Jul 01, 2009

LCGC North America

Volume 27, Issue 7, pg 534–540

When a least-squares linear regression is used to fit experimental data to a linear calibration curve, equal emphasis is given to the variability of data points throughout the curve. However, because the absolute variation (as opposed to %-error) is larger for higher concentrations, the data at the high end of the calibration curve tend to dominate the calculation of the linear regression. This often results in excessive error at the bottom of the curve. One way to compensate for this error and to give a better fit of the experimental data to the calibration curve is to weight the data inversely with the concentration, a process called curve weighting. The weighting of calibration curves often will lower the overall error of the method and, thus, improve the quality of the analytical results. Most LC calibration curves that span several orders of magnitude show increasing error with increasing concentration, whereas the relative error (percent relative standard deviation, %RSD) is reasonably constant. Curve weighting should be evaluated whenever the relative error is fairly constant throughout the calibration curve (5). There also are regulatory reasons why curve weighting should be considered. The FDA's guidelines for validation of bioanalytical methods (6) contains this statement (my italics): "Standard curve fitting is determined by applying thesimplest model that adequately describes the concentration–response relationship using appropriate weighting and statistical tests for goodness of fit." How are "simplest," "adequately," and "appropriate" determined? It seems to allow many interpretations, however, one key point is that weighting should be evaluated, and the evaluation of the impact of curve weighting allows for statistical tests to be applied.
A thorough evaluation of the appropriateness of curve weighting and selection of the weighting factor is best done at the end of method development or during method validation when a sufficiently large data set is available to calculate standard deviations at each calibrator concentration. However, because most LC calibration curves exhibit similar characteristics from the standpoint of weighting, we can use a few shortcuts for the present discussion. The first step is to determine if the standard deviation at the lower limit of the curve is significantly different from that at the upper end. This determination is based upon the The next step is to determine the proper weighting factor for the data. The calculations are based upon a fairly complex equation that can be found in references 5 or 7. For my own satisfaction, I programmed this into an Excel spreadsheet, but by the time I entered all the formulas and debugged them so that I could get the same answers as the textbook, I had spent an entire day. A much easier approach is to use the curve weighting options that are built into most data analysis software packages. These allow you to choose a weighting factor – 1/ You can compare the effectiveness of the various weighting schemes at reducing method error by calculating the sum of the absolute values of the relative error (ΣRE). The weighting factor that gives the smallest ΣRE is the best choice. You will find that the ΣRE will drop quickly as weighting is increased, then stabilize. I suggest using the least amount of weighting that minimizes the error, as is shown in the examples in the following section. This should satisfy the FDA's "simplest model that adequately describes the concentration–response relationship using appropriate weighting" (6). |