Bayesian estimation of inverse dose response

Inverse dose-response estimation refers to the inference of an effective dose of some agent that gives a desired probability of response, say 0.5. We consider inverse dose response for two agents, an application that has not received much attention in the literature. Through the posterior profiling technique (Hsu, 1995, The Canadian Journal of Statistics 23, 399-410), we propose a Bayesian method in which we approximate the marginal posterior distribution of an effective dose using a profile posterior distribution, and obtain the maximum a posteriori (MAP) estimate for the effective dose. We then employ an adaptive direction sampling algorithm to obtain the highest posterior density (HPD) credible region for the effective dose. Using the MAP and HPD estimates, investigators will be able to simultaneously calibrate the levels of two agents in dose-response studies. We illustrate our proposed Bayesian method through a simulation study and two practical examples.