Abstract

I appreciate the opportunity to respond to Dr. Armstrong's comments regarding my article' I) on sample size formulae for the research-minded industrial hygienist. Dr. Armstrong makes two points: First, the use of the "symmetric t-interval . . . can be substantially misleading in smaller sample sizes with moderate or large GSDs." Second, an investigator may wish to estimate a workplace long-term mean within plus or minus an "absolute precision" in which case the denominator of my Equation 6 is replaced by delta(2) where delta represents some "absolute precision" goal (e.g., in Dr. Armstrong's example, plus or minus 0.1 mg/m3). Regarding the first point, I agree with Dr. Armstrong that the symmetric t-interval can be misleading when the sample size is small and the observed GSD is large. For this reason I stated that one should use my Equation 6 "with some caution when (n)(pilot) is small and the estimated GSD is large" as the actual confidence level will be somewhat less than the nominal level. Still, the purpose of sample size formulae, as discussed in my paper, is to get one in the ballpark. Most workplaces are sufficiently dynamic that even an "exact" procedure for estimating sample sizes (based upon pilot study data) may yield a sample size that is no longer appropriate for the current exposure distribution. I agree with Dr. Armstrong that for small sample sizes (due to financial constraints) and large GSDs one should use either Land's exact interval procedure or the Modified Cox approximate interval formulae to construct confidence intervals about the sample mean. Towards this end, I am preparing a paper on the use of "graphical" procedures that will allow one to (a) easily construct Land's "exact" confidence interval around the sample mean and (b) easily construct an "exact" confidence interval around the" exceedance fraction" (Le., the fraction of exposures expected to exceed an occupational exposure limit [OEL]). Such confidence intervals can be directly interpreted and easily used for hypothesis testing (Le., the intervals can be directly compared to the OEL and an acceptable exceedance fraction to determine if one is significantly above or below the target value). Hopefully, this paper will soon see the light of day. I wonder, however, if Dr. Armstrong's concern is warranted when one actually collects the number of measurements estimated by Equation 6. I suspect that when f is set to 0.2 or 0.3 the standard symmetric confidence interval (about the mean) is unlikely to have a negative lower bound, as in Dr. Armstrong's example. Regarding Dr. Armstrong's second point, I have two observations. First, the "absolute precision" approach may make it difficult to determine "safe" (ie., acceptable) exposures at the low end of the exposure-response curve. In principle, when attempting to resolve a linear or monotonic exposure-response relationship one should strive to accurately estimate both the low end and the high end (and everything in between; which may be why principle is so little regarded). The "absolute precision" approach tends to accurately estimate high long-term exposures (large sample sizes) and poorly estimate low long-term exposures (due to small sample sizes). Consider the situation where the exposure and response can be described by a simple linear relationship. Imprecision at the low end will result in a highly variable intercept. Since it is usually the low end of an exposure-response analysis that excites so much interest and debate it would be unwise to devote the majority of the sampling effort (of a research project) to characterizing the high end. Second, if the "absolute precision" goal is a large fraction of the true mean exposure for each individual or exposure group (where an exposure group is a job, occupation, exposure zone, area, process or any other logical means of aggregating workers into similar risk categories) it is likely that small sample sizes and imprecise estimates will result. This may make it difficult to resolve true differences between individuals or exposure groups.