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Annex A: Maximum Likelihood Methods for Fitting the Weibull Model

Suppose that the probability P(d) of a tumor occurring at dose d follows the Weibull model

(a, b, k > 0) as in (2.5). We wish to estimate the unknown model parameters a, b and k on the basis of an experiment with s + 1 dose levels 0 = do < d1 <… < ds. Suppose that xi of the ni animals in group i = 0, 1,…,s develop tumors. Estimators of the unknown model parameters may be obtained by maximizing the binomial likelihood

where pi = P(di) and x = (x0, x1 …, xk). Numerical procedures for obtaining the maximum likelihood estimators (mle's) of the unknown model parameters, as well as the mle of the TD50 and its standard error, are described by Krewski & Van Ryzin (1981).

It is possible that this likelihood may not attain a global maximum, in which case the mle's of the unknown parameters do not exist. To illustrate, take s = 2, n0 = n1 = n2 = n, and suppose that x0 = x1 = x with x2 = y > x. The likelihood function L then satisfies the upper bound

Let c0 and c1 be defined by the equations

and



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APPENDIX F 160 original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as the authoritative version for attribution. ANNEX A: MAXIMUM LIKELIHOOD METHODS FOR FITTING THE WEIBULL MODEL Suppose that the probability P(d) of a tumor occurring at dose d follows the Weibull model (a, b, k > 0) as in (2.5). We wish to estimate the unknown model parameters a, b and k on the basis of an experiment with s + 1 dose levels 0 = do < d1 <… < ds. Suppose that xi of the ni animals in group i = 0, 1,…,s develop tumors. Estimators of the unknown model parameters may be obtained by maximizing the binomial likelihood where pi = P(di) and x = (x0, x1 …, xk). Numerical procedures for obtaining the maximum likelihood estimators (mle's) of the unknown model parameters, as well as the mle of the TD50 and its standard error, are described by Krewski & Van Ryzin (1981). It is possible that this likelihood may not attain a global maximum, in which case the mle's of the unknown parameters do not exist. To illustrate, take s = 2, n0 = n1 = n2 = n, and suppose that x0 = x1 ≡ x with x2 ≡ y > x. The likelihood function L then satisfies the upper bound Let c0 and c1 be defined by the equations and

OCR for page 160
APPENDIX F 161 original typesetting files. Page breaks are true to the original; line lengths, word breaks, heading styles, and other typesetting-specific formatting, however, cannot be About this PDF file: This new digital representation of the original work has been recomposed from XML files created from the original paper book, not from the retained, and some typographic errors may have been accidentally inserted. Please use the print version of this publication as the authoritative version for attribution. If k → ∞ and b → 0 or ∞ with bd2k = c1 held constant, then bd1k = c1(d1/d2) → 0 and L → L*. Thus, no finite mle of k exists in this case. This seems k intuitively reasonable, since data of the type under consideration are consistent with dose- response curves of arbitrarily large upward curvature (i.e., arbitrarily large values of k). Noting that (0 < p < 1), it follows that the mle of the TD100p is equal to d2 for any value of p in this case, an unpleasant conclusion. Other estimation methods such as least squares may be expected to perform in a similar manner. Of the 217 data sets considered by Krewski et al. (1990b), mle's were readily obtained for the 122 dose-response curves that were strictly increasing. The mle's for a further 69 data sets did not appear to exist because of nonmonotonicity as discussed above. The final 26 data sets involved only a control group and single nonzero dose, so that the shape parameter k could not be estimated. For the 122 data sets for which mle's could be obtained, an adjusted measure of carcinogenic potency given by was calculated using the factor f2/k discussed in annex C. This effectively adjusts all TD50 values to a two year standard rodent lifespan. By linear approximation (Rao, 1973), the variance of is given by