relating percent analytical relative error to error on concentration

[Mike Jercinovic]

Okay, I am trying to understand the statistics a bit.  If I calculate detection limits and sensitivity, I get a listing of "percent analytical relative error at 1-sigma.  This calculation is based on the standard error of the net peak as developed by Scott and Love.  What I would like to assess is the uncertainty in the estimate of composition, generally estimated using something like the Ziebold-Lifshin approach to obtaining sigma - concentration from sigma-k.  (Goldstein et al. 3rd ed. eq. 9.20). Seems like the percent analytical relative error when applied to the obtained concentration should give a value similar to the sigma-concentration.  I am probably not thinking this through properly (missing a step somewhere?), but I don't get comparable values.  Any insight would be helpful!
Mike J.

[John Donovan]

Hi Mike,
Not sure, but one thing you can do is turn on "debug mode" in the Output menu and then you can see the intermediate calculations that go into the calculation as seen here:

Line:  271 Intermediate Analytical Sensitivity Calculations (nominal beam =  1):
Element/Xray     ti ka   ti ka   ti ka   ti ka   ti ka
Bgd Count/Sec      1.0     2.6     3.4      .8     1.4
Bgd CountTime  1920.00 1920.00 1920.00 1920.00 1920.00
Bgd Raw Count  381085.1010769.1315427. 302595. 529365.
Peak Cnt/Sec       1.0     2.6     3.4      .8     1.4
Peak Cnt Time  1920.00 1920.00 1920.00 1920.00 1920.00
Peak Raw Count 380952.1014290.1315704. 302399. 531793.

And in case it helps, here is the code used for the calculation:

Code: [Select]

Function ConvertAnalyticalSensitivity2(datarow As Integer, chan As Integer, sample() As TypeSample) As Single
' Calculate percent error for a single element

ierror = False
On Error GoTo ConvertAnalyticalSensitivity2Error

Dim temp1 As Single, temp2 As Single
Dim temp3 As Single, temp4 As Single

ConvertAnalyticalSensitivity2! = 0#

' Init debug variables
bgdtime! = 0#
bgdcount! = 0#
peakcount! = 0#

' Calculate
If sample(1).BgdData(datarow%, chan%) < 0# Then Exit Function
If sample(1).OnTimeData!(datarow%, chan%) <= 0# Then Exit Function

' Determine background count time for unknown
If sample(1).BackgroundTypes%(chan%) <> 1 Then  ' 0=off-peak, 1=MAN, 2=multipoint

' 0=linear, 1=average, 2=high only, 3=low only, 4=exponential, 5=slope hi, 6=slope lo, 7=polynomial, 8=multi-point
If sample(1).OffPeakCorrectionTypes%(chan%) = 2 Then
bgdtime! = sample(1).HiTimeData!(datarow%, chan%)   ' high only
ElseIf sample(1).OffPeakCorrectionTypes%(chan%) = 3 Then
bgdtime! = sample(1).LoTimeData!(datarow%, chan%)    ' low only
Else
bgdtime! = sample(1).HiTimeData!(datarow%, chan%) + sample(1).LoTimeData!(datarow%, chan%)   ' all other off peak types
End If

Else
bgdtime! = sample(1).OnTimeData!(datarow%, chan%)   ' use on-peak time for MAN
End If
If bgdtime! = 0# Then Exit Function

' De-normalize unknown peak counts for time and beam (use corrected peak intensity + background for greater accuracy)
peakcount! = (sample(1).CorData!(datarow%, chan%) + sample(1).BgdData!(datarow%, chan%)) * sample(1).OnTimeData!(datarow%, chan%)
If peakcount! = 0# Then Exit Function
If Not sample(1).CombinedConditionsFlag% Then
Call DataCorrectDataBeamDrift2(peakcount!, sample(1).OnBeamData!(datarow%, chan%))    ' use OnBeamData in case of aggregate intensity calculation (average aggregate beam)
If ierror Then Exit Function
Else
Call DataCorrectDataBeamDrift2(peakcount!, sample(1).OnBeamDataArray!(datarow%, chan%))    ' use OnBeamDataArray in case of aggregate intensity calculation (average aggregate beam)
If ierror Then Exit Function
End If

' De-normalize unknown background counts for time and beam
bgdcount! = sample(1).BgdData(datarow%, chan%) * bgdtime!
If bgdcount! = 0# Then Exit Function
If Not sample(1).CombinedConditionsFlag% Then
Call DataCorrectDataBeamDrift2(bgdcount!, sample(1).OnBeamData(datarow%, chan%))    ' use OnBeamData in case of aggregate intensity calculation (use average aggregate beam)
If ierror Then Exit Function
Else
Call DataCorrectDataBeamDrift2(bgdcount!, sample(1).OnBeamDataArray!(datarow%, chan%))    ' use OnBeamDataArray in case of aggregate intensity calculation (use average aggregate beam)
If ierror Then Exit Function
End If

' Check for anomalies
If sample(1).OnTimeData!(datarow%, chan%) <= 0# Then Exit Function
If bgdtime! <= 0# Then Exit Function

temp1! = peakcount! / sample(1).OnTimeData!(datarow%, chan%) ^ 2
temp2! = bgdcount! / bgdtime! ^ 2

temp3! = peakcount! / sample(1).OnTimeData!(datarow%, chan%)
temp4! = bgdcount! / bgdtime!

If temp1! + temp2! < 0# Then Exit Function
If temp3! - temp4 = 0# Then Exit Function
ConvertAnalyticalSensitivity2! = 100# * Sqr(temp1! + temp2!) / (temp3! - temp4)

Exit Function

' Errors
ConvertAnalyticalSensitivity2Error:
MsgBox Error$, vbOKOnly + vbCritical, "ConvertAnalyticalSensitivity2"
ierror = True
Exit Function

End Function

Are you asking about trace elements because my take is that the analytical sensitivity calculation from Love-Scott is only useful for minor and major element concentrations?

[Mike Jercinovic]

Mainly I am trying to come up with a quick way to evaluate the standard deviation in wt.% (or ppm) for a single point in PfE.  I can work out a spreadsheet calculation with the standard and unknown info using the Ziebold-Lifshin formula, which seems to agree quite well with the Cameca formulation of StdDev. wt%.  Of course, this has to be all trace element business!

[John Donovan]

Hi Mike,
I understand. There's many ways to approach this for sure.

My preference is to utilize the t-test detection limit calculation from Goldstein et al. because it utilizes the actual measured variance and weights it for the number of points acquired. But that doesn't help you for evaluation of the single point detection limit. On the other hand one can compare the t-test (on a presumably homogeneous material) and compare that with the single point detection limit as shown here in bold:

Un   31 1920 sec on SiO2, Results in Elemental Weight Percents
 
ELEM:       Ti      Ti      Ti      Ti      Ti      Si       O
TYPE:     ANAL    ANAL    ANAL    ANAL    ANAL    SPEC    CALC
BGDS:      LIN     LIN     LIN     LIN     LIN
TIME:  1920.00 1920.00 1920.00 1920.00 1920.00     ---     ---
BEAM:   200.76  200.76  200.76  200.76  200.76     ---     ---

ELEM:       Ti      Ti      Ti      Ti      Ti      Si       O   SUM 
XRAY:     (ka)    (ka)    (ka)    (ka)    (ka)      ()      ()
   271 -.00003  .00039  .00003 -.00006  .00051 46.7430 53.2576 100.001
   272  .00010  .00039  .00022 -.00036 -.00030 46.7430 53.2570 100.000
   273  .00003  .00037  .00008  .00007  .00048 46.7430 53.2577 100.002
   274  .00002  .00016  .00015 -.00005  .00009 46.7430 53.2572 100.001
   275 -.00010  .00019 -.00002  .00016  .00009 46.7430 53.2572 100.001

AVER:   .00000  .00030  .00009 -.00005  .00017  46.743  53.257 100.001
SDEV:   .00007  .00011  .00010  .00020  .00034    .000    .000  .00067
SERR:   .00003  .00005  .00004  .00009  .00015  .00000  .00012
%RSD:  4057.70 37.3350 103.073 -404.32 193.525  .00000  .00050
STDS:      922     922     922     922     922     ---     ---

STKF:    .5621   .5621   .5621   .5621   .5621     ---     ---
STCT:   667.34 1600.07 1901.70  531.93  828.32     ---     ---

UNKF:    .0000   .0000   .0000   .0000   .0000     ---     ---
UNCT:      .00     .01     .00     .00     .00     ---     ---
UNBG:      .99    2.63    3.41     .79    1.38     ---     ---

ZCOR:   1.1969  1.1969  1.1969  1.1969  1.1969     ---     ---
KRAW:   .00000  .00000  .00000  .00000  .00000     ---     ---
PKBG:  1.00002 1.00271 1.00077  .99951 1.00155     ---     ---
BLNK#:      27      27      27      27      27     ---     ---
BLNKL: .000142 .000142 .000142 .000142 .000142     ---     ---
BLNKV: .000000 -.00125 -.00272 .000689 -.00040     ---     ---

Detection limit at 99 % Confidence in Elemental Weight Percent (Single Line):

ELEM:       Ti      Ti      Ti      Ti      Ti
   271  .00049  .00033  .00032  .00054  .00046
   272  .00048  .00033  .00032  .00054  .00046
   273  .00049  .00033  .00032  .00054  .00046
   274  .00049  .00033  .00032  .00054  .00046
   275  .00049  .00033  .00031  .00054  .00046

AVER:   .00049  .00033  .00032  .00054  .00046
SDEV:   .00000  .00000  .00000  .00000  .00000
SERR:   .00000  .00000  .00000  .00000  .00000

Percent Analytical Relative Error (One Sigma, Single Line):

ELEM:       Ti      Ti      Ti      Ti      Ti
   271  -655.0    40.4   585.3  -396.9    42.4
   272   238.0    40.3    66.2   -70.9   -72.1
   273   874.2    41.8   179.3   347.3    45.2
   274  1001.7    95.3   102.3  -547.3   250.4
   275  -226.3    80.7  -965.6   164.0   248.4

AVER:    246.5    59.7    -6.5  -100.8   102.9
SDEV:    707.2    26.4   574.8   373.8   141.9
SERR:    316.3    11.8   257.1   167.2    63.5

Detection Limit (t-test) in Elemental Weight Percent (Average of Sample):

ELEM:       Ti      Ti      Ti      Ti      Ti
  60ci  .00008  .00004  .00019  .00014  .00009
  80ci  .00013  .00007  .00032  .00022  .00015
  90ci  .00018  .00009  .00044  .00031  .00021
  95ci  .00024  .00012  .00057  .00040  .00028
  99ci  .00039  .00020  .00095  .00066  .00046

Analytical Sensitivity (t-test) in Elemental Weight Percent (Average of Sample):

ELEM:       Ti      Ti      Ti      Ti      Ti
  60ci     ---     ---     ---     ---     ---
  80ci     ---     ---     ---     ---     ---
  90ci     ---     ---     ---     ---     ---
  95ci     ---     ---     ---     ---     ---
  99ci     ---     ---     ---     ---     ---

So what I do is run a few test points on a homogeneous standard material and then hopefully that provides me with the confidence I need.

[Mike Jercinovic]

Right, that all is making good sense, but what is being asked by one of my users is not quite that.  If, you have a concentration of say 10ppm, and a detection limit of 1ppm, it is in principal detectable, but what is the std. dev. of the concentration?  So 10ppm +/- 4ppm? or is it 10ppm +/1 30ppm? (at some specified precision, 1-sigma, 2-sigma, etc.) makes a difference when comparing point to point along a line where you would like to apply some sort of error envelope.

[John Donovan]

Right, that all is making good sense, but what is being asked by one of my users is not quite that.  If, you have a concentration of say 10ppm, and a detection limit of 1ppm, it is in principal detectable, but what is the std. dev. of the concentration?  So 10ppm +/- 4ppm? or is it 10ppm +/1 30ppm? (at some specified precision, 1-sigma, 2-sigma, etc.) makes a difference when comparing point to point along a line where you would like to apply some sort of error envelope.

Yeah the detection limit is not really what you want for an uncertainty "envelope" I agree.

But let me just say something about your example above for others reading this that might be confused: if I have a 10 PPM concentration for a single line and a reported 3 sigma detection limit (which is what we report) of 10 PPM, then that 10 PPM concentration is 99% confidently detected, not merely "in principal detectable". And of course concentrations less than 10 PPM are less than 99% confidently detected, and concentrations over 10 PPM are more than 99% confidently detected.

But what would be wrong in applying the analytical sensitivity % to the concentration? So if the analytical sensitivity is 100% on a 10 PPM concentration, then wouldn't the "envelope" of uncertainty be +/- 10 PPM for 1 sigma confidence or +/- 30 PPM for 3 sigma confidence?

Am I missing something?

[Mike Jercinovic]

It absolutely seems like applying the % analytical sensitivity to the concentration should be fine.  However, it seems like it should roughly correspond to the concentration SD calculated through the Ziebold-Lifshin relationship, and this is where I am stuck at the moment.  I need to check my calculations again.

[John Donovan]

It absolutely seems like applying the % analytical sensitivity to the concentration should be fine.  However, it seems like it should roughly correspond to the concentration SD calculated through the Ziebold-Lifshin relationship, and this is where I am stuck at the moment.  I need to check my calculations again.

Hi Mike,
I'm not sure if the analytical sensitivity and standard deviation calculation results should be similar or not when dealing with trace elements.

However, if I examine statistics for some major elements, for example Si and Ca in my wollastonite standard, I see that the analytical sensitivity calculations agree pretty well with the relative standard deviations:

Un    6 Wollastonsite (Willsboro,NY) majors, random gr1, Results in Elemental Weight Percents
 
ELEM:       Ca      Si       O
TYPE:     ANAL    ANAL    CALC
BGDS:      LIN     LIN
TIME:    60.00   60.00     ---
BEAM:    20.01   20.01     ---

ELEM:       Ca      Si       O   SUM 
   193  34.354  24.581  41.721 100.656
   194  34.352  24.536  41.669 100.558
   195  34.219  24.548  41.629 100.396
   196  34.013  24.561  41.562 100.136
   197  34.352  24.481  41.606 100.439
   198  34.390  24.488  41.629 100.507

AVER:   34.280  24.533  41.636 100.449
SDEV:     .143    .040    .054    .178
SERR:     .059    .016    .022
%RSD:      .42     .16    .13
STDS:      358      14     ---

STKF:    .1693   .4101     ---
STCT:   2291.6 23654.2     ---

UNKF:    .3192   .2098     ---
UNCT:   4319.7 12104.0     ---
UNBG:    130.4    45.1     ---

ZCOR:   1.0740  1.1691     ---
KRAW:   1.8850   .5117     ---
PKBG:    34.14  269.68     ---

Percent Analytical Relative Error (One Sigma, Single Line):

ELEM:       Ca      Si
   193      .3      .1
   194      .3      .1
   195      .3      .1
   196      .3      .1
   197      .3      .1
   198      .3      .1

AVER:       .3      .1
SDEV:       .0      .0
SERR:       .0      .0

[John Donovan]

Here is the new analytical sensitivity code for the Scott and Love (1983) calculation:

Code: [Select]

Function ConvertAnalyticalSensitivity3(datarow As Integer, chan As Integer, sample() As TypeSample) As Single
' Calculate percent error for a single element using raw intensity arrays (to better handle aggregate data statistics)

ierror = False
On Error GoTo ConvertAnalyticalSensitivity3Error

Dim m As Integer
Dim temp1 As Single, temp2 As Single
Dim temp3 As Single, temp4 As Single

ConvertAnalyticalSensitivity3! = 0#

' Init debug variables
bgdtime! = 0#
bgdcount! = 0#
peaktime! = 0#
peakcount! = 0#

' Sanity checks
If sample(1).OnPeakCounts_Raw_Cps!(datarow%, chan%) < 0# Then Exit Function
If sample(1).OnTimeData!(datarow%, chan%) <= 0# Then Exit Function

' Determine peak time for unknown
peaktime! = sample(1).OnTimeData!(datarow%, chan%)

' Determine background count time for unknown
If sample(1).CrystalNames$(chan%) <> EDS_CRYSTAL$ Then

' Check if off peak element
If sample(1).BackgroundTypes%(chan%) = 0 Then  ' 0=off-peak, 1=MAN, 2=multipoint

' 0=linear, 1=average, 2=high only, 3=low only, 4=exponential, 5=slope hi, 6=slope lo, 7=polynomial, 8=multi-point
If sample(1).OffPeakCorrectionTypes%(chan%) = 2 Or sample(1).OffPeakCorrectionTypes%(chan%) = 5 Then
bgdtime! = sample(1).HiTimeData!(datarow%, chan%)                 ' high only

ElseIf sample(1).OffPeakCorrectionTypes%(chan%) = 3 Or sample(1).OffPeakCorrectionTypes%(chan%) = 6 Then
bgdtime! = sample(1).LoTimeData(datarow%, chan%)                  ' low only

Else
bgdtime! = sample(1).HiTimeData!(datarow%, chan%) + sample(1).LoTimeData!(datarow%, chan%)                 ' all other off peak types
End If

' MAN element
ElseIf sample(1).BackgroundTypes%(chan%) = 1 Then  ' 0=off-peak, 1=MAN, 2=multipoint
bgdtime! = sample(1).OnTimeData!(datarow%, chan%)   ' use on-peak time for MAN

' MPB element
Else
For m% = 1 To sample(1).MultiPointNumberofPointsAcquireHi%(chan%)
bgdtime! = bgdtime! + sample(1).MultiPointAcquireCountTimesHi!(datarow%, chan%, m%)
Next m%
For m% = 1 To sample(1).MultiPointNumberofPointsAcquireLo%(chan%)
bgdtime! = bgdtime! + sample(1).MultiPointAcquireCountTimesLo!(datarow%, chan%, m%)
Next m%
End If

' EDS element
Else
bgdtime! = sample(1).OnTimeData!(datarow%, chan%)   ' use on-peak time for EDS
End If
If bgdtime! = 0# Then Exit Function

' Load raw peak cps for unknown using raw data array
peakcount! = sample(1).OnPeakCounts_Raw_Cps!(datarow%, chan%)

' Load raw bgd cps for unknown using calculated peak to background ratio (peak to bgd are calculated on the fly so aggregate intensities are handled properly)
bgdcount! = peakcount! / tRowUnkPeakToBgds!(datarow%, chan%)

' Calculate actual raw counts
peakcount! = peakcount! * peaktime!
bgdcount! = bgdcount! * bgdtime!

' Check for anomalies
If peakcount! = 0# Then Exit Function
If bgdcount! = 0# Then Exit Function
If peaktime! <= 0# Then Exit Function
If bgdtime! <= 0# Then Exit Function

' Calculate analytical sensitivity from Scott and Love 1983
temp1! = peakcount! / peaktime! ^ 2
temp2! = bgdcount! / bgdtime! ^ 2

temp3! = peakcount! / peaktime!
temp4! = bgdcount! / bgdtime!

If temp1! + temp2! < 0# Then Exit Function
If temp3! - temp4 = 0# Then Exit Function
'ConvertAnalyticalSensitivity3! = 100# * Sqr(temp1! + temp2!) / (temp3! - temp4)
ConvertAnalyticalSensitivity3! = Abs(100# * Sqr(temp1! + temp2!) / (temp3! - temp4))         ' assume absolute value

Exit Function

' Errors
ConvertAnalyticalSensitivity3Error:
MsgBox Error$, vbOKOnly + vbCritical, "ConvertAnalyticalSensitivity3"
ierror = True
Exit Function

End Function

This code is now utilized in the v. 13.6.1 of Probe for EPMA.