In reviewing the literature on asbestos fiber counting, there appears to be considerable variability in counts from analyst to analyst within a given laboratory as well as between laboratories. As a consequence, a new observed count from a particular analyst in a particular laboratory may have considerable deviation from the true value. In an effort to provide a connection between observed and true counts and to characterize the uncertainty in the true count, Dulal Bhaumik and colleagues have extended the ideas of Gibbons and Bhaumik (2001) and Bhaumik and Gibbons (2005) to the case of a Poisson random variable, which is the appropriate distribution for rare-event count data. This appendix provides a brief sketch of a potential methodology for addressing the variability and includes an illustration of this methodology using data from the New York state inter-laboratory asbestos testing program.

Let *y*_{ij} be the *j*th observation from the *i*th laboratory, *j =* 1, … , *n*_{i}, and *i* = 1, … , *k*. We assume that the count variable *y*_{ij} follows a Poisson distribution with parameter *λ*_{ij}. To model the interlaboratory variability,

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B
Counting Strategies1
In reviewing the literature on asbestos fiber counting, there appears
to be considerable variability in counts from analyst to analyst within a
given laboratory as well as between laboratories. As a consequence, a
new observed count from a particular analyst in a particular laboratory
may have considerable deviation from the true value. In an effort to pro-
vide a connection between observed and true counts and to characterize
the uncertainty in the true count, Dulal Bhaumik and colleagues have
extended the ideas of Gibbons and Bhaumik (2001) and Bhaumik and
Gibbons (2005) to the case of a Poisson random variable, which is the
appropriate distribution for rare-event count data. This appendix provides
a brief sketch of a potential methodology for addressing the variability
and includes an illustration of this methodology using data from the New
York state inter-laboratory asbestos testing program.
POTENTIAL METHODOLOGY
Let yij be the jth observation from the ith laboratory, j = 1, … , ni,
and i = 1, … , k. We assume that the count variable yij follows a Poisson
distribution with parameter λij . To model the interlaboratory variability,
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.
93

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94 REVIEW OF THE NIOSH ROADMAP
we assume a mixed-effect Poisson regression model with random pa-
rameters 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 ∑ .
Hence, regarding the complete distributions of yij, our assumptions are as
follows:
yij ~ P(λij ) , and ln(λij ) = β 0i + β1i xij ,
where β 0i and β1i are respectively the random intercept and slope pa-
rameters for the ith laboratory and xij is the true count in the jth meas-
urement from the ith laboratory. Of course, we never know the true count
(i.e., xij ), but a reasonable substitute is a consensus estimate based on a
series of leading laboratories or analysts.
At this stage, we assume that all of the yij and the corresponding xij
are known. We estimate the model parameter ∑ by the method of mar-
ginal maximum likelihood (MML). The resulting estimate of ∑ denoted
ˆ
ˆ
by ∑ is consistent and MML also provides the standard error of ∑ . Given
MML estimates of the means and covariance matrix, we can obtain em-
ˆ ˆ
pirical Bayes estimates of β 0i and β1i , denoted by β 0i and β1i .
Let yil be a new observation from the ith laboratory. We do not know
the value of the corresponding true observation xil . Our goal is to esti-
mate xil and construct a confidence region for xil using the previous esti-
ˆˆ ˆ
mates ∑ , β , and β of ∑ , β , and β , respectively. We follow the
0i 1i
0i 1i
likelihood-based procedure to estimate xil (i.e., maximize the likelihood
ˆˆ ˆ
function of yil with respect to x using ∑ , β , and β ). Denote this esti-
il 0i 1i
ˆ ˆ
mate of xil by xil . The expression of xil is as follows:
ˆ
ln( yil ) − β 0i
xil =
ˆ . (1)
ˆ
β 1i
This estimate is valid provided yil > 0 . In the case of y il = 0 we
must set xil = 0 . Also note that the above estimate of xil becomes nega-

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95
APPENDIX B
ˆ
tive if ln( yij ) < β i 0 . In such a scenario we also set xil = 0 . x il is asymp-
ˆ
totically unbiased for large values of x′l x il , where x′ = (1 xij ) .
∑ il
i
ˆ ˆ
The standard errors of β 0i and β1i are obtained directly via MML. The
laboratory-specific estimate of the conditional variance of yil is
ˆ ˆ
exp(β 0i + β1i xij ) . Using the Delta method, we obtain the estimate of the
ˆ ˆ
variance of ln(yil) as exp(− β 0i − β1i xij ) . Hence an approximate expres-
ˆ
sion for the variance of xil is exp(x′ ∑ x ij / 2) /( SE ( β1i )) 2 . Thus, the
ˆ ij
ˆ
standard error of xil is SE( xil ) = exp(x ′ ∑ x ij / 2) /( SE ( β 1i )) 2 . Let us
ˆ ˆ ij
denote the 95 percent asymptotic confidence region of xil by ℜ1 ( xil ) ,
where
ℜ1 ( xil ) = {xil : −1.96 ≤ ( xil − xil ) / SE ( xil ) ≤ 1.96} .
ˆ ˆ (2)
The aforementioned confidence region of xil is based on the assump-
tion that we had samples from the ith laboratory and we estimated the
laboratory-specific parameters and also estimated ∑ borrowing strength
from all of the laboratories. However, if the new observation yil comes
from an arbitrary new laboratory and the estimates of its parameters are
not available, then we should estimate xil globally (i.e., based on the ex-
pected 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 ( xil ), and a 95 percent con-
fidence region for that true count. There are several useful things that we
can do with these quantities. First, we can now always provide an uncer-
tainty interval surrounding our best estimate of the true fiber count. Sec-
ond, 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., xil = 0 ) and then compute the upper 95 percent pre-
diction limit for the observed y. Any observed y greater than the predic-
tion limit will indicate that the true count is greater than zero. The

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96 REVIEW OF THE NIOSH ROADMAP
prediction limit for y given x = 0 can be computed via simulation using
the following expressions of the unconditional mean and variance of y:
E ( y ) = exp(γ 0 + σ 11 / 2) , and
V ( y ) = exp(γ 0 + σ 11 / 2) + [ exp(γ 0 + σ 11 / 2) ] [ exp(σ 11 − 1)] .
ILLUSTRATION
To illustrate the statistical methodology for inter-laboratory calibra-
tion of counts and to obtain a better feel for the magnitude of the vari-
ability 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 mi-
croscopy (PCM) were analyzed. For TEM, there were a total of 327
samples from 43 laboratories. For PCM there were a total of 9400 air-
borne 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 Labo-
ratory 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 struc-
tures/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 mini-
mum level described by Gibbons et al. (1997), Zorn et al. (1997) and
Gibbons and Coleman (2001), to obtain estimates of the critical level
(LC), detection limit (LD) and quantification limit (LQ) for these data
(see Currie, 1968, for an excellent review). The LC is a threshold used
to determine whether or not detection has occurred. The LD is the lowest

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97
APPENDIX B
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 LQ 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 reveals that the absolute variability is proportional to the
true (i.e., average) count. This is consistent with a Poisson random vari-
able 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 + γ1x) + (u0 + u1 x) where u has a normal distri-
bution 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.
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, indi-
cating that the laboratories exhibit considerable variability in their indi-
vidual 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. Fig-
ure B-2 confirms that there is considerable variability in the slopes of the
estimated calibration functions.

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98 REVIEW OF THE NIOSH ROADMAP
4000
3500
3000
observed counts
2500
2000
1500
1000
500
0
0 500 1000 1500 2000 2500
true counts (mean)
FIGURE B-1 Raw asbestos (AM) counts analyzed by TEM.
TABLE B-1 Marginal Maximum Likelihood Estimates of the Mixed-
effects—Poisson Regression Model for TEM Data
Parameters Estimates SE z-value p-value
γ0 5.8000 0.0718 80.73 < .0001
γ1 1.0128 0.0965 10.50 < .0001
Σ(1,1) 0.1792 0.0416
Σ(1,2) −0.2321 0.0552
Σ(2,2) 0.3388 0.0773
NOTE: SE = standard error.

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99
APPENDIX B
10000
predicted mean counts(displayed in a log scale)
1000
100
10
0 500 1000 1500 2000 2500
true counts (mean)
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 reveals that the critical level (LC) is 481 fibers, the detec-
tion limit (LD) is 1335 fibers and the quantification limit (LQ) is 3003
fibers. At the LQ, 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 fi-
bers.
Finally, Figure B-5 displays a plot of the relationship between aver-
age 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.

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100 REVIEW OF THE NIOSH ROADMAP
FIGURE B-3 Alternative minimum level model with 99 percent interval
for asbestos TEM samples in mm2.
FIGURE B-4 Standard deviation vs. concentration for asbestos TEM
samples in mm2.

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101
APPENDIX B
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).
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.
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 laborato-
ries. 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.

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102 REVIEW OF THE NIOSH ROADMAP
130000
120000
110000
100000
90000
observed counts
80000
70000
60000
50000
40000
30000
20000
10000
0
0 200 400 600 800 1000 1200 1400 1600 1800
true counts (mean)
FIGURE B-6 Raw PCM data.
TABLE B-2 Marginal Maximum Likelihood Estimates of the Mixed-
effects—Poisson Regression Model for PCM Data
Parameters Estimates SE z-value p-value
γ0 3.8559 0.0864 44.63 < .0001
γ1 0.6006 0.0498 12.07 < .0001
Σ(1,1) 0.5158 0.0526
Σ(1,2) -0.2128 0.0265
Σ(2,2) 0.1593 0.0182
Figure B-8 presents empirical Bayes estimates of the individual labo-
ratory calibration functions. Note that the y-axis is in log-scale. This fig-
ure confirms that there is considerable variability in the slopes of the
estimated calibration functions.

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103
APPENDIX B
2500
2000
observed counts
1500
1000
500
0
0 100 200 300 400 500 600 700 800 900 1000
true counts (mean)
FIGURE B-7 Raw PCM data excluding outliers.
FIGURE B-8 Individual laboratory estimated calibration functions for PCM.
NOTE: The y-axis is in a log scale with base = 10.

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104 REVIEW OF THE NIOSH ROADMAP
Next, we estimated detection and quantification limits for the PCM
data. The results are displayed graphically in Figure B-9. This figure re-
veals that the critical level (LC) is 127 fibers, the detection limit (LD)
is 589 fibers and the quantification limit (LQ) is 924 fibers. At the LQ,
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 predic-
tion intervals are conservative due to the large number of measurements.
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.
Finally, Figure B-11 displays a plot of the relationship between aver-
age 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-9 Alternative minimum level model with 99 percent predic-
tion interval for asbestos PCM samples in mm2.

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105
APPENDIX B
FIGURE B-10 Standard deviation vs. concentration for asbestos PCM
samples in mm2.
FIGURE B-11 Percent relative standard deviation vs. concentration for
asbestos PCM samples in mm2.

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106 REVIEW OF THE NIOSH ROADMAP
Discussion
Fiber-counting protocols must be considered as a contributor to vari-
ability. The AHERA method was produced in 1987 as a simplification of
the EPA Level II analysis (U.S. EPA, 1987). “Clusters” of fibers are
counted as one structure under the AHERA method, whereas a more de-
tailed and prescriptive method, ASTM D6281, requires the analyst to
count and measure individual fibers within clusters (ASTM, 2008). In
fact, a separate inter-laboratory study, which used AM filters from one of
the batches discussed here, produced a relative standard deviation of only
11 percent at a concentration of ~400 s/mm2 when the ASTM D6281
method was used. No inter-laboratory data are available for the NIOSH
7402 TEM method, where PCM-equivalent fibers (length >5-μm, width
>0.25 μm, aspect ratio >3) are counted (NIOSH, 1994a). Variability
would probably be similar to the ASTM method, but NIOSH 7402 does
not allow counting of fibers thinner than 0.25 μm, so this method would
not monitor the very thin fibers that are considered to be the most haz-
ardous.
Filter type and preparation techniques are other sources of variabil-
ity. MCE filters sometimes have surficial defects that cause skewed
deposition across the filter face, but the skewing is not obvious once the
filter is collapsed. Furthermore, differences in collapsing methods and in
etching rates (poorly defined and inconsistently calibrated) add to the
variability (Webber et al., 2007).
Another source of variation that cannot be de-coupled is the differ-
ence in filters received by each laboratory for each PT batch. In-house
validation of homogeneity of AM filters has been checked by analyzing
5 filters from each generation batch of 109 filters. Relative standard de-
viations for these counts, by the same analyst and same instrument, are
typically 10 percent around a concentration of 1,000 s/mm2.
In both the examples, asymptotic normality is used (see Tables B-1
and B-2) in order to arrive at the reported p-values. This is appropriate
since the sample sizes are large. However, if the number of laboratories
and/or the count data within each lab are sparse, asymptotic methods
for hypothesis testing may yield biased results. One option is to use small
sample asymptotic theory that has recently become available (see
Brazzale et al., 2007 and Bellio, 2003). The relevant theory would need
to be developed for the mixed effects Poisson regression model proposed
here and is likely to be quite valuable in the context of the Rocke-
Lorenzato model as well.

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107
APPENDIX B
Taken as a whole, the current analysis reveals that there is consider-
able variability in asbestos fiber counting under both TEM and PCM
methodologies. Although detection limits are smaller for PCM than for
TEM, PCM cannot be considered an alternative because it cannot detect
the thin fibers of most concern and it cannot even determine if a fiber is
asbestos. It is critically important for the analytic community to address
the issue of TEM variability so that more reliable exposure concentra-
tions can be determined.
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 ex-
pected 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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