Calculating RRFs with polynomial regression????

[mas23]

Hello All,

I have been qualifing a RP-HPLC using UV dual wavelength detection for impurities and assay. The impurities is run at 205 nm, while the assay is run at 293 nm (a maximum for the API). I have great linearity at the higher wavelength (80-120% linear range), but when I run linearity at the lower wavelength the best r^2s are from cubic regression lines.

The chalenge of this method was to get sensitivity from 0.005-15% the nominal sample concentration as one of the starting materials and one of the intermediates are genotoxic (AMES +).

I get accptable r^2s using linear regression, as well as good % recovery using the RRFs calculated from the linear regression lines for the linearity run.

However, upon running intermediate precision, my % recoveries were high (~114-115%) even for the API. But the w/w assays all came out around 100%. Thus, I went back and looked at my regression lines using log transformations on both axies and saw that the linear regression line deviating from the data points at the lower concentrations.

Phew...Now my question is this:
I am not a familiar with non-linear regression curves, but need to know if there is a way to calculate RRFs using cubic regression lines.

[tom jupille]

if there is a way to calculate RRFs using cubic regression lines.

No, because there is no such thing as *a* RRF in that case. RRF will vary as a quadratic function of the concentration.

Look at it this way (for a linear regression):
1. The relative response factor is the ratio of the response factors for two compounds.
2. The response factor for any compound is the slope of it's regression line
3. The slope is the first derivative of the fitting function. Since a linear function is a 1st degree polynomial, the slope is a 0th degree polynomial (i.e. an constant).
4. The derivative of a 3rd degree polynomial is a second degree polynomial, hence the "Response Factor" will not be a single factor, but rather a second-degree polynomial.

More specifically, the point to RF's (and RRF's) is to allow single-point calibration, which is invalid if you don't have a linear fit.

Your basic problem is that you are covering a very wide range, so that ordinary least squares fit is invalid in the first place (your data are heteroskedastic). This is a situation where you should look at weighted least squares, or a log-log fit.

[mas23]

That's what I thought as the linear regression is based off off averages which makes sense that the average areas would pull the line up at the lower concentrations. I have excel 2007. How do I do a weighted least squares regression line? The formulas that I have seen all involve partial derivitaves. Is there an easier way?

[tom jupille]

Many data systems include weighted least squares. There is at least one Excel add-in that I know of (although I've never used it):
http://mrflip.com/resources/ExcelFunctions/

There is more information in the NIST statistical handbook:
http://www.itl.nist.gov/div898/handbook ... pmd143.htm