Helsel & Cohn (1988)Helsel, D. R., & Cohn, T. A. (1988). Estimation of descriptive statistics for multiply censored water quality data. Water Resources Research, 24(12), 1997-2004. http://dx.doi.org/10.1029/WR024i012p01997. http://dx.doi.org/10.1029/WR024i012p0199...
|
ZDL |
25 |
500 |
Log-normal |
60 |
RMSE |
Mean |
MLE: Significant bias in the estimates of means and standard deviations |
HDL |
Mixture of two log-normals |
Bias |
Median |
DL |
Delta |
|
Standard deviation |
MLE |
|
|
Interquartile ranges |
ROS |
|
|
|
Kroll & Stedinger (1996)Kroll, C. N., & Stedinger, J. R. (1996). Estimation of moments and quantiles using censored data. Water Resources Research, 32(4), 1005-1012. http://dx.doi.org/10.1029/95WR03294. http://dx.doi.org/10.1029/95WR03294...
|
MLE |
10 |
5000 |
Log-normal |
20 |
RMSE |
Percentile 10,90 |
MLE: Suitable for estimating quantiles and interquartile ranges in highly censored data; |
ROS |
25 |
Mixture of two log-normals |
60 |
Mean |
ROS: Suitable for estimating means and standard deviations in medium to long time series with short to medium censoring |
|
50 |
Gamma |
80 |
Standard Deviation |
|
|
|
Delta |
|
Interquartile |
|
|
|
|
|
Ranges |
|
She (1997)She, N. (1997). Analyzing censored water quality data using a nonparametric approach. Journal of the American Water Resources Association, 33, 615-624. http://dx.doi.org/10.1111/j.1752-1688.1997.tb03536.x. http://dx.doi.org/10.1111/j.1752-1688.19...
|
HDL |
21 |
1000 |
Log-normal |
Three randomly between |
Bias |
Mean |
HDL: Best for CV = 1.00 and 2.00 |
KM |
Gamma |
10 and 80 |
Standard error |
Standard Deviation |
KM: Second-best technique, similar to MLE |
MLE |
|
|
|
|
MLE: Best for CV = 0.25, 0.50. |
ROS |
|
|
|
|
Means: Worse estimates for higher CV values |
Shunway et al. (2002)Shunway, R., Azari, R., & Kayhanian, M. (2002). Statistical approaches to estimating mean water quality concentrations with detection limits. Environmental Science & Technology, 36, 3345-3353. http://dx.doi.org/10.1021/es0111129. http://dx.doi.org/10.1021/es0111129...
|
MLE |
20 |
500 |
Log-normal |
50 |
Bias |
Mean |
ROS: No bias for the log-normal distribution, but larger standard error for highly asymmetrical series |
ROS |
50 |
Gamma |
80 |
Confidence interval |
Variance |
MLE: Recommended to use a bias corrector |
Hewett & Ganser (2007)Hewett, P., & Ganser, G. (2007). A comparison of several methods for analyzing censored data. The Annals of Occupational Hygiene, 51(7), 611-632. http://dx.doi.org/10.1093/annhyg/mem045. http://dx.doi.org/10.1093/annhyg/mem045...
|
HDL |
mai/19 |
100 |
Log-normal |
jan/50 |
Bias |
Mean |
MLE: Recommended for all scenarios |
LR2 |
20-100 |
Contaminated log-normal |
50-80 |
RMSE |
95th quantile |
ROS: Recommended for estimating averages |
DL |
|
|
|
|
|
KM: Presented poor estimates |
KM |
|
|
|
|
|
LD: Overestimated the mean and underestimated the 95th percentile |
MLE |
|
|
|
|
|
|
ROS |
|
|
|
|
|
|
Authors
|
Methods
|
Elements
|
Random Samples
|
Distribution
|
Censoring Percentage
|
Accuracy Measure |
Evaluated Stats
|
Conclusions Related to the Log-normal Distribution
|
Antweiller & Taylor (2008)Antweiller, R. C., & Taylor, H. E. (2008). Evaluation of statistical treatments of left-censored environmental data using coincident uncensored data sets: I. Summary statistics. Environmental Science & Technology, 42(10), 3732-3738. http://dx.doi.org/10.1021/es071301c. http://dx.doi.org/10.1021/es071301c...
|
ZDL |
34-841 |
44 |
No specific distributions |
Randomly between |
Bias |
Mean |
KM: Achieved the best results for censoring up to 70%, except when estimating the median |
|
HDL |
14 and 95 |
Percentile |
ROS and HDL: Yielded reasonable results |
|
DL |
|
25, 50 and 75 |
No method yielded suitable results for censoring greater than 70% |
|
KM |
|
Standard deviation |
|
|
MLE |
|
Interquartile range |
|
|
ROS |
|
|
|
Niemann (2016)Niemann, J. (2016). Statistical modelling of environmental data with non-detects. Retrieved in 2023, August 20, from https://www.causeweb.org/usproc/sites/default/files/usresp/2016/december/jennifer-niemann.pdf. https://www.causeweb.org/usproc/sites/de...
|
ZDL |
50 |
10000 |
Log-normal |
5 to 60 |
Bias |
Mean |
HDL, LR2: Good for ratings up to 30% |
|
HDL |
RMSE |
MLE: Exhibited significant bias and high RMSE |
|
LR2 |
Confidence interval |
HDL: Stood out for censorship rates exceeding 50%, providing unbiased estimates and low RMSE |
|
DL |
|
|
|
KM |
|
|
|
MLE |
|
|
Tekindal et al. (2017)Tekindal, M. A., Erdogan, B. D., & Yavuz, Y. (2017). Evaluating left-censored data through substitution, parametric, semiparametric, and nonparametric methods: a simulation study. Interdisciplinary Sciences, Computational Life Sciences, 9(2), 153-172. http://dx.doi.org/10.1007/s12539-015-0132-9. http://dx.doi.org/10.1007/s12539-015-013...
|
LR2 |
20 |
10000 |
Log-normal |
5 |
Bias |
Mean |
ROS: Recommended for estimating mean values; |
|
DL |
80 |
Exponential |
25 |
Median |
LR2: Exhibited less bias when estimating medians |
|
KM |
140 |
Weibull |
45 |
Standard deviation |
KM, DL: Demonstrated similar performance, with the overestimation of means and the underestimation of standard deviations |
|
MLE |
200 |
|
65 |
|
MLE: Worst scenario |
|
ROS |
260 |
|
|
|
|
Canales et al. (2018)Canales, R. A., Wilson, A. M., Pearce-Walker, J. I., Verhougstraete, M. P., & Reynolds, K. A. (2018). Methods for handling left-censored data in quantitative microbial risk assessment. Applied and Environmental Biology, 84(20), 1-10. http://dx.doi.org/10.1128/AEM.01203-18. http://dx.doi.org/10.1128/AEM.01203-18...
|
LR2 |
100 |
10000 |
Log-normal |
< 10 |
Bias |
Mean |
ROS: Performed better in series with a high percentage of censored data |
|
DL |
35 |
RMSE |
MLE: Showed poor performance, with a high RMSE, especially in series with pronounced asymmetry |
|
KM |
65 |
|
|
|
MLE |
90 |
|
|
|
ROS |
97 |
|
|
George et al. (2021)George, B. G., Gains-German, L., Broms, K., Black, K., Furman, M., Hays, M. D., Thomas, K. W., & Simmons, J. E. (2021). Censoring trace-level environmental data: statistical analysis considerations to limit bias. Environmental Science & Technology, 55, 3786-3795. http://dx.doi.org/10.1021/acs.est.0c02256. http://dx.doi.org/10.1021/acs.est.0c0225...
|
HDL |
20 |
1000 |
Log-normal |
30 |
|
Mean |
KM: Overestimated means and underestimated standard deviations, performing less poorly in highly skewed distributions |
|
MLE |
50 |
Moderately and highly Asymmetrical |
50 |
Standard deviation |
ROS: Demonstrated the best performance |
|
ROS |
|
|
80 |
|
HDL: Provided reasonable estimates for means but performed poorly for standard deviations |
|
KM |
|
|
|
|
MLE: Performed poorly in asymmetrical series |