Combination therapy is increasingly central to modern medicine. Yet reliable analysis of combination studies remains an open challenge. Previous work suggests that common methods of combination analysis are too susceptible to noise to support robust scientific conclusions. In this paper, we use simulated and real-world combination datasets to demonstrate that traditional index methods are unstable and biased by pharmacological and experimental conditions, whereas response-surface approaches such as the BRAID method are more consistent and unbiased. Using a publicly-available data set, we show that BRAID more accurately captures variations in compound mechanism of action, and is therefore better able to discriminate between synergistic, antagonistic, and additive interactions. Finally, we applied BRAID analysis to identify a clear pattern of consistently enhanced AKT sensitivity in a subset of cancer cell lines, and a far richer array of PARP inhibitor combination therapies for BRCA1-deficient cancers than would be identified by traditional synergy analysis.