Skip to main content

Thank you for visiting You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

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.

This is a preview of subscription content, access via your institution

Access options

Rent or buy this article

Get just this article for as long as you need it


Prices may be subject to local taxes which are calculated during checkout

Figure 1


  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.


  1. Kromhout, H., Symanski, E., and Rappaport, S. M. A comprehensive evaluation of within- and between- worker components of occupational exposure to chemical agents. Ann Occup Hyg. (1993) 37: 253–270.

    CAS  Google Scholar 

  2. Lyles, R. H., Kupper, L. L., and Rappaport, S. M. Assessing regulatory compliance via the balanced one-way random effects ANOVA model. J Agric, Biol Environ Stat. (1997) 2: 64–86.

    Article  Google Scholar 

  3. Peretz, C., Goldberg, P., Kahan, E., Grady, S., and Goren, A. The variability of exposure over time: a prospective longitudinal study. Ann Occup Hyg. (1997) 41: 485–500.

    Article  CAS  Google Scholar 

  4. Preller, L., Kromhout, H., Heederik, D., and Tielen, M. Modeling long-term average exposure in occupational exposure–response analysis. Scand J Work, Environ Health. (1995) 21: 504–512.

    Article  CAS  Google Scholar 

  5. Rappaport, S. M., Lyles, R. H., and Kupper, L. L. An exposure assessment strategy accounting for within- and between-worker sources of variability. Ann Occup Hyg. (1995) 39: 469–495.

    Article  CAS  Google Scholar 

  6. Searle, S. R., Casella, G., and McCulloch, C. E. Variance Components. Wiley, New York. 1992.

  7. Wang, C. M., and Iyer, H. K. Tolerance intervals for the distribution of true values in the presence of measurement errors. Technometrics. (1994) 36: 162–170.

    Article  Google Scholar 

  8. Williams, J. S. A confidence interval for variance components. Biometrica. (1962) 49: 278–281.

    Article  Google Scholar 

Download references


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

Author information

Authors and Affiliations


Corresponding author

Correspondence to CHAVA PERETZ.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

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

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI:


  • ANOVA estimator
  • bias adjustment
  • exposure assessment
  • hygiene surveys
  • repeated measures
  • variance component


Quick links