Linearity Standards and the R squared value

[MestizoJoe]

Hi everyone,

I've noticed that some of my collegues prepare linearity standard solutions by
diluting a stock to get level 1
diluting level 1 to get level 2
diluting level 2 to get level 3, etc...

They, in general get R squared values which are 0.9999 or better.

When I prepare linearity standards by taking aliquots of the same stock to prepare all 5 levels I rarely get 0.9999 for R squared. Usually it's more like 0.9995, or something like that.

Is there a mathematical explanation for this?

Also, my linearity standards are usually 50%, 80%, 100%, 120%, and 150% of some value. The serial dilutions cannot be this tight. Is this a reason for my R squared values?

Let's assume everyone has the same, good technique when preparing their solutions.

[Peter Apps]

Without knowing the details of the dilution schemes I would guess that this has a lot to do with the (in)accuracy with which different volumes can be measured and dispensed.

The serial dilution scheme (which your colleagues use) will need to measure and dispense larger volumes than your one-step scheme.

For example, a serial scheme that takes 1 ml of stock and dilutes it to 10 ml, then 1 ml of the 1/10 dilution and dilutes it 10 ml etc for five steps will end up with 10 ml of a 1 in 100 000 dilution (and 9 ml of each of the higher levels). If you try to make the 10 ml of 1/100 000 in one step you have to measure and dispense 10/100 000 ml - which is 1/10 of a microlitre.

The disadvantage of the serial dilution is that errors propagate down the series - if the first dilution is wrong then all the others will be as well. This generates bias rather then non-linearity and so can be very difficult to recognise.

Peter

[Alexandre]

I prefer independently prepared dilutions, each from stock. For some regulatory jobs we do at least 3 point calibration and check our stocks vs std of different origin. Unless you have very good std and/or a reference material to eliminate bias concern.

Some even suggest preparing 6 point average cal curve from independently prepared solutions each concentration point in independent triplicate (interpretation of a guideline).

How bad the calibration is should be investigated during validation. If there is no stat. significant difference for slope and intercept. Many methods can be simplified to a single point calibration with relevant QA/QC measures.

[HPLCaddict]

Just as a sid note, r squared should NOT be considered as a conformation of linearity. In order to really evaluate linearity other factors should be considered, too (e.g. sensitivity plot, Mandel test, etc.). From experience, r squared can safely only be used to confirm ABSENCE of linearity (when it's really low). A high value close to 1 might be a good indicator that you're calibration in fact is linear, but it's not a proof.

By the way, a calibration with r squared of 0.9999 or even 1.0000 is not automatically better than one with 0.9995 - can a calibration be "more linear" than the other one?

From a practical point of view - robustness of your method (including sample preparation) is MUCH MORE important than lifting r squared from 0.995 to 0.999999999!!

[krickos]

Hi

The only possible general statistical reason I recall on top of my head (see Miller & Miller or other decent book), is that for traditional linear regression analysis, the distance on the x-axis should be eqvidistant for correlation coefficeint evaluation. ie rather 50-75-100-125-150% than 50-80-100-120-150%. With the later the three middle point "weight" heavier, small variation there progress to a greater absolute error at the extremes (50/150%) if eqvidistant the weight is equal as I recall.

I suck at debating statitics in english but hope it made some sense.

[DR]

If you search these very forums, you will find a few threads dedicated to the subject of appropriate weighting of linear regression curves in chromatography.

[MaryCarson]

Also, my linearity standards are usually 50%, 80%, 100%, 120%, and 150% of some value. The serial dilutions cannot be this tight. Is this a reason for my R squared values?

The short answer is "Yes." One can get amazing R^2 values if your standard curve extends over 3-4 orders of magnitude. However, one needs to look at the residuals on the low end of these long curves. They can exceed 100% of nomimal value, even where R^2 is 0.9999.

[MestizoJoe]

Thank you for the great replies. I will definitely take your points into consideration for future work.

[sllimbri65]

MestizoJoe

I am in Merrillville, can I ask where you work?

[aldehyde]

Phew, this kind of math stuff is what I struggle with and you all did a great job of explaining it.