Estimating uncertainty of a calibration curve

[JWF239]

I am using FTIR to quantify gas contaminants in air and oxygen samples using a gas cell. I have my calibration curves completed and LODs and LOQs calculated using the blank determination method (LOD = ave of blanks + 3 * Std Dev of blanks and LOQ = ave of blanks + 10 * Std Dev of blanks). For some of the curves I have separate LCS samples at the maximum allowable concentrations for the certifications I am seeking and I have calculated uncertainty by compiling 42 scans for these components and then find a relative standard deviation (std dev / average), incorporate my k factor and then turn it into a % for percent uncertainty.

The problem is that I don't have an extra standard other than what I used to create the curves for a few of them. I have a blank determination data set, but if I try to turn it into a relative value and percentage then I end up dividing by a value that is supposed to be zero. Since it is very close to it I end up with RSDs of like 300% which is obviously not applicable to the whole curve.

My idea was to take every calibration point on these curves and compare the actual and calculated value and just compile these into a set to be used for uncertainty calculations. Is this a valid way, or is there a better way to estimate uncertainty in a curve without other standards?

[Peter Apps]

By coincidence (?) this question came up a couple of days ago; viewtopic.php?f=5&t=42281&p=202651#p202651.

Does that give you what you need ?

Peter

[JWF239]

By coincidence (?) this question came up a couple of days ago; viewtopic.php?f=5&t=42281&p=202651#p202651.

Does that give you what you need ?

Peter

Peter, thank you very much. This helped tremendously.

I used the S0 and S1 method on page 115 by taking the difference between the observed and input values for each point used to create the curve, squaring these values, and plotting them against the squared observed concentrations (The observed are what my instrument is actually calculating, this is what I am using as Xi, and the input values are what I told the computer each of those concentrations was to develop the curve).

Now I have a linear regression that I can use to estimate uncertainties at different concentrations that follows what I expected. This doesn't really take into account bias though, for instance, I know my CO2 curve has a bias of 14 ppm at 1000 ppm, and I have verified that my PT results would be more correct if I applied this bias. But I can't just say the whole curve has a bias of 14 ppm, because if I was at 100 ppm it would certainly be less. So now I am trying to figure out how to implement the bias into my data along with my uncertainty.

For some of my components I have compiled data at both 0 ppm, as well as another concentration. For instance, I have 42 scans used to compile a bias for CO at 0 ppm, and then another 42 scans of CO at 5 ppm. I was going to just plot these two and obtain another linear regression for my bias but I would only have two points, and some of my components I would only have 1 point.

Also, I used the blank determination method as described above to calculate LODs and LOQs, but lets say my ethane LOQ is 0.6 ppm, but my average of the blanks was -0.02. Does that mean my LOQ is REALLY 0.6 ppm, or 5.8 ppm?

The bias I believe is only really important if it would make an otherwise failing sample appear to pass so I could just say "The CO2 bias is 14 ppm at 1000 ppm, so any sample over 986 is considered failing". But I am trying to make the methods so that people that aren't as familiar with FTIR can run them and I would like them to do the same processing to all data, every time so if I can find a way to do a similar regression to have a formula so I can calculate bias it would be best.


It may not seem like it is much difference but my CO2 curve has a blank average of -100 ppm but a standard deviation of only 2.675. If I used those points in the uncertainty determination as described on page 115 then it would see that every individual measurement is around -100 ppm, and then compare it to 0 which would give a huge uncertainty at 0, but it wouldnt say anything to the ability to accurately hit -100.

Sorry for bombarding you with this, I will probably make another thread to discuss it but I am still working through it myself, just thought I'd throw it on here as well. Any bias information you could help me with would be greatly appreciated!

[Peter Apps]

Whether bias changes with concentration is something you really need to determine empirically by measuring the concentration in some reference materials.

On a pedantic note - any number for bias should have a + or - sign to show whether your results are high or low.

Peter

[JWF239]

Whether bias changes with concentration is something you really need to determine empirically by measuring the concentration in some reference materials.

On a pedantic note - any number for bias should have a + or - sign to show whether your results are high or low.

Peter

So what would I do if I did have reference standards? I have the ones used to create the curves, so those plus the blank would give me at least 3 points for every curve to make another bias regression, but isn't that what I am doing when I follow page 115 in the eurochem guide you referenced?

For instance, my CO2 blank values with n=4 have an average of -100 ppm but a standard deviation of only 2.675. So if I plotted those using that method, the actual would be 0 ppm but the measured would be -100 ppm.

This method I am using would make it look like I had an uncertainty of 100 ppm in the absence of the substance, but really shouldn't that come from the standard deviation of the points instead? It seems more like what I am plotting here is the bias because I am plotting what the curve calculates for a scan vs what I told it it was. Isn't that exactly bias?

I want to make it a percentage by taking the difference divided by the average measurement of it (in some cases this would just be one half of the difference), originally thinking it would be telling me bias, but wouldn't THAT actually be giving me % uncertainty?


Sorry for bombarding you, I have just been working on this for awhile and really want to submit it for accreditation.

[Peter Apps]

If a blank gives a -100 ppm result and a 1000 ppm standard gives a 986 ppm result your absolute bias bias in increasing with decreasing content. Which is unusual.

The other odd thing is that the rsd of the blank result is so low - usually an LOQ is defined as the concentration that generates results with an rsd of less than 10%. so zero ppm (or -100 ppm) is above your LOQ.

I fear that there might be something wrong with your calibration of signal vs CO2 content.

Peter

[JWF239]

If a blank gives a -100 ppm result and a 1000 ppm standard gives a 986 ppm result your absolute bias bias in increasing with decreasing content. Which is unusual.

The other odd thing is that the rsd of the blank result is so low - usually an LOQ is defined as the concentration that generates results with an rsd of less than 10%. so zero ppm (or -100 ppm) is above your LOQ.

I fear that there might be something wrong with your calibration of signal vs CO2 content.

Peter

Peter, I am using an IR region that is interfered with by water vapor. I have a 10 m path length cell and since CO2 absorbs so strongly in the 2200 cm^-1 region it saturates around 100 ppm. I have a separate CO2 curve for measuring levels under 100 ppm using that region that can much more accurately measure concentrations close to zero. I am using the region around 3800 cm^-1 for this curve to measure concentrations that would otherwise saturate on the low range CO2 curve. I have passed over 8 CAPT blind PT samples for high range CO2 using the curve with the -100 ppm blank average, as well as obtained the 987 average from a separate reference material with 1000 ppm concentration.

I am fairly certain the -100 ppm bias of the blank is due to natural interferences from water vapor, but since this curve isn't meant to quantify concentrations below 100 ppm I did not think the bias at 0 ppm really mattered since I can measure my range so well and could attribute it to a known interference. This is actually the only curve I am not applying the blank determination method for LODs and lOQs to since it doesn't apply to that concentration but I admit it does make the LOQ appear to be less than 0 before any bias is accounted for. I did not use any 0 ppm points in the creation of the calibration curve. For instance, using the S0 and S1 method described in the Eurochem manual gave me

U^2 = 0.0001(x^2) + 165.37

With my lowest point on the curve being 350 ppm and the highest point on the curve being 1400 ppm. So at 1000 ppm my uncertainty would be about 16 ppm, with a K factor of 2 that makes my reportable expanded uncertainty at 1000 ppm CO2, according to that method, +/-32 ppm. Where my requirement is +/- 50 ppm.

Separately, I have a certified standard at 1000 ppm that was not included in the creation of the curve so therefore not in the uncertainty linear regression done above. When I compiled n=42 runs of this standard together I have a standard deviation of 12 ppm, which with a K factor of 2 gives me an expanded uncertainty of +/- 24 ppm.

Considering this is a different method than used before and we got 24 ppm and 32 ppm, plus I can pass all the CAPT blind PTs I believe the curve does what I need it to.


So, in a final application for certification of this curve I now have a way to report uncertainty at any concentration covered by the curve. I also KNOW that this curve has a bias of -14 ppm at 1000 ppm seen from a certified reference material study of n=42, as well as verified in CAPT pt studies.

I just need a way to consider myself covered for bias and then I can play around with it more as I go but my boss wants me to have the application in asap.

Basically, would it be acceptable to say that from my reference material study this curve exhibits a bias of -14 ppm and that 14 ppm should be added to all results, and ignore the @ 1000 ppm caviet under the justification that the reference material study was performed at the pass/fail cutoff and therefore even if my bias at 700 ppm is really 6 ppm instead of -14 ppm, it doesn't really matter and is covered under my uncertainty?

[JWF239]

I know I am getting wrapped up in details and minute statistical definitions and I could spend forever continuing to, but really I am asking for help getting out of that and to a point of "completion", as far as having applied for certification of specific compounds at specific concentrations. Once I get to that point I can always continue to improve.

[Peter Apps]

The details are important, but only as long as the big things are in place to support them, and I need to take you back a couple of steps.

You can only work in the range between your highest and lowest standards, so the uncertainty and result from the blank are irrelevant to either curve.

What equation are you using to calculate CO2 content from analyzer outputs ?

The uncertainty of the calibration adds to the repeatability uncertainty of individual measurements.

Peter

[JWF239]

The details are important, but only as long as the big things are in place to support them, and I need to take you back a couple of steps.

You can only work in the range between your highest and lowest standards, so the uncertainty and result from the blank are irrelevant to either curve.

What equation are you using to calculate CO2 content from analyzer outputs ?

The uncertainty of the calibration adds to the repeatability uncertainty of individual measurements.

Peter

Peter,

I am testing both oxygen samples as well as breathing air samples. The oxygen samples have a CO2 limit of 5 ppm so I have one curve using the CO2 peak at 2200 cm^-1 with standards of 5 ppm and 10 ppm, as well as blanks. For this curve I used the blank determination method I described above to determine the LOD and LOQ as well as an uncertainty @ 0 ppm using the compiled blank scans with an n=42.

I then have a separate curve for the air samples which have a limit of 1000 ppm. This curve was created with standards of 500, 800, 1000, and 1200 ppm. This curve uses a combination band region near 3600 cm^-1 where water vapor also happens to absorb. Since I have a 10 m path length cell any little bit of water vapor is going to cause some sort of area to be present in blank scans using this curve, but the curve was not created using blank scans; I have 13 of these curves, my 42 scans I keep mentioning are just my blanks that I collected making these curves all compiled into one data file. I then just ran each curve against the set and compiled my data from there. I only included the blank study on this high range CO2 curve because it was so easy once I was doing the others and I was curious.

Here is a link to the calibration regions and curves with linearities and equations included. You can also get a sense of my calibrated regions.

http://imgur.com/a/kB4Jo

The equations displayed on the graphs are what I am using to solve for concentrations in unknowns, with the x being the value obtained from the computer integrating using my specified settings. The Y is what I would currently report but I am trying to figure out how to incorporate bias first.

The uncertainty I report is determined by the S0 and S1 method described in Eurochem page 115. My concentrations that I square (my x) are the calculated values (the Y's from equations in images) give me for each point used to create that calibration curve. The Y that I use in this instance is the calculated value (new x) minus the value that I told the computer to use when creating the curve.

My uncertainty equation is therefore then made completely from the points used to make the curve. When I was talking about comparing my two high range CO2 uncertainties, they were both examinations of the same high range curve, one through this uncertainty plotting of the calibration points, and the other through the examination of a certified standard at 1000 ppm.

For my low range CO2 curve, and for some others like ethylene, I only have the uncertainty determined through the comparison of curve points to the input values and the uncertainty of blank scans through my n=42 blank data set.

Here I have included the uncertainty determinations for these two curves. They don't look very linear because they are biased toward the regions I actually have standards for, as well as based on the uncertainty of the standards themselves. Never the less, it gives me a good example if the uncertainty changes with concentration positively as expected, or negatively because the effects on the blanks are bigger than any uncertainties in my standards.

http://imgur.com/a/3qhJ3

The one equation for high range CO2 uncertainty was verified through the separate reference standard that I described when discussing 32 or 24 ppm uncertainties.

This all leads me to believe these are valid estimates of my uncertainty when reporting concentrations that fall within the calibrated range, now I am just trying to wrap up a good way to include the bias' that I know would make my measurements look better, but mostly just to cover failures. When my 1000 ppm standard shows up as 987 ppm over 42 tries then I need my reports to show that 990 ppm is failing, correct? I feel like I have all the parts, I am just trying to combine them into system for obtaining a reportable number.

[JWF239]

So to clearly answer the question:

I am simply pressing a button to obtain my concentrations now; I am not using any equation to calculate content from analyzer outputs. The computer is doing that for me through the equation on the calibration curve images.

[Peter Apps]

Comments are in blue

The details are important, but only as long as the big things are in place to support them, and I need to take you back a couple of steps.

You can only work in the range between your highest and lowest standards, so the uncertainty and result from the blank are irrelevant to either curve.

What equation are you using to calculate CO2 content from analyzer outputs ?

The uncertainty of the calibration adds to the repeatability uncertainty of individual measurements.

Peter

Peter,

I am testing both oxygen samples as well as breathing air samples. The oxygen samples have a CO2 limit of 5 ppm so I have one curve using the CO2 peak at 2200 cm^-1 with standards of 5 ppm and 10 ppm, as well as blanks. For this curve I used the blank determination method I described above to determine the LOD and LOQ as well as an uncertainty @ 0 ppm using the compiled blank scans with an n=42.with only two points you do not know if the calibration is linear, and the negative result for the blank with its very low rsd suggests strongly that it is not. If you and the regulators are happy with a two-point curve then you can empirically estimate uncertainty of calibration + repeatability uncertainty by running multiple replicates at 5 and 10 ppm, but this will not account for line curvature. More points along the line would allow you to estimate uncertainty die to curvature or to fit a curved calibration whose residuals would be lower than for a straight line

I then have a separate curve for the air samples which have a limit of 1000 ppm. This curve was created with standards of 500, 800, 1000, and 1200 ppm.you can at least see if the line is straight or curved This curve uses a combination band region near 3600 cm^-1 where water vapor also happens to absorb. Since I have a 10 m path length cell any little bit of water vapor is going to cause some sort of area to be present in blank scans using this curve it is also going to cause extra signal in standards and samples, giving signals that are biased high. The impact on the results for CO2 depends on whether the water content in blanks, samples and standards is equal and always the same. If for e.g. you calibrate using a standard with more moisture than is in your sample the result for CO2 in the sample will be biased low. In the far more likely scenario of variable water content you will have increased variability in signal and results, as well as bias, but the curve was not created using blank scans; I have 13 of these curves, my 42 scans I keep mentioning are just my blanks that I collected making these curves all compiled into one data file. I then just ran each curve against the set and compiled my data from there. I only included the blank study on this high range CO2 curve because it was so easy once I was doing the others and I was curious.

Here is a link to the calibration regions and curves with linearities and equations included. You can also get a sense of my calibrated regions.

http://imgur.com/a/kB4Jo

The equations displayed on the graphs are what I am using to solve for concentrations in unknowns, with the x being the value obtained from the computer integrating using my specified settings. The Y is what I would currently report but I am trying to figure out how to incorporate bias first.is x the signal (IR absorption), or has the computer already done some calculations on it ?. From what you say in the next paragraph about the computer subtracting values it sounds is if you are processing the data twice - once to get x and again to get Y from x.

The uncertainty I report is determined by the S0 and S1 method described in Eurochem page 115. My concentrations that I square (my x) are the calculated values (the Y's from equations in images) give me for each point used to create that calibration curve. The Y that I use in this instance is the calculated value (new x) minus the value that I told the computer to use when creating the curve.now I am confused. In the previous paragraph x is the signal from the instrument. You should calculate concentrations from x by using the calibration equation that you got by plotting signal against standard concentration (you must know the concentration of the standard before you measure it). So what value of what are you telling the computer to use, and what are you subtracting it from ?

My uncertainty equation is therefore about comparing my two high range CO2 uncertainties, they were both examinations of the same high range curve, one through this uncertainty plotting of the calibration points, and the other through the examination of a certified standard at 1000 ppm.

For my low range CO2 curve, and for some others like ethylene, I only have the uncertainty determined through the comparison of curve points to the input values and the uncertainty of blank scans through my n=42 blank data set.

Here I have included the uncertainty determinations for these two curves. They don't look very linear because they are biased toward the regions I actually have standards for, as well as based on the uncertainty of the standards themselves. Never the less, it gives me a good example if the uncertainty changes with concentration positively as expected, or negatively because the effects on the blanks are bigger than any uncertainties in my standards.your calibration points are scattered on both axes - you have variability of both signal (as expected ) and CO2 content of the standards. Why are the standards varying ?, and how do you know (without measuring) what their CO2 content is ?

http://imgur.com/a/3qhJ3

The one equation for high range CO2 uncertainty was verified through the separate reference standard that I described when discussing 32 or 24 ppm uncertainties.

This all leads me to believe these are valid estimates of my uncertainty when reporting concentrations that fall within the calibrated range, now I am just trying to wrap up a good way to include the bias' that I know would make my measurements look better, but mostly just to cover failures. When my 1000 ppm standard shows up as 987 ppm over 42 tries then I need my reports to show that 990 ppm is failing, correct?No, not correct, if your method has a systematic relative bias of 1.3% there is something wrong with it, either to do with the measurements themselves, or with the calibration I feel like I have all the parts, I am just trying to combine them into system for obtaining a reportable number.

[Peter Apps]

So to clearly answer the question:

I am simply pressing a button to obtain my concentrations now; I am not using any equation to calculate content from analyzer outputs. The computer is doing that for me through the equation on the calibration curve images.

How you do the calculation is not important - what is important is how you get your x value in your previous post, and how you calculated the calibration equation that you use to get from x to Y.

Peter

[JWF239]

How you do the calculation is not important - what is important is how you get your x value in your previous post, and how you calculated the calibration equation that you use to get from x to Y.
Peter

Peter,

Thank you again for taking the time to go through all of this. I know it is a lot and I really appreciate it. I don't believe I am doing a good job clearly explaining myself here so I will attempt to be more concise in this post.

Concern"with only two points you do not know if the calibration is linear, and the negative result for the blank with its very low rsd suggests strongly that it is not. If you and the regulators are happy with a two-point curve then you can empirically estimate uncertainty of calibration + repeatability uncertainty by running multiple replicates at 5 and 10 ppm, but this will not account for line curvature. More points along the line would allow you to estimate uncertainty die to curvature or to fit a curved calibration whose residuals would be lower than for a straight line"

ResponseI am not using only 2 points as can be seen in the imgur post I shared. For the low range curve I have two certified standards plus the "100%" blanks. Each certified standard is then used to simulate many different concentrations by varying the pressure in my gas cell. This means I only have two certified sources, but over 50 points covering a range of concentrations.

Example: I collect a scan of my 5 ppm standard at 0 PSIg, this point is said to be 5 ppm. I then collect a scan of the 5 ppm standard at 2 PSIg. I then calculate a new simulated concentration Cs = 5 ppm * (14.7 PSIg + 2 PSIg)/14.7 PSI. So this point would be 5.68 ppm. I realize this also expands any error in the standards but the fact that I can vary the pressures like this from two different sources and it still be linear when including the blanks shows that the standards are what they claim to be and that varying the pressures to simulate concentrations doesn't add much uncertainty to the curve. My linearity for the curve is still 0.999.

So far everything in this post relates to a single curve used to measure CO2 in the range from 0 to 11 ppm using an integration region that is not interfered with by water. I used two different methods to check the uncertainty of this curve. One was compiling 42 100% scans together which gave me an uncertainty of +/- 0.07 ppm with a blank average of 0.046 ppm.

Separately, I did the S0 and S1 method. I took the points used in the creation of this calibration curve and for each point I plotted the squared value of the concentration obtained when I use the curve to calculate the concentration on the x axis, vs the squared difference between the concentration plotted on the x axis and the concentration I told the curve it was supposed to be. The result of this is the top image in my second imgur link. So every point on that curve corresponds to a point on my calibration curve. Specifically, it is plotting the calculated values for my calibration points vs the difference to what they are supposed to be. Using this method and solving for my uncertainty at 0 ppm (to compare to my other method for determining uncertainty) as described in The Eurochem manual page 115 gave me an uncertainty at 0 ppm of +/- 0.088 ppm. This is very similar to the other uncertainty value of +/- 0.07 ppm that was found using a completely different method and scans.



Concern"it is also going to cause extra signal in standards and samples, giving signals that are biased high. The impact on the results for CO2 depends on whether the water content in blanks, samples and standards is equal and always the same. If for e.g. you calibrate using a standard with more moisture than is in your sample the result for CO2 in the sample will be biased low. In the far more likely scenario of variable water content you will have increased variability in signal and results, as well as bias"

ResponseSo here we are switching gears to a completely different curve, still used to quantify CO2, but at much higher concentrations. The levels of CO2 I am interested in saturate the region used in the low range CO2 curve too quickly. Therefore I am using this region that has water interference because it was what gave me the best results experimentally. I determined this by compiling all my CAPT proficiency test samples and several SRM samples into a file. I'd then change the curve and recalculate values for all the CAPT samples and SRM concentrations. I used the region that gave me the best combination of R%Ds for all the samples and the best R^2 value for linearity.

The "far more likely scenario" you mention is of course what is actually happening here. Here is a very quick and generic image as an example.

http://imgur.com/a/0nM30

Now, if this theoretical curve had some interference, such as water, this would mean that the area for the 1000 ppm point would actually be greater than it should be. This would shift the point to the right, but not up. If this is the case for every point used to make the curve then this would give the curve a y-intercept lower than expected. This is exactly the case with my high range CO2 curve, which has a Y-intercept of -100. This number is very repeatable as seen from my extremely low standard deviation of under 3 ppm. You seemed really concerned with this in a previous post but I just don't see how this is a problem. This curve and the region associated with it are not mean to quantify values below 350 ppm, so the fact that the bias at 0 ppm is -100 ppm shouldn't matter considering it is a repeatable phenomenon and follows what was expected. I mean, isn't that exactly what bias is supposed to represent? The two CO2 curves, low range and high range, act more as a piecewise function.

This high range CO2 curve has been vary accurate at quantifying CO2 results in CAPT PT samples which are meant to simulate real world samples near these concentrations. Additionally, all of my methods have maximum allowable water concentrations reported as dew points. Every sample will have a dew point with it; if the dew point exceeds allowed values then I don't even need to run the CO2 because the sample already failed and I already know the high levels of water vapor will interfere with the high range CO2 readings.


Concern"is x the signal (IR absorption), or has the computer already done some calculations on it ?. From what you say in the next paragraph about the computer subtracting values it sounds is if you are processing the data twice - once to get x and again to get Y from x."

And

"How you do the calculation is not important - what is important is how you get your x value in your previous post, and how you calculated the calibration equation that you use to get from x to Y."

ResponseThis is a link to images of the actual calibration curves and actual equations used to calculate concentrations. http://imgur.com/a/kB4Jo

For the low range CO2 curve the equation I use to quantify CO2 is Y = 1.135x + 0.012; Where x is the integrated area from the scan and Y is the calculated concentration. This equation was constructed by the software. I put in my 50 or so calibration points, tell the computer what concentration each of those areas is supposed to represent, and it spits out this equation. The x in this case is literally just the area. The only calculations involved to this point are done to the interferogram to convert it into a spectrum. These include selected apodization (Happ-Ganzel), zero fill (8) and other spectral parameters. These obviously affect the resulting spectra but these spectral settings are always applied the same way. I don't believe these are considered data processing steps on the x variable as much as they are spectral settings.


Concern"now I am confused. In the previous paragraph x is the signal from the instrument. You should calculate concentrations from x by using the calibration equation that you got by plotting signal against standard concentration (you must know the concentration of the standard before you measure it). So what value of what are you telling the computer to use, and what are you subtracting it from ?"

Response I can see how this was confusing. We are switching gears again completely here. At this point the calibration curves are complete. They were indeed created from standards of known concentration by plotting the instrument signal on the x-axis and these known concentrations on the y-axis. I THEN created completely separate uncertainty graphs, seen here

http://imgur.com/a/3qhJ3

These are separate graphs from my calibration curves and are meant to tell me different things. Every point on these uncertainty curves corresponds to a specific point on the respective calibration curve. So the x-values on these uncertainty curves are actually the Y-value outputs of the calibration curves. In other words, I take a calibration point, say 5 ppm, and run that point back through my calibration curve. So for my low range CO2 curve, Y = 1.135x + 0.012, It is literally taking the area that I associated with 5 ppm, and then calculating a concentration for it. Possibly 5.3 ppm for instance. All of these "calculated" concentrations are then squared and used as my x values on the uncertainty charts. The Y-value for this point would be (5.3 ppm - 5ppm)^2.

So for the uncertainty graphs, every single point corresponds to a point on the respective curve. The X-values for the calibration curves are areas, the Y-values are input concentrations from known standards. For the uncertainty curves, the X-values are the calculated concentrations for these "known concentration" points, and the Y-values are simply the difference between this value and what was expected.

Concern"your calibration points are scattered on both axes - you have variability of both signal (as expected ) and CO2 content of the standards. Why are the standards varying ?, and how do you know (without measuring) what their CO2 content is ?"

Response

I assume you are referring to the uncertainty graphs here, and not the actual calibration curves themselves. Under that assumption, here are the uncertainty graphs again and an explanation.

http://imgur.com/a/3qhJ3

Looking again at the low range CO2 here, the Y-values aren't measured concentrations, they are differences between calculated and expected values. You know how I said I had two SRM standards for my low range CO2 curve, one at 5 ppm and one at 10 ppm? The variation in the x-axis is due to the slight pressure differences these scans were run at. So you can see clumps around 25 and around 100 on the x-axis. This is because these points were made using standards at 5 and 10 ppm and then varying pressure slightly. The values close to 0 on the Y-axis aren't points with little or no signal, they are points with little or no difference between what I calculated and what I input into the computer. Again, every point was made from a SRM with known concentration, so by knowing the pressure I can calculate expected values of these SRMs at different pressures to simulate more points. Of course doing this with only one standard would be linear, but here I have two standards and blanks of which are linear, I then just fill in between these three points with slight pressure variations.

You asked how do I know the CO2 content without measuring, and for these uncertainty charts (again, not the calibration curves) I don't really. My X-values (on the uncertainty charts) for each point are concentration values obtained by plugging the area associated with that point into the Y= mX+B equation from the calibration image. For low range CO2 this was the Y = 1.135x + 0.12 equation I keep coming back to. It called to use Xi in the Eurochem guide, where Xi is the individual measurements, correct? I had a similar bit of confusion with notation when looking through this because my calibration curves are Conc = M * (signal) + B, while in the eurochem guide they have it as signal = M* (Conc) + B.

They then mention the uncertainty of a predicted X due to the variability in Y, which is exactly how I have it set up now, right? When looking at the uncertainty graphs, I have some uncertainty on my concentration (x) due to variability in y (my uncertainty).


Concern"No, not correct, if your method has a systematic relative bias of 1.3% there is something wrong with it, either to do with the measurements themselves, or with the calibration"

I am not quite following here. Shouldn't any acceptable bias be determined by the method or certification being sought? And why is a 1.3% bias not acceptable if I can see it and correct for it? Isn't that exactly what a bias is? Would it be acceptable at 0.5%? And if I was correcting for it either way, why is one acceptable and the other not? If I literally only care if a CO2 level is above or below 1000 ppm with an uncertainty of +/- 50 ppm, and I know that 1000 ppm actually shows up as 987 ppm with an uncertainty of +/- 24 ppm I don't see how that isn't enough to certify concentrations above or below 1000 ppm.






I very much appreciate your help. It has been useful for me to sit down and try to explain everything I am working on to an outsider. It really forces me to think it through more clearly. Again, I am confident in these curves ability to meet the requirements of the certifications I am seeking, but I am just trying to pick a way to report bias and stick with it. I wanted to do bias by concentration like the uncertainty but it seems like I am going to just have to report a bias only when it would cause a sample to fail.

[Peter Apps]

With the further explanation of how you generate different concentrations for your calibrations I really think that you are making this much more complicated than it needs to be.

Forget all about the uncertainty of the blank. The blank is outside your calibration range and so you will never be able to report a result for a blank.

Use the Eurachem equations to calculate uncertainty of calibration.

Estimate bias by analyzing some standards (with a very strong preference for ones that you did not use to generate the calibration) and comparing your result with the known CO2 content. Correct your results from the calibration by adding or subtracting the bias as appropriate.

Peter