Evidence of Absence: Bayesian Way to Reveal True Zeros Among Occupational Exposures

Workplace exposure measurements typically contain some observations below limit of detection. The current paradigm for exposure data interpretation relies on the lognormal distribution, where censored observation are assumed to be present but not quantifiable. However, there are setting were such assumptions are untenable and true zero exposures cannot be ruled out. This issue can be non-trivial because decisions about compliance depend on the adequacy of the lognormal model.

We adapted previously described statistical models for mixture of true zeros and lognormal distribution to function within Bayesian procedure that overcomes historical limitations that precluded them from being used in practice. We compared the performance of the new models and the traditional lognormal model in simulation. Their implementation is illustrated in diverse datasets.

The approach we propose involves estimating the proportion of true zeroes, and the geometric mean and standard deviation of the lognormal component of the mixture. This can be implemented in practice either based on the truncated lognormal model fit to the observed data, or on the censored Bernoulli-lognormal mixture model, which has the advantage of allowing for multiple censoring points. Both models can be implemented via a free online application. In simulations, when none of the censored values were zeros, all estimation procedures led to similar risk assessment. However, when all or most of the censored values were zeros, the traditional approach that assumes lognormal distribution performed noticeably worse than newly proposed methods, typically overestimating noncompliance. Application to real data suggests that we cannot rule out presence of true zero exposures in typical measurement series gathered by occupational hygienists.

Forcing the usual lognormal model to data containing a large proportion of censored values can bias risk assessment if a substantial part of the censored points are true zeroes. The Bernoulli-lognormal model is a suitable and accessible model that can account for such challenging data, and leads to unbiased risk assessments regardless of the presence of true zeros in the data.