Annual sampling frequency and false-negative probability in water-quality threshold-exceedance detection: an exact conditional analysis of the Buranhém River, Bahia, Brazil

Abstract Routine monitoring captures conditions present during sampling, so sparse schedules may fail to encounter threshold exceedances recorded at other times in the same year. Using 106 Class 2 freshwater campaigns from five Buranhém River sites during 2015–2024, the analysis examined seven water-quality parameters and a composite any-exceedance outcome within site-years. Exact finite-population probabilities were calculated for each positive annual reference set to determine whether hypothetical schedules of one to four campaigns would include at least one recorded exceedance. Fixed common support defined the primary one- to three-campaign comparison, requiring at least three classifiable results for the target outcome; Monte Carlo simulation and independent enumeration served only as computational verification. Across 20 positive common-support site-years, mean conditional detection probabilities for the composite outcome were 58.3%, 87.9%, and 95.8% for one, two, and three campaigns, corresponding to conditional false-negative probabilities of 41.7%, 12.1%, and 4.2%. Annual simple random sampling yielded 53.9%, 79.5%, and 94.3%, indicating that temporal allocation affected the magnitude of the estimates but not the observed increase with sampling frequency. Twelve of the 20 site-years contained exactly three classifiable campaigns, so finite-population exhaustion partly influenced the three-campaign estimate; restricting the analysis to site-years with at least four campaigns yielded 89.6% at three campaigns. These results describe subsampling of the observed historical record and neither estimate unobserved events nor establish a universal minimum monitoring frequency.