluated. In a second case study, the method was applied to an empirical data set for arsenic emissions from a combustion source. The results show that with more censoring, the uncertainty of the non-detected portion of the fitted distribution increases but the range of uncertainty in the mean does not increase as much. For the empirical data set, the data contain multiple detection limits. The uncertainty of the mean for the arsenic emission factors was found to be -91% to +260%. Future work to further demonstrate this technique for dealing with censored data is recommended.