WHAT EXACTLY DO WE MEAN BY WEIGHTED REGRESSION?

[drhlrao]

Hi Everybody,

I am new to this site & am seriously looking for a clarification on WEIGHTED Linear Regression in LC-MS.

Suppose I have just a SINGLE set of 10 pairs of x (concentration ratio) and y (peak area ratio) for a calibration curve, what exactly do we mean by "1/x" weighting?

I am aware that unweighted linear regression follows the format y = ax +b. Can someone illustrate me the exact meaning of applying "1/x" weights on original data? Eg : Do we need to divide 'y' values by corresponding 'x' values? What would be the format of the final linear equation after applying weights?

Thanks in advance

[tayzyboy]

im not sure of everything, but I'm sure someone else will fill in...

The curve equation is still y=mx+C whatever weight is used.

the weight applies to the curve fit. A curve weighted "X" will have a bias to go through points with higher X values and so will fit more closely to higher points in the curve.
"1/X" weighting will weight more towards the lower end of the curve and fit there better.

Not sure how you would model this by hand (we use sigmaplot to do it automatically)

Hope thats some help to start with!

[tom jupille]

Second edit :

It's not relatively easy to do in Excel. However, I found a "freeware" Excel add-in function for weighted least squares at:
http://mrflip.com/resources/ExcelFunctions/

I'm not a statistician (not even close!); but there's a semi-understandable discusion of various approaches in the NIST "Engineering Statistics Handbook": http://www.itl.nist.gov/div898/handbook ... pmd452.htm

The gist of this seems to me to be that if you have wide-range chromatographic data, you may as well simply do a log-log fit (which is what I believe Uwe Neue said in another thread).

[Douglas Moore]

In general, the confidence limits at the upper and lower end of a calibration curve tend to vary more than at the middle. The weighted regression attempts to quantify and correct for this variance. I would recommend you review the article that appeared in LC/GC Europe which is linked below:
http://www.lcgceurope.com/lcgceurope/da ... rticle.pdf

[Mark Tracy]

The uncertainty of chromatographic data tend to be proportional to the peak size. That means that at 1ppm or 10ppm the %rsd is about the same. The catch is that the simple least-squares method assumes that the uncertainty is the same, independent of peak size. That is the %rsd is larger for small peaks; not a good assumption. Weighting by 1/X or 1/XX corrects for the wrong assumption. Of course each fixed weighting scheme has its assumptions too, and you should test to make sure that weighting really does improve the measurements. The best weighting scheme uses the actual variances measured at each calibration level, but hardly anyone except a physicist takes the trouble to do that.

[HW Mueller]

The way I saw this is that the values at the low conc. end just don´t figure in the addition of their squares, such that huge differences (errors) in these low concentration peaks do not show up (practically don´t add anything). So instead of assessing the data without this statistics we will forever fudge around with it. Need to keep people busy. (Just an opinion, inspired by Rutherford: "If your experiment needs statistics, then you ought to have done a better experiment). [Sorry for the repetition, one more, though: I do see the worth of statistics for defining "significant", etc.]

[bert]

Unfortunately, least squares linear regression equations do not fit every straight line . This is not due to statistics, but mathematics. Drhlrao, for the equations, used for weighted regression, search the web. One example is the following link:

http://www.mathworks.com/access/helpdes ... fitt5.html

Regards Bert

[HW Mueller]

Always thought that statistics was math, well, why not make things complicated if they are simple!

[Mark Tracy]

Statistics is a way of training and disciplining your intuition. It is difficult primarily because human psychology has a poor fit to quantitative phenomena. The highest and best use of statistics for the chemist is to prove that you have done your experiment right.

[Uwe Neue]

I did not want to weigh in on this, because my recollection of the details of the procedure is getting fuzzy. However, there is some excellent information on the subject on the NIST website.

From the practice of the chromatographer, the problem is rather simple. One injects a constant volume, and the response varies all over the place, depending on the concentration of the analyte. This is why we create our calibration curves by injecting the same volume. When you do that, all your errors in the response are related to volume errors, not concentration errors. As a consequence, the error at the high end of the calibration curve is proportionally the same as the error at the low end of the calibration curve. Therefore you MUST use a weighting of your calibration curve that accomodates this problem. The usual approach is to use 1/x weighting.

[Alex Buske]

Just an other view:

If you did a calibration over more than one order of magnitude, unweighted regression would over-represent the higher values.
For all your Calibration points you have a SD and a RSD (for one injection it ist still there, but cannot be determined).
So for a point with conc. 1 the SD could be 0.1 and the RSD would be 10%. For Conc. 100 the SD would be 10 and the RSD still 10% and for 10000 SD 1000 and RSD 10% respectively.
For regression usually the squares of the deviation (calc<->found) are minimized. Using the absolute deviations (comparable to the SDs above) in the example above the line would go through the 100 and 10000 -point.
By using the 1/x weighting the deviation at a certain point is divided by the value and you woulfd get something similar to the RSD. This regression is then based on the relative deviation (or their squares).

The whole thing is relativly easy to do in excel if the solver module is installed.
Just define Cells for your regression parameters (a and b), write down you experimental values (x and y in two rows) in the third row calculate the y-values based on your regression arameters. Fourth row is the difference between found and calculated y. Fifth row is the squares of fourth. then you will need a sum of fith row field. Now, using the solver, you can minimise the the "sum of error squares field) by variying the regression parameters
For a weigthed regression in the fourth row simply divide the diffwerence by the x-value.

alex