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

Biological limit values (BLV) are often determined from the occupational exposure limits (OEL) in modelling biological data obtained on a number of exposed subjects based on measurements of air exposure. In order to obtain such BLVs, biomonitoring studies are conducted collecting simultaneously biological and airborne measurements to these substances in exposed workers. One obstacle in the modelling of such data is the often large number of values below the limit of detection (LOD) for both biological and airborne measurements (left-censored measurements). A second difficulty, which is also a strength, is that multiple measurements are obtained for the same workers, leading to non-independence of the data. In this paper, we propose a statistical method based on Bayesian theory making use of measurements below the LOD for both dependent (biological) and independent (air exposure) data, and taking into account multiple measurements on the same worker. This method relies on the modelling of the airborne exposure measurements using standard random effect models adapted for values below LOD and the simultaneous modelling of the biological measurements assumed to be linearly (on the log scale) related to the airborne exposure while accounting for between-worker variability. This method is validated by a simulation study in which up to 50% of the measurements are censored for both variables in realistic settings. This simulation study shows that the proposed method is uniformly more efficient than the candidate alternative we considered (maximum likelihood estimation; MLE method) that did not make use of a data with airborne measurements below the LOD. When the method is applied on a real biomonitoring data set among electroplating workers exposed to chromium with 54% censored airborne measurements and 20% censored urinary measurements, the slope is steeper when incorporating these data using the proposed Bayesian method leading to different BLV estimations depending on the OEL used.