George et al. have recently discussed the problem of dealing with data sets which are missing values because some of the measurements are below the detection limit of the analytical method. (1) These are commonly called left-censored data sets, and the problem is what to do with the missing data if one is calculating the central tendency of the data set. In an elegant introduction, George et al. discuss various strategies that have been used to solve this problem over the years, and they discuss the pros and cons of each. For example, just ignoring the missing values and calculating the arithmetic mean of the remining values clearly leads to a value that is biased high. Other suggestions based on the substitution of a constant value, for example, one-half of the detection limit, for the missing values, are intellectually suspect because, as Helsel has said, one is “fabricating data”. (2) George et al. have addressed this issue by investigating 13 different approaches, some simple and some statistically sophisticated. They have applied these approaches to a “case study” consisting of 47 measurements of dibenzo[a,h]anthracene emissions from cookstoves. These data are shown here in Table 1. Note that there are 26 nondetects and 21 detects. The results of the 13 strategies studied by George et al. for dealing with this particular highly left-censored data set show means (in ng/μL) ranging from 0.343 ± 0.665 to 0.742 ± 0.684 (see George et al. Table 1). None of these 13 means are statistically distinguishable from one another, which suggests that it makes no difference what approach one uses to “correct” for this problem. Another approach that is often used with left censored data (an approach that George et al. did not investigate) is to calculate the median of the detected values. In fact, I have shown that the biases associated with medians are less than those associated with geometric means for data sets with <40% nondetects. (3) For the data set shown in Table 1, the median of the detected values is 0.60 ng/μL, which is about in the middle of the range of the 13 means reported in Table 1 by George et al. Of course, the median cannot be reported with an error limit, and this makes its use problematic in some cases. Another approach is possible if the data set is sufficiently numerous to reconstruct the probability distribution function. (4) In earlier work, I demonstrated that is was straightforward to fit a log-normal distribution to censored data and thus to exactly determine the true geometric mean and its error. This approach is demonstrated in Figure 1, which shows the fitted log-normal distribution (green line) for a PCB-128 data set in which about one-third of the measurement were below the detection limit. I suspect that George et al. did not study this method because of the paucity of good measurements in their data set. Figure 1. Histogram of PCB-128 concentrations measured in the atmosphere near the shores of the Great Lakes. Note the logarithmic concentration scale. The red line is the log-normal distribution fitted to the data without compensating for the nondetects; the mean of this distribution is 0.0163 pg/m³. The green line is the log-normal distribution fitted to the data compensating for the nondetects; the mean of this distribution is 0.0092 pg/m³. In this case, 35% of the 582 measurements were nondetects, and this is shown as the space between the red and green lines. This plot is modified from Hites. (4) Lastly, I suggest that when one has a data set with more than half of the measurements below the detection limit (26 out of 47 in Table 1), there is not much that can be done about it after the fact. When designing a measurement campaign, it is important to select analytical methods with sufficient sensitivity to prevent excessive nondetects. This may require a preliminary study to help select the measurement technology and the optimum sample size. In many cases, it is possible to increase the method detection limit by simply increasing the sample size. For example, instead of sampling 100 m³ of air. it may be possible to sample 1000 m³, which boosts the detection limit by a factor of 10. Replication helps, but remember that doubling the number of replicates only decreases the error by . Sometimes, the method detection limit is relatively high because of chemical noise (interferents) in the samples, and in these cases, a more specific analytical method may be called for (for example, LC/MS/MS instead of GC/MS). In the end, good statistics cannot compensate for poor analytical chemistry. This article references 4 other publications.

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评论“审查痕量级环境数据：限制偏差的统计分析考虑”

乔治等人。最近讨论了处理缺失值的数据集的问题，因为某些测量值低于分析方法的检测限。(1) 这些通常称为左删失数据集，问题是如果计算数据集的集中趋势，如何处理缺失的数据。在优雅的介绍中，乔治等人。讨论多年来用于解决此问题的各种策略，并讨论了每种策略的优缺点。例如，仅忽略缺失值并计算重新挖掘值的算术平均值显然会导致偏高的值。其他基于替换常数值的建议，例如，检测限的二分之一，作为缺失值，在智力上是可疑的，因为，正如 Helsel 所说，一种是“制造数据”。(2) 乔治等人。通过研究 13 种不同的方法来解决这个问题，有些简单，有些在统计上很复杂。他们已将这些方法应用于一项“案例研究”，该研究由 47 项二苯并[啊]来自炉灶的蒽排放。这些数据显示在表 1 中。请注意，有 26 个未检测到，21 个检测到。George 等人研究的 13 种策略的结果。处理这个特定的高度左删失数据集的平均值（以 ng/μL 为单位）范围从 0.343 ± 0.665 到 0.742 ± 0.684（参见 George 等人，表 1）。这 13 种均值中没有一种在统计上可区分，这表明人们使用什么方法来“纠正”这个问题没有区别。另一种常用于左删失数据的方法（George 等人未研究的方法）是计算检测值的中值。事实上，我已经证明，对于小于 40% 的未检测数据集，与中位数相关的偏差小于与几何平均值相关的偏差。(3) 对于表 1 所示的数据集，检测值的中位数为 0.60 ng/μL，大约处于 George 等人在表 1 中报道的 13 个平均值范围的中间。当然，中位数不能用错误限制来报告，这使得它在某些情况下的使用有问题。如果数据集足够多以重建概率分布函数，则另一种方法是可能的。(4) 在早期的工作中，我证明了将对数正态分布拟合到截尾数据是很简单的，从而准确地确定真正的几何平均值及其误差。图 1 展示了这种方法，其中显示了 PCB-128 数据集的拟合对数正态分布（绿线），其中大约三分之一的测量值低于检测限。我怀疑乔治等人。没有研究这种方法，因为他们的数据集中缺乏良好的测量。图 1. 在五大湖沿岸附近大气中测得的 PCB-128 浓度直方图。注意对数浓度标度。红线是拟合数据的对数正态分布，没有补偿未检测到；该分布的平均值为 0.0163 pg/m³ . 绿线是拟合数据的对数正态分布来补偿未检测到；该分布的平均值为 0.0092 pg/m ³. 在这种情况下，582 次测量中有 35% 未检测到，这显示为红线和绿线之间的空间。该图是从 Hites 修改而来的。(4) 最后，我建议当一个数据集有一半以上的测量值低于检测限（表 1 中的 47 个中的 26 个）时，事后对此无能为力。在设计测量活动时，重要的是选择具有足够灵敏度的分析方法以防止过多的未检出。这可能需要进行初步研究以帮助选择测量技术和最佳样本量。在许多情况下，可以通过简单地增加样本量来提高方法检测限。例如，而不是采样 100 m ³的空气。可能采样 1000 m ³，这将检测限提高了 10 倍。复制有帮助，但请记住，将重复次数加倍只会将误差减少。有时，由于样品中存在化学噪声（干扰物），方法检测限相对较高，在这些情况下，可能需要更具体的分析方法（例如，LC/MS/MS 代替 GC/MS）。归根结底，良好的统计数据无法弥补糟糕的分析化学。本文引用了 4 篇其他出版物。