"Error" on element by difference

[Mike Jercinovic]

Maybe this has come up before, but we occasionally need to analyze an element by difference.  The question is how to assess the error on the computed concentration of the element by difference.  Seems like this is effectively the error on the total, so for an oxide, you should compute the individual precision for each element in wt.% and convert those to oxide wt.%.  Then these precision values for each element need to be propagated (as a sum of oxides), so presumably then the square root of the sum of the squares.  Am I thinking about this right?

Mike J.

[Probeman]

I think that could work if you're asking about predicting error on a single analysis.

That value could then be compared, I think, to the std dev of the average of the element by difference on a homogeneous sample for a group of analyses.

[Nicholas Ritchie]

Mike,
  Computing the error on element by difference by only considering precision will greatly underestimate the true uncertainty. This technique will totally ignore the uncertainties introduced by the model-based uncertainties in the matrix correction which are critical when computing differences of mass-fractions. The precision error (Type A, counts) is likely to be swamped by the systematic error (Type B, matrix correction).  Since all the element mass-fraction uncertainties are correlated, the optimal way to perform this calculation is that described in Ritchie, N. W. (2020). Embracing uncertainty: Modeling the standard uncertainty in electron probe microanalysis—Part I. Microscopy and Microanalysis, 26(3), 469-483. and Ritchie, N. W. (2021). Embracing Uncertainty: Modeling Uncertainty in EPMA—Part II. Microscopy and Microanalysis, 27(1), 74-89..  Since implementing this analytical model is a nightmare, a Monte Carlo uncertainty propagation is far easier to implement.  You sample over the uncertainty distribution of MACs and other input parameters to estimate uncertainty in the matrix correction due to uncertainty in the input paramteters.  This neglects systematic errors due to the base model (XPP vs PAP vs CITZAF vs ...) but this can be included by running the Monte Carlo uncertainty propagation over various models.  However, in most cases, the simpler model discussed in Ritchie, N. W., & Newbury, D. E. (2012). Uncertainty estimates for electron probe X-ray microanalysis measurements. Analytical chemistry, 84(22), 9956-9962. which overlooks correlation between the elements is probably adequate.  It will certainly be a lot more realistic that just assuming precision errors.

[Probeman]

As Nicholas mentions there are counting uncertainties, and then there are MAC uncertainties, and of course that includes matrix correction uncertainties.

I think Mike was asking about calculating "counting" uncertainties for an element by difference by using the counting statistics of the measured elements, which is why I suggested comparing that estimate to measured average uncertainties for an element by difference in a homogeneous material.

If we're attempting to ascertain other uncertainties such as accuracy errors, then yes, we need to look at MACs and matrix correction uncertainties. In CalcZAF and Probe for EPMA there is a "Use All Matrix Corrections" checkbox which now has 11 different matrix corrections.

https://smf.probesoftware.com/index.php?topic=1362.msg9789#msg9789

Using that option one can see the variance across the various matrix corrections (some only included for historical comparisons), to get an idea on accuracy errors, which as Nicholas mentions can be the largest source of error in such calculations. 

I am reminded of the water by difference calculations that some geologists perform, where if one neglects to include the water by difference in the matrix correction for hydrous glasses, one can easily obtain an error in the water estimate of around 25% relative (say, 4% H2O vs. 5% H2O):

https://smf.probesoftware.com/index.php?topic=11.msg235#msg235

Then there's also the uncertainties of beam sensitivity effects, which we attempt to correct for using a TDI correction... in hydrous glasses (which are very beam sensitive), one must correct for both matrix effects on water by difference and also correct for TDI effects.

In CalcZAF there is an option to calculate error histograms for all matrix corrections and all MAC tables (11 * 7 = 77 error distributions:



for a standard material. This method was developed at the request of Paul Carpenter but I haven't really tried it out beyond that.

[Mike Jercinovic]

Thanks for the feedback John and Nicholas.  Yes, in this case I am just thinking of counting statistics for a single line (data point) as PfE does with the Percent Analytical Relative Error if I understand that correctly.  Of course, for assessing a full picture of the accuracy of a measurement, then we are into the amazing "embracing uncertainty" contributions that Nicholas has done.  As expected, that beautiful rigorous approach will expand to a very large total uncertainty which is hard to get a grip on in day-to-day probing.  We try to mitigate this somewhat by running secondary standards, consistency references, etc. to at least be able to say that our calibrations, etc., are reasonable. But the ultimate 'correctness' of a particular result is always subject to not only knowledge about the variables we at least think we understand along with the potential ones we might not even be aware of.  Analytical relative error us useful, and Cameca's Peaksight does something similar with a single point 3-sigma wt. % std. deviation. These are easy thing to put in a file and send to someone along with concentrations, but it's not always easy to convey an understanding that this is really only counting statistics and does not actually fully represent the total uncertainty.  Anyway, if we can look at a percent analytical relative error for measured elements, then maybe there is some value in assessing the precision on the total wt. % as it might relate to an element included by difference.

[Probeman]

Thanks for the feedback John and Nicholas.  Yes, in this case I am just thinking of counting statistics for a single line (data point) as PfE does with the Percent Analytical Relative Error if I understand that correctly. 

Yes, the analytical precision is based on the unk and std peak to background ratios for a single point (or pixel). Here is a discussion we had a while back on this:

https://smf.probesoftware.com/index.php?topic=1307.0

The question is how to assess the error on the computed concentration of the element by difference.  Seems like this is effectively the error on the total, so for an oxide, you should compute the individual precision for each element in wt.% and convert those to oxide wt.%.  Then these precision values for each element need to be propagated (as a sum of oxides), so presumably then the square root of the sum of the squares. 

As I suggested previously, I think you should try your idea (above) and compare it to the standard deviation calculated from a replicate set of element difference measurements on a homogeneous material.

The idea being that a replicate set of measurements should reproduce the counting statistics you calculate for an element by different on a single point.

[Nicholas Ritchie]

It might be helpful to change vocabulary.  Instead of referring to the `error` maybe `precision` might be a better term to describe this calculation - as in `estimating the precision of element-by-difference.`

[Mike Jercinovic]

Right Nicholas, I agree 100% +/- 0.0

[Probeman]

It might be helpful to change vocabulary.  Instead of referring to the `error` maybe `precision` might be a better term to describe this calculation - as in `estimating the precision of element-by-difference.`

Yes. That is why I first asked him if this was about "counting statistics" on a single point for the element by difference.

I still think the best bet is to perform replicate measurements of the element by difference on a homogeneous sample and then you have the answer, assuming a roughly similar composition...