Droplet-freezing experiments have been the workhorse of immersion-mode ice-nucleation research since the 1950s, yet the field has never agreed on how to quantify the uncertainty in their results. Now a team at the University of Leeds has borrowed a framework from an unlikely source, medical statistics, to bring rigour to the problem. In a study published in Atmospheric Measurement Techniques, Thomas Whale and colleagues show that droplet-freezing measurements can be recast as survival data, the same kind of event-time data that oncologists use to track patient outcomes. This reframing allows them to compute well-calibrated confidence intervals for freezing curves without any binning or model assumptions, and to introduce a new statistical test for deciding whether two freezing curves are genuinely different or merely products of sampling noise.
The data at the heart of these experiments are deceptively simple. A researcher cools many tiny droplets of water, often containing dust, pollen washing water, or other potential ice-nucleating particles, and records the temperature at which each droplet freezes. Plotting the frozen fraction against temperature produces the familiar fraction-frozen curve, the raw material for nearly every claim about how efficiently a material nucleates ice. Microlitre-scale assays typically involve 40 to 100 droplets per experiment, high-throughput plate systems several hundred wells, and microfluidic devices hundreds to thousands of droplets. Despite this long history, approaches to calculating confidence intervals have been inconsistent, ranging from binomial intervals and Poisson fiducial limits to various parametric bootstrap schemes, each resting on assumptions about the distribution of active sites or the stochasticity of nucleation that may not hold.
The Leeds team’s insight is that a droplet freezing at temperature T is formally equivalent to a subject experiencing an event at time t, since temperature decreases steadily during the experiment. The Kaplan–Meier estimator, first published in 1958, estimates the probability that a subject survives beyond a given time as a product of conditional survival probabilities across each observed event. Applied to droplets, the same product formula yields the fraction of droplets still unfrozen at any temperature, and the associated asymptotic variance, computed on a log-minus-log transformed scale to keep intervals within physical bounds, delivers pointwise 95 percent confidence intervals for the frozen fraction. Crucially, the method is non-parametric: it makes no assumption about the distribution of freezing temperatures, and unlike bootstrapping it requires no generation of synthetic experiments or choices about data handling.
The intervals also propagate naturally to the quantities researchers actually publish. Because cumulative ice-nucleation spectra such as K(T), ns(T), and nm(T) are direct logarithmic transformations of the frozen fraction, the confidence limits transform along with the curve. The transformation does magnify uncertainty wherever the frozen fraction approaches zero or one, a well-known sensitivity of logarithmic spectra, and intervals become formally undefined once every droplet has frozen. Differential spectra, by contrast, require additional binning, smoothing, or fitting steps, so the authors confine their treatment to frozen fractions and cumulative quantities, where the connection to the Kaplan–Meier estimator is direct and unambiguous.
To check that the intervals actually deliver their promised coverage, the researchers built synthetic experiments from smooth dual-Gaussian fits to three published freezing curves chosen for their contrasting shapes: an instrument background curve, a very steep curve for birch pollen washing water, and a broad, extended curve for BCS375 feldspar. They repeatedly sampled synthetic experiments of varying droplet numbers from these known distributions and counted how often the nominal 95 percent interval enclosed the true underlying value. The verdict was reassuring in the middle and honest at the edges. Coverage approached the nominal level for droplet numbers typical of real experiments and continued to improve as the number of droplets increased, but it deteriorated near the ends of the curve, where finite experiments simply contain little information about tail behaviour, an effect amplified after transformation to K(T). There is, the authors stress, no universal droplet number at which an experiment becomes statistically reliable; steep single-population curves need fewer droplets than broad or multi-population ones to reach the same precision.
The second half of the paper tackles a question asked constantly in ice-nucleation laboratories: are these two freezing curves different? Until now, answers have rested on visual inspection or ad hoc statistics, with no pre-specified false-positive rate. The team evaluated the log-rank test, the standard companion to Kaplan–Meier curves in clinical trials, which compares observed with expected freezing events at each temperature under the null hypothesis that the two experiments share the same underlying distribution. In simulations using droplet numbers of 5, 20, 50, 100 and 500 per curve, false-positive rates sat close to the expected 5 percent level for droplet numbers of 20 and above, though the ordinary log-rank test crept above 7 percent with only five droplets per curve, and the Kolmogorov–Smirnov test proved markedly conservative in that sparse regime.
The log-rank test’s Achilles heel emerged when the curves differed in shape rather than position. The test is most powerful when the ratio of conditional event rates is roughly constant across temperature, an assumption almost certainly violated by ice-nucleation data, where different populations of ice-active material may dominate different parts of the spectrum. Because the log-rank statistic sums signed differences between observed and expected events, excess freezing at high temperatures can cancel a deficit at low temperatures. In the starkest demonstration, comparing the steep birch pollen curve against the broad feldspar curve, the empirical-distribution tests detected the difference in all 1000 simulations at 100 droplets per curve, while the ordinary log-rank test rejected the null in only 69.5 percent of cases and the weighted FH(1,0) variant in just 54.1 percent.
To fill this gap, the authors introduced the Integrated Squared Difference test, a permutation test that measures the accumulated squared vertical separation between two empirical fraction-frozen curves over a specified temperature interval. Because all differences contribute positively, no cancellation is possible, and because the null distribution is built numerically by pooling the freezing temperatures and randomly relabelling them between groups, the test requires no proportional-hazards assumption and no asymptotic distribution theory. In a mixture-population scenario designed to mimic a sample whose warmer-freezing component grows from 10 to 30 percent of the total, the ISD test had the highest power of all methods, detecting the difference in 82.4 percent of simulations, while the ordinary log-rank test managed only 35.3 percent.
Yet the authors resist crowning a single winner. For simple shift-like differences, such as a 1 °C translation of the background curve, the ordinary log-rank test actually performed best, detecting the shift in 81.5 percent of simulations at 100 droplets, well ahead of the ISD test at 66.3 percent. The power of every test depended jointly on droplet number, the size of the difference, and the width of the underlying freezing distribution. For the broad feldspar curve, all six tests failed to detect a 1 °C shift in more than 45 percent of experiments even with 100 droplets per curve, and reliable detection required around 500 droplets. With five or fewer droplets per curve, the authors caution, results from any permutation-based test should be interpreted very cautiously, since the attainable p-values are coarse and power is inherently limited.
The practical guidance that emerges is straightforward. Researchers should report fraction-frozen curves with Kaplan–Meier pointwise confidence intervals and always state the droplet number, remembering that the shaded intervals are pointwise rather than simultaneous bands over the full temperature range. For general comparisons of two directly comparable experiments, the ISD permutation test is a sensible first choice, while the log-rank test remains useful when proportional hazards are expected or curves differ by a small shift, and weighted log-rank tests are best avoided absent a pre-specified reason for the weighting. Every test on offer is global: a significant result says the curves differ somewhere in the analysed interval, but not where or why, questions that still demand inspection of the curves and of normalised spectra. Given that mixed-phase cloud glaciation depends on precisely these measurements, a shared statistical language for freezing curves may ripple well beyond the laboratories that adopt it. The team has released Jupyter notebooks implementing the intervals and both tests, lowering the barrier for the field to standardise.
Subject of Research: Application of survival-analysis statistics to droplet-freezing ice-nucleation data
Article Title: Survival-analysis methods for droplet-freezing data: Kaplan–Meier confidence intervals and non-parametric curve comparison
Article References: Whale, T. F., Surawy-Stepney, T., & Barr, S. L. (2026). Survival-analysis methods for droplet-freezing data: Kaplan–Meier confidence intervals and non-parametric curve comparison. Atmospheric Measurement Techniques, 19(20), 6699-6718. https://doi.org/10.5194/amt-19-6699-2026
Image Credits: AI Generated
Keywords: ice nucleation, droplet-freezing assays, Kaplan-Meier estimator, survival analysis, confidence intervals, log-rank test, permutation test, atmospheric science, mixed-phase clouds, statistics, fraction-frozen curves, Survival-analysis
News Source: Russell Cooper. (October 23, 2026). Borrowed from medicine, a statistical fix for icy droplet data. Scienmag.
