Survival-analysis methods for droplet-freezing data: Kaplan–Meier confidence intervals and non-parametric curve comparison

Droplet-freezing assays underpin immersion-mode ice-nucleation research yet approaches to uncertainty quantification for fraction-frozen curves and derived active-site densities ( n _s ( T )) are inconsistent. Further, there is currently no rigorous method for significance testing the difference between fraction-frozen curves. To address these issues, we recast droplet-freezing measurements as survival data and apply analysis techniques typically used in medical statistics. Using the Kaplan–Meier estimator, we derive nonparametric pointwise confidence intervals for droplet fraction-frozen without binning or model assumptions. These intervals can also be propagated to cumulative ice-nucleation activity spectra such as  K ( T ), n _s ( T ) or  n _m ( T ). Simulations based on fitted literature freezing curves show that Kaplan–Meier intervals have close to nominal coverage over the central portion of fraction-frozen curves, but coverage deteriorates near the ends of the curve where finite experiments contain little information. We also compare several statistical tests for distinguishing between droplet-freezing curves. These tests quantify whether the observed separation between two finite-droplet curves is larger than expected from sampling variability alone. The log-rank test, often used alongside Kaplan–Meier confidence intervals in medical statistics, can be useful for approximately proportional-hazards-like alternatives, but can have low power when curves differ in shape or reflect multiple ice-active populations. We therefore introduce an integrated squared fraction-frozen distance permutation test, which we refer to as the Integrated Squared Difference (ISD) test. This test compares the accumulated squared separation between two empirical fraction-frozen curves directly and does not require proportional hazards. Simulations comparing the ISD test, log-rank and several standard empirical-distribution-function tests show that no single test is most powerful in all situations: performance depends on droplet number, curve shape, and the form of the difference being tested. We recommend reporting fraction-frozen curves with Kaplan–Meier pointwise confidence intervals and droplet numbers, and using the ISD test for general curve comparison. We note finally that a significant global test indicates that two curves differ somewhere over the analysed temperature interval, but does not identify where they differ or why; this still requires inspection of the curves and potentially of normalised spectra.