Calibration Curves

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Lysander
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Calibration Curves

Hey Guys

Question is the following

We know that R square gives the value of 0.99 or higher value we are taking in parameter in order to see if this is a good curve.
My question is the following
How would you catch mistakes of a " perfect " curve?

Let me elaborate

How would you perform a Zero-Intercept test?
How would you calculate residual plots in your curve to determine whether the results are the True values?

Thank you

Dr. Analytical
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Sorry I didn't get to your

Sorry I didn't get to your post earlier.  This is a good question that mot people don't really understand.

R-square is one measure of calibration quality, but as you imply, it is not perfect.  In some cases a good r-square can still result in wrong results, usually at the low concentration end of the curve.

A zero-intercept test will answer the question, "is my intercept statistically different from zero?"  If the answer is "no" then your curve passes the test and you have more confidence in your results.  The actual test is based on the standard deviation of the intercept and the appropriate t-value from a table.  Most statistics books can give you more details.  Excel calculates the range in the Analysis Toolpack. 

A residual is the difference between the measured data point and the regression line.  Negative residuals are below the line, positive values mean the point is above.  If you plot residuals vs. concentration of standards, you should get a random plot centered around zero.  If you see any trends (up, down, curved) in the residuals, then you may have used the wrong model for the curve fitting.  Excel also creates these plots.

I can give you more examples if you write back, but they are impractical to post here.

Lysander
Lysander's picture
Dr. A

Dr. A
Thank you so much for responding sir-
 
Here are some of the concerns that I have-Please correct me if I’m wrong-
 
Case I- Here is what I think Linearity can be explained on the sets of calibration points
I think we can set our lowest point to 50% of our assay % and go up to 150%, which will be the highest point. By going down to 50% We would like to ensure that the method will be able to calculate the results accurately below the ICH reporting limit.
Now then; if I have a Y- intercept- Have I made a mistake?
Should I bracket my standards in a narrower range limit to ensure more accurate results?
 
Case II- If we have a Y-Intercept and we have ruled out that we have not made any mistakes in sample prep and calibration( Hypotheically speaking)- Can we proceed with a single point calibration with a Drug substance?
 
 
Case III- At what phase of calibration we should consider Y-Intercept- rather a “significant” one?
Is it this when your 0-intercept test come to play?
 
 
Thanks a million  Dr. A
Regards
-Lysander

Dr. Analytical
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Let's look at this in a

Let's look at this in a slightly different way - what do you want to get from your analysis?  That is, what are your goals (or data quality objectives, or whatever you call them)?

First, if you are doing a trace analysis and samples can vary over a significant range, and you don't need high accuracy, you can use fewer standards over a wider range.  If you need higher accuracy, you will have to use more standards and reduce the range of the standards.  That's the general approach to setting up a calibration curve.

If you are doing a formulation analysis, and all samples are expected to be in a small range (e.g., 98 - 102%), then single point calibration is perfectly acceptable, assuming multiple injections of the standard, of course.  Normally you would need to demonstrate linearity and accuracy over a wider range, which is where the 50% and 150% values come from. 

One school of thought says that if you get acceptable accuracy over your tested range, then nothing else matters.  This is true to a point.  However, errors can develop in the calibration curve that might not be noticed using this approach.  You would expect them to be revealed in the accuracy readings, but that may not always happen.

Regarding the intercept, it will always be some value - it is never exactly zero.  But if it fails the zero-intercept test, then it is significantly different from zero and you have a problem with your calibration system, which could be due to either the standards or the instrument, or both.  This curve could produce significant errors, particularly at the low concentration end.

The zero-intercept test is really only important when you need to establish a calibration system over a wider range (say, more than a factor of three change).  You are doing this because the concentration in the samples is expected to occur over this range.  Linearity and LOD and LOQ are often required as well, because these parameters are important in producing quality data.  If you are not expecting a big change in concentration (formulation analysis), then the intercept is not important and the 50%/150% test can be used prior to single point calibration.

This was an indirect response to your examples.  Did it answer your questions?

Ardenay
Ardenay's picture
I'm just browsing, looking

I'm just browsing, looking for something else, but feel I must comment on two points.

1)  "r2 is a measure of relationship between *two random variables* [and] has no meaning in calibration under the conditions [usually encountered]..." : P&ApplChem 1998 70 993-104. In my opinion it should be *banned* from all analytical validation protocols and reports. The underlying reason is that the amounts/concentrations are assumed to be error free; in practice it's OK (good reference needed) to assimilate relatively small errors on the x axis with the measurement errors.

2)  Often, your intercept *is* significantly different from zero. This happens with LC/UV detection because of stray light and spectral bandwidth factors, when you calibrate between say 80 and 120% of nominal (the calibration curve starts to slope a little bit downwards at high values of absorbance). What we need is advice on working out to what extent the non zero intercept has an effect on the analytical result when single point calibration is used. In practice, since we often use duplicate standards, I'd want to know if they should be nearly identical or used to make a 2-point least squares calibration curve. The latter is perfectly correct but might give different confidence limits for the analytical result.

Dr. Analytical
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Ardenay:

Ardenay:
You make some good arguments, but perhaps the situation is not always so bad.

I agree that r2 is not the best way to evaluate a calibration curve.  It is one piece of information, that should be used together with other information, like analysis of residuals and the zero intercept test.  However, we live in a world that likes simple tests that give clear answers - the curve is good or bad.

I do not agree that it compares "random variables."  It is true that the statistics are based on the assumption that there is no error in the standards.   While we all know that this is not theoretically possible, it is possible to prepare standards with an error that is minimal compared to the response variations.  In practice, this approach works well for many systems, as long as you do not over-interpret the results.

The zero intercept test is only used when you have a curve that covers a wide range, and you expect your samples to have concentrations over this entire range.   If you are analyzing over a restricted range, as in formulation analysis, then the 50%/150% or 80%/120% standards should produce acceptable accuracy, and that is all the information that you need.   Since you are not near the LOD or LOQ, the zero intercept test is less useful. 

Finally, every different calibration scheme will have it's own set of confidence limits, but I am not a statistician and can not quote specific results.  If you are expecting all of your samples to be within a small range (you must define and validate what you mean by "small"), then a single point calibration with multiple injections should be acceptable. 

I do know from examining the equation for confidence intervals in linear regression that the error limits decrease more quickly from duplicate analysis of standards than from injection of more standards at different concentrations.  From this I would infer, but have not proven, that two injections at the same concentration would be better than one injection at two different concentrations.

Dr. Analytical
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One final thought:

One final thought:

I like to evaluate my curves by analyzing the standards as if they are samples.  The results should agree within some acceptable difference.  This approach will very easily identify problems at the low concentration end of the curve, which you mentioned, and may also indicate other problems.

Greg Pronger
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Merlin has given an excellent

Merlin has given an excellent explanaiton here, but from a practical perspective is whether your data system offers weighting regarding the calibration fit. It is my experience that at weighting of the inverse of concentration will significantly improve the curve fit at low concentration. My observation (and maybe Dr Analytical can give the statistical explanation) is that the unweighted fit will tend to fit the curve tighter to greater numeric values for the concentration while the inverse of concentration tends to more evenly balance the bias.

Greg

Dr. Analytical
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Greg:

Greg:
You give me more statistical credit than I deserve. 

I do know that the standard, unweighted least squares approach produces the minimum error at the average value for each axis.  That means, the location on the line that represents the average of the concentration values is the place where the errors are smallest.  They increase as you move away from this point in either direction.

Weighted least squares is not as common, because the assumptions are more complex and not as general, but my understanding is that this approach is only valid if the variability is not constant across the data points, and you know what that variability is - either higher or lower at one extreme.  Since most of us do not have this information (or want to take the time to generate it), this approach is rarely used.

 

Greg Pronger
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My experience with the

My experience with the weighted is based upon the USEPA changing from having calibrations for environmental use unweighted to weighted due to the issue of the fit at low concentrations. From more of an external observers perspective, my undstanding was that going to the weighted, removed the bias to higher values (or average). Their technical bulletin did not include the statistical reasoning for the change other than a recognition that unweighted was not fitting data as well as the weighted.

dawnrb
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Dr. Analytical

Dr. Analytical

I am trying to do what you have suggested.  What equation would I use to evaluate the std point as a sample and then calculate the % recovery to verify that the linear curve and data point are within limitation

Dr. Analytical
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Dawnrb:

Dawnrb:
Sorry for responding late.  The spam filters have been removing my notifications.

If you are evaluating a standard, then use the equaion for your fitted line (slope and intercept).  Simply put in the response for the standard injection and determine the concentration.  If that value is within an acceptable range, then you have verified that your calibration curve is still valid.