An introduction to the determination of the probability of a successful trial: Frequentist and Bayesian approaches

Determination of posterior probability for go-no-go decision and predictive power are becoming increasingly common for resource optimization in clinical investigation. There are vast published literature on these topics; however, the terminologies are not consistently used across the literature. Further, there is a lack of consolidated presentation of various concepts of the probability of success. We attempted to fill this gap. This paper first provides a detailed derivation of these probability of success measures under the frequentist and Bayesian paradigms in a general setting. Subsequently, we have presented the analytical formula for these probability of success measures for continuous, binary, and time-to-event endpoints separately. This paper can be used as a single point reference to determine the following measures: (a) the conditional power (CP) based on interim results, (b) the predictive power of success (PPoS) based on interim results with or without prior distribution, and (d) the probability of success (PoS) for a prospective trial at the design stage. We have discussed both clinical success and trial success. This paper's discussion is mostly based on the normal approximation for prior distribution and the estimate of the parameter of interest. Besides, predictive power using the beta prior for the binomial case is also presented. Some examples are given for illustration. R functions to calculate CP and PPoS are available through the LongCART package. An R shiny app is also available at https://ppos.herokuapp.com/.