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

Tran Huynh, Harrison Quick, Gurumurthy Ramachandran, Sudipto Banerjee, Mark Stenzel, Dale P. Sandler, Lawrence S. Engel, Richard K. Kwok, Aaron Blair, Patricia A. Stewart

Research output: Contribution to journalArticlepeer-review

38 Scopus citations


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 (X0.95) 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 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.

Original languageEnglish (US)
Pages (from-to)56-73
Number of pages18
JournalAnnals of Occupational Hygiene
Issue number1
StatePublished - Nov 12 2015

Bibliographical note

Publisher Copyright:
© 2015 The Author 2015. Published by Oxford University Press on behalf of the British Occupational Hygiene Society.


  • Bayesian
  • exposure assessment
  • left-censored data
  • β-substitution


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