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Estimation of average concentration in the presence of nondetectable values.

Hornung RW; Reed LD
Appl Occup Environ Hyg 1990 Jan; 5(1):46-51
Methods for estimating average analyte concentrations when some of the collected samples have concentrations below the limit of detection (LOD) were discussed. Because the concentrations of industrial contaminants are generally much lower than those in the past, the proportion of nondetectable samples in typical industrial hygiene data sets has increased. Two methods commonly used for estimating average concentrations in the presence of nondetectable values include the Hald method and the LOD/2 method. The Hald method involved using knowledge of the normal distribution to extrapolate back from the LOD to yield maximum likelihood estimates of the mean concentration and its standard deviation. The LOD/2 method assumed that samples with concentrations below the LOD could be assigned a value that was half of the LOD. A log transformation was then applied to the data and estimates of the geometric mean concentration and its standard deviation were obtained. The Hald method was very accurate but very complex requiring many laborious calculations. The LOD/2 method was very simple to use. Because it has assumed that data below the LOD follow a uniform distribution it can give incorrect estimates of the mean concentration and standard deviation. A new proposed method was presented. It was based on substituting the LOD divided by the square root of 2 for each concentration that was below the LOD. This followed from the assumption that all values below the LOD could be approximated by a triangle. The three methods were evaluated in a computer simulation. The Hald method was superior to the LOD/2 and proposed method if fewer than half of the concentrations were below the LOD. If the data were not highly skewed and the percentage of undetectable values was less than 30% the proposed method produced less bias than the LOD/2 method. The LOD/2 method produced less bias when the percentage of undetectable values was more than 30% but less than 50%. The authors conclude that the Hald method is best if a high degree of accuracy in the geometric mean and standard deviation are required. When the data are not highly skewed the new method can be used if fewer than 30% of the values are below the LOD. Otherwise the LOD/2 method should be used. None of the methods give good results when more than half of the values are below the LOD.
AIHYEX; NIOSH-Author; Mathematical-models; Statistical-analysis; Industrial-hygiene; Simulation-methods; Computers; Sampling-methods
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Applied Occupational and Environmental Hygiene
Page last reviewed: May 11, 2023
Content source: National Institute for Occupational Safety and Health Education and Information Division