Rational for Quadratic or Linear curves...

[tima]

Hello All,

I have a very basic question to ask that I have not seen touched upon here. This is my first posting. Our lab performs a number of tests in the field of occupational hygiene so we employ a number of different instruments (GC-FID, ECD, HP-IC, GC/MS and HPLC). The question that has been posed is regarding the use of the types of calibration curves that should be used throughout our laboratory.

Our laboratory manager insists we should use only linear calibration curves for all of our quantitative analysis and if we can draw upon some references or a compeling reason why quadratic curves are justifable we should provide this information. But just from my experience there have been a number of cases where I achieved better results with a quadratic calibration (e.g. the QC - spikes have better accuracy) as well as having better r^2 fittings.

Where can get a "bible" reference on this topic?? When people publish are they compelled only to use linear calibrations?? Any thoughts and insights would be greatly appreciate. Thanks

[tom jupille]

In fact, a standard linear regression is inherently unsuitable for wide-range chromatographic data. That's why LC-MS methods often resort to weighted least squares. To quote from the NIST/SEMATECH Engineering Statistics Handbook:

One of the common assumptions underlying most process modeling methods, including linear and nonlinear least squares regression, is that each data point provides equally precise information about the deterministic part of the total process variation. In other words, the standard deviation of the error term is constant over all values of the predictor or explanatory variables. This assumption, however, clearly does not hold, even approximately, in every modeling application.

In essense, that says that our regular least-squares fit assumes the same error across the whole range; chromatographic data usually has the same percentage error across the whole range (i.e., larger errors at higher concentrations).

I would turn the issue around and ask your manager to justify accepting a demonstrably inferior (linear) fitting function when a demonstrably superior function is available.

That said, it probably won't do you any good, because regulators as a group are naive about statistics, to the point where "linearity" appears in regulatory documents.

[MikeD]

Perhaps your manager needs you to demonstrate the repeatability of your non-linear calibration, even if a say a quadratic fit has good correlation.

If for example an ECD response was starting to top out at high concentration that might be repeatable.

On the other hand if it was poor desorption efficiency from the sampling media at low concentration that caused the detector response curvature then repeatability might not be as good. Time to adjust the procedure and move into the linear range if possible.

[DR]

One problem is that most of your detection equipment relies on Beer's law (or correspondingly similar principle for non UV), which puts you in a linear response mode - (analyte concentration is proportional to detector response). If you were using a detector whose standard response curve was decidedly nonlinear, you would be well advised to use a log or quadratic or whatever kind of curve best emulated the standard response.

Use of a non-linear (weighted or otherwise) r² further confounds things with the regulatory crowd as it is very easy to get a pretty good fit when using a second (or higher) order equation. If you allow a polynomial equation of a high enough order to be used, you can get r²>0.999 for a shotgun pattern.

[tom jupille]

One problem is that most of your detection equipment relies on Beer's law (or correspondingly similar principle for non UV), which puts you in a linear response mode - (analyte concentration is proportional to detector response). If you were using a detector whose standard response curve was decidedly nonlinear, you would be well advised to use a log or quadratic or whatever kind of curve best emulated the standard response.

I agree wholeheartedly. In fact, you could argue that "emulating a standard response" is exactly what Beer's Law does. Spectrophotometers do not measure absorbance; they calculate absorbance. What they measure is transmittance (T). Absorbance is -log(T). [Any one remember using a Beckman DU?]

What the instrument does is to take an inherently non-linear response and convert it to a linear form. It's been done that way for decades, and so we've all come to regard A as the "standard" output, and as a consequence are uncomfortable when we have to linearize the output of another system.

[tima]

Thanks for all the responses. The real concern I have is this is going to written in our laboratory SOP's. I come from the world of validated methods (pharmaceuticals) so if the method has been demonstrated to work and we establish acceptable criteria then thats what one goes for. Not the old school of presciption tests like those found with all the NIST, OSHA and ASTM published methods. That the old fashioned thinking that does not embrace best available technology or new ways of approaching analytical chemistry.

So if I run a batch of samples, in order to have confidence in the dataset then it must have acceptable surrogate recoveries (sample to sample), duplicates, spikes and a back calibration curve that is acceptable. So if it is done with a quadratic and it reaches acceptance criteria then my thinking is that its fine. I still need to find some references or recent publications in reputable journals to convince myself and my manager that using a quadratic calibration is acceptable. It does raise questions that frankly I don't have simple answers to. Thoughts??

[Uwe Neue]

There appear to be some additional complications in this game, but I am not a sufficient expert, and hope that others can give better advice.

I have been made aware that the response and the linearity in MS is also a function of coeluting components of the sample. With other words, you would get a nice linear response from a standard, but a standard in a plasma sample may give you a curved response, and a lower or higher response, mostly a downward curvature.

[Dan]

To write the SOP in such a restrictive manner so as to only allow for linear detector repsonse is to be setting up for trouble. Either the persons forcing such an SOP have seen the future and know that no method will ever have a non-linear response or they have accepted the fact that they will need to prepare a deviation/CAPA report and spend a lot of time arguing with QA when the non-linear response is encountered.

I have seen methods that make use of non-linear calibration curves in my years in the pharamaceutical industry. Off hand, I can think of AA as the technique where non-linear response is actually the norm and not the exception. I don't recall for sure, but this may also be true for ICP.

As to references for you, the only thiing I can think is to look at manuals/vendors for AA instruments.

Perhaps it is also useful to look at info for CDS (chromatography data system) software. All CDS software allows for use of non-linear calibration curves. They put that in the software for a reason.

Regards,
Dan