Statistical Methods for Estimating TD₅₀

For NCI/NTP bioassays and a few experiments from the general literature, TD₅₀ 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.

λ₀(t) is the tumor-incidence rate at zero dose. The parameter β and the function λ₀ 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 χ² 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.

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 χ² statistic tests the adequacy of the model.

The estimate of TD₅₀ 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 λ₀(t) (Sawyer et al., 1984). For either method of estimating TD₅₀, if the χ² 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 TD₅₀ 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 TD₅₀.

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 TD₅₀: 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