 Dataplot Vol 1 Vol 2

# LIMITS OF DETECTION

Name:
LIMITS OF DETECTION
Type:
Analysis Command
Purpose:
Perform a limits of detection (LOD) analysis based on the ASTM E-2677 standard.
Description:
There are a number of approaches to determining a limits of detection. This command implements the method given in

E2677 - 14 "Standard Test Method for Determining Limits of Detection in Explosive Trace Detectors," ASTM International, 100 Barr Harbor Drive, PO BOX C700, West Conshohoceken, PA 19428-2959, USA.

The mathematical basis for this method is given in

Rukhin, A. L. and Samarov, D. V., "Limit of Detection Determination for Censored Samples," Chemometrics and Intelligent Laboratory Systems, Vol 105, 2011, pp. 188-194.

Although this method was developed in the context of explosive trace detectors, its use is not limited to this application. We do not give the detailed mathematical formulation here (see the Rukhin and Samarov paper). The following is a brief discussion of this standard.

ASTM subcommittee E54.01 has developed a Standard Test Method for the determination of Limit of Detection (LOD) in trace explosive detectors. The Method was developed following ISO-IUPAC guidelines that harmonize concepts of detection limits and considers the observed behaviors of response signals in a wide range of trace detectors. Here, the LOD90 is defined as the lowest amount of a particular substance for which there is 90% confidence that a single measurement will have a true detection probability of at least 90% while the true non-detection probability of a realistic process blank is at least 90%. The LOD90 value is therefore a directly useful metric of trace detector performance and reliability, since the value reflects the practical detection capability of the detector system, influenced by the inherent sensitivity, selectivity, and response variability of the system under realistic deployment conditions.

The standard specifies the LOD90 value which is based on a false positive probability of 10%. You can specify an arbitrary false positive probability level by entering the command

LET ALPHA = <value>

The most typical values for ALPHA are 0.10 (LOD90), 0.05 (LOD95), or 0.01 (LOD99) with the default being 0.10.

Ideally, the data for the limits of detection should include a large number of process blank replicates as well as detector responses from a large number of replicates from two mass levels closely straddling the actual LOD90 mass level.

By this strategy, the Method is insensitive to many pitfalls that are encountered in commercial trace detectors, including detector response saturation, truncated response distributions, and response heteroscedasticity, i.e. changes in response variation with signal level. Practicalities dictate, however, that a limited number of replicates be analyzed and that mass levels be selected that are wide enough apart to guarantee the straddling of the unknown LOD90 value. By setting the minimum number of required replicates to ten, and by utilizing a short sequence of mass levels that increase by a factor of three, data for the calculation of an adequate LOD90 estimate may be obtained. This limits of detection method has been tested and validated with real and simulated data possessing several types of error structure.

The data for the limits of detection command must be replicated mass-response pairs with the following requirements:

1. at least 10 replications per level
2. at least three distinct levels, including one for process blank response (mass = 0)

Dimensional units for mass levels and responses, while not identified, must be consistent. The limit of detection is reported in the same dimensional units as the mass inputs.

Before calculating the limit of detection, the following data quality checks are performed (the limits of detection will not be calculated if these conditions are not satisfied):

1. Less than 10 replicates in any level
2. Less than three distinct levels
3. Absence of a process blank level (mass = 0)
4. Unequal number of values for the mass and the the repsonse
5. Responses in the highest mass level are not significantly different from those in the process blank level

If data quality passes, estimates will be returned for the limit of detection and the 90% upper confidence LOD90 limit (a measure of uncertainty). Results may also be returned with a message that data quality was marginal, and offer a suggestion for improving the quality.

The LOD90 value determined is the best estimate of the minimum mass of a particular analyte required to elicit a real and reliable response in the detector tested. This minimum detector response is the Critical Value (CV, an optional printed output), which should relate to the peak detection threshold value that can usually be set manually in the trace detector. The peak detection threshold value is usually set higher than the CV unless the LOD90 is determined using realistic chemical background.

Caveat: The LOD90 determination assumes normality for the non-zero measurements. If your data is not normal or contains significant outliers, the LOD90 value will still be calculated. However, non-normality or outliers can result in larges biases for the LOD90 estimate. Measurement data is often skewed right. In this case, you may want to take the log of the non-zero values before using the LIMITS OF DETECTION command (i.e., a lognormal model of the data might be more realistic than a normal model).

To check for outliers and non-normality, the following analyses and graphs may be useful.

1. Perform a Grubbs test for a single outlier for each level of the mass for the non-zero measurements.

2. Generate a scatter plot of the measurements against their associated mass levels.

3. Generate a multiplot showing a normal probability plot of the non-zero measurements for each distinct level of the mass.

In addition to the LOD value and confidence limit, two tables are generated.

1. The first table provides a summary of the number of zero and non-zero response values, the mean, and the standard deviation for each mass level.

2. The second table contains estimates obtained from the LOD computations for each mass level. The last 4 columns are of particular interest.

• Column 4 contains the LOD estimate based on the cumulated sample. The final LOD90 estimate should be one of the values from this column.

• Column 5 contains the standard error of the LOD estimate in Column 4.

• Column 6 contains the 90% upper confidence limit of the LOD value.

• Column 7 contains an upper tolerance limit on the LOD.
Syntax 1:
LIMITS OF DETECTION <y> <x>             <SUBSET/EXCEPT/FOR qualification>
where <y> is a response variable;
<x> is a variable containing the mass levels;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

Although it is typical for the data to be sorted by the value of the mass level, this is not required. The <y> and <x> variables must have the same length.

Syntax 2:
REPLICATED LIMITS OF DETECTION <y> <x> <batch>
<SUBSET/EXCEPT/FOR qualification>
where <y> is a response variable;
<x> is a variable containing the mass levels;
<batch> is a group-id variable;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

This syntax can be used when multiple analytes are being tested. Each distinct value of <batch> denotes a specific analyte for which a limit of detection will be computed. So if <batch> has three distinct values, three limit of detection computations will be performed.

Although it is typical that the data be sorted by the value of the group-id variable and for the data within a specific group to be sorted by the mass level, this is not required. The <y>, <x> and <batch> variables must have the same length.

The word REPLICATION is optional. If exactly three variables are specified, the REPLICATION option is assumed.

Syntax 3:
MULTIPLE LIMITS OF DETECTION <y1> ... <yk> <x>
<SUBSET/EXCEPT/FOR qualification>
where <y1> ... <yk> is a list of 1 to 30 response variables;
<x> is a variable containing the mass levels;
and where the <SUBSET/EXCEPT/FOR qualification> is optional.

This syntax can be used when multiple analytes are being tested. Note that this syntax assumes each analyte has the same mass variable. If this is not the case, the REPLICATED syntax should be used.

Although it is typical for the data to be sorted by the value of the mass level, this is not required. All the variables must have the same length.

For this syntax, the word MULTIPLE is required.

Examples:
LIMITS OF DETECTION Y X
REPLICATED LIMITS OF DETECTION Y X GROUP
LIMITS OF DETECTION Y X GROUP
MULTIPLE LIMITS OF DETECTION Y1 Y2 Y3 Y4 Y5 X
MULTIPLE LIMITS OF DETECTION Y1 TO Y5 X
ONE SAMPLE PROFICIENCY TEST Y LABID
TWO SAMPLE PROFICIENCY TEST Y1 Y2 LABID
Note:
The following commands can be used to specify variuous probability levels

 LET ALPHA = = probability of a false positive (no signal, alarm)
LET BETA = = probability of a false negative (signal, no alarm) LET GAMMA = = confidence level for the LOD and the tolerance bound LET PA = = the coverage level for the tolerance bound

The default for all four of the above is 0.10. Values greater than 0.5 are interpreted as 1 - the value (i.e., entering .9 is equivalent to entering 0.1).

Some instruments may provide a vendor supplied critical value. To specify a pre-defined critical value, enter

LET CRITICAL VALUE = <value>

If no critical value is given, the critical value is estimated from the data. This value should typically only be given if supplied by the vendor for a specific instrument.

The data quality checks are typically based on 90% of the values being either zero or 90% of the values being non-zero. To specify a different threshold percentage for the data quality checks, enter

LET THRESPR = <value>

where <value> is between 80 and 100. Values between 90 and 100 are most common and a value outside the (80,100) interval will be set to the default of 90.

The defaults for all of the above parameters are those specified in the E-2677 standard.

Note:
The following commands can be used to control what output is returned from the LIMITS OF DETECTION command

SET LOD SUMMARY TABLE <ON/OFF>
SET LOD TABLE <ON/OFF>

To specify whether the critical value is printed, enter

SET LOD PRINT CRITICAL VALUE <ON/OFF>

All of the above are ON by default.

Note:
There are two methods for computing the critical value from the data. One is based on quantiles of the data and the other is based on the quantile of a normal distribution. To specify which method to use, enter

SET LOD CRITICAL VALUE <QUANTILE/NORMAL>

The default is QUANTILE.

Note:
By default, Dataplot writes the values from the Summary Table to dpst1f.dat and the values from the LOD Table to dpst2f.dat. You can control whether the dpst1f.dat and dpst2f.dat files are generated by entering

SET LOD OUTPUT FILES <ON/OFF>

The default is ON. Note that if the REPLICATION or MULTIPLE option is used to generate multiple limits of detection analyses, only the last limit of detection analysis will be contained in dpst1f.dat and dpst2f.dat.

Note:
By default, Dataplot saves the following parameters

 LOD = The LOD value. LODSE = The standard error of the LOD value. LODCV = The critical value. LODUCL = The upper confidence limit for the LOD value.

If you use the MULTIPLE or REPLICATION options, these values are written to dpst3f.dat instead. Each row of dpst3f.dat will contain the values for a specific limits of detection analysis.

Note:
You can use the CAPTURE HTML command to generate these tables in HTML format. You can use the CAPTURE LATEX command to generate these tables in Latex format. You can use the CAPTURE RTF command to generate these tables in Rich Text Format (RTF).
Note:
A unique feature of this standard is that a web-based calculator is defined as part of the standard. This web-based calculator uses Dataplot as the computational back-end to generate the limits of detection analysis and to generate the supplementatry graphs and outlier analysis.

The URL for this web-based calculator is

Default:
None
Synonyms:
None
Related Commands:
 E691 INTERLAB = Perform an interlaboratory analysis based on E-691. PROFICIENCY TEST = Perform one sample or two sample proficiency test based on ASTM E-2489. GRUBBS TEST = Perform a Grubbs outlier test. NORMAL PROBABILITY PLOT = Generate a normal probability plot. CAPTURE HTML = Generate output in HTML format. CAPTURE LATEX = Generate output in LaTek format. CAPTURE RTF = Generate output in Rich Text Format (RTF) format.
Reference:
Rukhin, A. L. and Samarov, D. V., "Limit of Detection Determination for Censored Samples," Chemometrics and Intelligent Laboratory Systems, Vol 105, 2011, pp. 188-194.

E2677 - 14 "Standard Test Method for Determining Limits of Detection in Explosive Trace Detectors," ASTM International, 100 Barr Harbor Drive, PO BOX C700, West Conshohoceken, PA 19428-2959, USA.

Currie, L.A. (1999), "Detection and quantification limits: origins and historical overview," Analytica Chimica Acta 391, 103-134.

Applications:
Detection Limits
Implementation Date:
2009/08
2011/01: Allow user-specified crtical value
2012/01: Options for which outputs are printed
2014/03: E2677 adopted, a few tweaks in the output to be consistent with the standard
2018/07: Modified to accomodate negative response values
2018/11: Support for REPLICATION/MULTIPLE options
2018/11: A few minor tweaks in the output format
Program 1:

. Step 1:   Read the data
.
dimension 40 columns
skip 25
read std_lod.txt x y1 to y6
skip 0
.
. Step 2:   Run the limits of detection command
.           and the Grubbs test for outliers
.
set write decimals 4
limits of detection y1 x
print " "
print " "
replicated grubbs test y1 x  subset y1 > 0

The following output is generated
             Limits of Detection Analysis
(Based on ASTM E-2677 Standard)

Response Variable: Y1
Mass Variable:     X

Final Estimate:
Critical Value (Cv90):                          32.7600
Detection Limit (LOD90):                         1.6418
90% Upper Confidence Limit on LOD:               2.4654

Summary Table
--------------------------------------------------------------------------------------
Mass      Number of          Number of            Mean of             SD of
Values    Zero Values    Non-Zero Values    Non-Zero Values   Non-Zero Values
--------------------------------------------------------------------------------------
0.0000              0                 20            21.1700            6.0745
1.0000              0                 12            29.8250            8.8439
3.0000              0                 12            53.9083            6.5607
10.0000              0                 12           119.2417            7.1701
30.0000              0                 12           317.1667           13.1563
--------------------------------------------------------------------------------------

LOD Table
---------------------------------------------------------------------------------------------------------
Linear      StdDev of   LOD Estimate        1-Sigma                       90%-90%
Least Squares    Errors From       Based On    Uncertainty      90% Upper          Upper
Mass          Slope         Linear      Cumulated         on LOD     Confidence      Tolerance
Values      Estimates          Model         Sample       Estimate   Limit on LOD   Limit on LOD
---------------------------------------------------------------------------------------------------------
0.0000             **             **             **             **             **             **
1.0000         6.3649         8.8439         2.6813         1.0942        13.9859        23.8593
3.0000         9.7710         8.0447         1.6418         0.1801         2.4654         4.0708
10.0000         9.5957         7.6683         1.6215         0.1383         2.1200         3.4476
30.0000         9.7690         9.2329         1.7980         0.1321         2.2435         3.7266
---------------------------------------------------------------------------------------------------------

***** NOTE--
SUBSET VARIABLE = Y1
SUBSET MINIMUM  =  0.1000000000E-05
SUBSET MAXIMUM  =  0.3402823000E+39
INPUT  NUMBER OF OBSERVATIONS  =       68
NUMBER OF OBSERVATIONS IGNORED =        0
OUTPUT NUMBER OF OBSERVATIONS  =       68

Grubbs Test for Outliers: Test for Minimum and Maximum
(Assumption: Normality)

Response Variable: Y1
Factor Variable 1: X                             1.0000

H0: There are no outliers
Ha: The extreme point is an outlier
Potential Outlier Value Tested:                 34.1000

Summary Statistics:
Number of Observations:                              20
Sample Minimum:                                 14.6000
ID for Sample Minimum:                                3
Sample Maximum:                                 34.1000
ID for Sample Maximum:                               18
Sample Mean:                                    21.1700
Sample SD:                                       6.0745
Sample Skewness:                                 1.0107
Sample Kurtosis:                                 2.8222

Grubbs Test Statistic Value:                     2.1286

Percent Points of the Reference Distribution
-----------------------------------
Percent Point               Value
-----------------------------------
0.0    =          0.000
50.0    =          2.121
75.0    =          2.325
90.0    =          2.557
95.0    =          2.708
97.5    =          2.843
99.0    =          3.001
100.0    =          4.249

Conclusions (Upper 1-Tailed Test)
----------------------------------------------
Alpha    CDF   Critical Value     Conclusion
----------------------------------------------
10%    90%            2.557      Accept H0
5%    95%            2.708      Accept H0
2.5%  97.5%            2.843      Accept H0
1%    99%            3.001      Accept H0

Grubbs Test for Outliers: Test for Minimum and Maximum
(Assumption: Normality)

Response Variable: Y1
Factor Variable 1: X                             2.0000

H0: There are no outliers
Ha: The extreme point is an outlier
Potential Outlier Value Tested:                 43.2000

Summary Statistics:
Number of Observations:                              12
Sample Minimum:                                 16.9000
ID for Sample Minimum:                               29
Sample Maximum:                                 43.2000
ID for Sample Maximum:                               22
Sample Mean:                                    29.8250
Sample SD:                                       8.8439
Sample Skewness:                                 0.1240
Sample Kurtosis:                                 1.8189

Grubbs Test Statistic Value:                     1.5123

Percent Points of the Reference Distribution
-----------------------------------
Percent Point               Value
-----------------------------------
0.0    =          0.000
50.0    =          1.886
75.0    =          2.079
90.0    =          2.285
95.0    =          2.411
97.5    =          2.518
99.0    =          2.636
100.0    =          3.175

Conclusions (Upper 1-Tailed Test)
----------------------------------------------
Alpha    CDF   Critical Value     Conclusion
----------------------------------------------
10%    90%            2.285      Accept H0
5%    95%            2.411      Accept H0
2.5%  97.5%            2.518      Accept H0
1%    99%            2.636      Accept H0

Grubbs Test for Outliers: Test for Minimum and Maximum
(Assumption: Normality)

Response Variable: Y1
Factor Variable 1: X                             3.0000

H0: There are no outliers
Ha: The extreme point is an outlier
Potential Outlier Value Tested:                 66.7000

Summary Statistics:
Number of Observations:                              12
Sample Minimum:                                 46.4000
ID for Sample Minimum:                               44
Sample Maximum:                                 66.7000
ID for Sample Maximum:                               34
Sample Mean:                                    53.9083
Sample SD:                                       6.5607
Sample Skewness:                                 0.7109
Sample Kurtosis:                                 2.5374

Grubbs Test Statistic Value:                     1.9497

Percent Points of the Reference Distribution
-----------------------------------
Percent Point               Value
-----------------------------------
0.0    =          0.000
50.0    =          1.886
75.0    =          2.079
90.0    =          2.285
95.0    =          2.411
97.5    =          2.518
99.0    =          2.636
100.0    =          3.175

Conclusions (Upper 1-Tailed Test)
----------------------------------------------
Alpha    CDF   Critical Value     Conclusion
----------------------------------------------
10%    90%            2.285      Accept H0
5%    95%            2.411      Accept H0
2.5%  97.5%            2.518      Accept H0
1%    99%            2.636      Accept H0

Grubbs Test for Outliers: Test for Minimum and Maximum
(Assumption: Normality)

Response Variable: Y1
Factor Variable 1: X                             4.0000

H0: There are no outliers
Ha: The extreme point is an outlier
Potential Outlier Value Tested:                106.1000

Summary Statistics:
Number of Observations:                              12
Sample Minimum:                                106.1000
ID for Sample Minimum:                               47
Sample Maximum:                                131.8000
ID for Sample Maximum:                               53
Sample Mean:                                   119.2417
Sample SD:                                       7.1701
Sample Skewness:                                -0.0836
Sample Kurtosis:                                 2.4406

Grubbs Test Statistic Value:                     1.8329

Percent Points of the Reference Distribution
-----------------------------------
Percent Point               Value
-----------------------------------
0.0    =          0.000
50.0    =          1.886
75.0    =          2.079
90.0    =          2.285
95.0    =          2.411
97.5    =          2.518
99.0    =          2.636
100.0    =          3.175

Conclusions (Upper 1-Tailed Test)
----------------------------------------------
Alpha    CDF   Critical Value     Conclusion
----------------------------------------------
10%    90%            2.285      Accept H0
5%    95%            2.411      Accept H0
2.5%  97.5%            2.518      Accept H0
1%    99%            2.636      Accept H0

Grubbs Test for Outliers: Test for Minimum and Maximum
(Assumption: Normality)

Response Variable: Y1
Factor Variable 1: X                             5.0000

H0: There are no outliers
Ha: The extreme point is an outlier
Potential Outlier Value Tested:                337.6000

Summary Statistics:
Number of Observations:                              12
Sample Minimum:                                297.9000
ID for Sample Minimum:                               64
Sample Maximum:                                337.6000
ID for Sample Maximum:                               66
Sample Mean:                                   317.1667
Sample SD:                                      13.1563
Sample Skewness:                                -0.1358
Sample Kurtosis:                                 1.8576

Grubbs Test Statistic Value:                     1.5531

Percent Points of the Reference Distribution
-----------------------------------
Percent Point               Value
-----------------------------------
0.0    =          0.000
50.0    =          1.886
75.0    =          2.079
90.0    =          2.285
95.0    =          2.411
97.5    =          2.518
99.0    =          2.636
100.0    =          3.175

Conclusions (Upper 1-Tailed Test)
----------------------------------------------
Alpha    CDF   Critical Value     Conclusion
----------------------------------------------
10%    90%            2.285      Accept H0
5%    95%            2.411      Accept H0
2.5%  97.5%            2.518      Accept H0
1%    99%            2.636      Accept H0

.
. Step 3:   Generate the scatter plot of the data
.
character x all
line blank all
label case asis
y1label displacement 10
x1label Mass Level
y1label Measurement Response
tic offset units screen
tic offset 5 5
plot y1 x
label
tic offset units
tic offset 0 0 .
. Step 4:   Generate the scatter plot of the data but
.           restrict plot to levels around critical value
.
.           Determine smallest level with
.
.           1.  All response values greater than the critical value
.           2.  First mass level greater than LOD value
.
set let cross tabulate collapse
let yminv = cross tabulate minimum y1 x
let xd = distinct x
let xd = sort xd
let nd = size xd
let nval1 = 0
let nval1 = size yminv subset yminv < lodcv
let nval1 = nval1 + 1
.
let nval2 = nd
loop for k = 1 1 nd
let aval = xd(k)
if aval > lod
let nval2 = k
break loop
end of if
end of loop
.
let nval = max(nval1,nval2)
let xval = xd(nval)
let xval = max(xval,lod)
let lodrnd = round(lod,2)
let cvrnd = round(lodcv,2)
.
character x all
character hw 2.0 1.50 all
line blank all
label case asis
y1label displacement 10
x1label Mass Level
y1label Measurement Response
title Critical Value: ^cvrnd, LOD: ^lodrnd
tic offset units screen
tic offset 5 5
plot y1 x   subset x <= xval
line dash
line color red
drawsdsd 15 lodcv 85 lodcv
drawdsds lod 20 lod 90
line blank
line color black
label
title
character hw 1.0 0.75 all
tic offset units
tic offset 0 0 .
. Step 5:   Generate a normal probability plot for the levels
.
let ymin = minimum y1
let ytemp = y1
let xtemp = x
if ymin >= 0
retain ytemp xtemp subset ytemp > 0
end of if
let xdist = distinct xtemp
let xdist = sort xdist
let ndist = size xdist
.
let ntemp = sqrt(ndist)
let ntemp = int(ntemp+1)
let nrow = ntemp
let ncol = ntemp
if ndist <= 9
let nrow = 3
let ncol = 3
end of if
if ndist <= 6
let nrow = 2
let ncol = 3
end of if
if ndist <= 4
let nrow = 2
let ncol = 2
end of if
.
multiplot nrow ncol
multiplot scale factor nrow
multiplot corner coordinates 5 5 100 95
.
line blank all
character circle all
character fill on all
character hw 1.0 0.75 all
title offset 2
title case asis
justification left
.
loop for k = 1 1 ndist
let aval = xdist(k)
title Mass Level = ^aval
normal probability plot ytemp subset xtemp = aval
let cc = round(ppcc,3)
let ntemp = size ytemp subset xtemp = aval
let cccv = norppcv(ntemp,0.05)
move 17 85
text PPCC: ^cc
move 17 80
let cccv = round(cccv,3)
text PPCC 5% CV: ^cccv
end of loop
end of multiplot
justification center
move 50 97
case asis
text Normal Probability Plots of the Non-Zero Measurements Program 2:

skip 25
read std_lod.txt x y1 to y6
skip 0
.
set write decimals 4
multiple limits of detection y1 y2 y3 y4 y5 y6 x

The following ouput is generated
            Limits of Detection Analysis
(Based on ASTM E-2677 Standard)

Response Variable: Y1
Mass Variable:     X

Final Estimate:
Critical Value (Cv90):                          32.7600
Detection Limit (LOD90):                         1.6418
90% Upper Confidence Limit on LOD:               2.4654

Summary Table
--------------------------------------------------------------------------------------
Mass      Number of          Number of            Mean of             SD of
Values    Zero Values    Non-Zero Values    Non-Zero Values   Non-Zero Values
--------------------------------------------------------------------------------------
0.0000              0                 20            21.1700            6.0745
1.0000              0                 12            29.8250            8.8439
3.0000              0                 12            53.9083            6.5607
10.0000              0                 12           119.2417            7.1701
30.0000              0                 12           317.1667           13.1563
--------------------------------------------------------------------------------------

LOD Table
---------------------------------------------------------------------------------------------------------
Linear      StdDev of   LOD Estimate        1-Sigma                       90%-90%
Least Squares    Errors From       Based On    Uncertainty      90% Upper          Upper
Mass          Slope         Linear      Cumulated         on LOD     Confidence      Tolerance
Values      Estimates          Model         Sample       Estimate   Limit on LOD   Limit on LOD
---------------------------------------------------------------------------------------------------------
0.0000             **             **             **             **             **             **
1.0000         6.3649         8.8439         2.6813         1.0942        13.9859        23.8593
3.0000         9.7710         8.0447         1.6418         0.1801         2.4654         4.0708
10.0000         9.5957         7.6683         1.6215         0.1383         2.1200         3.4476
30.0000         9.7690         9.2329         1.7980         0.1321         2.2435         3.7266
---------------------------------------------------------------------------------------------------------

Limits of Detection Analysis
(Based on ASTM E-2677 Standard)

Response Variable: Y2
Mass Variable:     X

Final Estimate:
Critical Value (Cv90):                          12.8300
Detection Limit (LOD90):                         2.2424
90% Upper Confidence Limit on LOD:               3.5695

Summary Table
--------------------------------------------------------------------------------------
Mass      Number of          Number of            Mean of             SD of
Values    Zero Values    Non-Zero Values    Non-Zero Values   Non-Zero Values
--------------------------------------------------------------------------------------
0.0000              0                 20             0.2050            7.2844
1.0000              0                 12            14.2500            9.8204
3.0000              0                 12            28.4833           10.3888
10.0000              0                 12            86.7417           12.5594
30.0000              0                 12           262.1167           11.0680
--------------------------------------------------------------------------------------

LOD Table
---------------------------------------------------------------------------------------------------------
Linear      StdDev of   LOD Estimate        1-Sigma                       90%-90%
Least Squares    Errors From       Based On    Uncertainty      90% Upper          Upper
Mass          Slope         Linear      Cumulated         on LOD     Confidence      Tolerance
Values      Estimates          Model         Sample       Estimate   Limit on LOD   Limit on LOD
---------------------------------------------------------------------------------------------------------
0.0000             **             **             **             **             **             **
1.0000        11.4947         9.8204         1.7003         0.4406         4.4354         7.4830
3.0000         8.8679        10.0867         2.2424         0.2916         3.5695         5.9200
10.0000         8.4413        10.8222         2.4674         0.2039         3.3003         5.4811
30.0000         8.6232        10.8134         2.4140         0.1825         3.0279         5.0053
---------------------------------------------------------------------------------------------------------

Limits of Detection Analysis
(Based on ASTM E-2677 Standard)

Response Variable: Y3
Mass Variable:     X

Final Estimate:
Critical Value (Cv90):                         152.3000
Detection Limit (LOD90):                         4.3436
90% Upper Confidence Limit on LOD:               5.9310

Summary Table
--------------------------------------------------------------------------------------
Mass      Number of          Number of            Mean of             SD of
Values    Zero Values    Non-Zero Values    Non-Zero Values   Non-Zero Values
--------------------------------------------------------------------------------------
0.0000              0                 20            89.6000           35.6893
1.0000              0                 12           118.3333           33.9367
3.0000              0                 12           150.0000           18.6304
10.0000              0                 12           272.4167           33.8216
30.0000              0                 12           603.9167           35.5207
--------------------------------------------------------------------------------------

LOD Table
---------------------------------------------------------------------------------------------------------
Linear      StdDev of   LOD Estimate        1-Sigma                       90%-90%
Least Squares    Errors From       Based On    Uncertainty      90% Upper          Upper
Mass          Slope         Linear      Cumulated         on LOD     Confidence      Tolerance
Values      Estimates          Model         Sample       Estimate   Limit on LOD   Limit on LOD
---------------------------------------------------------------------------------------------------------
0.0000             **             **             **             **             **             **
1.0000        26.1403        33.9367         3.2526         1.2697        14.0966        21.9875
3.0000        19.9561        27.1842         3.8270         0.6656         6.3183         9.2591
10.0000        18.1982        29.2709         4.3436         0.5454         5.9310         8.7120
30.0000        17.1817        31.0788         4.7354         0.5461         6.0514         8.9284
---------------------------------------------------------------------------------------------------------

Limits of Detection Analysis
(Based on ASTM E-2677 Standard)

Response Variable: Y4
Mass Variable:     X

Final Estimate:
Critical Value (Cv90):                          29.8000
Detection Limit (LOD90):                         1.5189
90% Upper Confidence Limit on LOD:               2.1779

Summary Table
--------------------------------------------------------------------------------------
Mass      Number of          Number of            Mean of             SD of
Values    Zero Values    Non-Zero Values    Non-Zero Values   Non-Zero Values
--------------------------------------------------------------------------------------
0.0000              0                 20            20.6000            6.3029
1.0000              0                 12            30.1667            4.4890
3.0000              0                 12            47.6667            5.7102
10.0000              0                 12            62.4167            7.5131
30.0000              0                 12            70.7500            4.8265
--------------------------------------------------------------------------------------

LOD Table
---------------------------------------------------------------------------------------------------------
Linear      StdDev of   LOD Estimate        1-Sigma                       90%-90%
Least Squares    Errors From       Based On    Uncertainty      90% Upper          Upper
Mass          Slope         Linear      Cumulated         on LOD     Confidence      Tolerance
Values      Estimates          Model         Sample       Estimate   Limit on LOD   Limit on LOD
---------------------------------------------------------------------------------------------------------
0.0000             **             **             **             **             **             **
1.0000         8.1512         4.4890         1.5008         0.2975         2.9526         4.4843
3.0000         8.5105         5.0306         1.5189         0.1890         2.1779         3.2843
10.0000         4.4465         9.8315         4.2909         0.4176         6.0165         9.9587
30.0000         1.9318        16.6355        14.3901         1.3526        20.2616        35.5926
---------------------------------------------------------------------------------------------------------

Limits of Detection Analysis
(Based on ASTM E-2677 Standard)

Response Variable: Y5
Mass Variable:     X

Final Estimate:
Critical Value (Cv90):                          66.9000
Detection Limit (LOD90):                         3.6005
90% Upper Confidence Limit on LOD:               4.8802

Summary Table
--------------------------------------------------------------------------------------
Mass      Number of          Number of            Mean of             SD of
Values    Zero Values    Non-Zero Values    Non-Zero Values   Non-Zero Values
--------------------------------------------------------------------------------------
0.0000              0                 20            43.7000           16.6041
1.0000              0                 12            58.0833           18.1732
3.0000              0                 12            72.5833           17.2440
10.0000              0                 12           154.7500           11.8254
30.0000              0                 12           344.3333            3.3394
--------------------------------------------------------------------------------------

LOD Table
---------------------------------------------------------------------------------------------------------
Linear      StdDev of   LOD Estimate        1-Sigma                       90%-90%
Least Squares    Errors From       Based On    Uncertainty      90% Upper          Upper
Mass          Slope         Linear      Cumulated         on LOD     Confidence      Tolerance
Values      Estimates          Model         Sample       Estimate   Limit on LOD   Limit on LOD
---------------------------------------------------------------------------------------------------------
0.0000             **             **             **             **             **             **
1.0000        12.0090        18.1732         3.4519         1.5454        22.9214        36.8784
3.0000         9.1536        17.4612         4.4290         0.8898         8.2311        12.8500
10.0000        10.7118        15.9206         3.6005         0.4068         4.8802         7.4342
30.0000        10.0258        14.3593         3.6472         0.4079         4.6299         6.8946
---------------------------------------------------------------------------------------------------------

Limits of Detection Analysis
(Based on ASTM E-2677 Standard)

Response Variable: Y6
Mass Variable:     X

Final Estimate:
Critical Value (Cv90):                          44.0000
Detection Limit (LOD90):                         5.8926
90% Upper Confidence Limit on LOD:               7.4054

Summary Table
--------------------------------------------------------------------------------------
Mass      Number of          Number of            Mean of             SD of
Values    Zero Values    Non-Zero Values    Non-Zero Values   Non-Zero Values
--------------------------------------------------------------------------------------
0.0000             20                  0             0.0000            0.0000
1.0000             12                  0             0.0000            0.0000
3.0000              8                  4            51.0000            4.9666
10.0000              0                 12           111.8333           10.4693
30.0000              0                 12           310.8333           10.7605
--------------------------------------------------------------------------------------

LOD Table
---------------------------------------------------------------------------------------------------------
Linear      StdDev of   LOD Estimate        1-Sigma                       90%-90%
Least Squares    Errors From       Based On    Uncertainty      90% Upper          Upper
Mass          Slope         Linear      Cumulated         on LOD     Confidence      Tolerance
Values      Estimates          Model         Sample       Estimate   Limit on LOD   Limit on LOD
---------------------------------------------------------------------------------------------------------
0.0000             **             **             **             **             **             **
1.0000             **             **             **             **             **             **
3.0000        17.0000         4.9666         3.7488         0.1825         5.4695         6.7043
10.0000        11.3528        12.8120         5.8926         0.1892         7.4054         9.6081
30.0000        10.4629        13.3701         6.2753         0.0731         7.3204         9.4806
---------------------------------------------------------------------------------------------------------


NIST is an agency of the U.S. Commerce Department.

Date created: 11/06/2018
Last updated: 11/06/2018