For decades, the individual effective dose (IED) hypothesis proposed by Gaddum (1953) has been accepted as the sole explanation for the lognormal or log logistic model fitted to dose–response data. It is postulated that each individual has a unique individual tolerance, or an IED, beyond which it dies. Since the survival of an individual is determined by both the exposure intensity and the exposure duration, IED can also be taken as the exposure duration one can tolerate under any fixed dose. Instead of contributing survival to an innate characteristic of the individual, an alternative hypothesis explains survival as governed by a stochastic process occurring similarly in all individuals.

In this dissertation, we propose using the correlation coefficient to evaluate the relative contribution of IED and stochasticity. We consider experiments in which a group of testing organisms was twice exposed to the same concentration of a toxicant. Our models are fitted to five sets of interval censored times-to-death (TTD) data with proof censoring.

Based on the equal means and variances assumptions of TTDs during the two exposures, we develop a maximum likelihood estimator of the correlation based on a bivariate lognormal model. A graphical tool is developed to assess the normality of the unobserved marginal distribution of TTDs in the second exposure based on the conditional distribution and a scaled conditional function.

We also fit the same model under the Bayesian framework. We further propose relaxing the equal means and variances assumptions using a set of constrained informative priors. These models demonstrate robust inference about the correlation against 10–15% difference in median TTDs and ratio of standard deviations. Coupled with the powerful posterior predictive diagnostic tool, the Bayesian model has provided informative inference about the correlation.

Compared to previous findings, our Bayesian results lead to a similar conclusion for the NaPCP experiment, but not for the CuSO₄ and the NaCl experiments. We conclude that NaPCP is dominated slightly by a stochastic mechanism. However, there was 78% and 92% chance of IED dominance for CuSO4, and only 58% chance of IED dominance for NaCl. Our investigation ends with the exploration of an accelerated failure time model with shared frailty as an alternative to the bivariate lognormal model.