PubMed · 10130009
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N Kaufman. 1993. Managed care and structural change.. https://pubmed.ncbi.nlm.nih.gov/10130009/
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We review statistical methodology for estimating mean concentrations of potentially toxic pollutants in water, for small samples that are not normally distributed and often contain substantial numbers of nondetects, i.e. samples that are only known to be below some set of fixed thresholds. Maximum likelihood estimation (MLE) and regression on order statistics (ROS) are two main approaches that dominate the literature, with transformation bias under non-normality that increases with the severity of censoring being the main problem. We consider exact maximum likelihood estimators in conjunction with the Box-Cox transformation and propose the Quenouille-Tukey Jackknife as a method for bias reduction and variance estimation. Exact maximum likelihood estimators resulting from the expectation-maximization (EM) algorithm are exhibited in a simple heuristic form that also provides estimated values for the nondetects as subsidiary outputs. We show in simulationsthatthetwo main approaches perform well for the log-normal and gamma distributions as long as the jackknife is employed to reduce bias. Bias corrections to MLE used in the literature are shown to correct in the wrong direction under severe censoring. The jackknife is also used for estimating the variance of the both the MLE and ROS estimators. Robustness is improved by searching a class of power transformations (Box-Cox) for the best approximating normal distribution. We conclude that both the exact MLE and ROS procedures can be useful under varying experimental conditions. Limited simulations indicate that the ROS procedure is unbiased and has a smaller variance than the MLE under the log-normal distribution and is robust. The MLE performed better in simulations involving the gamma as the underlying distribution. We also compare the estimators for the mean and variance that one obtains from typical sets of water quality data, analyzing for copper, alumnium, arsenic, chromium, nickel, and lead.
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