What is the right model y=mx+b or y=mx for hplc and UV calibrations, why?

It depends on how many standards you use, when you validate your method and see no difference between calibrating on 1 standard (y = mx) or a serie of standards ( y = mx+b), you can just calibrate on 1 standard.

If I understand your question I think y=mx is the correct model for the majority of situations.

If you use y=mx+b this leads to amount=(response-intercept)/slope. In most situations I don't think the intercept has a physical meaning and therefore should not be corrected for.

In UV it is easier to see where the intercept can have a physical meaning. Let say you prep your standard in a diluent that is different than the solvent used for the samples. If the standard diluent has some absorbance at the wavelength used and you blank with the sample solvent then there is some absorbance present at all levels of the standard due to the diluent. Now, when you generate the calibration curve the intercept is the absorbance of the diluent and should be corrected for using y=mx+b.

I can't think of a situation in HPLC where you would subtract the intercept, but I'm sure someone else will. In most cases I think the small intercept is just a consequence of small random errors present at each level, so using y=mx+b would be the equivalent of saying that since our random errors are small let's just add some back at the end.

Tom, I have to disagree. A calibration curve is simply our way to to model the relationship between concentration and response. We really don't know if a linear relationship is actually correct. In my experience, it has not been unusual to encounter a non-linear relationship when a UV detector is calibrated over a broad (2-3 orders of magnitude) analyte concentration range. This non-linearity is what leads to a non-zero intercept. The use of y=mx (i.e., single point calibration) is generally only advisable when the area of the component of interest in the sample is very close to that of the area in the standard. Since you are essentially extrapolating the calibration line to zero (without actually verifying if the intecept is, in fact, zero), the farther you move away from the one measured point, the larger the error becomes.

That being said, I believe y=mx+b is the most correct model for the majority of situations, since it compensates (at least somewhat) for detector non-linearity over that portion of the response/concentration range where you are working.

Lab Rat,
You have two questions. HPLC and UV.
HPLC with a UV detector is not usual spectroscopy. Most detectors connect to data stations with built in integrators. Y=mx relationship rarely holds for an integrated signal because of peak dividing, skimming, and baseline determination.
Use y=mx+b if the standard deviation of y on x is about the same magnitude as the standard deviation for single concentrations of analyte. If it does not, even this model does not measure up.
Jim

Thanks everyone. I guess I had more questions than I thought. HPLC vs UV and physical model vs mathematical model.

Anon: Not necessarily related to the number of standards you can always force through zero.

Tom: Thanks for clearing up the physical model - I agree. An injection of zero amount should produce zero response. My impurity data quantitated fine using Beer's Law y=mx but not using the y=mx+b calibration curve (see below).

hinsbarlab: I disagree, this is what brought up this question. A linear approximation to a nonlinear curve is only going to be accurate over a small range. I was trying to quantitate assay and related substances with the same injection. I tried to quantitate off a single calibration curve and slightly exceeded the linear range of a very old system at the 120% level. Still passed the linearity criteria of the protocol r^2>0.999, but the small intercept caused large errors at low levels. Quantitation of impurity levels was fine if I used the average response factor or the 1% standard (y=mx).

Jim: what is peak dividing. I developed a good method, the critical resolution was 1.8. The standard deviations would be different at different levels wouldn't they?

Lab Rat: My point exactly. Linear approximation to a nonlinear curve using only one data point is going to be accurate over an even smaller range. Question. Are your impurity levels significantly less than your lowest calibration standard? If so, you're extrapolating outside the calibration range which we all know is a no-no for reasons just such as these.

Lab Rat,
With that resolution (1.8) and assuming that the peak resolved follows the analyte does not change in size run to run, sometimes you may get a y=mx curve.

When the integrator has to resolve merged peaks, the place for dividing them will change as the ratio of the two peaks change. With 1.8 resolution, your peaks overlap about 5 percent. If the peak of interest follows the resolved peak, the integrator will give the extra area to the peak of interest, driving the intercept of the calibration curve away from zero to a positive value. Other conditions will cause other offsets.

Another writer to your question mentioned a linear response. What may have been discussed is the linearity depends on your requirement for accuracy. Y=mx will be linear over a shorter range than y=mx+b, and y=mx+b will fail for normally, real world, integrations when the range becomes large. That is, very few calibrations curves function well at <10times the detection limit and 1000 times the detection limit, because the effects of finding the baseline, start, and end of the peak is more difficult and their values effect the integrated area calculation more serverely.
Jim

Sorry, I am at a total loss to understand this discussion. Someone please straighten me out if I misunderstood my math, learned years ago. We used to call y=mx+b the slope intercept form of a STRAIGHT line, sort of hinted on above. Then y=mx is also a straight line with the intercept = 0, that is the line goes through the origin of the graph. If m, the slope, is the same for the two curves (b= not 0 and b=0) then they are parallel. Or, the other way around, if you use y=mx+b to describe a straight going through the origin, then b=0... where is the problem? The intercept can be any number, + or -, how does this change linearity? If your curve is not linear then y=mx+b can not be fitted to your points. (y=mx is just a special case of y=mx+b with b=0, similar to, for instance, y=mx+2, another special case with b=2, a staight line going through y=2, x=0)

My suggestion: do everything possible to avoid nonlinear responses.

I think it is possible to generate a calibration curve where the intercept is forced to be 0 (y = mx). A "best-fit" line is generated using the calibration data points with the requirement that the line passes through 0. A second calibration curve could also be generated where the intercept is not forced to be zero but falls where the "best-fit" line generated using only the calibration data points intercepts the y-axis (y = m'x + b). Thus the values of the slopes would not necessarily be the same. Would they be significantly different? I guess it depends on the results you need and your definition of significant. Depending on your needs, it may be necessary to use a non-linear curve or multiple linear calibration curves depending on the concentration of the analyte in your sample.

There is a statistical definition of the word significant. A better way to determine if I need to use the intercept is to carry-out a t-test (Student) for the intercept under the hypothesis of the true value being exactly zero, with a 95% of confidence. The demonstration that intercept is not statistically different from zero during your method validation enables you to calibrate with a single point, that is the model Y =m X.

H W Mueller: Of course your math is correct. I am validating an HPLC assay and related substance method. My question was, given that my peak is baseline separated, what is the correct model for a calibration curve? Should it be a straight linear regression (y=mx+b) or should I force it through zero (y=mx). There is a strong bias for not forcing it through zero at my company. Instead, impurity level is quantitated off a lower level standard. The party line is that using the 1% or even 0.1% standard is better because it is closer to the expected level. This sounds good, but, the data does not support it. There is no statistical difference in the specific response between the 100% standard and the 1% standard. So if there is no difference in the specific response then the only reason we have been using the 1% standard is to quantitate the impurity level against a single point calibration (y=mx).

With regard to "do everything possible to avoid nonlinear responses", of course and I do. The problem is that even with linear data random error produces a little scatter and can produce a small intercept on linear regression. While this intercept is insignificant when compared to the 100% level, it is very significant compared to the 0.1% level.

hinsbarlab: my lowest level standard is a 0.1% standard, this is the specification level and 2X the reporting level(0.05%). What kind of limits do you use for your y-intercept? Our company commonly uses 0.5% or 1% of the 100% response. The problem is not that I'm performing a linear approximation to a non-linear curve. The problem is the small intercept produced by a linear regression of linear data when quantitating very low levels.

Marcelo: Thanks, I looked at 18 calibration curves that I have generated. 9 had small positive intercepts and 9 had small negative intercepts. I'm guessing that zero is probably within 2SD of the average.

I like Tom's idea of forcing our calibration curves throught zero unless we have identified a source of bias.

I like Russ's idea of processing the calibration standards twice once using a standard linear regression to ensure linearity (r or r^2) and no significant intercept and if this is true to calibrate using a linear regression that is forced trough zero.

How do others handle quantitating both assay and impurity levels off the same injection. Thanks for the comments

I am enjoying this discussion and would like to understand the problem in greater detail.

Is this correct?: a known compound is calibrated from 0.1% to some greater value. Then this same calibration curve is used to quantitate something unknown, and at lower concentrations than 0.1%, perhaps 0.01% or less?

If the above understanding is correct, why can't one simply add a calibration point at 0.01 or even 0.001% or whatever it takes to get a calibration point below the lowest level of interest? I never reported anything that was not between calibration points.

If the blank was blank, I always forced through zero, if the calibration was linear, which in UV it always was. It seems to me to be the most reliable point on the calibration curve.

Given the vast range of extinction coefficients possible (the range was 5,000 for things we did), all other assumptions seem insignificant compared to using a known's response factor to estimate an unknown.

Hi Bill,

Yes you understand the problem correctly. I completely agree with everything you said.

I work for a pharmaceutical company, and when a compound is early in development we may not have isolated and characterized all degradants. We do forced degradation studies but often don't have accurate response factors or degredation impurity standards. I agree regarding the response factor assumption and have raised this point myself.

I'm a little suprised that there is such a difference of opinion on how to calibrate this. You and Tom say to force it through zero, which I also agree with, Jim and hinsbarlab say not to. Most of my coworkers here also think I should not force the calibration through zero - but then tell me to do the functional equivalent by performing a single point calibration just for the degradants.

I could move the 0.1% standard lower but it wouldn't change anything. I'm not a statistician but our HPLCs tend to have system suitability precision of around 0.2% so it doesn't seem out of line that a linear regression of the calibration standards could have an intercept of 0.1% or even greater. This is obviously a problem if you are trying to quantitate at 0.1% or even less.

Thanks for all of the comments.

I haven’t been here for several days, and it is a pleasure to see a good discussion. Let me first comment on the use of 0 as a data point. If you can demonstrate that there is no response if you inject a blank, then 0 is a data point that should definitely be used. If you are working with a rather clean sample, then it should be possible to demonstrate the absence of any response when the sample is free from the analyte that you are trying to quantitate. If you are dealing with a plasma sample, this may not be possible. Under these circumstances it is valid to assume a non-zero intercept. Under most other circumstances, one can demonstrate that there is no response possible, if no analyte is injected. Then 0 is actually the strongest point on the calibration curve and one should do everything possible to force the line through 0. This may mean that one should also assume a non-linear relationship, if a linear fit indicates a problem with this. This is the situation that Hinsbarlab has pointed out. If I do a calibration over several orders of magnitude, it is not impossible that a non-linearity is involved. Then one should still fit through 0, but with a relationship that allows a curvature.
Jim Gorum has pointed out a case where a positive intercept can be generated due to the neighboring peak. This is something that can be corrected for. It is easiest, if the neighboring peak has a constant height, but it is also possible to do this with a variable height of the neighbor. I would advocate to subtract the contribution of the neighboring peak, and then force through 0. Of course, one should also discuss how such a procedure should be handled in a regulated environment, or if there are special issues related to this.
I disagree that one should do everything to avoid non-linear responses. They are a fact of life. With intelligence, they can be dealt with. It is my understanding that especially in the world of MS detection nonlinearity is an everyday event.
One of the items that has not yet been brought up is the errors in generating both the calibration curves and the response. Most of the time, the same injection volume is used with various concentrations of the standard. Under these circumstances, the dominant errors are the injection volume and possibly the HPLC integration. Therefore the relative error remains largely constant, but the absolute error increases with the size of the response. Such a situation should be dealt with using weighted calibration curves. But now I am slowly getting out of my league, and I wonder if somebody else would comment on the use of weighted calibrations.

To keep this short, just one example of what I meant with avoiding nonlinearity:
If you expect unknown concentrations outside the linear range of a UV-detector just dilute the sample to lie within the linear range to avoid using curved cal curves (ugh). If you get out of the linear range of a UV detector you can have horrendous variations in response.
One can imagine that there are situations where linearity is not existing, but nonlinearity has not yet been part of my life.

Uwe, I don´t understand that weighting, because of absolute error. Thought one uses weighting to give the very low values a chance to figure in "statistics". When I used ANOVA the first time on a calibration curve I was surprised that the correlation coefficient to the straight was super (0.9998 or so) even though my lowest concentration was too high by 100% (double of what it should have been due to carry-over). Weighting would have shown this in the correlation coefficient.

Marcello: Thanks for the reminder that there is a statistical definition for significant. I was thinking more along the lines of a "fitness-for-use" definition which can be variable. In this case, if the same fitness-for-use decision is made on the product regardless of which calibration method is used, while it might be technically and intellectually interesting to know which is more correct, aren't you spending a lot of time worrying about something that will have no effect on decisions made? If the variation in calibration curves makes a difference in the fitness-for-use decision, shouldn't you use multiple calibration points and use the calibration range bracketing the sample? Statistical treatments are definately useful but I guess my first emphasis is fitness-for-use. One example of the difference (from an admittedly fitness-for-use perspective) that I like is a company that demonstrates their product is in statistical control with tight control limits and plenty of control charts to show their process capability. There is just one problem: the product kills people.

H. W.
Here is the problem. Assume for simplicity that the only problem is the precision of the injection volume, and let us assume that the error is 3% RSD. If I prepare different concentrations of the standard over a concentration range of 2 orders of magnitude, every data point has the same relative error, but the absolute error decreases with the concentration. Now assume that I just take on data point everywhere on the curve. If I now do a linear curvefit to these results, the standard curvefitting procedure that we all learned in school does not know that the absolute precision of the data is substantially better at the low concentration than at the high concentration. It may just give me a large and unrealistic intercept that is much much larger than the datapoint at the low end of the calibration curve, definitely way outside the 3% precision of the low-end datapoint. This is the standard situation in most of our curvefitting. And this is where the weighted calibration is coming in.
There are supposedly weighted curvefitting procedures around where you can assign an error to every point on the curve. I have never done this, but such a procedure would fix the problem. The other possibility is a weighting of the error as a function of the value of the datapoint. This is called 1/x and 1/x^2 calibration. But here is where my knowledge ends.
I think that these weighted calibrations are the right choice for most of our applications, but I don't know enough about this. I would appreciate, if anybody could comment on this.
In the old times, I just used a logarithmic calibration curve. This takes care of the problem of the constant relative precision.

Uwe, you are right, it would be helpful to get input by a person more knowledgeable than us in statistics. For instance, I am at a loss to understand why one would apply weighting if you have the same relative RANDOM error at every data point. To me a 0.1 nmole error (random) at 1 nmole is just as bad as a 10 nmol error at 100 nmol. Now the case I mentioned had a systematic error (on top: same size error at all calib. points, absolurtely) which was huge, relatively, only at the lowest data points. My opinion: This sort of thing should be corrected, not mathematically treated. In that case it was corrected by removing a Rheodyne injection valve (I have mentioned manufacturers positively so it should be allowed negatively also) out of the system and using a homemade device instead (all important parts can be cleaned relatively easily) .
Now I would be highly appreciative if I was corrected on the following impression: a. The mathematics of curve fitting other than to a straight is less reliable than fitting to a straight. b. A non-straight standard curve USUALLY has a capricous underlying problem (as in my example, where the carryover is not constant).

I am not a statician but this is what I think is going on.

The linear regression produces a line that minimizes the sum of the square of the ABSOLUTE differences between the predicted line and the data points. Thus, with constant RELATIVE error the high levels of the calibration curve will dominate the regression because on an absolute scale 1% of 100 is 1 and 1% of 0.1 is 0.001.

Uwe is right that a wieghted regression can correct for this. However, determining a correct weighing factor of the form 1/ax^n would involve plotting the residuals and finding a weighing factor that would produce residuals equally scatered around zero for all levels. This would be covered in a statistical text under parametric curve fitting and goodness of fit. My guess is that 1/x^2 normalized to some value for the smallest x would yield good results.

On perhaps a more practical note, Lab Rat's problem could probably more easily be dealt with using two calibration curves. One for impurity level and one for assay level. 0.05%, 0.1%, and 0.2% for impurity level and 80%, 100%, and 120% for assay level. This way the linear regression for the impurity level isn't dominated by the residuals for the assay level.

Also, simply forcing the linear regression through zero (y=mx) would yield good accuracy at impurity levels, assuming no bias or interference.

Linear regressions are a good test of overall linearity of a data set however because of the squaring of residuals they are sensitive to outliers and unfortunately the correlation coefficient is too coarse a measurement to determine if a regression line is a good calibration curve. I think plotting of both the predicted and actual specific response at each level gives most chromatographers a much better idea of what is going on in a calibration. Thanks for the good discussion.

Aaaahhh ... after twenty posts the light finally shines. I couldn't figure out what you guys were talking about. I've been doing chromatography for 15 years and figured calibration was calibration. I'm glad I stuck this thread out, I think I learned something.

Tom, did you really read my posts?

Hans

Uwe, I apparently also didn´t read what you said carefully enough the first time. Apparently what you say is that in anticipation of an error, like in my example, etc., one should apply the weighting so that the lower values contribute a fair share to the calculated fit? We said the same thing on this?

Hans

Hans,

Sorry, I wasn't specifically responding to you, altough I did try to respond to the relative error issue. I'm not the person more knowledgeable in statistics that you were looking for. I'm still waiting for that person too!

I guess I was more commenting for Uwe, so he could correct my misconceptions, and trying to give Lab Rat a couple of ideas since he started this discussion.

I have always preferred average of response factors as the calibration method, and relative standard deviation of response factors as a measure of curve fit quality. Response factors will show nonlinearity at the beginning of the non linear range of a detector more clearly than a linear regression. When determing linearity a low RSD value for response factors means that the curve is linear and either passes through or very nearly through the origin.

I know that this is not a very popular method for calibration, but more than once I have found I have exceeded the linear range of the detector when a linear regression gave a good curve fit.

Hans,

Yes indeed, if the lower values have a higher absolute precision (at the same relative precision), a weighting should be applied that compensates for this.

Guys, there was an article by Hinshaw in the last US LC/GC on calibration in GC. Maybe we can get him to come here and discussthis with us.

I like Ron's method, since it shows the difficulties. However, If I understand this correctly, you will need multiple data points for every concentration to get the standard deviation. This is good, but it also is a lot of work.

Here is a spreadsheet that I think illustrates the problem.

Press F9 to recalculate a few times. The curves are based on Trial 1. The data in Trial 1-15 is simulated and is just the absorbance with a random error of 0-x%.

Calibration curves with seemingly good correlation coefficients can generate nonsensical specfic responses at low levels if they are based on unweighted linear regression.

I have been teaching our chemists to use these actual versus predicted specific response curves to evaluate calibrations. Somebody please correct me if I am wrong.

Quantitation Error.xls quantitation_error.xls (54 k)

Uwe,

You just need one data point for each concentration of the calibration curve. The response factor at each concentration is used to get a relative standard deviation of the response factors for the entire curve. Since the response factors are the instantaneous slope at each point, a small (<10%) RSD across the entire calibration range shows both good linearity and an intercept close to zero.

It's really good to read this discussion. I propose to open a new dicussion about calibration in order to put some more light.
Regards

Russ, Marcelo is not italian like Marcello although all my family came from Sicilia. I see your point and I agree that there is no single way to deal with every specific problem. I think that not always one should use the best or 'elegant' solution but the solution that works and solve the problem.

Marcelo

I was a little fast on hitting the button to post the previous button before I finished my thoughts. Going way back to the original posting, the average of response factors method is basically using a y=mx curve, since to get a good RSD of response factor you need an intercept near zero. As was pointed out in an earlier posting, this may not be the case in a complex sample matrix. As a general rule, I have found that in a relatively simple matrix there is generally a problem with either the instrument or the sample preparation if there is a large intercept value.

The response factor method has the advantage of not requiring weighting if the calibration range is large, leaving aside the issue of whether or not this is a good practice. In theory, and in pratice as long as the calibration curve does not cover much more that 2 to 3 orders of magnitude, the response factors should be almost constant as long as the intercept is close to zero.

I think we have two aspects:
1) Is my calibration curve linear?
2) In a daily routine, what kind of calibration should I use?

If I could show that a straight-line is a suitable model for my calibration range (my calibration data met the assumption I made for that model within the analytical range of my method) and the intercept is not significant (whatever that means, statistically speaking or from a practical view, i.e. intercept being less than 0.1% of the target response)I could calibrate my daily analysis by using Y=mX or response factor (one point, replicated bracketing, etc.).
In my experience this is applicable only for short ranges (50% - 150%). For 2 or more orders of magnitude, relative errors (CV%) increase but absolute errors (±SD) decrease as concetration decreases. Example 1-100 ng/ml
CV% at LOQ = 20% means SD=0.2 ng/ml. But CV%=2% at 100 ng/ml means SD=2 ng/ml > 0.2 (CV=20%)ng/ml at the level 1ng/ml.
The ordinary least square regression (OLS) minimizes the sum of the (Yobs - Ypred)^2. But the terms in the sum are in absolute scale, low conc. levels terms are much lower than those for high conc. levels. so that they are neglected in the sum of Squares. Therefore, the regression straight-line is essentially defined by the higher levels of conc.
Weighted least squares (WLS) re-scales terms of the sum of squares in order to compensate the lack of "weight" of low conc. terms.
I've got to go now but I'll try to show some real data tomorrow.

Marcelo

Uwe and Lab Rat,
On weighing.

Weighing gives emphasis to datum that incorrectly steers the slope of a curve. Weigh when you have proof of the problem. Not all weighing comes from changes of variance as the magnitude of the value increases or decreases. The error may be inherient in the experimental design or the even the number base of the arithemetic used. For example, use 4 points, (1,1), (2,2), (3,3), and (100,100); run a simulation by adding an error to a term, and then calculate the slope and intercept varying each point separately. The 100 point has a lever and can push around the slope and intercept more easily than the other points.

Weighing is a simple concept. You want to emphase a point? Multiply it by a million and add a million to the number you will devide by. For example, average 1 and 2, multiply 2 by a million, add the 1 and then devide by a million and 1. You see the answer is the average is 2. You must justify weighing, definitely a GLP consideration. Y=mx is a type of weighing, you must justify its use for your assay. It is not sufficient to justify its use on theoretical grounds.
Jim Gorum

Maybe it was good that I burned my fingers early with math (my example from above). Usually I now do this: Run a dilution series (successively halving conc.) over the expected concentrations and down to 0 conc., check whether the values fit the 1/2 of 1/2 of....., then occassionally run a single standard for the calibration and a blank as a check. Actually this seems to be a two point calibration since the PC uses y = mx. Now it appears to make sense that if your calib. curve really runs through ~ zero there is automatically little weighting and little error.

Dear Tom Mizukami

I agree. Ordinary least squares (OLS) can produce biased estimation at lower concentrations.
One of the assumptions for OLS is S.D. = constant, which is called homocedasticity of variance. That means if C=100 ng/ml and SD=1 ng/ml
CV%=1%, but C=1 ng/ml and S.D.=1 ng/ml yield CV%=100%. Usually CV% increases as conc. increases but empirically the increment is not so fast (Horwitz showed an empirical relation). I usually find CV% 15% - 20% at the LOQ. On the other hand, the problem I see in your calculation is that you are assuming a constant CV%=k and consequently SD=(k/100)*mean(Response). If we assume a linear proportion Response = m*X then SD=A*X, meaning that SD is a linear function of concentration, but this is a very strong dependence.
I think that a more realistic situation is CV% increasing but SD decreasing, as conc. decreases, as showed in my previous post.
You should try CV%= A/sqroot(X) and therfore SD=B*sqroot(X), that will require 1/x as weighing factor.
I also agree with Jim Gorum. I think 'keep it as simple as posible'. If you don`t need WLS don't use it. But if data show that weighing is necessary, that will justify its use.
Regards

Hi everybody,

let us not think about numbers, but let's think about what we are doing. We are diluting the sample, and we are injecting a constant injection volume. Under these circumstances, all data sets that I have seen have a roughly constant relative standard deviation at every concentration except for the very lowest concentrations, where the relative standard deviation increases. The exact same thing is happening when you look at data sets from SPE, and there the story is even clearer: the error is not buried in detector response and integration, but in the sample preparation or the injection volume.
Consequence of this is that one should use a weighing procedure as pointed out by Marcelo. I do not know if there is a special word for this type of behavior. It is not just general heteroscedasticity, but a proportionality of the error with the value measured.
According to the text books, this type of error can be overcome by manipulating the scale of the response to achieve homoscedasticity. If I understand this correctly, this means that I should take the logarithm and do a curvefit to the logarithmic data. This is somewhat the same as doing a Y=mX curvefit, but it may have an exponent different from 1 in X.
Maybe I should brush up, how the Y=mX curvefit works in detail...

Marcelo: I agree with Uwe, all of my data indicates relatively constant %RSD for all concentrations until you start approaching LOQ and noise becomes a limiting factor.

While I agree that a weighing procedure is probably the technically correct approach, I do not know if all data systems are capable of this.

In the vast majority of cases I have run where there was good resolution between the analyte and neighboring peaks and no matrix interference simply forcing the linear regression through zero was adequate to provide good accuracy across the approximately 2 1/2 orders of magnitude required to do what Lab Rat wanted to do ~0.05% - 100%.

Again, I think just looking at the actual and predicted specific responses (response/conc. or amt) tells the story. If your data has a constant specific response then it will be linear and a single point calibration or a multiple point linear regression forced through zero will be sufficient.

The problem is that the converse is not necessarily true. That is, just because your data is linear (r^2 NLT 0.999) does not mean that your data has a constant specific response. As Hans mentioned, you could have 100% error or even more in your low level standards due to some systematic error or even a dilution error and the regression statistics will still look fine. Most data systems will calculate response factors so it is just a matter of getting used to using them. I will try running some data through some simple 1/x and 1/x^2 weight factors and see what happens.

Jim: I don't understand your post. Weighing does not give emphasis to datum that incorrectly steers the slope of the curve. Weighing is trying to correct for this. The basic problem is that most chromatographers would feel that a 0.1% standard with a 1%RSD is just as "good" and valid as a 100% standard with a 1%RSD. Unfortunately the linear regression minimizes the square of the absolute deviations so the 0.1% standard does not have sufficient "power" to hold the intercept as close to zero as it "should" be. As far a compliance goes GLP labs will validate impurity level accuracy. All of this discussion is just about how to make this possible given Lab Rat's senario. There is no guidance that says (y=mx+b) = good and (y=mx) = bad. In the pharmacuetical industry we validate to demonstrate suitability for intended purpose. I am suggesting to Lab Rat that depending on the specific response at each level a simple single point calibration or a regression forced through zero (y=mx) may be suitable for the 0.1% - 100% range. You don't have to justify the use of y=mx any more than you have to justify the use of y=mx+b. You have to say what you did and present the supporting data, accuracy etc. I think you are doing Lab Rat a disservice by trying to inject unsupported compliance fear and uncertainty.

H W Mueller on dilutions and Tom on compliance.

Making standards and diluting them can lead to very bad estimates of accuracy for an assay in certain cases. For example, years ago I put amino acid analysis into a lab and settled on a dedicated Bechman system. They claim a 1% precission and accuracy for the equipment. When the tech prepared dilutions of their standard, the precision was 1%, when she prepared a set of standards where the amino acids varied independently of one another (but still within an order of magnitude) the aa assays had response factors from 5% to 60% for the worst pair. If the assay see no effects from the matrix, dilution works fine.

Compliance issues pose practical problems to a company working under the GMP. In reality, most work is never inspected closely unless a problem occurs in the field. When that problem occurs or when a sharp inspector visits the lab, certain rules become important:
1. Don't argue with the inspector.
2. Understand his observation and if you can work to show you are in compliance.
3. Run the lab in a way so that you can show compliance. Using the last rule causes the most stress. Some people will follow procedures that do not help the quality of the product but protect them from criticism (CYA), an expensive course for certain. Others do as little work as possible with the hope they can argue their way out of a problem if they are not lucky enough to avoid inspection, sometimes a very expensive course but also cheap when lucky. Another course is document your consideration of a possible problem, run an experiment and show the method or change did not cause a problem and put it to file. If later a problem does arise, the FDA looks favorably on the good faith effort to protect the public health. Relying on theory means considering the problem, not good faith effort. For example, if in Lab Rat's assay he could live with 100 +/- 10 units and he has performed the calculation with and without an intercept term and the difference of a reported value varies less than 1%, he can accept the simpler form of the equation, and he would have proof he considered a possible problem. Different than just saying it doesn't make any difference. That is, my point is not to get scared, it is do praxis not theory.

Jim Gorum

This doesn´t end!
Jim, you gave a beautiful example for a common experience: One can budge up everything.
One more on weighting: In my example the ANOVA was done to show correlation with a straight, not for the calibration curve. A weighting method would have shown the huge discrepancy in the lowest values, as noted above. Now, if I used my unweighted curve in the calculations of conc. it would have given more correct concentrations than a weighted curve (which would have given the wrong lower determinations more "say" in where the curve lies, besides giving a lower correlation coeff.).
Hopefully, a last remark on this: If you want to be sure of your chromatography you have to do a second one, as different from the first as possible, or an alltogether different 2nd method.

Wow, what a great thread. I really want to thank everybody that posted in response to my cryptic question.

Let me summarize to see if I got all of this straight. When I prepare a series of standards for a calibration curve my precision is probably injector limited so each level will have approximately the same %RSD. Now as I go up in concentration my absolute standard deviation is increasing. Thus when I perform a linear regression over a large range the deviations of the high levels dominate the regression. This is why random errors in these high level standards can yield intercepts that are very large relative to the lowest levels of my calibration curve. A weighted linear regression is the correct way to compensate for this but all data systems may not be able to perform weighted regressions and it may not be easy to determine the proper weights to use. I should be comparing the actual specific responses of each standard level to the predicted specific response of the proposed calibration curve and quantifying the difference. I can then choose the calibration model base on the specific response data and have a basis for my choice. I cannot believe how I was so ignorant about something that seems so basic like calibration. I really want to thank everybody again I didn't even know it was dark until you people turned on the light.

Dear Tom and Uwe

Regarding constant RSD the problem is the word "constant". If we talk about replicated standard perparation within the same conditions (analyst, day, etc.) by dilution with conc. being above 10xLOQ then RSD will fall under 1-0.8% and seems to be constant, but the fact is that the calculated RSD is just an estimated of the true variance. does it matter whether RSD = 0.4%, 0.2% or 0.7%? For most people this means constant, but under the error model assumptions RSD is a random variable we are trying to estimate, although its distribution is not uniform as is used in the Tom's XL spread sheet.
Threfore, I don't know the true relation between variance and concentration, all we can do is to build a model. If your model fit your data, OK!
Because of my work with biological samples, I'm not so lucky, I'm always between LOQ and 100xLOQ, preparation of standards usually involves clean up procedures, liq-liq extraction or SPE, etc. which affect the variance (error) structure.
Constant RSD means weighing by 1/x^2 rather than
1/x but 1/x^2 was too strong for my data.
That's only an opinion why we see different behaviors
Regards

Marcelo

Marcelo,

Apparently I got finally somebody who can explain to me the difference in the weighing procedures and the meaning of it. Until now, I have not yet been successful. I either found websides full of equations that were incomprehensible to a poor chemist like me, or I found stuff that basically says: press this button and it will be OK.

Would you mind enlightening me?

Best regards
Uwe

Uwe, let me say then that I'm a poor chemist too. If you and the other guys agree, I'll open a new discussion to share information and opinions regarding weighted regression and other topics about calibration, which has always been something very interesting for me.

Marcelo