Censored Trace Level Water Quality Data - Estimation of Distributional Parameters

Aug 28, 2026

Water quality datasets frequently contain observations reported as less than the detection limit, creating censored data that complicate estimation of key distributional parameters. This study verifies earlier simulation findings by applying eight estimation methods to actual water-quality measurements, identifying log-probability regression (LR) as best overall for mean and standard deviation, while a lognormal maximum likelihood approach (LM) performs best for median and interquartile range. It further develops confidence-bound estimation for censored-data parameters and assesses how accurately methods recover uncensored sample statistics from censored records.

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Introduction

Verification of Previous Simulation Study

Approach

# INTRODUCTION Water quality data often include observations measuredonly as less than the detection limit,resulting in censored datasets.Eight methods for estimating distributional parametersfor censored water quality data were evaluated by Gilliom andHelsel [this issue].Results of extensive Monte Carlo simula-tions,in which large numbers of small samples were generatedfrom 16 different parent distributions and censored to varyingdegrees,indicated that a log-probability regression method(LR)was the best method for estimating the mean and stan-dard deviation of censored data and that a lognormal maxi-mum likelihood method (LM)was best for estimating themedian and interquartile range.That study,hereafter calledthe simulation study,also showed that censored data setscould be effectively classified using a sample statistic called therelative quartile range(rqr),which is the interquartile range ofuncensored observations divided by the detection limit.Classi-fication of simulation data sets according to rqr indicated theprobable underlying distribution,and resulted in improvedestimates of the precision of distributional parameters as com-;pared to unclassified data sets. The purposes of this study are to(1)verify the findings fromthe previous simulation study by evaluating the same parame-ter estimation methods using actual water-quality data;(2)describe an approach for estimating confidence boundsaround parameter estimates made from censored water quali-ty data;and (3)evaluate how well the estimation methodscalculate uncensored sample statistics from censored data setsand compare their errors to those for estimating populationparameters. # VERIFICATTON OF PREVIOus StMULATION STUDY Evaluations of parameter estimation methods in the pre-vious simulation study are verified by applying the same typeof analysis to actual water quality data.The best performingparameter estimation methods for actual water quality dataare compared to the simulation study results.The rqr classifi- This paper is not subject to U.S.copyright.Published in 1986 bythe American Geophysical Union. Paper number SW4031. # Estimation of Distributional Parameters forCensored Trace Level Water Quality Data 2.Verification and Applications ·*, DENNIS R.HELSEL AND RoBERT J.GILLIOM U.S.Geological Suruey,Reston,Virginia Estimates of distributional paramelers(mean,standard deviation median,interquartile range)areoften desired for data sets containing censored observations.Eight methods for estimating these parame-ters have been evaluated by R.J.Gilliom and D.R.Helsel (this issue)using Monte Carlo simulations.Toverify those findings,the same methods are now applied to actual water quality data.The best method(lowest root-mean-squared error (rmse))over all parameters,sample sizes,and censoring levels is logprobability regression (LR),the method found best in the Monte Carlo simulations.Best methods forestimating moment or percentile parameters separately are also identical to the simulations.Reliability ofthese estimates can_be expressed as confidence intervals using nse and bias values taken from thesimulation results.Finally,a new simulation study shows that best methods for estimating uncensoredsample statistics from censored data sets are identical to those for estimating population paramelers.Thus this study and the companion study by Gilliom and Helsel form the basis for making the bestpossible cstimates of either population parameters or sample statisties from censored water quality dala,and for assessments of their reliability. cation system developed in the simulation study is tested bycomparing method performance for actual and simulated datawithin each rqr class.and by evaluating the ability of rqrclassification to separate water quality data sets having differ-ent root mean squared errors(rmses)of parameter estimates. # Approach Uncensored data sets with more than 50 observations forsuspended sediment.total phosphorus,total Kjeldahl ni-tr

FAQs

Why are water quality datasets considered “censored” in this study?

Because some observations are recorded only as less than the detection limit, producing censored data rather than fully observed values.

Which estimation method performs best for mean and standard deviation of censored data?

Log-probability regression (LR) yields the lowest overall root-mean-squared error across parameters, sample sizes, and censoring levels.

How does the study plan to assess reliability of parameter estimates?

Reliability is expressed using confidence intervals derived from simulation-based NSE and bias values, and confidence bounds are further addressed as part of the study’s objectives.

Aug 28, 2026

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