LOD when the Y intercept is negative

[Peter Apps]

Hello all

I have a slightly unusual query about estimating LOD.

Standard practise for estimating LOD (or LOQ) from the calibration line uses: LOD = intercept + 3 . sd of regression.

I have a set of calibrations with negative Y intercepts (some of them hovering around significantly different from zero), if I apply the equation above as given the negative intercept makes the LOD lower than it would be with a zero intercept, which is counterintutive, since a negative Y intercept implies a positive X intercept, in other words that a peak area of zero is generated by a non-zero quantity.

My immediate reaction is to use the absolute value of the Y intercept, but I cannot find anything in any of the official guidance to reassure me that this is the right way to go.

Am I on the right track, or is there another approach ?

Thanks Peter

[tlahren]

Peter,

For chromatography applications my adviser had us using (3*seY-b)/m; where "seY" is the standard error in the Y-intercept, "b" is the Y-intercept, and "m" is the slope. (10*seY-b)/m would be the LOQ. This is similar to diluting a standard until the S/N is at 3 for LOD or 10 for LOQ. I have seen this calculation get funny when you have a poor intercept though so it may not work well for your application. There is a very detailed article about calibrations and quant/detect limits here:
http://www.ltrr.arizona.edu/~jburns/Art ... asslav.pdf
However, I had a hard time deciphering this statistician's language. Maybe you will have more luck.

Ty

[MMJ88]

It seems like using 3*S/N will get you your LOD regardless of y-intercept. Depends on what software you're using, but Chemstation will generate S/N reports for you. Check what the S/N of your lowest standard is, use that to estimate the concentration that will give you 3*S/N. Make up spikes at that concentration and verify.

However, to be honest, this is just a practical method I use for myself to know the LOD. I can't provide a citation for it. Though, I don't even think the "official" methods make much sense. For example, the EPA method (EPA 40 CFR 136, Appendix B, revision 1.11) basically says to make 7 spikes at a concentration that will give you 2.5-5 S/N, analyze them, and take the standard deviation. 3*st.dev. gives you the LOD. I have found this to give artificially low LODs, but I suppose that's a discussion for another time.

[Peter Apps]

Peter,

For chromatography applications my adviser had us using (3*seY-b)/m; where "seY" is the standard error in the Y-intercept, "b" is the Y-intercept, and "m" is the slope. (10*seY-b)/m would be the LOQ. This is similar to diluting a standard until the S/N is at 3 for LOD or 10 for LOQ. I have seen this calculation get funny when you have a poor intercept though so it may not work well for your application. There is a very detailed article about calibrations and quant/detect limits here:
http://www.ltrr.arizona.edu/~jburns/Art ... asslav.pdf
However, I had a hard time deciphering this statistician's language. Maybe you will have more luck.

Ty

Thanks Ty

I was specifically interested in the ICH paragraph 6.3.2 which uses the standard deviation of the calibration slope. This easily gives negative LODs, but only when the calibration is really repeatable. I have repeatabilities for 0.1 ng, but they only tell me that the LOD is lower than that.

Peter

[Peter Apps]

It seems like using 3*S/N will get you your LOD regardless of y-intercept. Depends on what software you're using, but Chemstation will generate S/N reports for you. Check what the S/N of your lowest standard is, use that to estimate the concentration that will give you 3*S/N. Make up spikes at that concentration and verify.

However, to be honest, this is just a practical method I use for myself to know the LOD. I can't provide a citation for it. Though, I don't even think the "official" methods make much sense. For example, the EPA method (EPA 40 CFR 136, Appendix B, revision 1.11) basically says to make 7 spikes at a concentration that will give you 2.5-5 S/N, analyze them, and take the standard deviation. 3*st.dev. gives you the LOD. I have found this to give artificially low LODs, but I suppose that's a discussion for another time.

Thanks.

Three times the basline noise (or 3 times the blank s.d.) gives very optimistic LODs since it assumes that there is no variability anywhere else in the analysis. Running the whole analysis multiple times at lower and lower levels gives the most robust and conservative estimate, but it is impractically time consuming for many applications.

I am doing it all in Excel - so I can see what the software is up to before it spits out a value.

Peter

[MMJ88]

Three times the basline noise (or 3 times the blank s.d.) gives very optimistic LODs since it assumes that there is no variability anywhere else in the analysis. Running the whole analysis multiple times at lower and lower levels gives the most robust and conservative estimate, but it is impractically time consuming for many applications.

Yeah, I agree. What I meant was getting actual peaks where the peak is at least 3 times S/N, rather than extrapolating upwards from noise/blank, which I agree is definitely optimistic. So I guess I was suggesting pretty much what you put forth in the second part of your post, the weakness of that being, as you say, not always practical time-wise. But I do think that's the way to get a realistic LOD.

Best of luck with your stuff.

[lmh]

I missed, this, but I'm pretty sure you should not be adding the intercept when calculating the LOD anyway. One reason is that para 6.3 in Q2(R1) doesn't (in my fairly elderly print-out). But the other is this:

Imagine a situation where I have a peak at zero injection (i.e. my intercept is above zero), but adding increasing amounts of analyte doesn't actually increase the signal at all. My calibration curve is always above zero, but has a slope of zero. Using the equation LOD = intercept + 3 . sd of regression I now have a LOD that's above zero even though I have a method that doesn't measure anything at all and always gives the same result irrespective of the sample! This can't be right...

As regards optimism, I agree entirely: I wouldn't feel happy trusting a LOD unless I injected a few samples at the LOD and saw a peak. But even then, if my LOD aims to offer only a 1% error rate, I'm not going to inject 100 samples in hopes of being wrong once!

[lmh]

Incidentally, I'd be more worried about why my intercept was negative. If it's negative because the calibration curve (signal = y) is upwards-concave at low concentration but you're fitting a straight line, then it will overestimate amounts at low concentration. I suppose this happens when the analyte tends to stick to a limited number of binding sites in glassware and pipettes (and is titrated out at low conc.), but I suspect you know a lot more about it than I do. I never know how to handle LODs with non-linear calibration curves. But fortunately I work in academia where most people wouldn't know a LOD they tripped over it.

[Peter Apps]

Hi lmh

The calibrations are nicely linear (high r-squared and no trend in the residuals) and the range is only two orders of magnitude with seven points, so I think that the non-zero intercepts (I have another set with a different injection set-up where they are all significantly positive) simply reflect consistent bias on the x axis - the quantity injected is more or less than the expected value and so the line moves right or left, but stays parallel to where it "should" be.

The rationale behind adding the absolute value of the intercept when the intercept is negative is that there has to be that much analyte there before the signal gets to zero, and then enough on top of that to be detected vs the noise.

In your imaginary case of the non-responsive detector you will be dividing the SD of the intercept by the slope - and getting infinity, and if you add the intercept to that you still have infinity !

I think that LODs and LOQs as currently expressed are a case of modern analytical capabilities running away from simple minded statistical models - the instruments are now so good that an overwhelming portion of the variability is in the sample prep - even if it is only dilute and shoot, and then the only sensible estimate of either limit comes from actually running replicate samples with known analyte content.

Peter

[lmh]

Thanks for such a full response.

(1) ooh, yes, I see what you mean. I need to think about this, but I sort-of feel that whether you include the intercept or not should depend on the reason why the curve missed the origin. If, for example, it misses because 1pg of sample analyte always gets lost for some reason (titrated out by binding sites on glassware, or whatever), then absolutely, you are totally right, the intercept must be included or very easily you could claim to detect less than 1 pg. On the other hand if it's just that the detection system always, reliably, adds 100 units of peak area to all samples, under all conditions, then surely it's OK to ignore this. The question we're asking is whether population B (sample containing LOD-amount of analyte) is significantly different to population A (blank sample), and what offset both have shouldn't matter (a t-test looks only at the difference of means, not their absolute value).

(2) But I agree this is all a load of very enthusiastic theory, and doesn't account for real life. Fortunately I'm in an environment where mostly people are using biological replicates, and biological variation dominates. I only really get involved in LODs when the answer is "not-detected", and someone wants to be sure I had sufficient sensitivity to see a biologically relevant amount.

[Peter Apps]

Hi lmh

Good points as usual !

On 1) - I agree completely, but paradoxically the section of the calibration that really tells us what is going on is right at the lower end, where random variation is relatively the largest and so deviations from any particualr relationship are hardest to detect .

There is some good material on the statistics of trace analysis - if I can find the links again I'll post them.

My approach is that if I am scraping along at the method's LOD or LOQ most of the time then some method development might be in order.

Peter

[lmh]

Thanks again! I learn from threads like this...