Confidence Intervals for Probabilities of Exceeding Threshold Limits With Censored Log-Normal Data

Abstract

In enforcing safety regulations in the context of occupational health, the question of the probability of exceeding the so‐called threshold limit value (TLV) in homogeneous exposure groups (HEG) arises. Because data are usually few, a simple non‐parametric approach based on numbers of exceedances over the TLV cannot be usefully adopted. However, the distribution of airborne concentrations in the exposures within HEGs is often compatible with a log‐normal distribution. Therefore inference based on this distribution may be adopted. There remain two problems for which there has been no satisfactory solution: the treatment of left censoring due to the detection limits of the exposure determinations, and the absence of a confidence interval (CI) for the probability of exceedance. In the non‐censored case we present an exact confidence interval based on the non‐central T‐distribution and two Monte Carlo solutions based on the bootstrap and on the Gibbs sampler. The Monte Carlo methods are extended in various ways to treat the censored case. A large scale simulation study shows that the Gibbs Sampling CIs has coverage close to nominal in nearly all situations. The parametric bootstrap, after bias correction, performs nearly as well. We also present some extensions of the Gibbs sampling method to the analysis of fibre exposure data, and to the analysis of the effect of finite precision and measurement error. A worked example is presented.

Risk of lung cancer in workers producing stainless steel and metallic alloys

“…workers never exposed to levels 1, 2 or 3). As exposure measurements are likely to follow a log-normal distribution [40], intensity codes 1, 2 and 3 were recoded 1, 10 and 100, respectively, before computing the ORs.…”

“…This recoding was performed (1) in accordance with the magnitude of levels usually measured in factories (ratios of the highest/lowest exposure levels for any contaminant) [16,17,19,20,21,34], and (2) because the exposure measurements follow a log-normal distribution [40]. Regarding exposure to metals, the lack of measurements did not allow the experts to consider qualitative aspects of metal exposure in the JEM, i.e.…”

Inference for the Lognormal Mean and Quantiles Based on Samples With Left and Right Type I Censoring

“…This example is taken from Wild et al (1996), and the data represent oil mist measurements obtained from a machining workshop in France. Oil mist is an aerosol of machining fluids suspected to induce several respiratory diseases.…”

“…Oil mist is an aerosol of machining fluids suspected to induce several respiratory diseases. As noted in Wild et al (1996), for the purpose of enforcing safety regulations in the context of occupational health, a parameter of interest is the exceedance probability, that is, the probability of oil mist levels exceeding a specified threshold limit. Inference concerning this parameter actually reduces to inference concerning a percentile of the underlying distribution; see Based on a normal probability plot of the log-transformed data, we conclude that the sample is from a lognormal distribution.…”

“…Yet another option is to use bootstrap methods. For estimating an exceedance probability, that is, the proportion of a population exceeding a threshold value, Wild et al (1996) considered nonparametric bootstrap, parametric bootstrap, hybrid bootstrap, and Gibbs sampling methods for finding interval estimates when a leftcensored sample is available from a lognormal distribution. The numerical studies in their article indicate that Gibbs sampling is the only satisfactory method for the specific problem that they have addressed.…”