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OCR for page 164
Issues in Risk Assessment
"shrinking" the Yi toward the mean . The estimators of µi have the correct dispersion in that
In fitting the Weibull model in (A.1), we found that the estimate of the variance of the log10TD*50 based on (A.9) appeared to be excessively large in a small number of cases. In order not to underestimate the between-study variability based on (B.2), we used a trimmed mean S*si2/n*, in which the largest and smallest 10% of the observed values of si2 were omitted (Hampel et al., 1986, p. 79). Specifically, the summation S* covers only those n* = 153 observations falling in the central 80% of the distribution of the si2.
Annex C: Adjustment of Potency Values for Less than Lifetime Exposure
In order to ensure that TD50 values for different chemicals are comparable, some adjustment for differences in the duration of the experimental period is desirable. Gold et al. (1984) adjusted TD50 values by a multiplicative factor of f2, where f represents the fraction of a two year period encompassed by the study period. This effectively scales the TD50 values to a standard two year rodent lifetime. Specifically, we have
where the TD50 denotes the estimate of carcinogenic potency based on the observed data for the actual experimental period, and denotes the standardized value.
To motivate the use of the adjustment factor f2, consider the extended Weibull model
OCR for page 165
Issues in Risk Assessment
depending on both dose d and time t. Under this model, the TD50(t) evaluated at time t is given by
Thus, the ratio of TD50's at two distinct times t1 and t2 is
where f = t1/t2. In the CPDB, Gold et al. (1984) use a one-stage model with k = 1 and set p = 2 based on empirical observations reported by Peto et al. (1984), leading to their adjustment factor f2. In our applications of the Weibull model in (A.1), we will use a similar adjustment factor of f2/k to standardize TD50 values to a two year rodent lifetime.
For a multi-stage model of the form
allowing for the effects of both dose d and time t, the TD50 at time t is obtained as the solution of the equation
It follows that the standardized value of the TD50 is obtained as the solution of the equation
As with the Weibull model, we set p = 2 in the applications considered in this paper.
. The estimators 
of µi have the correct dispersion in that
based on (B.2), we used a trimmed mean S*si2/n*, in which the largest and smallest 10% of the observed values of si2 were omitted (Hampel et al., 1986, p. 79). Specifically, the summation S* covers only those n* = 153 observations falling in the central 80% of the distribution of the si2.
