Sample sizes required to estimate the protective efficacy of a vaccine when there is an unequal allocation of individuals across the vaccine and placebo groups

Abstract

The effectiveness of a vaccine is measured by means of protective vaccine efficacy, defined by $VE=1-\frac{ARV}{ARU}$ , where $ARV$ and $ARU$ are, respectively, the disease attack rates in the vaccinated and the unvaccinated population. For each of the cohoret and case-control designs, methods have been presented in the literature for calculating the required sample size when the desired width of the confidence interval and the probability of coverage are pre-specified, where an equal number of individuals were assumed to be allocated to the vaccine and placebo group. In this article, we present a method for calculating the required sample size with a specified degree of precision when there is an unequal allocation of individuals across the two groups. The sample size required to achieve a desired power for the relevant level $\alpha$ test has also been explored, keeping the unequal allocation proportion in mind. The fraction of individuals allocated to the placebo group ( $\rho$ ) can be so chosen that the total sample size or the expected number of people developing the disease or some other criteria of interest is minimized.