Tellinghuisen Joel
Department of Chemistry, Vanderbilt University, Box 1668 B, Nashville, TN 37235, United States.
J Chromatogr B Analyt Technol Biomed Life Sci. 2008 Sep 1;872(1-2):162-6. doi: 10.1016/j.jchromb.2008.07.043. Epub 2008 Aug 6.
The quality coefficient (Q) has frequently been used to select weighting formulae in calibration, and especially so in bioanalytical work, where there has been increasing awareness of the importance of data heteroscedasticity in recent years. However, this quantity is statistically flawed and should not be used for this purpose. The quality coefficient is computed from the differences between the apparent and true concentrations of the calibration samples as obtained from the least-squares calibration fit. Q is defined as the sum of either the squares or the absolute values of these differences, taken directly or as percentage (relative) deviations. It is calculated for several different trial weighting formulae, and the lowest Q value is then deemed to identify the best weighting choice. However, these Qs are predisposed to favor data consistent with their definitions-homoscedastic data for tests employing absolute differences, and data having proportional error (constant coefficient of variance) for tests using relative differences--because the Q in each case closely resembles the quantity actually minimized by the least-squares fit of the calibration data. The problem is examined and illustrated through Monte Carlo computations on data having either constant or proportional uncertainty and subjected to both tests. A modified Q based on the results of both the absolute and relative tests is much more reliable than either test alone but is still not recommended as a solution to the weighting problem, as other, statistically sound approaches are available and readily used.
质量系数（Q）在校准中经常被用于选择加权公式，在生物分析工作中尤其如此，近年来人们越来越意识到数据异方差性的重要性。然而，这个量在统计上存在缺陷，不应用于这个目的。质量系数是根据从最小二乘校准拟合得到的校准样品的表观浓度和真实浓度之间的差异计算出来的。Q被定义为这些差异的平方和或绝对值之和，可以直接取这些值，也可以取百分比（相对）偏差。它是针对几种不同的试验加权公式计算的，然后认为最低的Q值能确定最佳的加权选择。然而，这些Q值倾向于支持与其定义一致的数据——采用绝对差异进行测试时支持同方差数据，采用相对差异进行测试时支持具有比例误差（恒定方差系数）的数据——因为在每种情况下，Q值都与校准数据的最小二乘拟合实际最小化的量非常相似。通过对具有恒定或比例不确定性的数据进行蒙特卡罗计算，并对这两种测试进行分析来说明这个问题。基于绝对测试和相对测试结果的修正Q值比单独使用任何一种测试都更可靠，但仍不建议将其作为加权问题的解决方案，因为还有其他统计上合理的方法可供使用且易于采用。
