Statistical approaches to estimating mean water quality concentrations with detection limits

Area of Science:

• Environmental Science
• Statistics
• Water Quality Analysis

Background:

• Accurate estimation of toxic pollutant concentrations in water is crucial for environmental monitoring and public health.
• Small sample sizes, non-normal distributions, and the presence of nondetected values (censored data) pose significant statistical challenges.
• Existing methods like Maximum Likelihood Estimation (MLE) and Regression on Order Statistics (ROS) can suffer from transformation bias, especially under severe censoring.

Purpose of the Study:

• To review and enhance statistical methodologies for estimating mean pollutant concentrations in water, particularly for challenging datasets.
• To evaluate the performance of exact Maximum Likelihood Estimators (MLE) combined with the Box-Cox transformation and the Quenouille-Tukey Jackknife for bias reduction and variance estimation.
• To compare the effectiveness of MLE and ROS under different distributional assumptions (log-normal, gamma) and censoring levels.

Main Methods:

• Utilized exact maximum likelihood estimators with the Box-Cox transformation.
• Applied the Quenouille-Tukey Jackknife for bias reduction and variance estimation.
• Employed the Expectation-Maximization (EM) algorithm to derive exact MLEs and estimate nondetected values.
• Conducted simulations to assess performance under log-normal and gamma distributions with varying censoring.

Main Results:

• The jackknife method effectively reduces bias and improves variance estimation for both MLE and ROS, especially for log-normal and gamma distributions.
• Bias corrections in existing MLE literature were found to be counterproductive under severe censoring.
• Regression on Order Statistics (ROS) demonstrated unbiasedness and smaller variance than MLE for log-normal distributions, showing robustness.
• Maximum Likelihood Estimation (MLE) performed better for gamma distributions.

Conclusions:

• Both exact MLE and ROS procedures are valuable statistical tools for water quality data analysis, with performance varying by experimental conditions and underlying distributions.
• The jackknife is essential for robust bias reduction and accurate variance estimation in these methods.
• The choice between MLE and ROS depends on the specific data characteristics, including the assumed distribution and degree of censoring.