A Comparison of the β-Substitution Method and a Bayesian Method for Analyzing Left-Censored Data

Abstract

Classical statistical methods for analyzing exposure data with values below the detection limits are well described in the occupational hygiene literature, but an evaluation of a Bayesian approach for handling such data is currently lacking. Here, we first describe a Bayesian framework for analyzing censored data. We then present the results of a simulation study conducted to compare the β-substitution method with a Bayesian method for exposure datasets drawn from lognormal distributions and mixed lognormal distributions with varying sample sizes, geometric standard deviations (GSDs), and censoring for single and multiple limits of detection. For each set of factors, estimates for the arithmetic mean (AM), geometric mean, GSD, and the 95th percentile (X 0.95 ) of the exposure distribution were obtained. We evaluated the performance of each method using relative bias, the root mean squared error (rMSE), and coverage (the proportion of the computed 95% uncertainty intervals containing the true value). The Bayesian method using non-informative priors and the β-substitution method were generally comparable in bias and rMSE when estimating the AM and GM. For the GSD and the 95th percentile, the Bayesian method with non-informative priors was more biased and had a higher rMSE than the β-substitution method, but use of more informative priors generally improved the Bayesian method's performance, making both the bias and the rMSE more comparable to the β-substitution method. An advantage of the Bayesian method is that it provided estimates of uncertainty for these parameters of interest and good coverage, whereas the β-substitution method only provided estimates of uncertainty for the AM, and coverage was not as consistent. Selection of one or the other method depends on the needs of the practitioner, the availability of prior information, and the distribution characteristics of the • 56Ann. Occup. Hyg., 2016, Vol. 60, No. 1, 56-73 doi:10.1093/annhyg/mev049 Advance Access publication 24 July 2015 measurement data. We suggest the use of Bayesian methods if the practitioner has the computational resources and prior information, as the method would generally provide accurate estimates and also provides the distributions of all of the parameters, which could be useful for making decisions in some applications.

Bivariate Left-Censored Bayesian Model for Predicting Exposure: Preliminary Analysis of Worker Exposure during the Deepwater Horizon Oil Spill

“…Although methods such as β-substitution were developed to find 6 the best imputed single value, such imputative strategies may not properly account for the inherent uncertainty in the imputed estimates and may introduce bias if the estimate is based on a small sample size or a high percentage of censored observations. A discussion of such substitution methods, including β-substitution, maximum likelihood and simple Bayesian models using non-regression approaches has been presented in Huynh et al (2014Huynh et al ( , 2015.…”

“…Previous research by Huynh et al (2015) suggests that Bayesian models have advantages over frequentist methods such as β-substitution for censored analyses in occupational health studies.…”

“…After assessing these fre-7 quentist methods, Huynh et al (2015) compared Bayesian models to the β-substitution method.…”

Bivariate Left-Censored Measurements in Biomonitoring: A Bayesian Model for the Determination of Biological Limit Values Based on Occupational Exposure Limits

“…Up to now, in the industrial hygiene context, methods of data censored treatment have mainly focused on the univariate analysis of the airborne exposure measurements (Helsel 2005) (Hewett and Ganser 2007) (Ganser and Hewett 2010) (Huynh et al 2014) (Huynh et al 2016). In this context, (Jin et al 2011) performed a simulation study comparing different approaches to take account of up to 80% left-censored measurements while at the same time acknowledging a between-worker variability.…”

“…Several analyses of left-censored occupational exposure data have been developed in literature, like substitution methods, β-substitution methods (Ganser and Hewett 2010), Kaplan-Meier methods, multiple imputation based model, even Bayesian method (Huynh et al 2014;Huynh et al 2016). However, while β-substitution methods have been found optimal for parameter estimation in (Huynh et al 2014) when considering only airborne exposure measurements, none of these methods (except the Bayesian) can be directly extended for the purpose of establishing the relationship between two left-censored variables.…”