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6.3.4.2 Since some calculations, for example that of s, may lead to computation of a relatively small difference
between two relatively large numbers; it is critical that all previous calculations not be based on rounded numbers.
In general, such analyses should be performed using a spreadsheet or computer program.
6.3.4.3 The previous equations neither require nor prevent use of data from blank quantification. In order to use
blank data in the MDL quantification with WLS, the blank should be measurable, be likely to give a response which
is consistent with the pattern in the other calibration data, and provide a level of variability consistent with the
pattern seen in the investigated standards.
6.3.5 Alternative Calculations — Situations exist for which both prior alternatives (¶6.3.3 and ¶6.3.4) may be
inappropriate. For example:
(a) a calibration model may be nonlinear,
(b) a calibration model may be multivariate in nature,
(c) the implicit assumption of normality of residuals (errors in predictions about the calibration) may not be an
appropriate distributional assumption,
(d) the lowest level standard used in calibration may be too far above the true MDL.
6.3.5.1 In case (a), a nonlinear model can be used instead of the default linear model in performing a regression
analysis which quantifies the Upper Confidence Limit for individuals at X = 0 at a 3 sigma equivalent probability
level (and back-projects this result through the nonlinear calibration model to estimate the MDL). How to
accomplish this is the responsibility of those applying this alternative. In some situations, a transformation of scale
(X and/or Y) may linearize the model such that the previously provided calculations are again applicable.
SEMI C10-0305 © SEMI 1998, 2005 8
6.3.5.2 In case (b), a multivariate calibration model can be used instead of the default linear model in performing a
regression analysis which quantifies the Upper Confidence Limit for individuals at X = 0 at a 3 sigma equivalent
probability level (and back-projects this result through the multivariate calibration model using the observed point of
maximum interference in terms of the other concentrations in the range of calibration to estimate the MDL). How to
accomplish this is the responsibility of those applying this alternative.
6.3.5.3 In case (c), other error distributions than the normal, can be used wherever appropriate. The same
philosophical approach should be applied; a regression analysis which quantifies the Upper Confidence Limit for
individuals at X = 0 at a 3 sigma equivalent probability level (and back-projects this result through the calibration
model to estimate the MDL). How to accomplish this is the responsibility of those applying this alternative.
6.3.5.4 In case (d), it is more likely that one will obtain a higher MDL than is truly the case. This phenomena is
more likely to occur when the blank is not usable in the regression analysis for any reason. The only fix for this is to
reapply the procedure with a set of standards which is not so distant from the MDL, but still otherwise meets the
requirements provided herein.
6.3.5.5 In cases (a), (b), and (c), or any combination thereof, the use of appropriate computer/statistical tools is the
responsibility of those applying these alternatives. Additional information on regression analysis methods can be
found in Applied Regression Analysis, 3rd Edition, Norman R. Draper and Harry Smith, John Wiley and Sons, ©
1998.
6.3.6 Example — Results of Applying (see ¶6.3.3 and ¶6.3.4)
6.3.6.1 Limited result summaries for obtaining the MDL by OLS and WLS are provided for the calibration data in
Table 2. These results (Table 3) can be used to benchmark one's implementation of the computational methodology.
Table 2 Calibration Data
Ppb signal ppb signal ppb signal
1 5.03 5 19.87 10 39.90
1 5.00 5 21.20 10 38.28
1 4.91 5 20.45 10 40.93
Table 3 Result Summaries
Statistic OLS WLS
MDL 1.097 0.108
UCL 5.390 1.531
m 3.857 3.869
b 1.157 1.112
s 0.800 0.00997
6.3.6.2 Note the relatively large difference in the estimated MDL (1.10 vs. 0.11) resulting from passing the same
data through each of the OLS and WLS algorithms. This data has very different levels of variability, depending on
the concentration level of the standard (Table 4). Such data should be analyzed with the WLS rather than the OLS
algorithm.
Table 4 Signal Variation by Concentration Level
Concentration
Level
Standard
Deviation
Variance
1 0.062 0.0038
5 0.667 0.4449
10 1.336 1.7849