Review of calculation of conditional power, predictive power and probability of success in clinical trials with continuous, binary and time-to-event endpoints

Assessment of study success using conditional power (CP), the predictive power of success (PPoS) and probability of success (PoS) is becoming increasingly common for resource optimization and adaption of trials in clinical investigation. Determination of these measures is often a non-trivial mathematical task. Further, the terminologies used across the literature are not consistent, and there is no consolidated presentation on this. We have made a structured presentation on these measures for both trial success and clinical success: first, we have summarized the expressions of CP, PPoS and PoS in a general setting and subsequently, expressions for these measures are obtained for continuous, binary, and time-to-event endpoints in single-arm and two-arm trial settings. Many of these expressions are previously published; however, some of the expressions are very new including the expressions for testing median of time-to-event endpoint in a single-arm trial. We have also shown that 1/(no.ofevents)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1/(\text{ no. } \text{ of } \text{ events})$$\end{document} consistently underestimates the variance of log(median) and alternative expression for variance was derived. Examples are given along with the comparison of CP and PPoS. Expressions presented in this paper are implemented in LongCART package in R and in R shiny app https://ppos.shinyapps.io/public/.