The Finkelstein and Schoenfeld (FS) test is a popular generalized pairwise comparison approach to analyze prioritized composite endpoints (e.g., components are assessed in order of clinical importance). Power and sample size estimation for the FS test, however, are generally done via simulation studies. This simulation approach can be extremely computationally burdensome, compounded by an increasing number of composite endpoints and with increasing sample size. We propose an analytic solution to calculate power and sample size for commonly encountered two-component hierarchical composite endpoints. The power formulas are derived assuming underlying distributions in each of the component outcomes on the population level, which provide a computationally efficient and practical alternative to the standard simulation approach. The proposed analytic approach is extended to derive conditional power formulas, which are used in combination with the promising zone methodology to perform sample size re-estimation in the setting of adaptive clinical trials. Prioritized composite endpoints with more than two components are also investigated. Extensive Monte Carlo simulation studies were conducted to demonstrate that the performance of the proposed analytic approach is consistent with that of the standard simulation approach. We also demonstrate through simulations that the proposed methodology possesses generally desirable objective properties including robustness to mis-specified underlying distributional assumptions. We illustrate our proposed methods through application of the proposed formulas by calculating power and sample size for the Transthyretin Amyloidosis Cardiomyopathy Clinical Trial (ATTR-ACT) and the EMPULSE trial for empagliozin treatment of acute heart failure.