Estimation of Distributional Parameters for Censored Trace Level Water Quality Data: 1. Estimation Techniques

TLDR

Many metal and organic contaminant studies in ambient waters face a high proportion of concentrations below laboratory limits of detection. The authors compared eight estimation techniques for censored data, using Monte Carlo simulations of small samples from diverse parent distributions, and devised criteria based on uncensored observations to select the optimal method for each dataset. Across all simulation conditions, the log‑probability regression method yielded the lowest errors for mean and standard deviation, while lognormal maximum likelihood performed best for median and interquartile range, and pre‑classifying data by probable parent distribution further improved accuracy.

Abstract

A recurring difficulty encountered in investigations of many metals and organic contaminants in ambient waters is that a substantial portion of water sample concentrations are below limits of detection established by analytical laboratories. Several methods were evaluated for estimating distributional parameters for such censored data sets using only uncensored observations. Their reliabilities were evaluated by a Monte Carlo experiment in which small samples were generated from a wide range of parent distributions and censored at varying levels. Eight methods were used to estimate the mean, standard deviation, median, and interquartile range. Criteria were developed, based on the distribution of uncensored observations, for determining the best performing parameter estimation method for any particular data set. The most robust method for minimizing error in censored‐sample estimates of the four distributional parameters over all simulation conditions was the log‐probability regression method. With this method, censored observations are assumed to follow the zero‐to‐censoring level portion of a lognormal distribution obtained by a least squares regression between logarithms of uncensored concentration observations and their z scores. When method performance was separately evaluated for each distributional parameter over all simulation conditions, the log‐probability regression method still had the smallest errors for the mean and standard deviation, but the lognormal maximum likelihood method had the smallest errors for the median and interquartile range. When data sets were classified prior to parameter estimation into groups reflecting their probable parent distributions, the ranking of estimation methods was similar, but the accuracy of error estimates was markedly improved over those without classification.