Metal and Nonmetal Diesel Particulate Matter (DPM)

Standard Error Factors for 350 μg/m^3^ Total Carbon Exposure Limit

# Error Factor

As with all other exposure-based M/NM compliance determinations, MSHA
will address uncontrollable sampling and analytical errors (SAE) by
allowing a margin of error before issuing a citation for exceeding the
total carbon (TC) limit. In addition to using the sum of organic carbon
(OC) and elemental carbon (EC) as a measure of total carbon (i.e.
TC~(OC+EC)~) MSHA will employ an enforcement policy for the personal
exposure limit that will use elemental carbon as a measure of TC (i.e.
TC~(ECx1.3)~). This policy will assure that a citation based on the 350
microgram per cubic meter of air (μg/m^3^) limit of TC is valid and not
the result of interferences. MSHA has developed appropriate error
factors to account for variability in sampling and analysis from such
things as pump flow rate, filters, and the NIOSH 5040 method. If the
TC~(OC+EC)~ measurement is equal to or below 350 (μg/m^3^) times the
error factor, MSHA will not issue a citation.

If the TC~(OC+EC)~ measurement is above the error factor level of 396
μg/m^3^ (350 μg/m^3^ times 1.13), MSHA will look at the EC measurement
from the sample and apply a factor of 1.3 to produce a statistical
estimate of TC without interferences. As a matter of enforcement
discretion, MSHA will use the TC~(ECx1.3)~ measurement as a check to
validate that an overexposure is not the result of interferences.

The Agency will issue a citation only if the personal exposure
measurement demonstrates noncompliance with at least 95-percent
confidence. We will achieve this 95-percent confidence level by
comparing each TC and EC measurement to the concentration limit
multiplied by the appropriate error factor. The error factor (EF) is
calculated as
$\text{EF} = 1 + (1\text{.}\text{645} \times \text{CV}_{\text{total}})$
where *CV~total~* is the coefficient of variation of the combined
sampling and analytical method.[^1]^,2^

The error factor accounts for three uncontrollable variance components
represented by *CV~total~* and then applied to the eight-hour equivalent
full-shift measure­ment of TC~(OC+EC)~ or TC~(ECx1.3)~ using Method 5040.
The three components address: (1) pump performance relative to the
nominal airflow of 1.7 L/min, (2) variability of the deposit area of
particles on the filter, and (3) laboratory analysis of carbon within
the deposit area on the filter. These individual variance components are
denoted as CV~P~, CV~D~, and CV~A~ respectively. To determine CV~total~,
the individual components were estimated separately and then combined
according to a standard propagation of errors formula:

$\text{CV}_{\text{total}} = \sqrt{\text{CV}_{P}^{2} + \text{CV}_{D}^{2} + \text{CV}_{A}^{2}}$.

Appendix 1 presents MSHA's estimates of the three CV components. Because
CV~A~ varies according to the amount of carbon deposited on a
filter,[^2] the error factor will be different for different personal
exposure limits. Based on the estimates shown in the following section
and the filter loadings corresponding to TC~(OC+EC)~ and TC~(ECx1.3)~ at
the exposure limit, the error factors we will use for the final personal
exposure limit of 350 μg/m^3^ total carbon effective January 20, 2007
are:

-   EF = 1.13 for TC~(OC+EC)~; and

-   EF = 1.11 for TC~(ECx1.3)~.

This means MSHA will issue a citation for noncompliance with the 350
μg/m^3^ total carbon exposure limit if we obtain eight-hour equivalent
full shift concentration measure­ments of TC that are:

-   396 μg/m^3^ TC~(OC+EC)~ or greater and are also

-   389 μg/m^3^ TC~(ECx1.3)~ or greater.

We believe that our estimates of CV~total~ are based on the best
scientific data currently available and reflects current sampling and
analytical errors. However, MSHA recognizes that future improvements in
sampling and/or analytical technology may reduce the random variability
associated with measuring carbon concentrations. Therefore, MSHA may
update these error factors based on future experimental data.

#  

# Appendix 1. Determination of the Error Factor

The error factor (EF) is defined as

$\text{EF} = 1 + (1\text{.}\text{645} \times \text{CV}_{\text{total}})$,
where
$\text{CV}_{\text{total}} = \sqrt{\text{CV}_{P}^{2} + \text{CV}_{D}^{2} + \text{CV}_{A}^{2}}$.
In this appendix, we will present and explain MSHA's current estimate
for each of the three components contributing to CV~total.~ These
estimates are used to determine the EF for TC~(OC+EC)~ and TC~(ECx1.3)~.

[CV~P~: Variability in volume of air pumped through the
filter]{.underline}

Variability in the air volume depends on three factors: (1) variability
in the initial setting of the pump rotameter to a calibration mark when
sampling begins, (2) pump calibration errors, and (3) variability in air
flow during the sampling period. Based on Bowman et al. (1984)[^3], MSHA
estimates that uncertainty due to the combined effects of calibration
errors and flow rate variability is represented by a coefficient of
variation (CV) no greater than 3%. Based on the experimental results
described by Tomb (1994)[^4], MSHA estimates that the CV component
associated with variability in setting the rotameter ball is
approximately 3%. Since variability in the initial flow rate is
independent of calibration of the pump rotameter and variability in flow
rate during sampling, these two uncertainty components can be combined
as follows to yield the CV representing uncertainty in total volume of
air pumped:

$\text{CV}\  = \ \sqrt{(0\text{.}\text{03})^{2} + (0\text{.}\text{03})^{2}} = 0\text{.}\text{042}$

[CV~D~: Variability in area of dust deposited on filter]{.underline}

Variability in SKC sampler performance is manifested as variability in
the area and uniformity, or density, of the particulate matter deposited
on the filter. Variability in the density of the deposit is included in
the estimated value of CV~A~ and is discussed below. The variability in
the total deposit area is addressed using CV~D~.

The manufacturer of the sampling device (SKC, Inc. Eighty Four, PA.) has
made a number of improvements designed to reduce variability in the
deposit area. The CV~D~ is based on a recent study assessing the
performance of the improved SKC sampler.[^5] The increased precision of
the deposit area was determined to be 0.005 using the data collected on
94 improved SKC samplers, therefore the CV~D~ is 0.5 (i.e. 0.005 times
100).

[CV~A~: Analytical measurement imprecision]{.underline}

Analytical measurement imprecision refers to the random variability of
repeated measurements of an analyte, performed on different punches
taken from the same filter, within the same or different laboratories.
In addition to imprecision in the instrumentation, this encompasses
random variability in the punch area and in the density of the deposit,
but not in the deposit area. Variability in the deposit area (a form of
sampling variability) is quantified separately as CV~D~ above. To
estimate CV~A~, MSHA used data obtained from the "paired punch
comparison" carried out as part of the Joint MSHA/Industry Study:
Determination of DPM Levels in Underground Metal and Nonmetal Mines
(Report on the 31-Mine Study, January 6, 2003). A full description of
the paired punch comparison is presented in that report, which has,
along with all of the data collected in connection with the study, been
placed into the public record for the Diesel Particulate Matter Exposure
of Underground Metal and Nonmetal Mines rule (RIN 1219-AB29). Although
the study mainly addressed variability in TC measurements, EC was
separately measured in the course of the laboratory analysis of each
punch. Consequently, both TC and EC analytic results are in the DPM
rulemaking record.

In the paired punch comparison, 621 filters were analyzed using two
standard punches taken from each filter. One punch (labeled "A") was
always analyzed in MSHA's laboratory. The second punch from the same
filter (labeled "B") was either analyzed in MSHA's laboratory or in one
of three other laboratories.[^6]

A repeated measures, random effects Analysis of Variance (ANOVA) was
performed to derive composite estimates of the intra- and
inter-laboratory components of analytic measurement imprecision, based
on the available data from all four laboratories. To stabilize the
variance in this analysis, a square-root transformation was first
applied to each TC~(OC+EC)~ and EC measurement. Appendix 2 contains
further justification for using this transformation and explains how it
can be used to estimate CV~A~ as a function of the filter loading.

The model used in the ANOVA was:

$$\sqrt{X_{\text{jA}}} - \sqrt{X_{\text{ijB}}} = \Delta_{i} + \lambda_{\text{ij}}(i \neq \text{MSHA}) + \varepsilon_{\text{ij}}$$

where

> *X* is the analytic result from punch A or B reported in μg/cm^2^;

*i* indexes the laboratory analyzing Punch B;

*j* indexes a specific filter;

> ∆*~i~* is a fixed effect, representing the systematic difference
> between MSHA's punch A results and the punch B results at laboratory
> *i*;
>
> *λ~ij~* is a random, Normally distributed, inter-laboratory effect
> with mean = 0 and variance = $\sigma_{\lambda}^{2}$;
>
> *ε~ij~* is a random, Normally distributed, intra-laboratory error with
> mean = 0 and variance = $\sigma_{\varepsilon}^{2}$.

Since MSHA was the only laboratory to have analyzed more than one punch
from the same filter, it was necessary to assume that intra-laboratory
imprecision, represented by $\sigma_{\varepsilon}^{2}$, was the same in
all four laboratories.

Although ∆*~i~* is a necessary part of the ANOVA model, it represents
bias that equally affects all results in the same laboratory. Therefore,
MSHA expects ∆*~i~* to be canceled out when control filters are used to
adjust the calculated concentrations. Consequently, for exposure
measurements that are appropriately corrected by means of control
filters, the composite estimate of analytical measurement uncertainty,
including both intra- and inter-laboratory imprecision, is represented
by
${\hat{\sigma}}_{T}^{2} = {\hat{\sigma}}_{\lambda}^{2} + {\hat{\sigma}}_{\varepsilon}^{2}$.
The estimates of $\sigma_{\lambda}^{2}$, $\sigma_{\varepsilon}^{2}$, and
$\sigma_{T}^{2}$ for TC~(ECx1.3)~ or TC~(OC+EC)~ based on this ANOVA are
shown in the following two tables.[^7]

Estimated analytical imprecision of EC measurements.

  -------------- -------------------------- ------------------------------ -------------------- --------------
                 $$\sigma_{\lambda}^{2}$$   $$\sigma_{\varepsilon}^{2}$$   $$\sigma_{T}^{2}$$   *σ~T~*

  Estimate       0.01642                    0.04917                        0.06559              0.256

  Standard Error 0.00751                    0.00624                        0.00418              N/A

  95% UCL        0.02877                    0.05943                        0.07246              0.269
  -------------- -------------------------- ------------------------------ -------------------- --------------

Estimated analytical imprecision of TC~(OC+EC)~ measurements.

  -------------- -------------------------- ------------------------------ -------------------- --------------
                 $$\sigma_{\lambda}^{2}$$   $$\sigma_{\varepsilon}^{2}$$   $$\sigma_{T}^{2}$$   *σ~T~*

  Estimate       0.07713                    0.03746                        0.11459              0.339

  Standard Error 0.00872                    0.00476                        0.01196              N/A

  95% UCL        0.09147                    0.04529                        0.13426              0.366
  -------------- -------------------------- ------------------------------ -------------------- --------------

As shown in Appendix 2, for a carbon measurement (*X*, expressed in
μg/cm^2^) based on a single punch, the coefficient of variation in
analytical error is

$\text{CV}_{\mu}\lbrack X\rbrack = \sigma_{Τ}\sqrt{\frac{2}{\mu}}$

where *μ* is the true carbon loading (μg/cm^2^) on the filter. However,
to reduce analytical measurement uncertainty, MSHA will average the
results (*X~1~* and *X~2~*) from two punches taken from each exposed
filter and then subtract the corresponding result (*B*) from the control
filter. The adjusted measurement based on averaging *X~1~* and *X~2~*
can be expressed as

$Y = \frac{X_{1} + X_{2}}{2} - B$.

To simplify the notation in what follows, *σ* will be used to represent
*σ~Τ~*. As shown in Appendix 2, Var\[*X~i~*\], the variance of *X~i~*,
is 2*σ^2^μ*. Similarly, Var\[*B*\] = 2*σ^2^*E\[*B*\], where E\[*B*\] is
the expected or mean density of carbon measured on a control filter.
Therefore, assuming independent analytical measurement errors for
*X~1~*, *X~2~*, and *B*,

$\begin{array}{r}
\text{Var}\lbrack Y\rbrack = \left( \frac{1}{2} \right)^{2}2\ \text{Var}\lbrack X\rbrack\  + \ \text{Var}\lbrack B\rbrack \\
\sigma^{2}\mu\  + \ 2\sigma^{2}E\lbrack B\rbrack \\
\sigma^{2}\left\lbrack \mu + 2E\lbrack B\rbrack \right\rbrack \\
\sigma^{2}\left\lbrack \left( E\lbrack Y\rbrack + E\lbrack B\rbrack \right) + 2E\lbrack B\rbrack \right\rbrack \\
\ \text{since E}\lbrack Y\rbrack\text{  is  }\mu - E\lbrack B\rbrack \\
\sigma^{2}\left\lbrack E\lbrack Y\rbrack + 3E\lbrack B\rbrack \right\rbrack
\end{array}$

It follows that:

$\begin{array}{r}
\text{CV}_{A} = \text{CV}\lbrack Y\rbrack \\
\frac{\sqrt{\text{Var}\lbrack Y\rbrack}}{E\lbrack Y\rbrack} \\
\frac{\sigma\sqrt{E\lbrack Y\rbrack + 3E\lbrack B\rbrack}}{E\lbrack Y\rbrack}
\end{array}$

Based on data compiled from MSHA's analysis of carbon measurements on
available control filters between 12/04/2002 and 10/11/06, the E\[*B*\]
is 0.1 μg/cm^2^ for EC and 6.0 μg/cm^2^ for TC~(OC+EC)~. Furthermore,
using an airflow rate of 1.7 L/min, a deposit area of 8.0425 cm^2^, and
a nominal sampling duration of 480 min:

> E\[*Y*\] ≥ 27.39 μg/cm^2^ for EC filter concentration levels at or
> above the exposure limit of 350 μg/m^3^ total carbon;
>
> E\[*Y*\] ≥ 35.51 μg/cm^2^ for TC~(OC+EC)~ filter concentration levels
> at or above the exposure limit of 350 μg/m^3^ total carbon.

Therefore, substituting the estimated value of *σ~Τ~* for *σ* in the
formula for *CV~A~* and noting that *CV~A~* decreases as E\[Y\]
increases, it is evident that:

> *CV~A~* ≤ 0.049 for EC ≥ 270 μg/m^3^;
>
> *CV~A~* ≤ 0.070 for TC~(OC+EC)~ ≥ 350 μg/m^3^.

# 

The net effect of including revised values for E\[B\] and CV~D~ results
in an increase in overall precision (CV~Total~) of 0.01 when applying
TC~(OC+EC)~ and TC~(ECx1.3)~ as measures of total carbon.

# Appendix 2. Use of Variance-Stabilizing Transformation for Analysis of TC~(OC+EC)~ and EC Measurement Variability

Let *i* index a specific filter, and let *X~i1~* and *X~i2~* denote the
two carbon measurements (μg/cm^2^) made using the two punches from that
filter. As noted in the documentation for Method 5040, the variance of a
carbon measurement made using this method (Var\[*X~i~*\]) is roughly
proportional to the carbon loading (µg/cm^2^) on a filter.[^8] This
relationship can be expressed as

$\text{Var}\lbrack X_{i}\rbrack = \lambda^{2}\mu_{i}$ (Eq.1)

where *λ^2^* is a constant and *µ~i~* is the true loading on the
*i*^[th]{.underline}^ filter. Since *µ* varies but *λ* is constant, it
follows that the coefficient of variation (*CV~μ~*\[*X*\]), which
quantifies measurement variability relative to any given loading,
decreases as *µ* increases:

$\text{CV}_{\mu}\lbrack X\rbrack = \frac{\sqrt{\text{Var}\lbrack X\rbrack}}{\mu} = \frac{\lambda\sqrt{\mu}}{\mu} = \frac{\lambda}{\sqrt{\mu}}$
(Eq. 2)

To estimate *λ*, and thereby to calculate *CV~µ~*\[*X*\] as a function
of filter loading, a variance-stabilizing square-root transformation was
applied to each measurement. Using the standard propagation of error
formula applicable to Eq. 1,

> $$\text{Var}\left\lbrack \sqrt{X_{i}} \right\rbrack \approx \frac{\lambda^{2}}{4}$$

for a carbon measurement at any filter loading.[^9] Based on this
approximation, and assuming independent measurement errors in *X~i1~*
and *X~i2~*,

$\text{Var}\left\lbrack \sqrt{X_{i1}} - \sqrt{X_{i2}} \right\rbrack = 2 \times \text{Var}\left\lbrack \sqrt{X_{i}} \right\rbrack = \frac{\lambda^{2}}{2}$
(Eq. 3)

Consequently,

$\begin{array}{r}
\lambda = \left( 2 \times \text{Var}\ \left\lbrack \sqrt{X_{i1}} - \sqrt{X_{i2}} \right\rbrack\  \right)^{\frac{1}{2}} \\
\  = \sigma\sqrt{2}
\end{array}$ (Eq. 4)

where *σ* denotes the standard deviation of the
differences$\sqrt{X_{i1}} - \sqrt{X_{i2}}$.

From Equations 1 and 4 it follows that
$\text{Var}\lbrack X_{i}\rbrack = 2\sigma^{2}\mu_{i}$, and combining
Equations 2 and 4 yields the formula used to quantify TC measurement
variability at a given filter loading:

$\text{CV}_{\mu}\lbrack X\rbrack = \sigma\sqrt{\frac{2}{\mu}}$ (Eq.5)

[^1]: The constant 1.645 is a 95-percent 1-tailed confidence
    coefficient.

[^2]: As explained in the appendix, CV~A~ (the analytical component of
    CV~total~) increases as the filter loading (i.e., density of
    deposited carbon) decreases. Therefore, since the loading will
    generally be lower at lower carbon concentrations, CV~A~ (and hence
    CV~total~) will increase as carbon concentration levels decrease.

[^3]: Bowman et al. (1984), Precision of Coal Mine Dust Sampling, CDC
    (NIOSH); NTIS No. PB-85-220-721

[^4]: Tomb (1994) Memorandum dated Sept. 1 to Chief, Division of Health,
    CMS&H, MSHA, Subject: Determination of the Precision of Setting the
    Rotameter Ball to a Calibration Mark on Personal Respirable Dust
    Sampling Pumps. (available from CMS&H single-sample rulemaking
    record)

[^5]: Noll, J.D., R.T. Timko, L. McWilliams, P. Hall, and R. Haney.
    "Sampling Results of the Improved SKC Diesel Particulate Matter
    Cassette," Journal of Occupational and Environmental Hygiene,
    2:29-37, January 2005.

[^6]: Because of the particular experimental design employed, the
    results combine purely analytical imprecision with variability in
    the density of the particulate deposited on the filter and with
    variability in the way the two punches were handled prior to
    analysis. Therefore, the estimate of CV~A~ presented here covers all
    three of these uncertainty components.

[^7]: Restricted Maximum Likelihood Estimates (REML) of the parameters
    were obtained using Module 3V of the BMDP statistical software
    package. The REML restriction is to the class of unbiased
    estimators.

[^8]: NIOSH Method of Analytical Methods, Fourth Edition. Method 5040,
    Issue 3 (interim), Sept. 30, 1999. p. 4.

[^9]: Ku, H.H. "Notes on the Use of Propagation of Error Formulas",
    *Precision Measurement and Calibration*, NBS Special Publication
    300, Vol. 1, 1969. pp. 331-341.