Uncertainty for Data with Non-Detects: Air Toxic Emissions from Combustion

ABSTRACT

Air toxic emission factor datasets often contain one or more points below a single or multiple detection limits and such datasets are referred to as “censored.” Conventional methods used to deal with censored datasets include removing non-detects, replacing the censored points with zero, half of the detection limit, or the detection limit. However, the estimated means of the censored dataset by conventional methods are usually biased. Maximum likelihood estimation (MLE) and bootstrap simulation have been demonstrated as a statistically robust method to quantify variability and uncertainty of censored datasets and can provide asymptotically unbiased mean estimates. The MLE/bootstrap method is applied to 16 cases of censored air toxic emission factors, including benzene, formaldehyde, benzo(a)pyrene, mercury, arsenic, cadmium, total chromium, chromium VI and lead from coal, fuel oil, and/or wood waste external combustion sources. The proportion of censored values in the emission factor data ranges from 4 to 80%. Key factors that influence the estimated uncertainty in the mean of censored data are sample size and inter-unit variability. The largest range of uncertainty in the mean was obtained for the external coal combustion benzene emission factor, with 95 confidence interval of the mean equal to minus 93 to plus 411%.

Key Words:

• urban air toxics
• emission factor
• censored datasets
• maximum likelihood estimation
• bootstrap simulation

ACKNOWLEDGMENTS

The work was supported by U.S. Environmental Protection Agency Science to Achieve Results (STAR) Grant No. R826790 to the Department of Civil, Construction, and Environmental Engineering at North Carolina State University. Yuchao Zhao conducted this work as graduate research assistant at NC State University and is now a postdoctoral fellow at the USEPA. This article has not been subject to any USEPA review and therefore does not necessarily reflect the views of the Agency, and no official endorsement should be inferred.

This article not subject to US copyright law.

Notes

^a Fuel type; C = coal, W = wood waste, FO = fuel oil.

^b Sample size.

^c Percentage of data samples that are censored.

^d NDL = number of detection limits.

^e Variability factor of the detected values, represented by the largest detected value divided by the smallest detected value.

^f Relative maximum detection limit, represented by the largest detection limit divided by the largest detected value.

^g R = USEPA qualitative ratings representing the data quality of the emission factor, taking into account test methods, sample size, and judgment regarding representativeness of the data.

^a Fuel type; C = coal, W = wood waste, FO = fuel oil.

^b Candidate parametric distributions, the preferred one is shown in bold.

^c Best estimated mean based on the average of the means of the 500 replicates of the CDF.

^d Lower and upper levels of the 95% confidence interval relative to the mean value, as a percentage difference compared to the mean value.

^e Width of the 95% confidence interval, equal to the sum of the absolute value of the lower and upper levels of the 95% confidence intervals.

^f For Cases 3, 5, and 8, results are shown only for distributions for which parameter estimates were within valid constraints.

^a Fuel type: C = coal, W = wood waste, FO = fuel oil.

^b Estimation of mean based on conventional methods: 1 = removal of non-detects; 2 = replace non-detects with zero; 3 = replace nondetects with DL/2; 4 = replace non-detects with DL.

^c The difference of the estimated means from different conventional methods lies in the third significant digit and thus not shown with two significant digits. The third significant figure is not statistically significant in all cases.

^a Ratio of the estimated mean of modified data to that of censored data based on the MLE/bootstrap method.

^b Ratio of the width of the 95 percent confidence interval estimated based on modified data to that estimated based on the MLE/bootstrap method applied to censored data.