Hexavalent chromium (Cr(VI)) is toxic, carcinogenic, and mutagenic substances. Oral exposure to Cr(VI) is thought to be

Overview

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Key numbers

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primarily from drinking water. However, under the certain reporting limit (~0.1 µg/L), percentage of Cr(VI) concentration
in mineral water products under the reporting limit were estimated higher than 50%. Data whose values are below certain
1000 randomly generated data subsets (n = 150) with the obtained parameters. The evaluation shows that MCMC is the best
10-90%. The mean concentration, which was estimated with MCMC, was 0.289×10 −3 mg/L and this value was sufficiently
lower than the regulated value of 0.05 mg/L stipulated by the Food Sanitation Act.
PDF: probability density function, RIQR: relative inter quartile range, RL: reporting limit, RL/2: substitution of half of RL, RSD: rela-
nature2). Continuous exposure to Cr(VI) may cause pulmo- should be used for data with censored percentage of <50%;
failure3). Once Cr(VI) enters inside the cell, it undergoes 80% censored percentage; 3: MLE should be used for large
age from 50 to 75% under the certain reporting limit (~0.1 data20). Thus, we expect that Bayesian framework may give a
(LOD) and the limit of quantification (LOQ)9–13). Data lack- been applied to left-censored data for microbiological con-
ing certain values which are lower than LOD and LOQ, are tamination data22), hexabromocyclododecane diastereomer
of LOD or LOQ for left-censored data on the safety side. data analysis25–30), the reports comparing the performance
ods may cause a severe overestimation that is meaningless limited. Huynh et al31) evaluated the estimation performance
In 2006, EPA guidance supported the use of 0, LOD/2, or also evaluated by comparison of Bayesian and likelihood-
LOD substitution in data with less than 15% nondetects14). based estimation for left-censored data using simulations.
of censored data is less than 5% and if the data are mildly and Rain33), and Feroze and Aslam34) performed a simulation
et al (2017)8), which has over 50% nondetects and no speci- using the EnvStats package. The minimum value of detected
observed data are reported in the paper, appropriate analysis mean equals the area beneath the CDF35). It is widely used
than 50% of data are nondetects or if fewer than eight detec- estimate. The reason is that an EAP reflect the information
of the algorithms of the MCMC method, is adopted. Various 40% nondetects, we considered any value falling below the
GSD), we estimated RL with using MCMC (see Supporting 90%. Then 6 parameters ( m̂ , ŝ , µ̂ , σˆ , L̂ , and Û ) were
Table 1. Summary statistics and estimated parameters from left-censored data based on six inferential methods.
b: Lower and upper 95% confidence interval of the mean. For MCMC, 95% credible interval of posterior predictive distribution for
We calculated the coverage probability of 95% CIs for
95% CIs for means estimated by DN and RL/2 were calcu-
Although the 95% CI of µ̂ RL/2 overlapped with that esti-
tions). For the MCMC method, 95% credible intervals (CrI)
for the mean were calculated (see line 43 of Fig. S1). Then the
was included within the 95% CI or 95% CrI (see Supporting
inter-comparison. Table 1 shows summary statistics and µ̂ ROS (0.257×10 −3) was relatively close to what was ob-
3.1.5 Maximum likelihood estimation mg/L stipulated by the Food Sanitation Act42). The mean
GSD, and RL) and generated quantities (means and SD) except for DN, but the 97.5th percentile value was larger
The 95% CrI for the mean by MCMC was wider than other to determine the most appropriate methods. We compared
mary statistics after discarding nondetects. MCMC tended to sored percentage was in the range of 10 to 90%. From these
from 10 to 90% are shown in Fig. 4. The CPs decreased
10%, and rapidly decreased around 70%. In ROS and MLE,
the CP showed similar values (75 to 85%) depending on the
Fig. 3. Bias and mean squared error (MSE) for GM and GSD estimated by DN, RL/2, KM, ROS, MLE, and MCMC from 1000 randomly
generated left-censored data subsets (n = 150) which follow a lognormal distribution with certain parameters (GM = 0.082×10 −3; GSD =
4.9) over a 10 to 90% censoring range. For MCMC, the expected a posteriori was used for further calculation.
Fig. 4. Coverage probability of 95% CI or 95% CrI for mean estimated by DN, RL/2, KM, ROS, MLE, and MCMC from 1000 randomly
generated left-censored data subsets (n = 150) which follow a lognormal distribution with certain parameters (GM = 0.082×10 −3; GSD =

Methods (brief)

normal distribution, geometric standard deviation, IQR: interquartile range, KM: Kaplan-Meier, LCI: lower confidence interval, LOD:
limit of detection, LOQ: limit of quantification, m̂ : estimate of geometric mean. The letters on the upper right indicate the estimation
nary congestions, liver damage, skin irritation, and kidney 2: ROS should be used for small sample size (<50) and 50-
a rapid metabolic reduction and is converted ultimately to sample size (≥50) and 50-80% censored percentage. It can
mated daily intake of Cr(VI) from consumption of mineral sample size, however it is also reported that MLE shows poor
concentrations ranged from 0.1 to 3.4 µg/L, and the data framework result in a reduced sample size, and Bayesian
(LOD) and the limit of quantification (LOQ)9–13). Data lack- been applied to left-censored data for microbiological con-
ing certain values which are lower than LOD and LOQ, are tamination data22), hexabromocyclododecane diastereomer
ence, many studies used substitution (010,11), LOD/210,13), or so on.
LOD9,12)) for left-censored data on the exposure estimation. While there are various reports on comparisons among
of LOD or LOQ for left-censored data on the safety side. data analysis25–30), the reports comparing the performance
In 2006, EPA guidance supported the use of 0, LOD/2, or also evaluated by comparison of Bayesian and likelihood-
LOD substitution in data with less than 15% nondetects14). based estimation for left-censored data using simulations.
substitution by LOD/2 should be used only if the percentage data in the specific area except for food safety science, Hady
lower LOQ was calculated and data below LOQ are marked KM is a non-parametric version of maximum likelihood
as “Tr.”. However, no description about the value of LOQ estimation, and this method estimates the percentiles, or cu-
The KM method can provide useful estimates when sample
when the sample size is small (less than 50) and where data normal distribution with the parameters estimated in section

Implications

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Update history

The five most recent substantive edits to this page, classified major (evidence or structure moved), correction (a published value or statement was wrong and has been fixed), or minor (narrative rewritten without changing the underlying evidence). Each description is derived from what the edit did to this page; the linked commit is the authoritative record, routine regeneration passes are excluded, and the full version history lives in git. When DOI minting comes online (see schema docs), each entry below will also link to a version-pinned DataCite DOI.

Commit	Date	Change	Description
b01ec52c	2026-08-04	major	2 sections added
d49e450f	2026-08-03	major	5 sections added; narrative text revised