Weighting for calibration, 1/x^1/30 ?

[dresdentl]

At low concentration often weighting as 1/x or 1/x^2 are used.

In my case, I get better results at the 3 lowest concetration, if I use following weighting: 1/x^1/30.

Is it mathematically corect to use such a weighting (1/x^1/30)?

Nevertheless, I get a good linarity from Level 1 to 5 and the accuracy for the lowest concetration is better.

But at the highest concentrations the results (accuracy) are worse than without weighting.

[Uwe Neue]

Weighting factors should have a reason. Random weighting factors make no sense.

[dresdentl]

To my knowledge, the use of weightings for calibration has always a reason. It has nothing to do with random. Very often, at low concentration the curve differs from that at higher concentration. For this reason, weightings are often recommended at low concentrations. This is the only reason for using weightings. Therefore, the use of weightings has nothing to do with random.

Normally, weightings like 1/x^2 deliver better results (see publications). In my case, I used a rather uncommon weighting.

On the other hand, the procedure of analyses, the way of doing your measurements, has something to do with random. The measured data are always distributed radomly below and above average (cv 15%). To assess the analyses method correctly, you need are larger set of data.

If the measured data are linear, you get exactly a straight line with this weighting and it should be mathematically correct therefore.

[dresdentl]

If the measured data are linear you get always a straight line and the same results, whatever kind of weighting you use or not. Based on a larger set of data, you should choose this weighting (or unweighting), which delivers the best results (best accuracy).


If you have a good linearity over the whole concentration range, you can use therefore weighting or not weighting and you get the same result. Weighting or not weighting doesn't matter, in this case. But in reality, the calibration curve isn't completely linear over the whole concentration range and then weighting does matter.

During the validation procedure it should be tested which weighting (or unweighted) delivers the best results.

[HW Mueller]

If the non-linearity is due to faulty chromatography (for instance, carryover) you should correct the chromatography, not fudge with math. What you did might not be random, but it might be equivocal, especially if you don´t bother to find out the reason for the non-linearity.

[dresdentl]

The linearity is very good R^2 > 0,999.

Nevertheless, near LLOQ you can get false high values.

An easier and better solution can be, to adjust the calbration curve to the expected concentrations. A low concentration curve for low concentration and a high concentration curve for high concentration. This is the procedure recomended by validation guidlines.

The number of measured data will also play a role. If you have a large collection of data, it should get more reliable. If you only have a small set of data, then random play a role.

My intention was more to generally discuss the appliance of weightings for calibration curves. I know that it is better to use unweighted curves.

[H.Thomas]

The linearity is very good R^2 > 0,999.

r^2 is not a measure of linearity.

I know that it is better to use unweighted curves.

I also tend to disagree on this one. The statistical reason for weighting is the difference in absolute variances at the lower and at the higher end of the calibration. Often the relative variances are more or less constant, and for a calibration over two orders of magnitude, the variance of the highest standard is also roughly 2 orders of magnitue higher than that of the lowest.

If you do not weight you data, you will get a very high error for concentrations in the low range.

It is not just a matter of "tweaking" the calibration, it's statistically sound.

[Peter Apps]

How did you arrive at that particular weighting factor ? If it was by trying a range of values and selecting what gave you the results that you wanted then you are on the slippery slope of post hoc data manipulation, and at the bottom lies cooking the results. There are some very long threads in the archives on this subject.

Peter

[dresdentl]

According to some publications weightings like 1/x, 1/x^2, 1/x^1/2 are used.
Sometimes 1/x^2 is recomeded for low concentrations. In analogy to this, I tried to use weightings with higher powers. It looks like the points were pushed more together. I tried to study different weightings. For the lowest point, it was the most suitable one. 1/x^2 was not suitable vor the lowest value.

For analyses, I will use the strategy to use a low concentration curve and a high concentration curve. I think this is better. Only the lowest value gave a better accuracy with this weighting, but I think with some additional data, it will get better and I can avoid theses weightings completely.

More theoretically, I would be interested in these weighting method.

[Uwe Neue]

Yes, weightings for calibration curves have a reason. In the case that you have described, I suspect a low-level impurity to overlap with your peak of interest. To ignore such a possible case is more than a slippery slope!

[H.Thomas]

More theoretically, I would be interested in these weighting method.

Shaun Burke: Regression and Calibration
http://chromatographyonline.findanalyti ... rticle.pdf

Weighted Least-Squares Regression in Practice: Selection of the Weighting Exponent
http://chromatographyonline.findanalyti ... rticle.pdf


Chapter 5.10 Weighted regression lines, in:
Miller, J. N.: Statistics and chemometrics for analytical chemistry

Agustin G. Asuero and Gustavo Gonzalez: Fitting Straight Lines with Replicated Observations by Linear Regression. III. Weighting Data
Critical Reviews in Analytical Chemistry, 37:143–172, 2007

[dresdentl]

From more theoretic point of view, weightings like 1/x^2 give better results at low concentrations because the calibration curve is reversed. On the other hand, at high concentrations the results will be worse.

But in my case, I noticed that I get the best results with a low and high concentration curve (unweighted). Even easier is the use of a automatic integration software. In this case, I don't need two calibration curves and I get the right results at all concentrations. But I noticed that the automatic integration software concentration calulation is more complex than by calculating with one linear equation. Does someone know how the software on the HPLC instrument mathematically works?

[Don_Hilton]

You talk about the curve being "reversed" at low concentrations - A curve may show a slight hook at the low end because of losses due to activity somewhere in the system. And the solution that people use for that hook is the curve order for regression - not the weighting. Curve order makes calibration curve curve (or not curve). Weighing allows for the changes in relative standard deviation at varying conentrations.

[carls]

Most chromatography data systems use the same equations we are all familiar with. One option that many offer is a point to point or bracketed calibration where the 2 calibration points closest to the sample (typically on each side) are used to interpolate the value for the sample.

You'll need to be more specific about the type of fitting that is used if you want equations

[dresdentl]

Concerning the hook, do they use nonlinear regression to solve this problem?
Is it also possible that the hook is caused by impurities or signal noise? Then the hook should go in the opposite direction, as if it is caused by substance loss.

In my case, I use following concentration range: From 0,039 to 10 µg/ml (9 calibration points). Maybe this is a too large range for only one linear calibration curve and then the best solution is to use a low and a high calibration curve, as I already mentioned.

Does someone has any clue about, how large a concentration range can be, in order to get excact results over the whole range (UV detection) with only one linear equation?