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
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