Determination of Sample Size in Vaccine Trials

Abstract

Sample size determination is an important as well as delicate problem in clinical trials. Often, prior information available before the beginning of the trial is used to estimate the sample size. Hence, it is of importance to quantify the prior information available in terms of an Effective Sample Size (ESS). The problem of finding Effective Sample Size (ESS) in Phase II clinical trials, where toxicity and efficacy are the two components of the treatment response vector, is considered. In particular, one of the components is assumed to be binary, and the other is assumed to be continuous. Theoretical expressions for calculating ESS are obtained, and the methods are evaluated and compared by simulation studies. Once the problem of finding ESS was solved, we deflected our attention to the actual problem of sample size determination, which basically gives an idea about the number of individuals to be recruited in the Phase III vaccine trials. It is to be noted that the problem of sample size determination in vaccine trials is slightly different from the usual drug trials, and both should be given equal weight. In the Phase III trial of a vaccine, a group of individuals are allocated to the vaccine or placebo for a fixed period of time. The effectiveness of the vaccine in preventing a particular disease in comparison to the placebo is measured by means of protective vaccine efficacy, defined as VE=1-ARV/ARU, where ARV and ARU are, respectively, the disease attack rates in the vaccinated and unvaccinated populations. For each of the cohort and case-control designs, the sample sizes are calculated by fixing the maximum width of the 100(1-α)% confidence interval (CI) of VE and finding the minimum sample size required to achieve that width. The majority of the literature focuses on obtaining sample sizes when an equal number of individuals are allocated to the vaccine and the placebo groups. In our work, we calculate the required sample sizes when there is an unequal allocation of individuals across the two groups. The fraction of individuals allocated to the placebo group (⍴) is chosen in such a way that some relevant criterion of interest, like the total sample size or the expected number of people developing the disease, is optimized. It was observed that the sample sizes and the optimal allocation proportions come out to be a function of the parameters VE and ARU, which are unknown and to be estimated by the data obtained from the trial. Hence, one has to rely on the guesses about the unknown parameters to get an insight about the sample size, and hence completely wrong or uncertain guesses can give misleading results. The Bayesian paradigm comes to the rescue in such a case, which helps us deal with the uncertainties associated with the assumed values of the unknown parameters. This motivated us towards our third work, which provides a Bayesian alternative to the sample size determination problem in vaccine efficacy (both cohort and case-control) studies. Both the frequentist and Bayesian sample sizes were then compared among themselves and with those taken in a real-life trial. To estimate VE, we follow all the participants in the trial for a fixed period of time. It is quite evident that the estimates of ARV, ARU, and subsequently that of VE, vary with the change in time to follow-up. Moreover, the datasets obtained from vaccine trials are of time-to-event (time to development of the disease) nature, and survival models give a better fit than the conventionally used Binomial model. In the survival set-up, the vaccine efficacy is redefined as VEH,(t) = 1-λB(t)/λA(t), where λA(t) and λB(t) are respectively the hazard rates of the unvaccinated and vaccinated populations at time t. We then approach the sample size determination problem under the survival framework after presuming appropriate distributions on the survival and censoring times. A semi-parametric approach using the Cox Proportional Hazards model, which does not require any distributional assumptions on the survival times, has also been considered. Sample sizes and optimal allocation proportions have been obtained, and their superiority over the Binomial sample sizes has been pointed out.