Bayesian estimation of vaccine efficacy

Abstract

Protective vaccine efficacy, VES, is measured as one minus the incidence rate ratio (IRR) or the relative risk (RR) in the vaccinated group compared with the unvaccinated group. In this paper, we systematically present Bayesian estimation of protective vaccine efficacy based on the Poisson and binomial distributions. We also propose a new tool, the vaccine efficacy acceptability curve, to represent the uncertainty for the estimate of the vaccine efficacy graphically. It is very useful, especially when there is no universal agreement on the acceptable vaccine efficacy. The vaccine efficacy acceptability curve is defined as the posterior probability that the measure of vaccine efficacy VES ≥ k for each acceptable value k. When a vaccine is highly efficacious, the number of vaccinated susceptibles being infected is likely to be very small or even zero. Then the assumptions of normality and log-normality of IRR or RR usually do not hold well. Although frequentist exact methods provide good estimates of the confidence interval, they are overly conservative and are computationally difficult to extend to estimate the vaccine efficacy acceptability curve. In this paper, our focus is on Bayesian estimation of protective vaccine efficacy, its highest probability density credible set, and the vaccine efficacy acceptability curve through Markov chain Monte Carlo (MCMC) methods. We illustrate the methods using the data from two pertussis vaccine studies and the H. influenza Type B preventive trial.