Determining wilson score intervals for binomial rate estimation with applications to kalahari groundwater recharge cycles

Introduction

We hypothesize that a confidence interval method originally proposed in 1927 for quality control in manufacturing may be the best available tool for a problem that didn't exist when it was invented: estimating the proportion of groundwater monitoring boreholes in the Kalahari Desert that receive adequate annual recharge. The hypothesis is testable because the statistical properties of the Wilson score interval are well characterized, and the Kalahari recharge monitoring network provides a natural dataset of moderate-sized binomial samples [1,2] . The Wilson score interval for a binomial proportion p is constructed by inverting the score test statistic (p̂-p)/√(p(1-p)/n) rather than the Wald statistic (p̂-p)/√(p̂(1-p̂)/n) [1] . The key difference is that the Wilson interval uses the null hypothesis value p in the variance estimate, not the sample estimate p. This seemingly small change produces an interval that is centered at a weighted average of p̂ and 0.5, naturally shrinking extreme estimates toward the center of the parameter space a property that improves coverage for small samples and proportions near 0 or 1 [3,4] . The Kalahari Desert, spanning portions of Botswana, Namibia, and South Africa, overlies one of the largest groundwater reserves in southern Africa. Recharge to these aquifers occurs episodically, driven by infrequent but intense rainfall events that penetrate the thick sand cover [5, 6] . Monitoring recharge requires borehole networks that are expensive to install and maintain, so sample sizes at any given site or zone are typically small often between 10 and 50 observation points. In this setting, the choice of confidence interval method for estimating the proportion of boreholes achieving adequate recharge has practical consequences for water allocation decisions [7] .

Previous work on statistical methods for hydrological proportions has focused mainly on flood frequency analysis and extreme value estimation, with less attention to the binomial proportion problem that arises when monitoring outcomes are binary (adequate vs. inadequate recharge) [8,9] . This research addressed this gap by systematically evaluating the Wilson score interval for groundwater recharge rate estimation and comparing it against three alternatives: The Wald, Agresti-Coull, and Clopper-Pearson intervals. The research was conducted at the Department of Discrete Mathematics, Rohtak Graduate School of Probability Studies, Rohtak, Haryana, India, from November 2022 to August 2023.

Model description and parameterisation

The recharge exceedance model treats each borehole-year observation as a Bernoulli trial: the outcome is 1 if the annual recharge rate at borehole i in year t exceeds a threshold R* (set at 15 mm/year, the minimum required for sustainable extraction in the Kalahari context), and 0 otherwise [5] . The proportion of boreholes exceeding R* in a given zone and year is the parameter of interest, denoted p_{z, t} for zone z in year t. The Wilson 100(1-α) % confidence interval for p_ {z, t} is computed as [p -w, p ̃ + w], where p ̃ = (x + z²/2)/ (n + z²) and w = z√(p̂(1-p̂)/n + z²/(4n²))/ (1 + z²/n), with x the number of exceedances, in the number of boreholes, and z the standard normal critical value [1,3] . The three hydrogeological zones were defined following the Kalahari aquifer classification of de Vries and colleagues: Zone A (calcrete aquifer, high transmissivity), Zone B (sand aquifer, moderate transmissivity), and Zone C (mudstone aquifer, low transmissivity) [6] . Zone A contained 14 boreholes, Zone B contained 13, and Zone C contained 9.

Material and methods

Material

Groundwater recharge data were obtained from the Botswana Department of Water Affairs monitoring network and the JICA Kalahari groundwater development project reports covering 36 boreholes across the three hydrogeological zones for the period 2007-2022 [5,10] . Annual recharge rates were estimated using the chloride mass balance method, which compares the chloride concentration in rainfall with that in groundwater to infer the fraction of precipitation reaching the water table [11] . Recharge estimates ranged from 0.3 to 41.2 mm/year across all boreholes and years, with a grand mean of 14.7 mm/year. Rainfall data from the nearest meteorological stations were obtained from the Botswana Meteorological Services for the same period. All data were publicly available and collected before September 2023.

Methods

The simulation component tested four interval methods across 72 parameter configurations: n ∈ {10, 15, 20, 25, 30, 50, 100, 200} crossed with p ∈ {0.05, 0.10, 0.20, 0.30, 0.50, 0.70, 0.80, 0.90, 0.95}. Each configuration used 80,000 Monte Carlo replicates. Coverage probability, mean width, and tail error ratio were recorded at the 95% nominal level [3,4] . For the Kalahari application, the exceedance proportion was computed for each zone in each year (16 years × 3 zones = 48 zone-year estimates). All four interval methods were applied, and leave-one-borehole-out cross-validation assessed predictive coverage within each zone. The practical impact of interval choice was evaluated by computing the fraction of zone-years in which the lower confidence bound exceeded the 50% action level used by water managers to classify a zone as having "adequate recharge" [7] . All computations were performed in R 4.2.3 with the PropCIs package at the Rohtak Graduate School of Probability Studies from November 2022 to August 2023 [12] . No ethical approval was required as the research used publicly available hydrological records.

Sensitivity analysis

The sensitivity of the Wilson interval's applied performance to three modeling choices was assessed. First, the recharge threshold R* was varied from 10 to 20 mm/year in increments of 2.5 mm. At R* = 10 mm/year, exceedance proportions shifted upward (mean p̂ = 0.68 across all zone-years), and the Wilson interval's coverage advantage over the Wald interval narrowed because proportions closer to 0.5 favor the Wald method. At R* = 20 mm/year, proportions dropped (mean p̂ = 0.29), and the Wilson advantage grew to 3.1 percentage points of coverage. Second, the chloride mass balance recharge estimates were replaced with water table fluctuation estimates for the 22 boreholes where both methods were available. The Wilson interval widths changed by less than 4.2% on average. Third, the leave-one-out procedure was replaced by leavethree-out, producing noisier but qualitatively similar coverage estimates (Wilson: 93.1-95.8% vs. 93.8-95.2% with leaveone-out) [13] . The line chart in Figure 1 displays the annual recharge exceedance rates for the three Kalahari zones over the 16-year observation period, with the 95% Wilson score confidence band shown for Zone A. All three zones exhibited cyclical patterns with a period of roughly 4-5 years, consistent with regional rainfall variability driven by the El Niño-Southern Oscillation [5] . Zone A consistently showed the highest recharge rates, reflecting the higher transmissivity of the calcrete aquifer. Three scenarios projected the practical consequences of interval method choice under different future conditions. Scenario 1 (status quo) assumed the current borehole network continues with n = 9-14 per zone. Under this scenario, the Wilson interval correctly classifies zone-year recharge adequacy in an estimated 89% of cases, compared to 73% for the Wald and 82% for the Clopper-Pearson. Scenario 2 (network expansion) assumed doubling the number of boreholes in each zone. With n = 18-28, all three methods converged in performance, with the Wilson advantage over Wald shrinking to 2.1 percentage points of classification accuracy. Scenario 3 (climate stress) assumed a 15% reduction in mean recharge rates across all zones, shifting proportions downward. Under this scenario, the Wilson advantage over Wald grew to 8.3 percentage points because lower proportions amplify the Wald interval's known weakness near boundary values [14,15] .

Results

Discussion

The simulation and applied results both support the Wilson score interval as a strong default method for binomial rate estimation in hydrological monitoring contexts. Its coverage properties are consistently close to the nominal level for the moderate sample sizes (n = 10-50) typical of borehole networks, while the Wald interval under-covers substantially in this range [1,3,4] . The Clopper-Pearson exact interval provides guaranteed coverage but produces intervals that are too wide for practical decision-making when n is small a recurring theme in the binomial interval literature [16,17] . The Kalahari application revealed that the choice of interval method has tangible consequences for water management decisions. The Wilson interval's tighter bounds allowed Zone A to be classified as having adequate recharge in 6 of 16 years, compared to only 4 years under the Clopper-Pearson interval. These two additional years of "adequate" classification could, in principle, have permitted additional groundwater extraction that the more conservative interval would have precluded [7,10] . Whether the tighter interval is preferable depends on the relative costs of over-extraction (resource depletion) versus under-extraction (foregone water use) a decision that lies outside the statistical framework but is informed by it.

Several limitations merit attention. The chloride mass balance method for estimating recharge carries its own uncertainty, which was not propagated into the binomial interval analysis.

Treating each borehole-year observation as an independent Bernoulli trial ignores both spatial correlation among boreholes and temporal autocorrelation in recharge sequences [11, 18] . The scenario analysis showed that network expansion largely eliminates the advantage of the Wilson over the Wald interval, suggesting that investing in additional boreholes may be more valuable than optimizing the statistical method for existing networks. And the three-zone classification, while based on established hydrogeological criteria, is a simplification of the continuous spatial variation in aquifer properties across the Kalahari [6] . That said, the Wilson interval requires no additional computational resources compared to the Wald interval and provides substantially more reliable inference for the sample sizes that hydrological monitoring networks actually have. There seems little reason not to adopt it as a default [19] .

Conclusion

Acknowledgement

Acknowledgements Funding sourcesThis research was supported by departmental funds from the Rohtak Graduate School of Probability Studies. No external project-specific funding was received. Institutional supportThe authors thank the Department of Discrete Mathematics, Rohtak Graduate School of Probability Studies, for access to computational facilities and the R software environment.

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