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

[carls]

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?

For UV the upper limit will be determined by the linearity of your particular detector. A general rule (from spectroscopy theory) is that maximum absorbance values below 1 AU (1000mAU) will still be in the linear region. Some vendors claim linearity above 1 AU but I have not tested this claim. Maybe someone else has. The low end will also be detector specific and is determined by the S/N. A S/N of at least 20 at the low end of your curve and absorbance below 1AU should give you a linear curve. However, by definition/theory prediction %error will be highest at the low end of the curve if no weighting is used.

I have had good success with 1/x^2 weighting. When looking at fit errors you must focus on %error and NOT absolute error. With 1/x^2 weighting typically the %error across the entire calibration range is similar unless there is some non-linearity in your system. Non-linearity could arise from several different sources, e.g. interference at low end or detector saturation at high end.

[dresdentl]

Concerning weightings, I think I have messed up something.

Weightings are calculated from standard deviation (s). I found that the initial weightings are 1/s^2. Then this is multiplied by the number of calibration points and divided by the sum of all weights. Is this the same as 1/A^2 or 1/C^2 weighting?
The larger the concentration range is, the more important should get the weightings. If you have the range 1 - 100 ng/ml and the coefficient of variation (CV) is the same for all concentration points, then the absolute standard deviation will increase 100-fold. But wouldn't it be easier and safer to use a low and high calibration curve?

I have to apologize, what I mentioned in the begining has nothing to do with weightings because if you replace the x and y values in y=bx +c with 1/x, 1/Y ; 1/x^2, 1/y^2 or 1/x^1/30, 1/x^1/30 then it represents a nonlinear equatition like Lineweaver-Burk. This would be a nonlinear calibration.

[Don_Hilton]

If the CV is the same at all concentrations, you should not need weighting. Typically the CV increases at low concentration - if it did not, we would not have to worry about a lower limit of quantitation or detection.

[tom jupille]

If the CV is the same at all concentrations, you should not need weighting.

Actually, Don, if the CV (%RSD) is constant, that's when you *do* need weighting. Ordinary least squares implicitly assumes that the absolute errors are constant across the entire range. If CV is constant, then absolute errors increase with the amount of analyte.

[dresdentl]

According to following articles,

haun 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

with the same cv for all calibration points, and c(high)/c(low)=10, the standard deviation increases 10-fold, and therefore the highest points will dominate the linear equation. But according to this article this is the case if the standard deviation (failure) is caused by volumetric measurement.
But in my case, I think things are different. I have 9 calibration levels and the slope of the level 1-5-linear equation is a little higher than the slope of the 1-9-linear equation is. But I guess that this has nothing to do with volumetric failure and it is caused by substance loss in the system as Don mentioned in his first article. The percentaged loss will by higher at low concentration. Therefore the slope is higher at low concentration (x=area, y=concentration). This failure can't be compensated by weights. I tested weightings with the HPLC-instrument-software and if I weight with 1/x^2 the analysis at lower concentrations may improve but at higher concentrations it get worse. Because of the higher weight of the low points R^2 worsens.
In my case, it may be better to use the umweighted curve and to improve the accuracy with 2 calibration curves for high and low concentration.

[Peter Apps]

Deviations from linearity (that generate low r squared) can arise from two sources: an x:y relationship that is genuinely non-linear, and random scatter of measurements around a genuinely linear relationship.

From the sound of it, dresdentl has a non-linear relationship between analyte quantity and detector signal, inevitably with some random measurement variability superimposed. Weighting will not help with the non-linearity, but it might witht he random scatter.

Peter

[Don_Hilton]

That series in LC-GC Europe is very good. The moment I saw Tom's reply to my incorrect statement about CV's and weightings, another article in that series immediately came to mind but I was not able to go looking for it at the time...

[carls]

with the same cv for all calibration points, and c(high)/c(low)=10, the standard deviation increases 10-fold, and therefore the highest points will dominate the linear equation.

This is what I have pointing out and is the reason to use weighting.

But I guess that this has nothing to do with volumetric failure and it is caused by substance loss in the system as Don mentioned in his first article. The percentaged loss will by higher at low concentration. Therefore the slope is higher at low concentration (x=area, y=concentration).

According to Beer's law:
y = absorbance (or area in your case) and x = concentration.

The slopes of 2 curves, one spanning a narrow high concentration range and one spanning a narrow low conc range, will no doubt be different since the high conc points no longer dominate the low conc points in the one data set regression and the range spanned by each curve is greatly reduced.

This failure can't be compensated by weights. I tested weightings with the HPLC-instrument-software and if I weight with 1/x^2 the analysis at lower concentrations may improve but at higher concentrations it get worse. Because of the higher weight of the low points R^2 worsens.

How does the %error at high and low concentration compare with weighting? If the %error is acceptable with one weighted calibration curve why use 2 unweighted curves? The practice of using weighted calibration is very common in bioanalytical MS applications where the concentration range of the calibration often spans 3 to 4 orders of magnitude.

[Uwe Neue]

You need to think about the origin of the error and then choose a weighting procedure that corresponds to the error.

A higher absolute error at high response is commonly caused by a procedure of constant injection volume. With other words, the error is due to the injection volume. In such cases, a 1/X^2 treatment of the data is appropriate.

Hope this helps in the understanding of the link between the practical procedure and the correct error handling. If you were to inject a variable volume, the error of the injection volume at high mass injected would decline, and no weighting would be needed (unless other elements of your system, such as the integration, would be an additional source of error).

[carls]

A higher absolute error at high response is commonly caused by a procedure of constant injection volume. With other words, the error is due to the injection volume. In such cases, a 1/X^2 treatment of the data is appropriate.

Perhaps I dont understand your post but higher absolute error (i.e. residual) is expected at high concentration. It is an artifact of the least squares fitting procedure.

[Uwe Neue]

No Carl, it is not a question of the fitting procedure, but a question of the injection procedure. Most chromatographers use a constant injection volume, and then the absolute error increases with the amount injected, while the relative error remains constant.

An alternative is to use a fixed concentration and an increasing injection volume. Now the relative error of the volume becomes smaller, as you increase the amount injected. The consequence is a smaller absolute error compared to the case above.

[carls]

No Carl, it is not a question of the fitting procedure, but a question of the injection procedure. Most chromatographers use a constant injection volume, and then the absolute error increases with the amount injected, while the relative error remains constant.

Are you suggesting the injection error changes with analyte concentration?

Do you have a literature reference explaining this more fully?

An alternative is to use a fixed concentration and an increasing injection volume. Now the relative error of the volume becomes smaller, as you increase the amount injected. The consequence is a smaller absolute error compared to the case above.

Injecting variable volumes introduces a number of additional sources of variability including injector accuracy, volume loading effects, etc.

Also, how is the injection volume for the sample chosen?

[Uwe Neue]

Carl, all what you have to do is think...
Your calibration curve is based on peak area, which is proportional to injected mass. If you inject a constant volume with the associated constant error, your mass error increases directly proporitonal with concentration...

(with other words, the injection error does not change with concentration, but the mass error increases with increasing concentration)

I am fully aware that the increase in sample volume by using a constant concentration adds additional complications, but it does alleviate to some degree (until the complicating factors kick in) the issue discussed above with constant volume but varying concentrations.

[carls]

Since in each case you must inject the same mass to get the same response, the only way this calibration approach can help is if the precision of the variable injector at high volume is better than the precision at an intermediate volume. My recollection is the precision is similar once the volume is above a “critical volumeâ€

[dresdentl]

I have tested now a larger validation data series where have calculated s for each calibration point. Then I accomlished a weighted (1/s^2) linear regression. Now I get good linearity and accuracy nearly over the whole concentration range.



Therefore I think, weightings deliver good results if you you use mean values of a larger data set. And then only one calibration curve may be satisfactory for the whole concentration range.