**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.

**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 = d_{o} < d_{1} <… < d_{s}. Suppose that x_{i} of the n_{i} animals in group i = 0, 1,…,s develop tumors. Estimators of the unknown model parameters may be obtained by maximizing the binomial likelihood

where p_{i} = P(d_{i}) and x = (x_{0}, x_{1} …, x_{k}). Numerical procedures for obtaining the maximum likelihood estimators (mle's) of the unknown model parameters, as well as the mle of the TD_{50} 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, n_{0} = n_{1} = n_{2} = n, and suppose that x_{0} = x_{1} = x with x_{2} = y > x. The likelihood function L then satisfies the upper bound

Let c_{0} and c_{1} be defined by the equations

and

**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.

If k → ∞ and b → 0 or ∞ with bd_{2}^{k} = c_{1} held constant, then bd_{1}^{k} = c_{1}(d_{1}/d_{2})^{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 TD_{100p} is equal to d_{2} 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 f^{2/k} discussed in annex C. This effectively adjusts all TD_{50} values to a two year standard rodent lifespan. By linear approximation (Rao, 1973), the variance of

is given by