Embed Size (px)
Citation preview
On-Line Supplements for
“Quasi NonParametric” Upper Tolerance Limits forOccupational Exposure Evaluations
Charles B. Davis, PhD *Principal Statistician
EnviroStatLas Vegas, NV
Paul F. Wambach, CIHConsultant
Rockville, MD
Exposition word count: 4921
* Corresponding author
Keywords: Upper Tolerance Limits, Exposure Limits, Censored Data, Uncensored Data,
Supplement 1
DISTRIBUTIONS USED IN THE STUDY
Distribution Families
Three families are used for contaminant concentrations in evaluating the performance of QNP UTLs in this study: Lognormal (LN), Gamma (GA), and Weibull (WE), defined for values ≥ 0.0. For each family there are two parameters: a shape parameter controlling skewness and a scale parameter controlling spread. A location parameter shifting the entire distribution is sometimes considered, but not here. Metrics of central tendency (means and medians), spread, and percentiles depend on both parameters.
The Lognormal family. The LN family is often assumed for environmental contaminant data: logs of data have normal distributions. This may be for convenience, as it often allows using familiar table, but there is some theoretical justification: if one imagines the concentration of a contaminant at a location to be a random fraction of a random fraction of … of the concentration at a source, on a log scale this looks like a sum of random variables. Distributions of sums of random variables tend toward normal distributions under rather general assumptions, according to the “Central Limit Theorem” of statistical theory.
The LN distribution shape is governed by the standard deviation σ of the log data. As σ → 0 the distribution becomes symmetric, approaching a normal distribution. As σ becomes large, the distribution becomes increasingly skewed (asymmetric), with a peak near zero and a long right tail. FIGURE 1.1 shows LN distributions with σ = 0.5, 1.0, 1.5, 1.75, 2.0, and 2.3. The last four are used in the significance level and power evaluations of Supplement 3; the first three are included here for visual comparison. The left plot emphasizes the region around zero; the right plot emphasizes the “middle” of the distributions. All have median (50th percentile) = 1.0 for comparability. The scale parameter is eµ, where µ is the mean of the logs of data values.
FIGURE 1.1. Lognormal (LN) distributions with varying values of the shape parameter σ. µ = 0 in each case, making the medians of the distributions all 1.0 for visual comparability. The left plot emphasizes the region around zero.
TABLE I gives selected percentiles and the mean value (balance point of the distribution) for various σ. For the conservative LN 2.0 distribution used in deriving QNP UTLs, the 95th percentile is over 26 times the median and 3.6 times the mean; the mean is the 84th percentile.
The Gamma family. The GA family of distributions is preferred by some as a model for environmental contaminant data; see Singh, Armbya, and Singh(1), for example. In this family, the closer to zero the shape parameter Φ is, the more peaked and positively skewed is the distribution shape. When Φ is a multiple of 1/2, these distributions are (re-scaled) chi-square distributions with 2Φ degrees of freedom. The second most skewed GA distribution used is somewhat similar to the LN 2.00 distribution used to construct QNP UTLs.
FIGURE 1.2. Gamma (GA) distributions with varying values of the shape parameter Φ, with medians set 1.0 for visual comparability. The left plot emphasizes the region around zero.
The Weibull family. The WE family of distributions is also used for non-negative, positively skewed data. In this family also, the closer to zero the shape parameter Φ is, the more peaked
TABLE I. LN Percentiles and Means, with Median = 1.0σ 10% 25% 75% 90% 95% 99% mean
0.50 0.53 0.71 1.40 1.90 2.28 3.20 1.131.00 0.28 0.51 1.96 3.60 5.18 10.2 1.651.50 0.15 0.36 2.75 6.84 11.8 32.8 3.081.75 0.11 0.31 3.26 9.42 17.8 58.6 4.622.00 0.08 0.26 3.85 13.0 26.8 104.9 7.392.30 0.05 0.21 4.72 19.1 44.0 210.7 14.1
TABLE II. GA Percentiles and Means, with Median = 1.0Φ 10% 25% 75% 90% 95% 99% mean
3.00 0.41 0.65 1.47 1.99 2.35 3.14 1.121.00 0.15 0.42 2.00 3.32 4.32 6.64 1.440.60 0.06 0.27 2.58 4.94 6.84 11.4 1.900.25 0.00 0.06 5.97 17.2 27.7 55.7 5.720.15 0.00 0.01 16.18 71.3 132.4 309.6 24.1
and skewed is the distribution shape. The second most skewed distribution is somewhat similar to the LN 2.0 distribution used to construct QNP UTLs, in the sense that the ratios between the 95th percentiles and medians are similar.
FIGURE 1.3. Weibull (WE) distributions with varying values of the shape parameter Φ, with medians set 1.0 for visual comparability. The left plot emphasizes the region around zero.
Nondetect Proportions
Another way to view the skewness of these distributions is through the nondetect rates for distributions which are borderline with respect to the hypothesis being tested. To examine this, the 95th percentile is set equal to the RC. Two typical RLs are used here, a tenth and a quarter of the RC, typical of the beryllium facility surveys. Even when 5 percent of the data are expected to exceed the RC, the “nondetect” rates for the conservative and very conservative distributions are around 80 percent when RL = RC/4, and around 70 percent when RL = RC/10.
TABLE III. WE Percentiles and Means, with Median = 1.0Φ 10% 25% 75% 90% 95% 99% mean
3.00 0.53 0.75 1.26 1.49 1.63 1.88 1.011.00 0.15 0.42 2.00 3.32 4.32 6.64 1.440.45 0.02 0.14 4.67 14.4 25.9 67.2 5.600.35 0.00 0.08 7.25 30.9 65.5 223.8 14.3
TABLE IV. Nondetect Rates When 95th Percentile = RC σ or Φ RL = RC / 4 RL = RC / 10
LN 0.50 13.0% 0.2%1.00 60.2% 25.5%1.50 76.4% 54.4%1.75 80.3% 62.9%2.00 82.9% 68.9%2.30 85.1% 74.0%
GA 3.00 21.0% 2.6%1.00 52.7% 25.9%0.60 64.0% 41.2%0.25 77.3% 63.6%0.15 82.4% 73.0%
WE 3.00 4.6% 0.3%1.00 52.7% 25.9%0.45 79.9% 65.5%
0.35 84.2% 73.8%
Discussion
The preceding plots and tables illustrate just how highly skewed the conservative distributions are by comparing the upper percentiles and the means with a common median of 1.0, and considering the nondetect rates for borderline distributions (where the 95th percentile equals the RC). In practice it is difficult to identify distributions or distinguish among distribution families when datasets are small and distributions are very skewed, since observations that are far in the tails are quite variable, can be quite extreme, and appear rarely. An isolated extreme value that would actually be consistent with, say, a LN 2.30, GA 0.15, or WE 0.35 distribution would often be treated as a special case “outlier” if not dismissed outright.
Analytical Variation
A major issue addressed by Davis, Field, and Gran(2) and included in the performance evaluation of QNP UTLs is that of analytical variation. These are the distributions of measurements when the underlying distribution of concentrations is a LN (or other) distribution.
As discussed, distributions of analytical variation appear to be reasonably represented by distributions with two components of variation, one with constant standard deviation and the other with standard deviation proportional to concentration (i.e., constant relative standard deviation = RSD = Coefficient of Variation = CV). See Rocke and Lorenzato(3) and Davis, Field, and Gran(2) for detailed discussion. Unlike Rocke and Lorenzato, Davis, Field, and Gran take both components to be normally distributed, based on Central Limit Theorem considerations and empirical observation. Four normal distributions for analytical variation were used, as listed in TABLE V. For example, with “B” (standard deviation)2 = 0.0022 + (0.02 * concentration)2.
The constant standard deviation term used was typical for the Be facility studies discussed in Supplement 3, with RC = 0.2. In “A” there is no analytical variation; the laboratory precision is perfect. “B” is more typical of good laboratory performance. Variation “C” corresponds to laboratory performance not quite meeting typical standards, since its overall RSD would never quite get down to the 10 percent often considered to be the requirement for acceptable large-concentration precision. “D” corresponds to less than acceptable laboratory performance; its three-standard-deviation precision bounds are between 10 and 190 percent of actual concentration even for large concentrations.
The following plots below give glimpses of the effect that this analytical variation can have on the distribution of measurements. The “A” curve is for the distribution of concentrations themselves. For these examples the 95th percentiles are set to 0.05, a quarter of the RC; these facilities are relatively clean. Common RLs in this situation would be around 0.02 to 0.05.
TABLE V. Normal Distributions for Analytical Variation StDev of Component with Constant
Standard Deviation Relative Standard DeviationA none 0 0%B good 0.002 (1% of RC) 2%C fair 0.005 (2.5% of RC) 10%D poor 0.01 (5% of RC) 30%
FIGURE 1.7. Distributions of measurements when the concentration C has one of the underlying distributions used in the evaluations, and analytical variation ranges from A (none) to D (poor). B is typical of good labs.
Note the increasingly high proportion of negative values as one progresses from “A” to “D”. As discussed in the body of the paper, it is encouraging that the performance of the QNP UTL procedure is nearly unaffected by analytical variation ranging from “A” to “C”, and indeed not affected that much even by “D”. This is not the case with other UTL procedures relying on censored-data statistical analysis, as discussed by Davis, Field, and Gran(2).
References for Supplement 1
1. Singh, A. and A.K. Singh: ProUCL Version 5.0.00 Technical Guide: Statistical Software for Environmental Application for Data Sets with and without Nondetect Observations, EPA/600/R-07/041, September 2013.
2. Davis, C.B., D. Field, and T.E. Gran: A Model for Measurements of Lognormally Distributed Environmental Contaminants, Report DOE/NV/25946—733, July 2009, revised January 2012; available on request.
3. Rocke, D.M. and S. Lorenzato: A Two-Component Model for Measurement Error in Analytical Chemistry, Technometrics 37(2): 176-184, May 1995.
Supplement 2
CONCERNING THE UPPER BOUND σ ≤ 2
QNP UTLs are derived assuming that distributions are LN with shape parameter σ at most 2.0, but their performance is seen to be insensitive or conservative with respect to that assumption. Supplement 2 reviews empirical evidence that suggests that this is a reasonably conservative upper bound. This bound may be exceeded in some situations; the analyses presented in this paper suggest that the target confidence level (95 percent) is not grossly violated even if σ is as high as 2.3 or so.
Three lines of evidence are presented. The first comes from meta-analyses of studies published in the IH literature. The second comes from examination of a large database of field data from the DOE survey of surface concentrations of beryllium (Be) at the Nevada Test Site (now the Nevada Nuclear Security Site). Those data, and the lessons learned from having many uncensored data (i.e., values reported as measured, without “less-thans”), were the original motivation for the development of QNP UTLs, as discussed by Davis, Field, and Gran(1). These two lines are discussed in detail in this Supplement. Finally, there is the fact that QNP UTLs based on this assumption match the AIHA “8 less than a tenth” rule-of-thumb(2) so well.
IH History and Literature
In 1993 Kromhout, Symanski, and Rappaport(3) analyzed a data set of 13,945 monitoring results they divided by job title and location into 165 groups. Their Table A1 lists σ for the within- and between-worker variance components in occupational exposure levels estimated in their analysis of variance. These two components were combined to a total standard deviation using the formula σt = √(σw
2 + σb2). Only 6 of the 165 (3.7 percent) of σt estimates exceed 2.0; these
involved only 131 (< 1 percent) of the monitoring results. The distribution of their σt estimates is shown in FIGURE 2.1.
Subsequently, Symanski, Maberti, and Chan(4) did a meta-analysis of 60 newly published reports that included 49,807 monitoring results for 571 job groups. In their analysis the authors used estimates of 95, the ratio of the 97.5th percentile to the 2.5th percentile of the distributions of the
individual worker’s shift-long exposures. For LN models 95 = exp(3.92*σ). These were computed from the analysis of variance giving both within- and between-worker variance components, and were summarized by percentiles for various worker categories, as given in TABLE I which is abstracted from their Tables 4 and 5. TABLE I presents the 95th percentiles of these distributions, and also combines the 95th percentiles of the resulting within- and between-worker standard deviations to provide upper estimates of the total measurement standard deviation σt. The percentile that should be associated with σt is unknown, as the authors do not report the joint distributions of their between- and within-worker 95 estimates; it is at least 90 percent. For only the omnibus category “across jobs/across locations” does the upper σt estimate exceed 2.0.
FIGURE 2.1. Estimates of the total standard deviation (σt) presented by Kromhaut et al.(3). The highest three estimates are based on datasets of only N = 12, 12, and 13 measurements.
Table I. Variance Metrics from Symanski et al.(4)
CategoriesBetween Workers Within Workers
Totalσ#
Groups 95th percentile #
Groups 95th percentile
Ȓ95 Sigma Ȓ95 Sigma
AerosolsChemical 35 28.5 0.85 33 65 1.06 1.37
Non-chemical 124 74.6 1.10 119 287.3 1.44 1.82
Dermal agentsChemical 47 46.6 0.98 45 159.6 1.29 1.62
Non-chemical 30 91.8 1.15 30 60.1 1.04 1.56
Gases/vaporsChemical 48 25.3 0.82 42 799.2 1.71 1.89
Non-chemical 22 5.4 0.43 20 27.8 0.85 0.95
SamplingRandom 87 84.4 1.13 88 118.2 1.22 1.66
Non-random 37 41.2 0.95 37 170.6 1.31 1.62
Period1 year or less 136 82.9 1.13 136 200.9 1.35 1.76
>1 year 44 26.6 0.84 43 699.5 1.67 1.87#
workers5–8 161 104.9 1.19 155 346.7 1.49 1.91>8 145 36.3 0.92 134 445.6 1.56 1.81
#measurements
10–25 175 104.9 1.19 165 412 1.54 1.94>25 131 35.6 0.91 124 445.6 1.56 1.80
Groupingcriteria
By job/by location 306 44.7 0.97 289 445.6 1.56 1.83By job/across location 124 102.8 1.18 103 108 1.19 1.68Across jobs/by location 115 330.7 1.48 107 140.2 1.26 1.94
Across jobs/across locations 22 1536.8 1.87 20 215.9 1.37 2.32
The published analyses of variance (ANOVAs) summarized in these two meta-analyses were primarily performed to support estimates of exposure levels of individual workers at risk for occupational diseases. Symanski et al. report that they screened publications to eliminate reports when the percentage of non-detects were likely to bias ANOVAs. Kromhaut et al. does not discuss censoring. Presumably the exposure distributions the two papers summarized contained
measurements well removed from RLs where analytical uncertainty was not a significant contributor to variance and could be ignored. In contrast, the QNP UTL will be used primarily to verify compliance when most monitoring results are below RLs. While it is possible that compliant conditions will be more skewed and analytical error a more important source of variance than those analyzed by Kromhaut et al. and Symanski et al., non-compliant distributions should be similar. Monitoring is performed to avoid a Type-I error of falsely concluding that working conditions are safe. Underestimating variance in compliant exposure distributions does not create Type-I errors because the non-compliant conditions the monitoring is meant to rule out would be less likely to have low-end variance inflated by analytical variation. Since non-compliant conditions should have exposure distributions similar to those analyzed by these two meta-analyses, they provide strong evidence that the assumption that the variability in occupational exposure monitoring results will have σ less than 2 is conservative for this application. As long as this is true, the QNP UTL will constrain the probability of falsely concluding working conditions are safe to less than 5 percent following the 1977 Leidel, Busch, and Lynch(5) decision protocol. The QNP UTL can be used to guide risk management decisions when parametric UTL methods can’t be used and when collecting 59 or more samples is not practical.
The DOE/NV Worker Environment Beryllium Characterization Study
This large study of surface concentrations of beryllium (Be) was conducted on the Nevada Test Site (now the Nevada National Security Site) and ancillary locations from 2002-2009. Over 240 facilities were sampled; these range in nature from large urban office buildings to shops in forward areas to experimental facilities to dormitories. Details may be found in the Final Report from that study(6). The exposure limit here is the DOE Release Criterion RC = 0.2 µg/100 cm2.
In a major departure from usual data reporting practice, nearly all of the data obtained during that study are uncensored. That means that, although the labs provided Reporting Limits (RLs) derived using typical calculations, data values less than their RLs were reported as measured, even if negative. The data examined here involve over 9500 such measurements, along with associated field blank measurements.
Examining so many uncensored data has been quite educational. One finding, discussed in detail by Davis, Field, and Gran(1), is that the usual statistical models assumed for environmental contamination data are simplistic, likely overly so. Although concentrations of environmental contaminants are necessarily non-negative, it does NOT follow that their measurements are also necessarily non-negative, and in fact about a third of the measurements examined here are negative. This serves as a reminder that one makes regulatory or IH decisions using measurements, not actual concentrations; the latter are in principle never knowable.
The Lognormal-Normal Model (LNNM)
Here is an illustrative example of uncensored data.
FIGURE 2.2. Blank-corrected Be measurements from Dataset 22. Of 356 measurements, 85 are negative and 78 are above a typical RL of 0.02 (= RC/10). The left plot shows all data; the right plot shows the region around 0.
Lognormal models are very often assumed for statistical distributions of environmental contaminants. In these the logs of the data have normal distributions; we denote the (natural) log-scale mean and standard deviation by the customary µ and σ. Lognormal distributions are inherently non-negative, and range from nearly symmetric (as the shape parameter σ → 0) to very skewed with long right tails (as σ becomes large). There is some theoretical justification for this model: if the concentration at a particular location at a particular time is a random fraction of a random fraction of … of the concentration at a source, on a log scale this becomes a sum of random quantities. The Central Limit Theorem of statistics says that under rather general conditions the distributions of sums of random quantities approach normal distributions.
However, concentrations are not measured precisely, but rather with inevitable analytical variability (measurement “error”). This can create negative measurements when the true concentration C is close to zero; after all, if the instrument is calibrated to provide approximately unbiased measurements, the average when C = 0 should be about 0, and if there is analytical variation, some measurements must be expected to be negative.
A Quick Look at Distributions of Field Blank Measurements
One way to look at this is through the distributions of field blanks: wipes not exposed to potential contamination that go through the entire analysis process “blind”. The data examined here were analyzed by two labs. Lab D used standard techniques (ICP-AES using the 313.042 nm emission line). Lab B data are mostly ICP-AES, but using the 234.861 nm emission line, which is subject to interferences from an adjacent iron emission line (not well documented at the time). The Lab B data include an ad-hoc adjustment for the iron peak, where it was not too high, and substitutes samples analyzed by ICP-MS otherwise, as documented in the DOE/NV Final Report(6). FIGURE 2.3 shows the overall distributions of field blank measurements by lab.
FIGURE 2.3. Distributions of Field Blank measurements, by Lab. A few high values, which would be “detects” with a typical RL of 0.02, are omitted from computations. Fitted Normal distributions are also shown.
Here are summary statistics for these; outliers are omitted from the computations.
The mean of blanks for Lab D is nearly zero; 60 percent are negative. Omitting one outlier, the normal probability plot correlation is R = 0.977; a normal model for the measurement variability of these blanks fits nicely. For Lab B, though, the mean of blanks is somewhat positive (about a quarter of way from zero to the typical RL of 0.02 = RC/10). Possibly these differences are related to the fact that instruments are generally calibrated using spiked reagent water, whereas even blanks contain interferences. For Lab B, omitting three outliers, the ProbPlot R = 0.991; a normal model fits these data nicely as well.
The data evaluated in the rest of this Supplement are blank-corrected: the mean of an appropriate subset of field blank measurements has been subtracted from the measurements. The appropriate subset uses the same instrument in the same lab, the same wipe medium, and common time period. The reason for doing the blank correction is to make data from different instruments, labs, media, and time periods more comparable than they might have been otherwise.
TABLE II. Distributions of Field Blank MeasurementsLab N* Mean StDev ProbPlot R # < 0D 297 -0.000845 0.002324 0.977 177B 221 0.005582 0.003255 0.991 12
* One value > 0.02 omitted for Lab D, three omitted for Lab B
Back to the LNNM
The nearly normal distribution of blank measurements, where the true concentration is presumable close to 0, gives insight into the actual nature of measurement “error” distributions. Rocke and Lorenzato(7) examined this issue in 1995, constructing a model consisting of two statistically independent sources (components) of variation, one being normal with fixed standard deviation, and the other being lognormal with standard deviation proportional to concentration. Davis, Field, and Gran(1) adopted a similar approach, though with both components normally distributed. Their overall distribution of measurements is gotten by taking the distribution of concentrations C to be lognormal. The distribution of measurements for each value of C is taken to be normal, but with standard deviation depending on C.
There are in principle six parameters:
µ: the log-scale mean of the LN distribution of actual concentrations;σ: the log-scale standard deviation of the LN distribution of actual concentrations;α: the mean analytical response when the actual concentration is 0; ß: the large-concentration ratio of mean analytical response to true concentration
(i.e., slope of the recovery line);γ: the standard deviation of analytical variation when the actual concentration is 0;
and δ: the large-concentration ratio of standard deviation to true concentration
≈ Relative Standard Deviation (RSD) if ß = 1.
The LNNM is formed as follows. Let C be the actual concentration, having a LN(µ,σ) distribution. Then each measurement is Y = α + ßC + (γ2 + (δC)2)1/2 Z, where Z has a standard normal distribution. The resulting probability density function (PDF) for measurements is the following, where t = µ + σz and φ is the standard normal distribution PDF:
This PDF (= likelihood function) is itself an integral, which makes fitting this model to data computationally cumbersome. It is not possible to simplify this expression.
The effects of parameters µ and σ are discussed in Supplement 1. The remaining four parameters determine the distribution of analytical variation. However, this model is not identifiable (i.e., the parameters are not unique for a given fitted model). This non-identifiability is resolved by setting ß = 1 (i.e., assuming ~100 percent recovery on average for large concentrations), equivalent to defining recoverable concentration in terms of instrument capability. One is tempted to similarly assign α = 0, reflecting the blank correction; this simplification is not supported by the data, however, particularly data from dustier facilities.
FIGURE 2.4 shows these components for the Dataset 22 data seen in FIGURE 2.2.
FIGURE 2.4. Fitted LNNM for Dataset 22, with its components. C (blue) is the distribution of true concentrations. The red curve shows the distribution of measurements when C = 0.0; the violet curve shows the distribution of measurements when C = 0.1.
The fitted parameter values for this dataset are α = −0.0021, ß = 1, γ = 0.0031, δ = 0.1 (see the discussion to follow for more on the role of δ), µ = −5.06, and σ = 1.49; there are N = 356 values, of which 85 are negative and 78 are above a typical RL of 0.02. TABLE III gives summary statistics for both the empirical and fitted distributions.
Fitting the LNNM to Datasets
LNNMs have been fit to each of 52 datasets to provide the results which follow. These datasets were amalgamated from data from 244 facilities as follows. First, facilities were grouped by lab and location or type of location. For each group, a preliminary one-way analysis of variance was performed on logs of shifted data (data were shifted so that the minimum values were around 0.003). Based on this, along with visual examination of data plots, the initial groups were subdivided into datasets with similar data distributions. The purpose of this initial triage is as follows. With five free parameters that can vary in fitting the LNNM, it is quite possible to over-fit idiosyncrasies that show up in small datasets. On the other hand, if data from dissimilar facilities are grouped together, it is quite possible that the resulting fitted log-scale σ can include
TABLE III. Dataset 22 Descriptive Statistics Mean StDev 95%ile Pr(Y > RC)
Fitted 0.0172 0.0556 0.0717 0.0102Empirical 0.0167 0.0420 0.0664 0.0113
facility-to-facility variation in addition to the within-facility variation relevant for the intended use of the method. Hence there is interest in having large datasets, providing more precise estimates, so long as those datasets are reasonably homogeneous.
The actual LNNM fitting is done by a manually guided numerical search. Initial values (α = 0, ß = 1, γ = 0.002, δ = 0.1, µ = −6, σ = 1.5) are assigned to the parameters, after which an automated search on values of µ and σ is conducted to maximize the Log Likelihood (LL), which is the log of the product of the likelihood functions (PDFs) for the data. Then a manually guided search is conducted on α and γ, seeking to find values which maximize LL. The maximizing values and LL are recorded. Then δ is allowed to vary in steps of 0.01 with the same objective; once the maximizing value of δ has been identified, the search over α and γ is repeated. With a couple of datasets the last two steps (searching on δ, then maximizing over α and γ) was repeated.
A discussion of the role of δ is in order. This is the large-C relative standard deviation (RSD) of measurement “error”. This is a primary metric used in evaluating laboratory performance; the “Quantitation Limit” (Currie’s “Determination Limit”(8)) is often thought of as a concentration C above which RSD is 10 percent or less. Hence, in fitting the LNNM δ is initially set to 0.1. When two models are fit, one being a sub-model of the other as is the case here, twice the increase in LL has approximately a chi-square distribution with degrees of freedom equal to the number of parameters constrained in the sub-model (one in this case). In this case, then, the LL should increase by 1.92 or so to indicate a statistically significantly improved fit to the data.
TABLE IV (at the end of this Supplement) contains the results of this fitting for all 52 datasets. For discussion, here is the portion of TABLE IV containing the Dataset 22 results.
Table IV excerpt. LNMM Fits for Dataset 22Data
Lab N< > Fitted parameters
Descriptionset 0 0.02 α τ δ µ σ LL22 B 356 85 78 -0.0021 0.0031 0.10 -5.060 1.490 1108.48 Base camp FO SH ST -0.0020 0.0031 0.09 -5.080 1.510 1108.51
With this dataset the increase in LL resulting from allowing δ to vary is only 0.03, far below the critical value of 1.92. As it turns out, in only two of the 52 datasets does allowing δ to vary result in a statistically significant increase in LL. In both of those the fitted values of δ, when allowed to vary, are not consistent with one’s conceptual model of the measurement process. In one case the fitted value of δ is 0.00, too low, and in the other it is 0.43, rather too high. (A δ of 0.43 implies that the measurement process will provide positive values at most around 99 percent of the time, regardless of how high C is! One surely would hope for better precision than this.)
Estimates of σ
In the initial fitting σ, the parameter of major concern for the performance of QNP-UTLs, was constrained to be no greater than the target value 2.0. If the fitted value was equal to 2.0 in the initial fitting, a second round of guided fitting was performed allowing σ to increase as high as it wanted. This occurred with 10 of the 52 datasets, as seen in the plots to follow. In no case was the improvement in fit statistically significant, however; the largest increase in LL was 1.0483, with a p-value of around 0.15.
The following plots show the fitted σ values. FIGURE 2.5 shows the fitted σ values by Lab and N. For small N there is a considerable variation in the fitted σ, which settles down by N ≈ 120. That value is taken, empirically, as the cut-off point for “small N” in these plots. Most of the σ estimates for larger N values are below 2.0; those that are higher do not exceed around 2.3. FIGURE 2.6 presents these results in histogram form.
FIGURE 2.5. Maximum Likelihood Estimates of σ by Lab and N.
Finally, FIGURE 2.7 shows the fitted σ values both by N category and proportion of data values larger than 0.02 (proportion of “detects” using a typical RL of 0.02). The more erratic fitted σ estimates come with small N with data more clustered around low values. Among the 26 datasets with N > 120, the fitted σ values range from 0.91 to 2.28; only three exceed 2.0, all with small proportions of “detects”. Among the 26 datasets with small N, 6 have fitted σ values above 2.0, but only 2 of those are with datasets with over 10 percent “detects”.
In studying the properties of this sort of MLE fitting of the LNNM, Davis, Field, and Gran(1) found that estimates of individual parameters were quite variable, with substantial correlations (trade-offs) among the fitted values of the parameters; there was less variation in the fitted models themselves (focusing in particular on the resulting estimates of the 95th percentile). The results presented here, FIGURE 2.7 in particular, suggest that the bound σ ≤ 2 is reasonably conservative, keeping in mind that the confidence level achieved by the QNP UTL does not deteriorate much if σ is a high as 2.3, as seen in FIGURE 3.2a of Supplement 3, and also that for none of the datasets was there a statistically significant improvement in the fit (as indicated by increase in LL) when the fitted σ was allowed to increase above 2.0.
FIGURE 2.6. Fitted σ values, by Lab and N category. “Small N” means N < 120; see FIGURE 2.5.
FIGURE 2.7. Fitted σ values by Lab, N category, and proportion of values > 0.02 (“detects”, using a typical RL = 0.02).
Conclusions
This Supplement provides support for the assumed bound σ ≤ 2 for an underlying lognormal distribution of contaminant concentrations. Three lines of evidence have been supplied: one from the IH literature, from exposure data acquired during worker protection investigations; one from surveys of Be surface concentrations at the Nevada Test Site and ancillary locations; and the fact that the QNP UTLs derived using this assumption agree nicely with the AIHI rule-of-thumb(2). In each case the evidence does support assuming this bound, recognizing that σ values mildly higher than that bound incur minor decreases in the confidence level attained by the QNP UTLs when indeed the underlying distribution of concentrations is lognormal.
References for Supplement 2
1. Davis, C.B., D. Field, and T.E. Gran: A Model for Measurements of Lognormally Distributed Environmental Contaminants, Report DOE/NV/25946—733, July 2009, revised January 2012.
2. Mulhausen, J., J. Damiano, and E.L. Pullen: Further Information Gathering, in A Strategy for Assessing and Managing Occupational Exposures, Third edition, J.S. Ignacio and W.H. Bullock (eds), Fairfax, VA, American Industrial Hygiene Association, 2006, pp. 415-421.
3. Kromhout H., E. Symanski, and S.M. Rappaport: A comprehensive evaluation of within- and between-worker components of occupational exposure to chemical agents. Ann Occup Hyg. 37(3):253-270 (1993).
4. Symanski E., S. Maberti, and W. Chan: A meta-analytic approach for characterizing the within-worker and between-worker sources of variation in occupational exposure. Ann Occup Hyg. 50(4):343-57 (2006).
5. Centers for Disease Control and Prevention: Occupational Exposure Sampling Strategy Manual Leidel, N.A., K.A. Busch, and J.R. Lynch, ( NIOSH 77-173), National Institute of Occupational Safety and Health, 1977.
6. National Security Technologies: Worker Environment Beryllium Characterization Study, prepared for the U.S. Department of Energy, Nevada Site Office, Report DOE/NV/25946—850, December 2009.
7. Rocke, D.M. and S. Lorenzato: A Two-Component Model for Measurement Error in Analytical Chemistry, Technometrics 37(2): 176-184, May 1995.
8. Currie, L.A.: Limits for Qualitative Detection and Quantitative Determination: Application to Radiochemistry, Analytical Chemistry 40(3): 586-593, March 1968.
Table IV. LNMM FitsData
Lab N< > Fitted parameters
Descriptionset 0 0.02 α τ δ µ σ LL1 D 82 71 0 -0.0035 0.0006 0.10 -7.270 1.211 425.79 Base camp OF -0.0035 0.0006 0.30 -7.230 1.171 425.80 2 D 80 64 0 -0.0013 0.0012 0.10 -10.350 1.999 418.21 Base camp FO
-0.0013 0.0012 0.49 -10.350 1.999 418.213 D 490 209 4 -0.0003 0.0012 0.10 -8.980 2.000 2441.27 Base camp OF FO EX
-0.0003 0.0012 0.23 -8.970 2.000 2441.28-0.0002 0.0012 0.10 -9.520 2.280 2441.80
-0.0002 0.0012 0.15 -9.520 2.280 2441.80 4 D 374 37 6 0.0004 0.0006 0.10 -7.800 1.830 1741.46 Base camp OF LA
0.0004 0.0006 0.15 -7.790 1.820 1741.465 B 29 18 4 -0.0058 0.0011 0.10 -5.920 1.820 104.01 Base camp OF -0.0058 0.0011 0.09 -5.920 1.820 104.01 1 high outlier omitted6 B 154 81 5 -0.0019 0.0023 0.10 -7.070 1.740 620.62 Base camp OF LA
-0.0019 0.0023 0.13 -7.070 1.740 620.627 D 319 226 2 -0.0036 0.0020 0.10 -6.600 0.990 1413.78 Urban OF -0.0036 0.0020 0.14 -6.590 0.980 1413.78 8 D 463 57 3 0.0010 0.0013 0.10 -8.300 1.710 2249.33 Urban OF
0.0010 0.0013 0.12 -8.300 1.710 2249.339 D 233 58 0 -0.0001 0.0011 0.10 -7.760 1.334 1168.14 Urban OF FO -0.0001 0.0011 0.41 -7.600 1.264 1168.16
10 B 739 371 1 -0.0006 0.0025 0.10 -9.470 1.994 3302.88 Urban OF-0.0006 0.0025 0.42 -9.470 1.994 3302.90
11 D 190 37 10 -0.0004 0.0011 0.10 -6.550 1.540 803.19 Base camp QU -0.0004 0.0011 0.13 -6.550 1.540 803.20
12 D 45 0 22 -0.0054 0.0000 0.10 -3.710 0.620 124.30 Base camp QU-0.0060 0.0000 0.11 -3.680 0.600 124.31
13 B 232 89 16 -0.0008 0.0023 0.10 -7.190 2.000 908.85 Base camp QU ST-0.0008 0.0023 0.34 -7.160 2.000 908.88-0.0008 0.0023 0.10 -7.240 2.060 908.89
-0.0008 0.0023 0.19 -7.240 2.060 908.89 14 D 317 169 15 -0.0014 0.0010 0.10 -7.240 1.800 1323.98 Base camp OF SH
-0.0014 0.0010 0.13 -7.240 1.800 1323.98 1 low outlier omitted15 D 575 48 29 0.0005 0.0016 0.10 -6.320 1.400 2340.48 Base camp SH FO ST 0.0005 0.0016 0.11 -6.320 1.400 2340.48 4 low outliers omitted
16 B 37 18 4 -0.0047 0.0007 0.10 -5.600 1.420 137.25 Base camp OF-0.0046 0.0006 0.00 -5.630 1.410 139.34 1 high outlier omitted
17 B 89 49 5 -0.0020 0.0021 0.10 -7.490 2.000 363.26 Base camp FO-0.0019 0.0021 0.39 -7.520 2.000 363.34-0.0015 0.0022 0.10 -9.000 2.970 364.23
-0.0015 0.0022 0.13 -9.010 2.990 364.23 18 B 107 52 0 -0.0059 0.0020 0.10 -5.240 0.410 464.48 Base camp FO SH
-0.0059 0.0020 0.11 -5.240 0.410 464.4819 B 37 25 0 -0.0200 0.0020 0.10 -4.050 0.105 159.89 Base camp OF -0.0200 0.0020 0.10 -4.050 0.105 159.89
20 B 47 28 2 -0.0009 0.0015 0.10 -8.510 2.000 218.70 Base camp SH-0.0008 0.0015 0.53 -8.570 2.000 218.78-0.0006 0.0015 0.10 -11.160 3.580 219.34
-0.0005 0.0015 0.00 -12.800 4.590 220.34
Data Lab N < > Fitted parameters Descriptionset 0 0.02 α τ δ µ σ LL21 B 42 15 2 -0.0004 0.0022 0.10 -8.240 2.000 179.40 Base camp ST
-0.0004 0.0022 0.60 -8.210 2.000 179.500.0001 0.0023 0.10 -12.740 4.560 180.45
0.0001 0.0022 0.00 -13.030 4.500 181.36 22 B 356 85 78 -0.0021 0.0031 0.10 -5.060 1.490 1108.48 Base camp FO SH ST
-0.0020 0.0031 0.09 -5.080 1.510 1108.5123 B 168 63 8 -0.0038 0.0016 0.10 -5.280 0.960 626.06 Base camp SH -0.0038 0.0016 0.11 -5.280 0.960 626.06
24 B 51 8 20 -0.0018 0.0012 0.10 -4.500 1.820 121.30 Base camp ST-0.0018 0.0012 0.29 -4.460 1.790 121.31
25 D 436 234 7 -0.0012 0.0010 0.10 -7.230 1.510 2050.12 Urban SH OF ST FO -0.0012 0.0010 0.08 -7.230 1.510 2050.12
26 B 81 3 23 0.0071 0.0043 0.10 -5.520 1.790 234.77 Urban SH ST0.0071 0.0043 0.15 -5.520 1.790 234.77
27 D 206 30 64 -0.0030 0.0010 0.10 -4.440 1.330 555.34 Forward FO SH -0.0030 0.0010 0.15 -4.440 1.330 555.35
28 B 409 155 76 -0.0032 0.0023 0.10 -5.410 1.690 1273.41 Forward SH-0.0031 0.0023 0.09 -5.440 1.720 1273.47
29 B 347 48 176 -0.0036 0.0006 0.10 -3.660 1.830 563.43 Forward SH FO -0.0036 0.0006 0.12 -3.660 1.830 563.43
30 D 184 20 0 0.0009 0.0014 0.10 -7.400 1.275 874.43 Forward OF EX0.0010 0.0014 0.64 -7.360 1.175 874.58
31 B 169 92 4 -0.0017 0.0027 0.10 -7.620 1.990 675.73 Forward OF -0.0016 0.0027 0.37 -7.640 1.990 675.81
32 B 264 153 5 -0.0036 0.0012 0.10 -6.030 1.150 1099.03 Forward OF EX LA-0.0043 0.0009 0.43 -5.650 0.870 1101.45
33 B 115 31 15 -0.0004 0.0036 0.10 -6.390 2.000 378.94 Forward FO SH-0.0003 0.0036 0.36 -6.370 2.000 379.000.0001 0.0037 0.10 -6.870 2.380 379.32
0.0001 0.0037 0.14 -6.870 2.380 379.32 34 B 110 21 41 -0.0028 0.0017 0.10 -4.330 1.710 249.36 Forward SH
-0.0028 0.0017 0.09 -4.330 1.710 249.3635 D 202 33 4 0.0000 0.0007 0.10 -7.170 1.540 949.58 Forward OF SH FO 0.0000 0.0007 0.06 -7.170 1.540 949.58
36 D 118 84 1 -0.0023 0.0009 0.10 -6.900 1.210 567.24 Forward OF-0.0023 0.0009 0.08 -6.900 1.210 567.24
37 B 143 100 10 -0.0044 0.0012 0.10 -6.530 2.000 548.40 Forward OF FO-0.0044 0.0012 0.32 -6.490 2.000 548.45-0.0042 0.0013 0.10 -6.780 2.240 548.73
-0.0042 0.0013 0.14 -6.780 2.240 548.73 38 B 42 7 27 -0.0062 0.0000 0.10 -3.270 1.420 63.12 Forward FO SH
-0.0061 0.0000 0.28 -3.240 1.400 63.1439 B 169 47 54 -0.0036 0.0020 0.10 -4.650 1.880 406.39 Forward EX -0.0036 0.0020 0.08 -4.650 1.880 406.39
40 D 15 0 7 0.0012 0.0000 0.10 -4.160 1.380 36.38 Forward FO 0.0012 0.0000 0.28 -4.120 1.340 36.39
Data Lab N < > Fitted parameters Descriptionset 0 0.02 α τ δ µ σ LL41 B 100 29 38 -0.0035 0.0006 0.10 -4.580 1.850 248.90 Forward SH QU FO -0.0035 0.0006 0.26 -4.550 1.830 248.90
42 B 45 8 31 -0.0051 0.0000 0.10 -3.070 1.990 39.54 Forward EX-0.0051 0.0000 0.32 -3.010 1.990 39.65
43 B 44 5 27 -0.0034 0.0000 0.10 -3.460 1.680 67.19 Forward FO ST -0.0034 0.0000 0.26 -3.430 1.660 67.20
44 B 270 148 17 -0.0042 0.0020 0.10 -5.960 1.590 978.55 Forward FO SH-0.0042 0.0020 0.11 -5.960 1.590 978.55 1 low outlier omitted
45 D 423 61 2 0.0018 0.0025 0.10 -9.500 1.993 1892.67 Other OF 0.0019 0.0025 0.87 -9.730 1.993 1893.05
46 D 130 2 11 -0.0007 0.0009 0.10 -5.040 0.910 475.09 Other ST-0.0007 0.0009 0.13 -5.040 0.910 475.09
47 D 23 2 5 -0.0036 0.0000 0.10 -4.430 0.850 73.13 Forward FO-0.0036 0.0000 0.14 -4.430 0.840 73.14
48 B 115 47 0 -0.0001 0.0028 0.10 -9.300 2.000 498.45 Urban LA-0.0001 0.0028 0.68 -9.310 2.000 498.490.0001 0.0028 0.10 -11.500 3.010 498.64
0.0001 0.0028 0.47 -11.350 2.950 498.64 49 D 30 0 11 0.0052 0.0038 0.10 -4.740 2.000 65.50 Urban LA
0.0054 0.0038 0.37 -4.700 2.000 65.56 1 low outlier omitted0.0066 0.0041 0.10 -5.220 2.580 65.970.0066 0.0041 0.11 -5.220 2.570 65.97
50 D 34 0 2 0.0035 0.0012 0.10 -7.000 1.806 145.97 Urban LA 0.0035 0.0012 0.08 -7.010 1.806 145.97
51 B 60 43 1 -0.0045 0.0013 0.10 -6.310 1.300 253.07 Urban EX -0.0045 0.0013 0.09 -6.310 1.300 253.07
52 B 30 11 2 0.0003 0.0031 0.10 -7.490 2.000 115.14 Other FO0.0003 0.0031 0.43 -7.450 2.000 115.160.0010 0.0033 0.10 -9.620 3.250 115.39
0.0013 0.0033 0.00 -12.590 4.990 115.99 Statistically significant improvement resulting from varying δFor σ, fitted value moderately above 2For τ or δ, fitted value inconsistent with conceptual model of the measurement process, likely due to over-fitting.Fitted value of σ well above 2
Supplement 3
EVALUATING THE PERFORMANCE OF QNP UTLS
QNP UTLs are derived assuming a LN distribution with shape parameter σ = 2.0. When used to test the null hypothesis that the 95th percentile of a distribution is at least as high as the OEL or RC, one wants the significance level to be at most 5 percent (= 1 – confidence level). There are several questions: what happens to the statistical performance if
the distribution is LN, but with σ larger than 2.0 (regarding the integrity of the significance level) or smaller (regarding statistical power);
the underlying distribution of concentrations is not LN but a member of some other distribution family; and/or
substantial amounts of analytical variation are present?
A simulation study was performed to study these. LN, gamma (GA), and Weibull (WE) distributions with several shape parameters were used for the concentrations, with varying amounts of analytical variation added, as described in Supplement 1.
Numbers of observations N equal to 10, 15, 20, 30, 45, and 59 were used. To evaluate significance levels, the scale parameter was adjusted to make the 95th percentile of the distribution of concentrations equal to the RC (taken as 0.20 for this study). To evaluate statistical power (the chance of correctly rejecting the null hypothesis and deciding that the 95th percentile is less than RC), the scale parameter was adjusted to make the 95th percentile equal to 0.15, 0.10, 0.05, and 0.02 (three-fourths, half, one-fourth, and a tenth of the RC). Typical Reporting Limits in this situation would be 25 percent to 10 percent of RC. Results are presented graphically. 10,000 randomly generated samples were used in each case.
For example, FIGURE 3.1 shows the measurement distributions obtained by starting with the conservative LN 2.0 distribution with 95th percentile set equal to 0.05, then adding varying amounts of normally distributed analytical variation. TABLE I gives the details of the normally distributed analytical variation added. There are two independent components. For example, the standard deviation of B (good) analytical variation is
StDev2 = 0.0022 + (0.02 * C)2, (4)
where C is the actual concentration of the contaminant. With A there is no analytical variation. B represents performance expected of good labs, and is typical for the study described in Supplement 2. With C the large-C standard deviation almost meets the 10 percent RSD criterion often thought of as representing adequate precision for quantitative analyses, and with D the large-C standard deviation is so large that measurements are almost, but not quite, guaranteed merely to be positive for large concentrations. The large majority of values are well below the RC; in cases C and D analytical variation is a major contributor to overall variability.
FIGURE 3.1. Distributions of measurements when the underlying distribution of concentrations C has a LN 2.00 distribution and varying amounts of normally distributed analytical variation are added.
Significance Levels
The first evaluation criterion is whether or not QNP UTLs reasonably honor the target significance level of (at most) 5 percent for the LN 2.0 distribution as well as a variety of other distributions. FIGURE 3.2a shows the empirical significance level against N for selected LN distributions. QNP UTLs are derived assuming σ = 2.0, seen in Supplement 2 to be a reasonable upper bound for data observed in practice. LN 2.3 is included in the evaluations to see how bad things might get with even more skewed LN distributions.
As N approaches 59 the significance levels all approach the nominal 5 percent, as expected since the QNP UTL becomes the NPUTL when N = 59. For smaller N the significance level is somewhat sensitive to the actual distribution. In this and FIGURES 3.2b and 3.2c there are error bars at N = 5; 95 percent of simulated empirical significance levels should be within these bars if the true significance level is actually the desired 5 percent.
The QNP UTL is calibrated to maintain a 5 percent significance level for the LN 2.0 distribution with no analytical variation. If the underlying distribution is not as skewed as LN 2.0, the significance level and hence power may be lower than desirable, perhaps requiring additional samples to declare a “pass”. On the other hand, the significance level for the very conservative
TABLE I. Normal Distributions for Analytical Variation StDev of Component with Constant
Standard Deviation Relative Standard DeviationA none 0 0%B good 0.002 (1% of RC) 2%C fair 0.005 (2.5% of RC) 10%D poor 0.01 (5% of RC) 30%
LN 2.3 distribution rises only to around 8 percent with the smallest N, and remains less than 6 percent for N ≥ 30 or so, which is itself reasonable agreement with the target 5 percent.
FIGURE 3.2a. Empirical significance levels for QNP UTLs for LN distributions with varying σ and amounts of analytical variation. QNP UTLs are derived assuming a LN 2.00 distribution without analytical variation, shown as the bold green line.
FIGURES 3.2b and 3.2c show the same sort of results for selected GA and WE distributions, again with varying amounts of analytical variation. In each plot the most conservative distribution in each family has significance levels not worse than those of the very conservative LN 2.3 distribution.
It is reassuring that the empirical significance levels seem quite insensitive to laboratory precision for each distribution family. This is very desirable, and is related to the fact that the QNP UTL is based only on the largest value. Other UTL procedures studied by Davis, Field, and Gran(1), particularly those relying on parametric censored-data estimation techniques, are much more sensitive to the nature of the low-end analytical variation, which is nearly always unobservable, being hidden below RLs.
On the other hand, the significance levels for the least skewed of the GA and WE distributions are very low for N < 45. This raises a concern that the QNP UTLs might be excessively conservative with these distributions, even more so than with LN distributions with σ < 2.0, possibly leading to less than satisfactory statistical power.
FIGURE 3.2b. Empirical significance levels for QNP UTLs for GA distributions with varying shape parameters Φ and amounts of analytical variation.
FIGURE 3.2c. Empirical significance levels for QNP UTLs for WE distributions with varying shape parameters Φ and amounts of analytical variation.
Statistical Power
Statistical power curves show the empirical significance level (when 95th percentile = RC = 0.2) or power (when 95th percentile < RC) against the 95th percentile. Ideally, the significance level would be ≈ 5 percent when the 95th percentile = RC, but high when 95th percentile < RC. As expected, power is better with larger N. There is a separate power curve for each combination of {distribution, number of observations, and amount of analytical variation}. For these evaluations the 95th percentiles are set equal to 10, 25, 50, 75, and 100 percent of RC respectively. Error bars at 0.01 show the 95 percent ranges where one expects the empirical power level to be with 10,000 samples if the true significance/power levels are actually 5, 50, or 95 percent.
FIGURE 3.3a gives power curves for the LN 2.0 distribution used to construct QNP UTLs. As with the significance level plots, it is reassuring that the precision of the analytical variation has little impact; so long as it is A, B, or C, those curves are nearly indistinguishable.
FIGURE 3.3a. Empirical power curves for QNP UTLs when the underlying distribution is LN 2.00, used for developing QNP UTLs, with varying amounts of analytical variation added.
FIGURES 3.3b through 3.3d show empirical power curves for the other LN distributions used; FIGURES 3.4a through 3.4d show them for the GA distributions; and FIGURES 3.5a and 3.5b show them for the WE distributions. (The GA 1.0 and WE 1.0 distributions are the same, being the familiar exponential distribution.)
FIGURE 3.3b. Empirical power curves for the very conservative LN 2.30 distributions.
FIGURE 3.3c. Empirical power curves for the not-so-conservative LN 1.75 distributions.
FIGURE 3.3d. Empirical power curves for the least conservative LN distribution used, LN 1.50.
FIGURE 3.4a. Empirical power curves for the very conservative GA 0.15 distributions.
FIGURE 3.4b. Empirical power curves for the conservative GA 0.25 distributions.
FIGURE 3.4c. Empirical power curves for the not-so-conservative GA 0.6 distributions.
FIGURE 3.4d. Empirical power curves for the less conservative GA 1.0 = WE 1.0 = exponential distributions.
FIGURE 3.5a. Empirical power curves for the very conservative WE 0.35 distributions.
FIGURE 3.5b. Empirical power curves for the conservative WE 0.45 distributions.
Another way to look at the power curves
These next plots allow for easier comparison of power curves among the various distributions. For these only the “A” (absent) analytical variation is shown, as FIGURES 3.3 through 3.5 demonstrate that the amount of analytical variation present plays at most a minor role in the performance of QNP UTLs. FIGURE 3.6 shows power curves side-by-side for the same number of observations N. The heavy green line in the left plots is for the LN 2.0 distribution used to develop QNP UTLs.
FIGURE 3.6. Side-by-side power curves for the same N, allowing easy comparisons for different distributions. These all are from simulations with analytical variation absent (case “A”); in previous plots, the amount of analytical variation is seen to have little effect on significance levels and power.
Discussion
With the exception of the two least skewed GA distributions (and also WE 1.0 which is the same as GA 1.0) for N ≤ 30, the low significance levels seen with some of the GA and WE distributions seem not to have a large adverse effect on the power curves, which are generally similar to that of the LN 2.0 distribution if not better. Rather, the power curves are a bit lower for small N, but rather higher for large N, particularly for the GA distributions.
But recall the iterative application intended; if the available data do not yet allow one to declare either a “pass” or a “fail”, one compares (largest value/RC) to the table of critical values, augments the dataset, and repeats the comparison. Accordingly, having lower power for small N may delay declaring a “pass” for a clean facility, but not necessarily preclude doing so.On the other hand, the power curves fall short of 100 percent even for N = 59 (the NPUTL). With skewed distributions such as these there will often be datasets with at least one value higher than the RC. But recall also the alternative part of the decision rule commonly employed which will declare a “fail” if any observation greater than the RC is obtained. This alternative part of the decision rule will affect any UTL procedure used, not just the QNP UTL.
Finally, in principle, if one had more precise information about the nature of the underlying distribution of concentrations involved in a specific application, one could achieve better power by deriving QNP UTLs from a less conservative model than LN 2.0. Such information is elusive, however, particularly when only censored data are reported.
Reference for Supplement 3
1. Davis, C.B., D. Field, and T.E. Gran: A Model for Measurements of Lognormally Distributed Environmental Contaminants, Report DOE/NV/25946—733, July 2009, revised January 2012; available on request.
