Generalizing bonferroni correction methods for multiple testing control with applications to australian outback rainfall variability

Introduction

The challenge is deceptively simple: you run 64 statistical tests, one per rain gauge station, and some come back significant. How many of those significant results are real? The Bonferroni correction says to divide your threshold by 64, which works but is famously heavy-handed it misses real effects to avoid false ones [1] . The Benjamini-Hochberg procedure takes a different philosophy, controlling the expected proportion of false discoveries rather than the chance of any single false discovery, and it is more permissive [2] . But neither approach uses the fact that rain gauge stations have a spatial structure: nearby stations share weather patterns, and their test statistics are correlated. Incorporating this structure is the aim of generalised Bonferroni methods. The basic insight is that when tests are positively dependent, the effective number of independent tests is smaller than the nominal count, and the Bonferroni threshold can be relaxed accordingly [3] . The difficulty lies in quantifying the dependency. This research uses the adjacency matrix of the station network an object from graph theory to estimate the effective number of tests and derives a generalised threshold that is tighter than classical Bonferroni but still controls the family-wise error rate [4] . The Australian outback provides a large-scale test bed. Its rainfall is dominated by sporadic, intense events driven by monsoonal troughs, tropical cyclones, and localised convection, creating a pattern of high spatial variability even within relatively small distances [5] . Detecting long-term trends against this noisy background requires careful multiple testing control. The research analyses monthly rainfall records from 64 stations in Queensland, the Northern Territory, and Western Australia, covering the period 1960 to 2022. The heat equation on the p-value surface and Markov chain pvalue sequences supply additional diagnostic tools [6] .

Model description and parameterisation

The generalised Bonferroni correction is parameterised by the adjacency matrix A of the station network, where A_ij = 1 if stations i and j are within 200 km and 0 otherwise. The effective number of independent tests is estimated as m_eff = m / (1 + (m-1)*rho_bar), where m = 64 is the number of stations and rho_bar is the mean pairwise correlation of test statistics estimated from the adjacency structure. The generalised significance threshold is alpha_gen = alpha / m_eff rather than alpha / m. The heat equation du/dt = kappa * laplacian_A(u) governs the diffusion of significance across the network, where laplacian_A is the graph Laplacian of A and kappa = 0.1 per year is a diffusion constant calibrated to match the observed spatial correlation decay with distance. The Markov chain model represents the sequence of ordered p-values as transitions between rejection and non-rejection states, with transition probabilities depending on the spacing distribution [4] .

Materials and methods

Materials

Monthly rainfall records for 64 stations distributed across the Australian outback were obtained from the Bureau of Meteorology (BoM) Climate Data Online service, covering January 1960 to December 2022. Stations were selected for record completeness (less than 5 percent missing months) and spatial coverage, with roughly equal representation from Queensland (22 stations), the Northern Territory (21 stations), and Western Australia (21 stations). Seasonal totals were computed for summer (December-February), autumn (March-May), winter (June-August), and spring (September-November). All analyses were performed at the Tucuman Faculty of Mathematical Analysis, Tucuman, Argentina, between November 2022 and June 2023. No ethical approval was required; only publicly available meteorological data were used [7] .

Methods

At each station, a non-parametric Mann-Kendall trend test was applied to annual and seasonal rainfall totals, yielding 64 p-values per series. The classical Bonferroni correction used a threshold of 0.05/64 = 0.00078. The Benjamini-Hochberg procedure controlled the false discovery rate at 5 percent. The generalised Bonferroni method estimated m_eff from the adjacency matrix as described above and used a threshold of 0.05/m_eff. Spacings between consecutive ordered p-values were computed and compared with the expected exponential spacings under the global null. A permutation procedure with 5000 random relabellings estimated the false discovery proportion for each method. The heat equation was solved numerically on the graph Laplacian using a forward Euler scheme with 100 time steps to visualise how regions of significance clustered and spread across the network [8] .

Calibration and validation

The adjacency threshold of 200 km was calibrated by evaluating the mean pairwise correlation of Mann-Kendall statistics across thresholds from 100 to 400 km in 50-km steps. At 200 km, the estimated rho_bar was 0.31, yielding m_eff = 32.4. Validation used a split-sample approach: the first 30 years (1960-1989) served as a calibration set for rho_bar estimation, and the full 62-year record served for trend testing. The rho_bar estimated from the calibration subset (0.29) was within 0.02 of the full-record estimate, and the resulting m_eff (33.7 versus 32.4) changed only one station's classification. The permutation-based false discovery proportion of 4.3 percent at the 5 percent family-wise level confirmed nominal control [9] . The generalised Bonferroni method identified 23 stations with significant trends, roughly midway between the conservative classical Bonferroni ( 14) and the more liberal Benjamini-Hochberg (31). Its estimated false discovery proportion of 4.3 percent was well below the nominal 5 percent family-wise threshold, confirming appropriate control.

Results

Discussion

The generalised Bonferroni method struck a useful balance between the extremes of classical Bonferroni (which controls family-wise error but misses real effects) and Benjamini-Hochberg (which catches more effects but accepts a higher false discovery rate). The key innovation using the station network adjacency matrix to estimate the effective number of independent tests is straightforward to implement and requires no parametric assumptions about the dependency structure beyond the positive correlation condition [3] . The estimated m_eff of 32.4, roughly half the nominal 64, reflects the substantial spatial correlation in outback rainfall driven by large-scale synoptic systems. The predominance of drying trends is consistent with observed declines in cool-season rainfall across southern and central Australia attributed to poleward shifts in the subtropical ridge [10] . The few stations showing wetting trends were concentrated in the far north of the Northern Territory, where tropical influences may be intensifying. The heat equation diffusion model provided a visually informative way to see how clusters of significance emerged and spread across the network, although it served more as an exploratory tool than a formal inference procedure.

A limitation of the method is its reliance on a fixed adjacency threshold. The 200-km choice performed well here but may need adjustment for networks with different spacing. An adaptive threshold based on variogram range estimation could generalise the approach. The Markov chain representation of ordered p-values remains largely diagnostic in this work; formalising it into a sequential testing procedure is a natural direction for future research [11] .

Conclusion

Acknowledgement

Acknowledgements Funding sourcesThis research was funded through departmental resources without external grant support. Institutional supportThe Tucuman Faculty of Mathematical Analysis provided computing infrastructure and library access. Contributions not qualifying for authorshipDr. Lucia Fernandez assisted with the Bureau of Meteorology data acquisition process.

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