# Statistical Methods for Estimating TD_{50}

For NCI/NTP bioassays and a few experiments from the general
literature, TD_{50} has been estimated using lifetable
data. The symbol “:” appears in the plot for lifetable
data. The lifetable methods which we have used to analyze the
experimental data have been described elsewhere (Sawyer *et al.,* 1984). Briefly, a proportional
hazards model (Cox, 1972) is assumed for the
time-to-tumor data, in which *λ(t, d)*, the
tumor-hazard rate at age *t* for a specific site, is linearly
related to *d*, the administered dose-rate of test chemical in
mg/kg body wt/day, as

**Equation 1.**

**λ(t,d) = (1 + β ·
d)λ**_{0}(t).
*λ*_{0}(t) is the tumor-incidence rate at
zero dose. The parameter *β* and the function
*λ*_{0} are estimated using maximum likelihood
methods. The likelihood ratio statistic tests the hypothesis that
the chemical has no carcinogenic effect (i.e., *β = 0*),
and a *χ*^{2} goodness-of-fit statistic tests the
validity of the linear relationship between dose and tumor
incidence expressed by Equation 1. In fitting the model, no attempt
is made to distinguish between tumors found in a fatal context and
tumors found in an incidental context. Thus the time-to-tumor
occurrence is taken to be the time to death of the animal, whether
death results from the tumor of interest, or from some other cause,
including terminal sacrifice (Peto *et al.,*
1984, Sawyer *et al.,* 1984)

For summary incidence data, we fit by maximum likelihood methods
the comparable model

**Equation 2.**

**p**_{d} = 1 - exp{-(a + bd)},
where *a > 0* and *b > 0*
and *p*_{d} is the probability that an animal exposed
at dose *d* for its lifetime develops a tumor. This model is
linear at low doses and is often referred to as the “one-hit
model.” Here, the number of animals developing tumors at dose
*d* is assumed to follow a binomial distribution with
parameters *n*_{d} and *p*_{d}, where
*n*_{d} is the number of animals initially exposed at
dose *d*. As with lifetable data, the likelihood ratio
statistic is used to test whether the compound is carcinogenic,
i.e. whether *b = 0*, and a *χ*^{2}
statistic tests the adequacy of the model.

The estimate of TD_{50} based on summary incidence data
is simply *log(2)/b*, where *b* is the maximum likelihood
estimate (MLE) of *b*. For lifetable data, the estimate is a
more complex function of the MLEs of *β* and
*λ*_{0}(t) (Sawyer *et
al.,* 1984). For either method of estimating
TD_{50}, if the *χ*^{2} goodness-of-fit
test indicated statistically significant departure from linearity,
(*p*<0.05) and this departure was downward, the analysis
was repeated eliminating the highest dose group. The purpose of
this procedure was to remove the effects of toxicity in summary
incidence analyses and to remove the effects of dose saturation in
the lifetable analyses. If the goodness-of-fit test indicated an
upward departure from linearity, no groups were eliminated when
fitting the model.

In the CPDB we have estimated 99% confidence intervals for
TD_{50} from lifetable data, when available, and otherwise
from summary incidence data. The method for calculating these
intervals from lifetable data is described in Sawyer *et al.,* (1984). For summary incidence
data, the 99% likelihood-ratio-test-based confidence limits are
obtained for *b* and are then transformed to limits for
TD_{50}.

## Bibliography

- Cox, D.R. Regression models and life
tables (with discussion)
*J. R. Stat. Soc. Brit. Series B* 34:
187-220 (1972).
- Peto, R., Pike, M.C., Bernstein, L.,
Gold, L.S., and Ames, B.N. The TD
_{50}: A proposed
general convention for the numerical description of the
carcinogenic potency of chemicals in chronic-exposure animal
experiments. *Environmental
Health Perspectives* 58: 1-8 (1984). Abstract Text
- Sawyer, C., Peto, R., Bernstein,
L., and Pike, M.C. Calculation of carcinogenic potency from
long-term animal carcinogenesis experiments.
*Biometrics* 40:
27-40 (1984). Abstract Text

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Last updated: September 20, 2004

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