A model is presented for applying Bayesian statistical techniques to the problem of determining, from The usual limited number of exposure measurements, whether the exposure profile for a similar exposure group can be considered a Category 0, 1, 2, 3, or 4 exposure. The categories were adapted from the AIHA exposure category scheme and refer to (0) negligible or trivial exposure (i.e., the true X0.95 .1%OEL), (1) highly controlled (i.e., X0.95 .10%OEL), (2) well controlled (i.e., X0.95 .50%OEL), (3) controlled (i.e., X0.95 .100%OEL), or (4) poorly controlled (i.e.,X0.95 >100%OEL) exposures. Unlike conventional statistical methods applied to exposure data, Bayesian statistical techniques can be adapted to explicitly take into account professional judgment or other sources of information. The analysis output consists of a distribution (i.e., set) of decision probabilities: e.g., 1%, 80%, 12%, 5%, and 2% probability that the exposure profile is a Category 0, 1, 2, 3, or 4 exposure. By inspection of these decision probabilities, rather than the often difficult to interpret point estimates (e.g., the sample 95th percentile exposure) and confidence intervals, a risk manager can be better positioned to arrive at an effective (i.e., correct) and efficient decision. Bayesian decision methods are based on the concepts of prior, likelihood, and posterior distributions of decision probabilities. The prior decision distribution represents what an industrial hygienist knows about this type of operation, using professional judgment; company, industry, or trade organization experience; historical or surrogate exposure data; or exposure modeling predictions. The likelihood decision distribution represents the decision probabilities based on an analysis of only the current data. The posterior decision distribution is derived by mathematically combining the functions underlying the prior and likelihood decision distributions, and represents the final decision probabilities. Advantages of Bayesian decision analysis include: (a) decision probabilities are easier to understand by risk managers and employees; (b) prior data, professional judgment, or modeling information can be objectively incorporated into the decision-making process; (c) decisions can be made with greater certainty; (d) the decision analysis can be constrained to a more realistic "parameter space" (i.e., the range of plausible values for the true geometric mean and geometric standard deviation); and (e) fewer measurements are necessary whenever the prior distribution is well defined and the process is fairly stable. Furthermore, Bayesian decision analysis provides an obvious feedback mechanism that can be used by an industrial hygienist to improve professional judgment. For example, if the likelihood decision distribution is inconsistent with the prior decision distribution then it is likely that either a significant process change has occurred or the industrial hygienist's initial judgment was incorrect. In either case, the industrial hygienist should readjust his judgment regarding this operation.
Corrigenda: In volume 3, issue 10 of Journal of Occupational and Environmental Hygiene, some incorrect text appeared in Rating Exposure Control Using Bayesian Decision Analysis by Paul Hewett, et al., pages 568-581. The correct text is shown below.
Page 571: The Bayesian approach to statistics and decision making is based on the equation developed by the Reverend Thomas
Bayes and published in 1763.
Page 572: The posterior distribution probability P(lnGi, lnDi data) is calculated using Eq. 1 and represents the mathematical combination of the prior distribution and the likelihood distribution, and reflects our final decision probability regarding the ith exposure profile.
Page 573: For this article we used the following integration boundaries:
Dmin = 1.05 Dmax = 4
Gmin = 0.0005 · LTWA Gmax = 5 · LTWA
Page 580: Similarly, for Gmin our recommended default setting is 1/2000th the cutoff for the OEL.
Page 581: Appendix I - WinBUGS code
# Set a uniform prior on the mean of the lognormal distribution between bounds of ln(0.05) = -0.3 and ln(500) = 6.2
mu ? dunif(?3.0,6.2)
# For a Limit of 1, set a uniform prior for the geometric mean of the lognormal distribution between bounds of ln(0.0005) = -7.6 and ln(5) = 1.6
mu ? dunif(-7.6,1.6)
Paul Hewett, Exposure Assessment Solutions, Inc., 1270 Kings Road, Morgantown, WV 26508