POTENTIAL METHODOLOGY

Let y _ij be the jth observation from the ith laboratory, j = 1, … , n _i ,and i = 1, … , k. We assume that the count variable y _ij follows a Poissondistribution with parameter λ _ij . To model the interlaboratory variability, we assume a mixed-effect Poisson regression model with random parameters β₀ and β₁. We further assume that the joint distribution of β₀ and β₁ is bivariate normal, with mean vector γ ₀ and covariance matrix Σ. Hence, regarding the complete distributions of y _ij , our assumptions are asfollows:

where β _{0 i} and β _{1 i} are respectively the random intercept and slope parameters for the ith laboratory and x _ij is the true count in the jth measurement from the ith laboratory. Of course, we never know the true count (i.e., x _ij ), but a reasonable substitute is a consensus estimate based on aseries of leading laboratories or analysts.

At this stage, we assume that all of the y _ij and the corresponding x _ij are known. We estimate the model parameter Σ by the method of marginal maximum likelihood (MML). The resulting estimate of Σ denoted by is consistent and MML also provides the standard error of . Given MML estimates of the means and covariance matrix, we can obtain empirical Bayes estimates of β _{0 i} and β_{1 i} , denoted by and .

Let y _il be a new observation from the ith laboratory. We do not know the value of the corresponding true observation x _il . Our goal is to estimate x _il and construct a confidence region for x _il using the previous estimates , , and of Σ, β _{0 i} , and β _{1 i} , respectively. We follow thelikelihood-based procedure to estimate x _il (i.e., maximize the likelihoodfunction of y _il with respect to x _il using , , and ). Denote this estimate of x _il by . The expression of is as follows:

(1)

This estimate is valid provided y _il > 0. In the case of y _ij = 0 wemust set x _il = 0. Also note that the above estimate of x _il becomes nega tive if . In such a scenario we also set x _il = 0. is asymptotically unbiased for large values of , where . The standard errors of and are obtained directly via MML. The laboratory-specific estimate of the conditional variance of y _il is. Using the Delta method, we obtain the estimate of the variance of ln(y _il ) as . Hence an approximate expression for the variance of is . Thus, the standard error of is . Let us denote the 95 percent asymptotic confidence region of x _il by , where

(2)

The aforementioned confidence region of x _il is based on the assumption that we had samples from the ith laboratory and we estimated the laboratory-specific parameters and also estimated Σ borrowing strength from all of the laboratories. However, if the new observation y _il comesfrom an arbitrary new laboratory and the estimates of its parameters are not available, then we should estimate x _il globally (i.e., based on the expected values of the laboratory-specific parameters).

Based on these methods, we can now obtain a point estimate of the true number of asbestos fibers in the sample (x _ij ), and a 95 percent confidence region for that true count. There are several useful things that we can do with these quantities. First, we can now always provide an uncertainty interval surrounding our best estimate of the true fiber count. Second, we can determine if the lower confidence limit is greater than zero. If it is, then we can have 95 percent confidence that the true number of asbestos fibers in the sample is greater than 0. Third, we can determine the detection limit, which is the smallest observed count for which the true count is greater than zero. To do this, we begin by setting the true count to zero (i.e., x _ij = 0 ) and then compute the upper 95 percent prediction limit for the observed y. Any observed y greater than the prediction limit will indicate that the true count is greater than zero. The prediction limit for y given x = 0 can be computed via simulation using the following expressions of the unconditional mean and variance of y:

ILLUSTRATION

To illustrate the statistical methodology for inter-laboratory calibration of counts and to obtain a better feel for the magnitude of the variability within and between laboratories, we obtained de-identified data from the New York State inter-laboratory asbestos testing program, which were graciously provided by Dr. James Webber of the Wadsworth Center of the New York State Department of Health. Results based on both transmission electron microscopy (TEM) and phase contrast microscopy (PCM) were analyzed. For TEM, there were a total of 327 samples from 43 laboratories. For PCM there were a total of 9400 airborne asbestos samples analyzed by several hundred laboratories, though participation in a single round ranged from 100 to 150 laboratories. The data were collected as a part of the New York State Environmental Laboratory Approval Program (ELAP) based on the semiannual proficiency testing of laboratories analyzing airborne asbestos, based on the Asbestos Hazard Emergency Response Act (AHERA) criteria and of laboratories analyzing airborne fibers by the NIOSH 7400 method (NIOSH, 1994b). For TEM, our analysis focuses on cummingtonite-grunerite (amosite; AM) counts from 11 rounds. Two additional rounds were not considered because they contained impractically high counts (>6,000 structures/mm²). (All data are expressed in structures/mm².)

In order to apply the previously described methodology based on a mixed-effects Poisson regression model, we must obtain an estimate of true count for each sample. To this end, we used the overall mean count over all of the laboratories that analyzed each sample. In addition to fitting the Poisson regression model, we also used the alternative minimum level described by Gibbons et al. (1997), Zorn et al. (1997) and Gibbons and Coleman (2001), to obtain estimates of the critical level (L_C), detection limit (L_D) and quantification limit (L_Q) for these data (see Currie, 1968, for an excellent review). The L_C is a threshold used to determine whether or not detection has occurred. The L_D is the lowest level for which there is simultaneous high confidence that: (a) detection will occur if the true value is at the detection limit; (b) there will NOT be detection if the true value is zero. The L_Q is the lowest level at which a specified (estimated) relative standard deviation is achieved, typically 10 percent, 20 percent, or 30 percent.

TEM Analyses

Figure B-1 displays the raw TEM asbestos counts on the y-axis and the mean asbestos counts on the x-axis (i.e., best available estimate of the true count).

FIGURE B-1

Raw asbestos (AM) counts analyzed by TEM.

Figure B-1 reveals that the absolute variability is proportional to the true (i.e., average) count. This is consistent with a Poisson random variable and can be modeled either via a Poisson regression model or a model that allows for non-constant variability in the calibration function as described by Gibbons and colleagues (1997, 2001). We next fit a mixed-effects Poisson regression model to the TEM AM asbestos data. The model is ln(λ) = (γ₀ + γ₁ x) + (u ₀ + u ₁ x) where u has a normal distribution with mean 0 and a variance-covariance matrix Σ. x is the true count divided by 1,000 (done to obtain parameter estimates in a metric of reasonable magnitude for the purpose of interpretation since the model is for the log count as shown above). The parameter estimates, standard errors and tests of significance are displayed in Table B-1.

TABLE B-1

Marginal Maximum Likelihood Estimates of the Mixed-effects—Poisson Regression Model for TEM Data.

The term Σ(2,2) is the variance of the slopes of the inter-laboratory calibration curves, which reveals a standard deviation of 0.58, which is 58 percent of the mean slope (1.0128) of the calibration curve over all of the 45 laboratories. This is an enormous relative standard deviation, indicating that the laboratories exhibit considerable variability in their individual calibration curves (i.e., differential sensitivity to changing numbers of particles from lab to lab).

Figure B-2 presents empirical Bayes estimates of the individual laboratory calibration functions. Note that the y-axis is in log-scale. Figure B-2 confirms that there is considerable variability in the slopes of the estimated calibration functions.

FIGURE B-2

Individual laboratory estimated calibration functions. NOTE: The y-axis is in a log scale with base = 10.

Next, we estimated detection and quantification limits from these data using the AML method described by Gibbons and colleagues (1997). The results are displayed graphically in Figure B-3.

FIGURE B-3

Alternative minimum level model with 99 percent interval for asbestos TEM samples in mm².

Figure B-3 reveals that the critical level (L_C) is 481 fibers, the detection limit (L_D) is 1335 fibers and the quantification limit (L_Q) is 3003 fibers. At the L_Q, the relative standard deviation is still reasonably large (i.e., 23 percent).

Figure B-4 presents the variance function, for which the best fit was based on the Rocke and Lorenzato Model (Rocke and Lorenzato, 1995) and reveals that the variability increases linearly from 100 to 2,500 fibers.

FIGURE B-4

Standard deviation vs. concentration for asbestos TEM samples in mm².

Finally, Figure B-5 displays a plot of the relationship between average counts and the relative standard deviation (RSD). This figure reveals that considerable uncertainty exists in asbestos counts throughout all of the samples investigated, regardless of the number of fibers. However, the RSD stabilizes at around 20 percent for true concentrations around 2,000.

FIGURE B-5

Percent relative standard deviation vs. concentration for asbestos TEM samples in mm².

PCM Analyses

In contrast to TEM, there were several extreme values (in excess of counts of 5,000) associated with the PCM data, despite the fact that the highest average concentration never exceeded 800 counts (see Figure B-6).

FIGURE B-6

Raw PCM data.

Exclusion of the 12 extreme counts reveals a more consistent pattern in the raw data (see Figure B-7). Results of the analysis of the raw PCM data excluding outliers are presented in Table B-2. For PCM, the true counts were divided by 100 to place the estimates on a scale that is more easily interpreted.

FIGURE B-7

Raw PCM data excluding outliers.

TABLE B-2

Marginal Maximum Likelihood Estimates of the Mixed-effects—Poisson Regression Model for PCM Data.

The inter-laboratory standard deviation is 0.40, which is 67 percent of the mean slope (0.60) of the calibration curve over all of the laboratories. This is an even larger relative standard deviation than obtained for TEM, indicating also that the laboratories exhibit considerable variability in their individual calibration curves (i.e., differential sensitivity to changing numbers of particles from lab to lab) for PCM.

Figure B-8 presents empirical Bayes estimates of the individual laboratory calibration functions. Note that the y-axis is in log-scale. This figure confirms that there is considerable variability in the slopes of the estimated calibration functions.

FIGURE B-8

Individual laboratory estimated calibration functions for PCM. NOTE: The y-axis is in a log scale with base = 10.

Next, we estimated detection and quantification limits for the PCM data. The results are displayed graphically in Figure B-9. This figure reveals that the critical level (L_C) is 127 fibers, the detection limit (L_D) is 589 fibers and the quantification limit (L_Q) is 924 fibers. At the L_Q, the relative standard deviation is still quite large (i.e., 32 percent). Figure B-9 also reveals that outliers still remain in the data; however, the prediction intervals are conservative due to the large number of measurements.

FIGURE B-9

Alternative minimum level model with 99 percent prediction interval for asbestos PCM samples in mm².

Figure B-10 presents the variance function, for which the best fit was based on the Rocke and Lorenzato Model (Rocke and Lorenzato, 1995). The figure reveals that the variability is constant below 100 fibers and then increases linearly from 100 to 800 fibers.

FIGURE B-10

Standard deviation vs. concentration for asbestos PCM samples in mm².

Finally, Figure B-11 displays a plot of the relationship between average counts and the relative standard deviation. The figure shows that considerable uncertainty exists in PCM asbestos counts throughout all of the samples investigated, regardless of the number of fibers. However, the RSD stabilizes at around 30 percent for fiber counts around 500.

FIGURE B-11

Percent relative standard deviation vs. concentration for asbestos PCM samples in mm².

THE PROBLEM OF NON-DETECTS

A complication in the statistical analysis of environmental data in general and asbestos in particular is the presence of non-detects. Even if the measured concentrations have a known distribution (e.g., normal, lognormal, Poisson) the overall distribution may not because of a mass of probability associated with a count of zero, or samples in which the material has not been detected. In the case of an asbestos count, it may be the case that there are more zeros (i.e., non-detects) than are expected based on a Poisson distribution. In this case, one may consider extensions to the Poisson model, such as a zero-inflated Poisson model (Lambert, 1992). General discussions of the treatment of nondetects in environmental data analysis can be found in Helsel (2005) and Gibbons et al. (2009).

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Footnotes

1

We would like to thank Professor Dulal K. Bhaumik of the University of Illinois at Chicago and his student Yoonsang Kim for help with statistical derivations and analyses, and Dr. James Webber of the Wadsworth Center of the New York State Department of Health for providing the data and helpful comments on an earlier draft. We would also like to thank the statistical reviewer for making several helpful comments and helping us expand the scope of the statistical discussion.