A Comparison of Several Methods for Analyzing Censored Data
1 Exposure Assessment Solutions, Inc., Morgantown, West Virginia
2 Department of Mathematics, West Virginia University, Morgantown, West Virginia
* Author to whom correspondence should be addressed. e-mail: phewett_2006_07{at}oesh.com
The purpose of this study was to compare the performance of several methods for statistically analyzing censored datasets [i.e. datasets that contain measurements that are less than the field limit-of-detection (LOD)] when estimating the 95th percentile and the mean of right-skewed occupational exposure data. The methods examined were several variations on the maximum likelihood estimation (MLE) and log-probit regression (LPR) methods, the common substitution methods, several non-parametric (NP) quantile methods for the 95th percentile and the NP Kaplan–Meier (KM) method. Each method was challenged with computer-generated censored datasets for a variety of plausible scenarios where the following factors were allowed to vary randomly within fairly wide ranges: the true geometric standard deviation, the censoring point or LOD and the sample size. This was repeated for both a single-laboratory scenario (i.e. single LOD) and a multiple-laboratory scenario (i.e. three LODs) as well as a single lognormal distribution scenario and a contaminated lognormal distribution scenario. Each method was used to estimate the 95th percentile and mean for the censored datasets (the NP quantile methods estimated only the 95th percentile). For each scenario, the method bias and overall imprecision (as indicated by the root mean square error or rMSE) were calculated for the 95th percentile and mean. No single method was unequivocally superior across all scenarios, although nearly all of the methods excelled in one or more scenarios. Overall, only the MLE- and LPR-based methods performed well across all scenarios, with the robust versions generally showing less bias than the standard versions when challenged with a contaminated lognormal distribution and multiple LODs. All of the MLE- and LPR-based methods were remarkably robust to departures from the lognormal assumption, nearly always having lower rMSE values than the NP methods for the exposure scenarios postulated. In general, the MLE methods tended to have smaller rMSE values than the LPR methods, particularly for the small sample size scenarios. The substitution methods tended to be strongly biased, but in some scenarios had the smaller rMSE values, especially for sample sizes <20. Surprisingly, the various NP methods were not as robust as expected, performing poorly in the contaminated distribution scenarios for both the 95th percentile and the mean. In conclusion, when using the rMSE rather than bias as the preferred comparison metric, the standard MLE method consistently outperformed the so-called robust variations of the MLE-based and LPR-based methods, as well as the various NP methods, for both the 95th percentile and the mean. When estimating the mean, the standard LPR method tended to outperform the robust LPR-based methods. Whenever bias is the main consideration, the robust MLE-based methods should be considered. The KM method, currently hailed by some as the preferred method for estimating the mean when the lognormal distribution assumption is questioned, did not perform well for either the 95th percentile or mean and is not recommended.
Received March 5, 2007; in final form August 17, 2007
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