ToxSci Advance Access originally published online on April 18, 2006
Toxicological Sciences 2006 92(1):346; doi:10.1093/toxsci/kfj200
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Published by Oxford University Press 2006.
TO THE EDITOR
National Center for Risk Assessment, Quantitative Risk Management Group, Office of Research and Development, U.S. Environmental Protection Agency (Mail Code 8623D), 1200 Pennsylvania Avenue, NW, Washington, District of Columbia 20460
E-mail: fox.john{at}epa.gov
Gaylor et al. (2004)
described a method of estimating the "zero-equivalent dose" (ZED) and a lower confidence limit for it (LZED), based upon a quadratic polynomial dose-response model. They also suggested a way to use the Environmental Protection Agency (EPA) benchmark dose software to estimate these values, which was to select the continuous-response polynomial model and to use the "Point" benchmark response (BMR) type, setting it equal to the average response for the control group.
The suggestion to select the Point BMR has the effect of treating the control response as if it were known, when it is an estimate of an unknown parameter. As a consequence, LZED would be overestimated. Use of the Point BMR, µ(BMR) = 
, assumes that the response at zero dose is known to have the exact value . Instead, LZED should be calculated using the "absolute deviation" BMR choice, which solves the relation 
This will fail when is set exactly equal to zero but will succeed when is some small number relative to the means, e.g., 0.00001. It is prudent to try a series of successively smaller values of to find a "stable" value for the lower confidence limit.
The lower confidence limit (LZED) has an exact solution when the polynomial model is correct over the range of doses used, when errors are normal, and variances are equal for the dose groups (satisfying these assumptions might require a transformation of the response means or weighted least squares could be used when variances are unequal).
The exact lower confidence limit for µY(XZED) is
 |
and y(x) is the sample mean response at dose x, all quantities being estimates. Note that one must first find XZED as the positive root of the polynomial 
where P(X) is the estimated polynomial. Then one must obtain the variance estimate, compute LCLYZED, and solve for the root X = LZED corresponding to LCLYZED using the estimated P(X).
I implemented this method in S-Plus (S-PLUS 2000 Professional Release 1, 1988–1999, Mathsoft, Inc.) to check its coverage and to compare the LZED with that produced by EPA's benchmark dose software, which uses an asymptotic method. I simulated 5000 experiments, using doses (0, 0.5, 1.1, and 2.0), mean responses (10, 4, 4.72, and 22), SDs of response all equal 2, and dose group sample sizes of 6. The mean responses correspond to expected values from a quadratic model 
which has a ZED of 1.5 and a response of 10 at zero dose. In 5000 repeated applications of the exact method, coverages were 94.66 and 95.82% for the lower and upper confidence limits, respectively.
For these mean responses and SDs, treated as observations from a single experiment, EPA's benchmark dose software (version 1.4) reported estimates of 1.5 and 1.44455 for the ZED and LZED, respectively, when a Point BMR of 10 (the response at zero dose) was selected. When BMR type "Abs. Dev." was set to 0.1, 0.001, and 0.00001, ZED estimates were reported as 1.50554, 1.50006, and 1.5, and LZED estimates were reported as 1.40179, 1.39534, and 1.39528, respectively. The exact method produced an LZED of 1.36896 and a ZED of 1.5 (to within seven decimal places).
I see no good reason to prefer the exact method over the BMDS method (profile likelihood) for general use because, while it is exact under its assumptions, these assumptions are likely to be violated in some degree, including specification of the correct model function. In that event, it is not obvious which approach would be better. Rather, the exact method was used here as a means to explain and illustrate why the absolute deviation BMR is the correct choice for estimating the LZED using EPA's benchmark dose software.
This comment is intended to clarify a statistical point and not to endorse or dispute the use of LZED. However, it is worth noting that purely statistical observations based on one study should be interpreted cautiously until corroborated by toxicological and mechanistic analyses.
NOTES
Disclaimer: The views expressed in this letter are those of the author and do not necessarily reflect the views or policies of the U.S. Environmental Protection Agency.
REFERENCES
Gaylor, D. W., Lutz, W. K., and Conolly, R. B. (2004). Statistical analysis of nonmonotonic dose-response relationships: Research design and analysis of nasal cell proliferation in rats exposed to formaldehyde. Toxicol. Sci. 77, 158–164.
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