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Improved non-negative estimation of variance components for exposure assessment


Hygiene surveys of pollutants exposure data can be analyzed by analysis of variance (ANOVA) model with a random worker effect. Typically, workers are classified into homogeneous exposure groups, so it is very common to obtain a zero or negative ANOVA estimate of the between-worker variance (σB2). Negative estimates are not sensible and also pose problems for estimating the probability (θ) that in a job group, a randomly selected worker's mean exposure exceeds the occupational exposure standard. Therefore, it was suggested by Rappaport et al. to replace a non-positive estimate with an approximate one-sided 60% upper confidence bound. This article develops an alternative estimator, based on the upper tolerance interval suggested by Wang and Iyer. We compared the performance of the two methods using real data and simulations with respect to estimating both the between-worker variance and the probability of overexposure in balanced designs. We found that the method of Rappaport et al. has three main disadvantages: (i) the estimated σB2 remains negative for some data sets; (ii) the estimator performs poorly in estimating σB2 and θ with two repeated measures per worker and when true σB2 is quite small, which are quite common situations when studying exposure; (iii) the estimator can be extremely sensitive to small changes in the data. Our alternative estimator offers a solution to these problems.

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  1. TLV=threshold limit value; a health-based concentration to which nearly all workers may be exposed without adverse effect.

  2. TLV–TWA=threshold limit value, with respect to 8-h time-weighted average, that should not be exceeded during any part of the working day.


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We acknowledge Liesbeth Preller from Wageningen University, the Netherlands, for providing the pig farmers exposure data set.

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Correspondence to CHAVA PERETZ.

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PERETZ, C., STEINBERG, D. Improved non-negative estimation of variance components for exposure assessment. J Expo Sci Environ Epidemiol 11, 414–421 (2001).

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