# Issues in Risk Assessment(1993)

## Chapter: ANNEX A: MAXIMUM LIKELIHOOD METHODS FOR FITTING THE WEIBULL MODEL

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Suggested Citation:"ANNEX A: MAXIMUM LIKELIHOOD METHODS FOR FITTING THE WEIBULL MODEL." National Research Council. 1993. Issues in Risk Assessment. Washington, DC: The National Academies Press. doi: 10.17226/2078.
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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

Page 161
Suggested Citation:"ANNEX A: MAXIMUM LIKELIHOOD METHODS FOR FITTING THE WEIBULL MODEL." National Research Council. 1993. Issues in Risk Assessment. Washington, DC: The National Academies Press. doi: 10.17226/2078.
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If k → ∞ and b → 0 or ∞ with bd2k = c1 held constant, then bd1k = c1(d1/d2)k → 0 and L → L*. Thus, no finite mle of k exists in this case. This seems 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

Page 160
Suggested Citation:"ANNEX A: MAXIMUM LIKELIHOOD METHODS FOR FITTING THE WEIBULL MODEL." National Research Council. 1993. Issues in Risk Assessment. Washington, DC: The National Academies Press. doi: 10.17226/2078.
×
Page 161
Suggested Citation:"ANNEX A: MAXIMUM LIKELIHOOD METHODS FOR FITTING THE WEIBULL MODEL." National Research Council. 1993. Issues in Risk Assessment. Washington, DC: The National Academies Press. doi: 10.17226/2078.
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Next: ANNEX B. SHRINKAGE ESTIMATORS OF THE DISTRIBUTION OF CARCINOGENIC POTENCY »
Issues in Risk Assessment Get This Book
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The scientific basis, inference assumptions, regulatory uses, and research needs in risk assessment are considered in this two-part volume.

The first part, Use of Maximum Tolerated Dose in Animal Bioassays for Carcinogenicity, focuses on whether the maximum tolerated dose should continue to be used in carcinogenesis bioassays. The committee considers several options for modifying current bioassay procedures.

The second part, Two-Stage Models of Carcinogenesis, stems from efforts to identify improved means of cancer risk assessment that have resulted in the development of a mathematical dose-response model based on a paradigm for the biologic phenomena thought to be associated with carcinogenesis.

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