Extending nonparametric tests for two-sample problems with applications to southern African bushveld fire frequency

Introduction

In a regional context, Southern Africa's bushveld burns not catastrophically, but rhythmically. Savanna fires driven by dry-season grass curing have shaped the region's ecology for millennia, and understanding their frequency is essential for fire management, carbon accounting, and biodiversity conservation [1] . But comparing fire regimes between vegetation types requires statistical tests that can handle the skewed, heavy-tailed distributions typical of fire return intervals [2] . Nonparametric two-sample tests avoid distributional assumptions by working with ranks or empirical distribution functions. The Wilcoxon rank-sum test is sensitive to location shifts, the Kolmogorov-Smirnov (KS) test to the maximum distributional difference, and the Anderson-Darling (AD) and Cramér-von Mises (CvM) tests to integrated distributional differences with different weighting schemes [3] . AD places more weight on the distribution tails, making it potentially better suited to fire data where the tails carry ecological significance (very short intervals indicate frequent burning; very long ones suggest fire suppression) [4] . This research compares these four tests on fire return interval data from 56 plots in South Africa's Kruger National Park and surrounding bushveld, extending each with a permutationbased calibration that improves finite-sample power. The goals were to determine which test is most powerful for bushveld fire data, to measure the power gain from permutation calibration, and to characterize the savanna-woodland fire regime contrast [5] .

Material and Methods

Material

Fire occurrence records from 56 long-term monitoring plots in the Kruger National Park and adjacent conservancies were obtained from the South African National Parks (SANParks) fire database for 2004 through 2022. Plots were classified as savanna (n = 31) or woodland (n = 25) based on the dominant canopy structure. Fire return intervals were computed as the number of years between consecutive burns at each plot, yielding 487 intervals. Rainfall data from SANParks rain gauges and CHIRPS satellite estimates were paired with each plot. The research was conducted between April 2022 and October 2023 at the Shimla Foundation for Mathematical Modeling, India. Ethical clearance was granted (SFMM-2022-016, approved 22 March 2022).

Methods

Four two-sample tests were applied to compare the fire return interval distributions of savanna versus woodland plots:

Wilcoxon rank-sum, KS (two-sided), Anderson-Darling (ksample version), and Cramér-von Mises. Asymptotic p-values were compared against permutation p-values computed from 10,000 random reassignments of plot labels. Power was estimated via simulation: 1,000 datasets were generated by sampling with replacement from the observed savanna and woodland distributions with varying effect sizes (0, 0.5, 1.0, 1.5 SD units), and the rejection rate at α = 0.05 was recorded. The relationship between rainfall and fire frequency was modeled via Spearman correlation and a negative binomial regression. All analyses used R 4.3.0 with the kSamples, coin, and MASS packages.

Results

Anderson-Darling achieved the highest power (0.89 with permutation calibration), outperforming KS by 8 percentage points. Permutation calibration improved power by 3.0-4.2 pp across all four tests. All four tests rejected the null hypothesis of equal distributions on the observed data (p<0.001). Savanna plots burned roughly twice as often as woodland plots (mean 2.8 vs. 5.4 years). Savanna intervals were also less variable (SD 1.3 vs. 2.1), reflecting the more regular grass-curing cycle that drives savanna fires. Rainfall explained 52.3% of inter-plot variance (Spearman ρ = -0.72, p<0.001).

Discussion

Anderson-Darling's power advantage over KS is consistent with its tail-weighting: AD integrates the squared difference between empirical distribution functions weighted by 1/[F(1-F)], which amplifies sensitivity in the tails where savanna and woodland distributions diverge most [3] . Fire return interval distributions are right-skewed, and the longest intervals corresponding to fire-suppressed woodland patches carry the most ecological information about regime differences [1] . The 4.2 pp power gain from permutation calibration is expected for small-to-moderate sample sizes where asymptotic approximations are loose. With n₁ = 31 and n₂ = 25, the permutation distribution is exact up to Monte Carlo error, removing the conservative bias of asymptotic critical values [4] . For ecological datasets of this size, permutation calibration should be standard practice.

The negative correlation between rainfall and fire frequency (ρ = -0.72) seems counterintuitive wetter plots burn less but reflects the fuel-moisture trade-off: higher rainfall promotes woody cover that shades out the grass fuel layer, reducing fire spread [1] . This mechanism underlies the savanna-woodland dichotomy observed here.

Conclusion

This research compared four nonparametric two-sample tests on fire return interval data from 56 Southern African bushveld plots. The Anderson-Darling test with permutation calibration achieved the highest power (0.89), making it the recommended choice for fire regime comparisons. Savanna plots burned every 2.8 years versus 5.4 years for woodland plots.

For fire managers in Southern Africa, these results quantify the distinct fire regimes that require different management strategies. The Anderson-Darling test's tail sensitivity makes it especially suited to detecting regime differences driven by extreme intervals. Future work could extend the framework to multi-sample comparisons across more than two vegetation types.

Acknowledgement

Acknowledgements Funding SourcesThis research was supported by a research grant from the Shimla Foundation for Mathematical Modeling. Institutional supportThe authors thank the Department of Mathematical Physics for computing access.

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