EPA-650/4-74-022
                COLLABORATIVE STUDY
                           OF
METHOD FOR  THE  DETERMINATION  OF  PARTICULATE
 MATTER  EMISSIONS FROM STATIONARY SOURCES
             (MUNICIPAL INCINERATORS)
                            by
                       Henry F. Hamil
                      Richard E. Thomas

                EPA Contract No. 68-02-0626
                SwRI Project No. 01-3462-002


                        Prepared for
               Methods Standardi/ation Branch
   Quality Assurance and Environmental Monitoring Laboratory
            National Environmental Research Center
               Environmental Protection Aj^ncy
             Research  Triangle Park, N. C. 27711

          Attn: M. Rodney Midgett, Research Chemist
         Section Chief, Stationary Source Methods Section

                         July 1, 1974
             SOUTHWEST  RESEARCH  INSTITUTE
             SAN ANTONIO      CORPUS CHRISTI      HOUSTON

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     This report has been reviewed by the Office of Research and Development, EPA, and approved for pub-
lication. Approval does not signify that the contents necessarily reflect the views and policies of the Environ-
mental Protection Agency,  nor does mention of trade names or commercial products constitute endorsement
or recommendation for use.

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           SOUTHWEST  RESEARCH  INSTITUTE
           Post Office  Drawer 28510, 8500 Culebra Road
                   San Antonio, Texas 78228
                COLLABORATIVE STUDY
                            OF
METHOD  FOR THE DETERMINATION  OF  PARTICULATE
 MATTER EMISSIONS FROM STATIONARY SOURCES
              (MUNICIPAL  INCINERATORS)
                            by
                        Henry F. Hamil
                      Richard E. Thomas

                EPA Contract No.  68-02-0626
                SwRI Project No. 01-3462-002


                        Prepared for
               Methods Standardization Branch
   Quality Assurance and Environmental Monitoring Laboratory
            National Environmental Research Center
               Environmental Protection Agency
              Research Triangle Park, N. C. 27711
                           n
          Attn:  M. Rodney Midgett, Research Chemist
         Section Chief, Stationary Source Methods Section
                             App roved:
                             John T. Goodw in
                             Director
                             Department of Chemistry
                             and Chemical Engineering

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                               SUMMARY AND CONCLUSIONS
     This report presents the results obtained from a collaborative test of Method 5, a test procedure promul-
gated by the Environmental Protection Agency for the determination of particulate emission levels from station-
ary sources. Method 5 specifies that particulate matter be withdrawn isokinetically from the source and its
weight be determined gravimetrically after removal of uncombined water.

     The test was conducted at a municipal incinerator using four collaborative teams. A total of 12 runs were
made over a two week period, and 47 individual concentration determinations made by the four collaborators.
From these, the values which conformed to the standards of 60 scf of gas collected and ±10% of isokinetic
sampling were used in the analysis.  The resultant working sample was 11 runs and a total of 32 individual
observations. These were submitted to statistical analysis to obtain precision estimates for Method 5.

     The precision is expressed in terms of within-laboratory, between-laboratory and laboratory bias compo-
nents.  For  purposes of statistical treatment, the determinations are grouped into blocks. Two separate furnace
trains were  used at the incinerator, the No. 1 unit during the first week of testing, and the No. 2 during the
second week. The values obtained from each stack were grouped as a block for this test. The statistical analysis
is based on  the assumption that the true emission concentration remains essentially constant over the course
of each week's runs.  No independent method for determining the  concentration was available during the test
to substantiate this assumption, but a preliminary statistical test on the determinations detected no significant
differences  among the runs that would indicate a changing mean value over the test period.

     The precision components are estimated in terms of standard deviations, which are shown to be propor-
tional to the mean of the Method 5  determinations, 8, and can be summarized as follows.

     (a)   Within-laboratory:  The estimated within-laboratory standard deviation is 25.3% of 5  and has
           24 degrees of freedom associated with it.

     (b)   Between-laboratory:  The estimated between-laboratory standard deviation is 38.7% of 5, with
           3 degrees of  freedom.

     (c)   Laboratory bias:  From the above, we can estimate a laboratory bias standard deviation of 29.3%
           of6.

     The above precision estimates reflect not only operator variability, but, to an extent, source variability
which cannot be separated from these terms.  Since the results summarized above  were obtained from a single
test, further testing would, of course, be necessary  to obtain conclusive results.

     Recommendations are made for the improvement of the precision of Method 5, and considerations given
for the use  of the method in field testing.
                                                 in

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                                    TABLE OF CONTENTS

                                                                                             Page

LIST OF ILLUSTRATIONS	     vi

LIST OF TABLES	     vi

I.    INTRODUCTION	     1

II.   COLLABORATIVE TESTING OF METHOD 5   	     2

     A.   Collaborative Test Site    	     2
     B.   Collaborators	     6
     C.   Philosophy of Collaborative Testing	     6

III.  STATISTICAL DESIGN AND ANALYSIS	     8

     A.   Statistical Terminology    	     8
     B.   Collaborative Test Plan    	     9
     C.   Collaborative Test Data	     9
     D.   Precision of Method 5	     10

IV.  COMPARISONS WITH OTHER STUDIES	     13

V.   RECOMMENDATIONS	     14

APPENDIX A-Method 5-Determination of Particulate Emissions From Stationary Sources	     15

APPENDIX B-Statistical Methods    	     19

     B.1   Preliminary Analysis of the Original Collaborative Test Data	     21
     B.2   Significance of the Port Effect	     21
     B.3   Transformations	     23
     B.4   Empirical Relationship Between Mean and Standard Deviation	     23
     B.5   Unbiased Estimation of Standard Deviation Components	     25
     B.6   Weighted Coefficient of Variation Estimates	     27
     B.7   Estimation of Precision Components   ....     	     29

LIST OF REFERENCES	     33

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                                   LIST OF ILLUSTRATIONS







Figure                                                                                           Page




    1      Flow Diagram of Holmes Road Incinerator Plant	     3




    2      Sampling Port Configuration	     4




    3      Average Velocity Profiles	     	     5




    4      Holmes Road Incinerator Test Site	     	     6




    5      Control Console Operation	     7




    6      Impinger Train Operation	     7




  B.I      Interlaboratory Run Plot	    24




  B.2      Intralaboratory Collaborator Block Plot	    25





                                        LIST OF TABLES




Table                                                                                            Page




    1      Test Site Data	     6




    2      Particulate Collaborative Test Data Arranged in Blocks, Ib/scfX 107	    10




  B.I      Original Particulate Concentrations, Ib X scfX 107	    21




  B.2      Significance of Port Effect	    22




  B.3      Data Transformation To Achieve Run Equality of Variance	    23




  B.4      Interlaboratory Run Summary	     .   .    24




  B.5      Intralaboratory Collaborator Block Summary	    25




  B.6      Run Beta Estimates and Weights	     	    30




  B.7      Collaborator Block Beta Estimates and Weights	    30
                                                  VI

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                                           I.  INTRODUCTION
     This report describes the work performed and results obtained on Southwest Research Institute Project
No. 01-3462-002, Contract No. 68-02-0626, which includes collaborative testing of Method 5 for particulate
emissions as given in "Standards of Performance for New Stationary Sources."^*

     This report describes the collaborative testing of Method 5 at a municipal incinerator, the statistical analysis
of the data from the collaborative tests, and the conclusions and recommendations based on the analysis of the
data.
*Superscript numbers in parentheses refer to the List of References at the end of this report.

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                       I.  COLLABORATIVE TESTING OF METHOD 5
A.   Collaborative Test Site

     Arrangements were made for collaborative testing of Method 5 at the Holmes Road Incinerator of the
Division of Solid Waste Management, Department of Public Works, City of Houston, Texas.  Facilities were in-
stalled to provide for simultaneous sampling by four collaborators, each collaborative team working on separate
ports on a platform on the stack.

     The incinerator was visited in November, 1972, to inspect the facilities and conduct  preliminary sampling
to determine site suitability.  At that time, details of site preparation, including addition of platform extensions
and additional sampling ports were made.  Site modifications were completed in May,1973, and another site
visit was made to assure suitability for collaborative testing.

     The 5-yr old facility consists of two parallel furnace trains, with a capacity of 400 tons of refuse per furnace
per day.  The furnaces are a triple grate continuous feed design. Refuse is transferred from the storage bins to the
furnace-charging hoppers by two traveling bridge cranes.

     From the charging hoppers, the refuse feeds continuously through water-cooled chutes onto traveling
grate stokers in the furnaces.  When the residue reaches the end of the combustion grate, it drops through a chute
to either of t\vo ash conveyers. The residue is quenched and conveyed to rotating screen separators, which dis-
charge into loading-out hoppers for trucking to the disposal area adjacent to the incinerator plant.

     Gases leaving the furnaces are cooled in water spray chambers and then enter the flue gas scrubbers  to
remove the fly ash. The  gases then pass through the induced draft fans and out the stacks. A flow diagram of
one of the incinerator units is shown in Figure 1.

     Four sampling ports were available on both the Number 1 (east) and Number 2 (west)  units.  The sample
ports were at 90 deg to each other and were offset 6 in. vertically from port to port to avoid probe interference.
Access to the sample ports was from  a 360-deg platform around the stacks, which were 6.5 ft inside diameter
and 148 ft high. The sample  ports were 102 ft above grade and 57 ft (8.8 diameters) above the nearest upstream
flow disturbance. This allowed the use of 12 traverse points,  6 on each diameter traverse.  Sample port layout
is shown  in Figure 2, and a velocity profile is shown in Figure 3.  The velocity profiles were obtained by averaging
the velocities obtained at each traverse point by  two laboratories on all velocity traverses.  A view of the test site
and Number 1 stack is in Figure 4.

     Pertinent information concerning stack dimensions, sample  poit location with respect  to flow disturbance,
and traverse point locations is tabulated in Table 1.

     Traversing for the test itself was accomplished as follows (see diagram p. 6): Sampling  teams at ports A and
B began sampling at points 1  and l', respectively. After ten minutes, the team at port A moved to point 2, and
the team  at port C began sampling at point 1.  Simultaneously, the team at port  B moved to  point 2' and the
team at port D moved to point l' and began sampling. This pattern was followed until the diameter traverse
was completed (it should be noted that the sampling period of the teams at ports A and B  was displaced in time
by ten minutes from that of the teams at ports C and D, the assumption being that stack conditions were  suffi-
ciently constant to justify this displacement in time.  This displacement in time was necessary to avoid interfer-
ence from two probes sampling the same traverse point simultaneously). Each team then moved its sampling unit
to the port on its right (as one faces the stack), and the traversing procedure was repeated. These two diameter
traverses then constituted a run. Port heights were varied (see Figure 2) so that interference of the probes was
prevented.

     The first week of the test was conducted on the Number 1  unit.  Failure of the moving grates on that unit
necessitated testing on the Number 2 unit during the second week of the test.

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H
O

o
ft!
in
OS
u
o
a
OS
p
a

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     1
     6"
FIGURE 1. SAMPLING PORT CONFIGURATION

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£   55
       0
       A
       B
                      12
                                    24
    36             48

Stack Diameter, inches


   Profile, East Stack
                                                                                                72
78
C
D
e   55
    50
    45
                      12
                                    24
       A
       B
                                             D
                                             O
   36             48

Stack Diameter, inches


  Profile, West Stack

   Axis through ports A, C
   Axis through ports B, D
                                                                                                72
                                                          78
C
D
                                    FIGURE 3. AVERAGE VELOCITY PROFILES

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                                          TABLE 1.  TEST SITE DATA
                     Inside stack diameter-

                     Distance of sampling site from downstream disturbance:

                     Distance of sampling site from upstream disturbance:

                     Number of traverse points on a diameter:

                     Sampling time at each traverse point:
                                     78 inches

                                     8.8 diameters

                                     >2 diameters

                                     6

                                     10 minutes
Traverse
 Point

   1
   2
   3
   4
   5
   6
Distance from outside wall to
   traverse point, inches:

           3-3/8
          11-1/2
          23
          55
          66-1/2
          74-5/8
 FIGURE 4. HOLMES ROAD INCINERATOR TEST SITE
              B.   Collaborators

                   The collaborators for the Holmes Road Incinera-
              tor test were Mr. Mike Taylor and Mr. Rick Hohmann
              of Southwest Research Institute, Houston Laboratory,
              Houston, Texas; Mr. Charles Rodriguez and
              Mr. Ron Hawkins of Southwest Research Institute,
              San Antonio Laboratory, San Antonio, Texas;
              Mr. Quirino Wong, Mr. Randy Creighton, and
              Mr. Steve Byrd, City of Houston, Department of Pub-
              lic Health; Mr. John Key, Mr. James Draper,
              Mr. Tom McMickle, Mr. Tom Palmer, Mr. Michael Lee,
              and Mr. Charles Goerner, Air Pollution Control Services,
              Texas State Department of Health.*

                   Collaborative tests were conducted under the gen-
              eral supervision of Mr. Nollie Swynnerton of Southwest
              Research Institute. Mr. Swynnerton had the overall
              responsibility for  assuring that the test was conducted
              in accordance with the collaborative test plan, and that
              all collaborators adhered to Method 5 as written in
              the Federal Register " '. Collaborators for the test
              were selected by Dr. Henry Hamil of Southwest
              Research Institute.
      In Figures 5 and 6, members of the collaborative teams are shown in the operation of impinger trains
and a control console during one of the test runs.

C.    Philosophy of Collaborative Testing

      The concept of collaborative testing followed in the tests discussed in this report involves conducting the
test in such a manner as to simulate "real world" testing as closely as possible. "Real world" testing implies
that the results obtained during the test by each collaborator would be the same results obtainable if he were
sampling alone, without outside supervision and without any additional information from outside sources,
i.e., test supervisor or other collaborators.

*Throughout the remainder of this report, the collaborative laboratories are referenced by randomly assigned code numbers as
Lab 101, Lab 102, Lab  103, and Lab 104  These code numbers do not correspond to the above ordered listing of collaborators.

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   FIGURE 5. CONTROL CONSOLE OPERATION
                                                         FIGURE 6. IMPINGER TRAIN OPERATION
      The function of the test supervisor in such a testing scheme is primarily to see that the method is adhered
to as written and that no individual innovations are incorporated into the method by any collaborator. During
the test program, the test supervisor observed the collaborators during sampling and sample recovery.  If random
experimental errors were observed, such as mismeasurement of volume of impinger solution, improper rinsing
of probe, etc.. no interference was made by the test supervisor.  Since  such random errors will occur in the
everyday use of this method in the field,  unduly restrictive supervision of the collaborative test would bias the
method with respect  to the field test results which will be obtained when the method is put into general usage.
However, if gross deviations were observed of such magnitude as to make it clear that the collaborator was not
following the method as written, the deviations would be pointed out  to the collaborator and corrected by the
test supervisor.

      While most of the instructions in the Federal Register are quite explicit, some areas are subject to inter-
pretation. Where this was the case, the individual collaborators were allowed to exercise their professional
judgement as to the interpretation of the instructions.

      The overall basis for this so-called "real-world" concept of collaborative testing is to evaluate the subject
method in such a manner as to reflect the reliability and precision that would be expected of the method in
field testing.

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                           III.  STATISTICAL DESIGN AND ANALYSIS


A.    Statistical Terminology

      To facilitate the understanding of this report and the utilization of its findings, this section explains the statistical
terms used in this report. The procedures for obtaining estimates of the pertinent values are developed and justified in
the subsequent sections.

      We^say that an estimator, 6, is unbiased for a parameter 0 if the expected value of 0 is 0, or expressed in notational
form, E(6) = 9. Let Xj, x2,. . • , xn be a sample of n replicates from the population of method determinations.

                I   n
           —_      \. '"^
      (1)   x = — 2_i xi as tne sample mean, an unbiased estimate of the true mean, 6, of the determinations, the
               " «=1
           center of the distribution. For an accurate method, 6 is ^, the  true stack concentration.

                  1   "
      (2)   s2  =	 y (xj - x )2 as the sample variance,an unbiased estimate of the true variance, a2 .  This term
           gives a measure of the dispersion in the distribution around §.

     (3)   s = \fs2 as the sample standard deviation, an alternative measure of dispersion, which estimates 0, the true
           standard deviation.

     The sample standard deviation, s, however, is not unbiased for a,( 7) so a correction factor needs to be applied.
The correction factor for a sample of size n is an , and the product of an and s is unbiased for a. That is, E(ans) = a.
As « increases, the value of an decreases, going for example from a3 = 1.1 284, a4 = 1.0854 to a,0 = 1.0281.

     We define
as the true coefficient of variation for a given distribution. To estimate this parameter, we use a sample coefficient of
variation, j3, defined by

                                                 . = ^s
                                                     X

where |3 is the ratio of the unbiased estimates of a and 6 , respectively.  The coefficient of variation measures the per-
centage scatter in the observations about the mean and thus is a readily understandable way to express the precision
of the observations.

     The experimental plan for this test calls for 12 runs. On each run, the collaborative teams were expected to col-
lect simultaneous samples from the stack in accordance with Method 5. Since the actual particulate emission con-
centration in the stack fluctuates, one can in general expect different true concentrations for each run.  To permit
a complete  statistical analysis, the individual runs are grouped into blocks, where each block has approximately the
same true emission concentration level.

     We can apply the statistical terms of the preceding paragraphs both to the collaborators' values during a given
run, and to each collaborator's values in a given block.  In this report, statistical results from the first situation are
referred to as run results. Those from the second situation are referred to as collaborator-block results. For example,
a run mean is the average of each collaborator's concentration level for the run as obtained by Method 5.  A

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collaborator-block coefficient of variation is the ratio of the unbiased standard deviation to the sample mean for all the
collaborator's runs grouped in the block.

      The variability associated with a Method 5 concentration determination is estimated in terms of the •within-
laboratory and the between-laboratory precision components.  In addition, a laboratory bias component can be esti-
mated.  The following definitions of these terms are given with respect to a true stack concentration, ju'-

      •    Within-laboratory-The within-laboratory standard deviation, a, measures the dispersion in replicate
           single determinations made using Method 5 by one laboratory team (same field operators, laboratory
           analyst, and equipment) sampling the same true concentration, ju. The value of 0 is estimated from
           within each collaborator-block combination.

      •    Between-laboratory-The between-laboratory standard deviation, a^, measures the total variability in a
           concentration determination due to simultaneous Method 5 determinations by different laboratories
           sampling the same true stack concentration, ju.  The between-laboratory variance, a\, may be expressed as

                                              al=al+ a2

           and consists of a within-laboratory variance plus a laboratory bias variance, o2L. The between-laboratory
           standard deviation is estimated using the run results.

      •    Laboratory bias—The laboratory bias standard deviation, OL = \fa^ — a2 , is that portion of the total
           variability that can be ascribed to differences in the field operators, analysts and instrumentation, and
           due to different manners of performance of procedural details left unspecified in the method.  This term
           measures that part of the total variability in a determination which results from the use of the method
           by different laboratories, as well as  from modifications in usage by a single  laboratory over a period of
           time.  The laboratory bias standard  deviation is estimated from the within-  and between-laboratory esti-
           mates previously obtained.

B.    Collaborative Test Plan

      The sampling was done through the four ports on each stack described previously, with each laboratory sam-
pling from two ports during each run.  At the end of the first hour of sampling, the teams  rotated in a counterclockwise
direction to the next adjacent port.  The starting ports for the teams during each run were chosen through  a random-
ization technique, and these  will be shown in Table 2.

      The incinerator operated only  with one stack at a time.  After the first week of testing, the plant shifted their
operation from the Number  1 unit to the Number 2 unit. Thus, the first five samples were taken from stack Number 1,
and the remaining seven from Number 2.  In the absence of operating characteristics or any means of determining a
true concentration level, the unit from which the determinations were made was used as the blocking factor. The
resulting design, then, was two blocks  of size 5 and 7, respectively, as shown in Table 2.

C.    Collaborative Test Data

      The data used in the statistical analysis of the collaborative test are presented in Table 2, along with the port at
which sampling was begun.  These determinations were subjected to preliminary calculation checks to ensure that all
values were determined by using the proper formulas and conversion factors. The data as presented are consistent
with the formulas prescribed in Method 5, and the details of the recalculation are given in Appendix B.I, along with
the determinations as reported by  the collaborators.

      No determination was  reported by Lab 104 in Run 1  due to equipment malfunction. In  order to evaluate
Method 5 for field testing, the analysis is done using only values which satisfy the requirements for a compliance
test result. These requirements include (1) that there be a minimum sampling volume of 60 cubic feet, and
(2) that the sample  be drawn between 90 and  110 percent of isokinetic  sampling.   The values marked with

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                     TABLE 2. PARTICULATE COLLABORATIVE TEST DATA ARRANGED
                                          IN BLOCKS, Ib/scf X 107
Tt]or\f
DIUClV
1




2






Run
IxUIl
i
,2
3
4
5
6
7
8
9
10
11
12
Lab 101
Data
219.1
230.2
202.7
278.2
298.4
267.5$
245.4
468.9
260.0*
197.2
232.2
250.4
Port
A
B
D
D
A
D
D
D
A
C
D
A
Lab 102
Data
170.6*
192.6
207.8
303.5*
236.4
183.8$
171.1*$
201.4
202.2*
121.5*$
217.4*$
205.6
Port
B
A
A
A
B
C
C
B
C
A
B
C
Lab 103
Data
93.6
163.6
157.8
183.6
151.2
125.9
137.9
179.1
157.0*
123.0
168.5
158.5
Port
C
D
B
C
C
A
A
A
B
B
C
D
Lab 104
Data
O.Of
187.6$
380.7
82.7
125.7$
187.5
171.7
249.0*$
228.5*
177.3
229.9
189.8$
Port
D
C
C
B
D
B
B
C
D
D
A
B
*Determmation was made with less than 60 ft3 of gas. corrected to standard conditions.
fNo value reported for this run.
$ Determination was not obtained between 90 and 110 percent of isokinetic.
Note:
EPA policy is to express all measurements in Agency documents in metric units.
When implementing this practice will result in undue cost or difficulty in clarity,
NERC/RTP is providing conversion factors for the particular non-metric units used
in the document. For this report, the factor is:
10'' Ib/scf = 1.6018 X 103 Mg/m3.
a double dagger are those with an unacceptable isokinetic variation factor, and those with an asterisk were samples
of less than 60 cubic feet, corrected to standard conditions.

      In these cases, the values were eliminated from the analysis, and no attempt was made to adjust these or sub-
stitute for them. While a substitution would yield a larger data base, the values would not be actual Method 5 deter-
minations, and so could inordinately bias the results, particularly when such a large number of substitutions would be
needed. This leaves a total of 32 valid determinations out of a possible 48.

      Although sample ports are placed in a manner to be as nearly equivalent as possible, the unknown factors of
gas flow patterns and variations can lead to a particular port showing a consistently higher or consistently lower
emission concentration level over the course of the test.  A test for the significance of a port effect is performed in
Appendix B.2.  Using a common rank test, no consistent tendency can be found for any of the ports to give a high
or a low emission level throughout  the runs. As a result of this, and the fact that two ports were sampled on each
run, no port factor is included in any further analysis.

D.    Precision of Method  5

      There are no techniques currently available  for determining  the true particulate emission concentration level
from a stack, and thus there is no way to determine the accuracy of the method. Further, no ancillary tests can be
run to separate the analytical phase of the test from the field phase. Thus the only means available for evaluating
Method 5 is to examine the precision  that can be expected from a field test result.

      As a preliminary to evaluating the precision of the method,  the determinations are  tested for equality of vari-
ance using Bartlett's test.  '  In addition, the determinations are passed through two common variance stabilizing
transformations, the logarithmic and the square root, and Bartlett's test is again  applied.  The use of transformation
                                                    10

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serves two purposes. First, it can put the data into an acceptable form for an analysis of variance; and second, it can
provide information concerning the true nature of the distribution of the sample points.


     In Appendix B.3, it is demonstrated that for the run data, the logarithmic transform provides the best fit.  As has
been demonstrated in a previous study by Hamil and Camann/4)  this is  a strong indication of a proportional relation-
ship between the mean and standard deviation for the run data.

     To further this argument, a regression line is fit to the paired sample means and sample standard deviations for
the run data in Appendix B.4.  The graph in Figure B.I shows the least squares fit to the points having been forced
through the origin.  The degree of fit is measured by the coefficient of determination, r2 , which has a value of 0.85 1 5
for the run data. This indicates that over 85 percent of the variation in the magnitude of the standard deviation is
attributed to the variation in the magnitude of the mean.
      The correlation coefficient, r = Vr2 . f°r the run data is 0.9228. Using tables given in Dixon and Massey/2) this
value is significant at the 5-percent level, which justifies the use of the no intercept regression line as a model for these
data points.

      A similar argument is presented for the collaborator-block data. For these paired values, the relationship is not
as strong as it was for the run data. The value of r2 for the least squares fit is 0.5343, which gives a value of r of
0.7309. Again referring to significance tables for r, we can determine that this value is significant at the 5-percent
significance level.

      The conclusion that we draw from the above  results is that there is a proportionality between the mean and the
standard deviation for both the run data and the collaborator-block data.  This implies that
and

                                                 ob =

where a and 05 are the within- lab and between-lab components respectively, C and Q, are constants, and 5 represents
the  true mean  determination  level. This is equivalent to saying that while the standard deviations change according
to the emission concentration, the ratio of the standard deviation to the mean, the coefficient of variation, remains
constant.  Thus, we may rewrite the above expressions as


  i                                               0/6=0

and

                                                 o/,/6 = ft,

where (3 is the true within-laboratory coefficient of variation, and fo is the true between-laboratory coefficient of
variation.

     On the basis of the previous argument, then, we will estimate  our precision components not directly, but by
estimating the appropriate coefficients of variation and expressing the standard deviations as percentages of the mean.
In Appendices B.5 and B.6, the  technique for obtaining estimates of (3 and |3^ is discussed, and it is shown that the
resulting estimates are unbiased for the standard deviation components. Our estimates of a and aj, are defined with
respect to the mean determination, S.
                                                    11

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and
      The actual estimates of 0 and /3j are obtained in Appendix B.7 from the collaborative test data.  The within-lab
coefficient of variation, taken from the collaborator-block data, is |3 = 0.253. Using this, we then estimate the within-
lab standard deviation as

                                                a = (0.253)6,

with 24 degrees of freedom.

      The between-lab coefficient of variation, 0£, is estimated from the values across any particular run. From
Appendix B.7, we have a value of 0b = (0.387), which gives an estimated between-lab standard deviation of

                                              ab = (0.387)6

at a true determination mean, 6 .  This estimate has 3 degrees of freedom associated with it.

      From the formula presented in section A above, the laboratory bias standard deviation may be estimated as
                                        = V(0.387)262  -(0.253)262
                                        = V(0.086)52

                                        = (0.293)6

at a true determination mean, 6.

      These values indicate a lack of precision in the method, especially with regard to the laboratory bias compo-
nent. The differences between the labs in this study were of such a magnitude to suggest that each laboratory was
obtaining a value whose usual range was essentially lab dependent. If such is the case, then there is an apparent
discrepancy arising from some technique or procedure in the method which is not defined in  the necessary detail.
This could include the methods for sample recovery from both the filter and the probe wash, for which more
detailed procedure should be specified and more care in handling should be recommended.

      The within-laboratory precision could also be aided by such detail, since it could help to avoid the probable
causes of unusual values, namely contamination of the probe tip by scraping against the inside of the port and loss
of particulates due to probe handling and cleaning.  Although not a major problem at this site, it has been a notice-
able factor in a previous test/5)
                                                   12

-------
                          IV. COMPARISONS WITH OTHER STUDIES


     Two other collaborative tests of Method 5 have been conducted/3'5^ at a cement plant and a fossil-fuel fired
steam generator. The following comparisons can be made with the results of those tests.

     •     The within-laboratory standard deviation estimate for this study is lower than the values obtained in the
           other studies.  The difference is due in some measure to the fact that the high value phenomenon that
           can effect the within-laboratory estimate is less prevalent in this test  than in the other two.

     •     The between-laboratory standard deviation estimate for this study is comparable to that of the power
           plant, and lower than the value obtained from the cement plant. However, the high values obtained in
           the cement plant test were localized in a single laboratory's results. This caused an inflation of the
           laboratory bias component, which accounts for the difference between that study and the other two.

     •     The laboratory bias standard deviation is higher than the value obtained in the power plant test. The
           difference in the manner of probe cleanup and sample recovery is probably the major cause of the
           variability in the results  from lab to lab.  In this study, the infrequency of high values led to the reduc-
           tion of the within-laboratory estimate, but the between-laboratory estimate remained  high. This
           suggests that a solution to the high value problem will not, by itself, result in a great improvement in
           the precision demonstrated  for the method but that work must  be done to add the details in the
           technique that will enable distinct crews to perform the method in a more nearly identical manner.
                                                  13

-------
                                        V. RECOMMENDATIONS

     On the basis of the conclusions and results presented and from observations by personnel in the field, the fol-
lowing recommendations can be made concerning Method 5:

     (1)   Frequency of calculation errors in the collaborative test data and differences among labs in the number of
           significant digits carried can be a major problem in evaluating field test results.  It is recommended that a
           standard Method 5 computer program be written to calculate compliance test results from raw field data.

     (2)   More detail should be specified in  the technique for sample recovery from the probe.  The probe collection
           and handling are the probable causes of the extreme high and low values which greatly contribute to the
           lack of precision in the method.

     (3)   At present, there is no standard technique for cleaning the filter apparatus, which undoubtedly is a major
           contributor to the high laboratory bias. It is recommended that a detailed procedure be established and
           incorporated into the method.

     (4)   When using the method in a stack  with high moisture content, filters tend to blind off rapidly, causing
           them to have to be changed during the course of the run. This has the effect of magnifying the handling and
           sample recovery requirements, thus resulting in more  opportunity for errors. Sampling teams using Method 5
           are encouraged to utilize as large a filter as they can accomodate when working at this type of site to
           minimize this problem.
                                                  14

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                       APPENDIX A

METHOD 5-DETERMINATION OF PARTICULATE EMISSIONS FROM
                  STATIONARY SOURCES

               Federal Register, Vol. 36, No. 247
                     December 23, 1971
                             15

-------
                                                   RULES AND  REGULATIONS

                                               2.1.4  Filter  Holder—Pyrex'  glass  with
                                             heating system capable of maintaining mini-
                                             mum temperature of 235" P.
                                               2.1.5  Implngers / Condenser—Four Itnpln-
                                             gera connected In Berles with glase boll  Joint
                                             fittings. The first, third, and  fourth Impln-
                                             gers  are  of the  Greeniburg-Smith  design,
                                             modified by replacing the tip with a !/2-Lnch
                                             ID  glass  tube extending to  one-half  Inch
                                             from the bottom of the flask The second 1m-
                                             plnger Is of  the  Greenburg-Smltli design
                                             with the standard tip. A condenser may be
                                             •used In place of the Implngers provided that
                                             the moisture content of  the  stack  gas can
                                             still be determined.
                                               2.1.6  Metering  system—Vacuum  gauge,
                                             leak-free  pump,  thermometers capable  of
                                             measuring temperature to within 6" F., dry
                                             gas meter with 2%  accuracy,  and  related
                                             equipment,  or equivalent,  as  required  to
                                             maintain an Isoklnetlc sampling rate and to
                                             determine sample volume.
                                               2.1.7  Barometer—To measure atmospheric
                                             pressure to ±0.1 Inches Hg.
                                               2.2  Sample recovery.
                                               2 2.1  Probe  brush—At  least as  long as
                                             probe.
                                               8.2.2  Glass wash bottles—Two.
                                               2.2.3  Glass sample storage containers.
                                               2.2.4  Graduated cylinder—26O ml.
                                               2.3  Analysis.
                                               2.3.1  Glass weighing dishes.
                                               2.3.2  Desiccator.
                                               2.3.3  Analytical balance—To measure to
                                             ±0.1 mg.
                                               2.3.4  Trip balance—300 g. capacity,  to
                                             measure to ± 0.06 g.
                                               3 Reagentt.
                                               3.1  Sampling.
                                               3.1.1  Filters—Glaes fiber, MSA 1106 BH1.
                                             or equivalent,  numbered  for Identification
                                             and prewelghed.
                                               3.1.2  Silica  gel—Indicating  type,  6-16
                                             mesh, dried at  178' C. (360' F.) for 2 hours.
                                               3.1.3  Water.
                                               3.1 4  Crushed Ice.
                                               3.2  Sample recovery.
                                               3 2.1   Acetone—Reagent grade.
                                               3.3  Analysis.
                                               3.31   Water.

                                                  MP4NGER TRAIN OPTIONAL  MAY BE REPLACED
                                                        BY AN EQUIVALENT CONDENSER
                                                                         HEATED AREA  FILTER HOLDER /  THERMOMETER   CHECK
                                                                                                                   ^VALVE
METHOD 6—DETERMINATION  or PARTXCULATC
   EMISSIONS  FROM STATIONARY SOURCES

   1. Principle  and  applicability
   1.1  Principle. Paniculate matter Is with-
drawn ifiokinetically from the source and Its
weight Is determined gravlmetrically after re-
moval of uncomfolned water.
   1 2  Applicability This meflhod Is applica-
ble for the determination of partlculate emis-
sions  from stationary  sources only when
specified by the test procedures for determin-
ing compliance with New Source  Perform-
ance Standards
  2  Apparatus.
  2 1   Sampling  train The design  specifica-
tions of  the partlculate  sampling train used
by EPA (Figure 5-1) are described In APTD-
0581.  Commercial models  orf this  train are
available                               *
  211  Nozzle—Stainless  steel (316)  with
sharp, tapered leading edge.
  212  Probe—Pyrex' glass with a heating
system capable of  maintaining a  minimum
gas temperature  of 250' F. at the exit  end
during  sampling  to prevent  condensation
from  occurring.  When  length limitations
(greater  than about 8 ft) are encountered at
temperatures less than 600* F., Incoloy 825 1,
or equivalent, may be used. Probes for sam-
pling  gas streams at temperatures  In excess
of 600° F. must have been approved by the
Admiiustaator.
  2.1.3  Pitot  tube—Type  S, or equivalent,
attached to probe  to  monitor stack  gas
velocity.
                                                     PROBE    
-------
                                                    RULES  AND  REGULATIONS
       SAMU (OIK

       WTCH KM M0._
       HUM *H-
                                                                 AMBIENT TtMPCMTUftE
                                                                 ASSUKD MOISTUV. *

                                                                 HEATH KX 1ETTINQ.

                                                                 ntW UNQTH.»
                    ntOMHtATtftSETTlNO	
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TOTAL
lAVUm
TIM
(«(.•**













AvtnAue
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PKSSUM
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nenuK
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ACMM
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OA5UJMJ
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(VnoH?














GAS SAURJ nuet**nm
AT on GAS UETEA
muT
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A*fl.
OUTUT
rr.^i.'F












Ai*.
A.9-
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°F













1
rEMPERATURE
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LEAVING
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   4.2  Sample recovery. Exercise care In mov-
 ing the collection train Trom the test site to
 the  sample  recovery  area  to minimize  the
 IOSB  of  collected  sample  or  the  gain  of
 extraneous paniculate matter.  Set aside  a
 portion of the  acetone used In the sample
 recovery as a blank for analysis. Measure the
 volume of water from the first  three 1m-
 plngers,  then discard.  Place the  samples In
 containers as follows.
   Container  No. 1  Remove the  filter from
 Its holder, place In this container, and seal.
   Container  No. 2. Place  loose  paniculate
 matter   and  acetone  washings  from  all
 sample-exposed surfaces prior  to the filter
 In this container and seal. Use a razor blade,
 brush, or rubber policeman to lose adhering
 particles.
   Container  No. 3  Transfer the  silica  gel
 from the fourth Unplnger to the original con-
 tainer and seal  Use a rubber policeman  as
 an aid  In removing  silica  gel  from the
 Implnger
   4 3  Analysis. Record the data required on
 the  example  sheet shown In Figure 5-3.
 Handle each sample container as follows:
   Container  No. 1. Transfer  the  filter and
 any loose paniculate matter from the sample
container  to a  tared  glass weighing  dish,
desiccate, and dry to a constant weight Re-
port results to the  nearest 0 6 mg.
   Container  No.  2  Transfer  the  acetone
washings to a tared beaker  and evaporate to
dryness at ambient temperature and  pres-
sure. Desiccate and dry to a constant weight.
Report results to the nearest 0 5 mg
  Container No. 3.  Weigh the spent silica gel
and report to the  nearest gram.
  6  Calibration.
  Use  methods  and equipment  which have
been  approved  by  the  Administrator  to
calibrate  the  orifice  meter,  pilot  tube, dry
gas  meter, and probe heater.  Recalibrate
after each test series
  8  Calculations.
  61  Average dry gas meter temperature
and average orifice pressure drop. See data
sheet (Figure 6-2).
  6.3  Dry gas volume. Correct  the sample
volume measured  by  the  dry gas meter  to
standard conditions (70° P., 29.92 Inches Hg)
by using Equation  fi-1.
v».,H=v.
            T
-------
     APPENDIX B




STATISTICAL METHODS
          19

-------
                                      STATISTICAL METHODS
     This appendix consists of various sections which contain detailed statistical procedures carried out in the analysis
of the particulate matter collaborative study data. Reference to these sections has been made at various junctures in
the Statistical Design and Analysis part of the body of this report. Each Appendix B section is an independent ad hoc
statistical analysis pertinent to a particular problem addressed in the body of the report.

B.1  Preliminary Analysis of the Original Collaborative Test Data

     Preliminary recalculations were made on the originally reported data to ensure that the proper formulas were used
and that the concentrations used in the analysis were correctly computed. There is some question about the propriety
of this in light of the  philosophy of collaborative testing expressed previously. However, no computation is made that
is not the direct result of the actual data obtained by the collaborator. Recalculation only ensures that there is some
consistency to the number of significant digits carried and that the concentration used in the analysis does not contain
a calculation error that will bias  the results. The data originally reported are shown in Table B.I, and comparison with
Table 2 in Section III indicates the need for such a step.
    TABLE B.I. ORIGINAL PARTICULATE
       CONCENTRATIONS, Ib/scf X 107
Run
1
2
3
4
5
6
7
8
9
10
11
12
Collaborator
Lab 101
219.0
2300
203.0
278.0
298.0
2680
245 0
469.0
260.0
197.0
232.0
251.0
Lab 102
170.9
192.6
208.0
303.0
236.0
183.8
171.1
201.5
202.0
121.5
217.4
205.6
Lab 103
93.6
163.6
157.8
183.7
153.2
125.9
137.9
179.1
157.0
123.0
168.5
158.5
Lab 104
Missing
187.6
380.7
82.7
125.6
187.5
171.7
249.0
228.5
177.3
229.9
189.8
                                                    Each collaborator reported a concentration for all runs, with
                                              the exception of Lab 104 in Run 1.  This resulted from a malfunc-
                                              tion in the temperature indicator which caused the collaborator to be
                                              unable to complete the run. In numerous other cases, however, the
                                              determinations were unacceptable with respect to the requirements
                                              listed in the Federal Register,'*' a minimum sampling volume of 60
                                              standard cubic feet and an isokinetic factor of 90 to 110 percent.
                                              The end result was that of the 47 total determinations made, only
                                              32 conformed to the  definition of a replication for a compliance
                                              test, and only these were used in the analysis.

                                                    Since there is no technique available for determining the true
                                              particulate emission concentration, the procedure for grouping the
                                              runs into fairly homogeneous blocks is somewhat arbitrary. The
                                              only readily available means for establishing blocks was to divide the
                                              runs according to the two weeks during which the test was conducted.
                                              The assumption that is made under these circumstances is that there
is an essentially constant level of particulate emissions over each week's operation. This assumption may not be valid
due to the variety of materials burned and normal fluctuations; however, a preliminary statistical test indicated no reason
to believe there were large differences in the true emission level during the two weeks of testing. Therefore, this block-
ing scheme was accepted as suitable.

      There were several values in the collaborators' data that seemed inconsistent with the remainder of the data. These
included the low values of Lab 103 in Run 1 and Lab  104 in Run 4, and the high values of Lab 104 in Run 3 and
Lab  101 in Run 8. The presence of these high and low values is a consistent problem with the use of Method 5.  There
has been some discussion of the possible causes of these occurrences, with careless probe handling and probe contamina-
tion being the most likely candidates. There is not sufficient cause to eliminate these from the analysis as outlying values,
however, since their presence is apparently characteristic of the type of results one can expect from the method in field
testing.

B.2  Significance of the Port Effect

      The test was designed to offset and eliminate any effect of the possible differences  in observed concentration
levels resulting from the pattern of gas flow in the stack. The possibility does exist, however, that a particular port will
consistently show a higher or lower value than the others.
                                                    21

-------
      A non-parametric analysis of variance is used to test the hypothesis of no port effect. The test is that of Kruskal-
Wallis/2^ and the resulting statistic is given by
                                           12
                                        N(N
                                                       	3(/V + 1)
where
                                                                                     ,-th
                                   RJ- the sum of the ranks of the observations from the i   port

                                    rif- the number of valid determinations made from the r   port
and
                                   N = ^  «,—the total number of valid determinations from the test
                                        / = 1
                                   k— the number of ports.
     The usable data points are presented in Table B.2, by port, along with the rank of each value. The values are as-
signed rankings in the collective sample, with the highest determination in each week given rank 1.  The sum of the
ranks for each port and the number of determinations are shown at the bottom of each column. Separate tests arc
performed for the four ports from the first stack and for the four from the second stack.
                                TABLE B.2. SIGNIFICANCE OF PORT EFFECT

Block
1





Port





*i
"z
A
219.1
192.6
207.8
298.4


(6)*
(9)
(7)
(2)
24
4
B
230.2
157 8
82.7
236.4


(5)
(12)
(15)
(4)
36
4
C
93.6
380.7
183.6
151.2


(14)
(1)
(10)
(13)
38
4
D
163.6
202.7
278.2



(11)
(8)
(3)

22
3
H = 1.5167
2











Ri
"i
125.9
137.9
179.1
229.9
250.4


(16)
(15)
(10)
(5)
(2)
48
5
187.5
171 7
201.4
123 0



(9)
(12)
(7)
(17)

45
4
197.2
168.5
205.6




(8)
(13)
(6)


27
3
2454
468.9
1773
232 2
158.5


(3)
(1)
(11)
(4)
114)
33
5
//= 1.9941
*Number in parentheses is rank of value in combined sample.
      For the first week's test, the value of H was 1.517, while for the second week,//had a value of 1.994.  For a
value of//to be significant, it must exceed a value taken from a table of the chi-square distribution with k — 1 =
4-1 = 3 degrees of freedom. The tabled value at a 5-percent level of significance is 7.81.  Thus, the difference be-
tween the determinations due to a port effect is essentially nonexistent, and on this basis, the port factor is eliminated
from further analysis.
                                                    22

-------
 B.3  Transformations
      In order to gain information concerning the distributional nature of the Method 5 determinations, the observations
 are passed through two common variance stabilizing transformations, the logarithmic and the square root. To determine
 the  adequacy of the transformations, Bartlett's test for homogeneity of variance *•  •* is used to measure the degree of
 equality achieved.

      The results of the test are shown in Table B.3 for the two transformations, as well as for the data in  their original
 form (linear). Bartlett's test statistic is presented along with degrees of freedom and the associated significance level for
                                                    each transformation.  The significance levels are obtained from
  TABLE B.3.  DATA TRANSFORMATION TO ACHIEVE    a chi-square distribution with the degrees of freedom shown.
           RUN EQUALITY OF VARIANCE
                                                          Clearly, at any usual significance level, all three forms of
                                                    the data provide an acceptable model from  the equality of vari-
                                                    ance aspect. However, the highest degree of equality is ob-
                                                    tained through the logarithmic transformation, and this is taken
                                                    to be the appropriate model.

                                                          In a previous study by Hamil and Camann,   ' this has
                                                    been shown to be  an indication that there is a proportionality
 between the mean and standard deviation for the run data. This same model has been shown to be appropriate for data
 from particulate matter emissions in a study at a powerplant as well.  '

 B.4 Empirical Relationship Between Mean and Standard Deviation

      In Appendix B.3, an underlying proportionality between the mean and standard deviation for the  run data is indi-
 cated.  In this section, we will attempt to establish this relationship empirically  from the determinations obtained at the
 Holmes Road Incinerator study.  Let

                  be the concentration reported during run k in  block/ by lab z
Transformation
Linear
Logarithmic
Square Root
Test
Statistic
8.678
5.923
6.505
Degrees of
Freedom
10
10
10
Significance
0.56
0.82
0.77
  1  P
= ~ y_]
  P i= 1
                         xijk  be the mean for run k, block/ across collaborators, forp collaborators
             s,k
                                            be the run standard deviation
      Table B.4 gives the values of the sample statistics obtained at the Holmes Road site. There is a general tendency
for the standard deviation in the run data to rise as the mean level of concentration rises. To further this idea, the
paired statistics are plotted, and a least squares regression line is fit to the points. The model used is the no intercept
model, or a line forced through the origin, since a mean of zero could only occur in the event that each determination
equalled zero.  The model used, then, is
                                                     s,k = b xjk


where b is a constant.  The individual points and the regression line thus obtained are shown in Figure B.I.  The measure
of the degree of fit obtained is the coefficient of determination, r2, which for the no intercept model is obtained by the
formula^
                                                   23

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    TABLE B.4. INTERLABORATORY RUN
                SUMMARY
Block
1




2






Run
1
2
3
4
5
6
7
8
9*
10
11
12
Mean Paniculate
Concentration,
Ib/scf
156.3
195.5
237.2
181.5
228.7
156.7
185.0
283.1
0.0
165.8
210.2
204.8
Std. Dev.,
Ib/scf
88.7
33.4
98.2
97.8
73.9
43.6
55.0
161.3
0.0
38.4
36.1
46.0
*No usable results in Sample 9.
                                                            n     n
                                                                             n        n
                                                                                        *fk
                                                           i=l   i=l      i=l     i=l

                                                  For the run data, the value of r2 is 0.8515, indicating that ap-
                                             proximately 85 percent of the variation in the magnitude of the
                                             standard deviation is attributed to variation in the magnitude of
                                             the mean. The correlation coefficient, r = \/P, between x^ and
                                             Sj/f is 0.9228. From a table of significant values of/- presented in
                                             Dixon and Massey/2^ this value  indicates that a significant amount
                                             of correlation exists between the sample mean and sample standard
                                             deviation for the no intercept model used.

                                                  A similar analysis can be used on the collaborator-block values.
                                             If we let nij be the number of valid determinations in block/ by
                                             collaborator /, then we have
 Slk
200
150
100
  50
                    X,;  =
Run Standard Deviation
    Ib/scf X 107

                                                    as the mean for collaborator; in block /
               50
            100
                                    150
                                 200
250
                                                                     300
                                            Run Mean,  Ib/scf X  107

                               FIGURE B.I. INTERLABORATORY RUN  PLOT
                                                  24

-------
and
                Si, = „	
(xijk ~ xij.y as the collaborator-block standard deviation.
                              k = 1
    TABLE B.5. INTRALABORATORY COLLAB-
          ORATOR BLOCK SUMMARY
Block
1



2



Collaborator
Lab 101
Lab 102
Lab 103
Lab 104
Lab 101
Lab 102
Lab 103
Lab 104
Mean Participate
Concentration,
Ib/scf
245.7
212.3
150.0
231.7
278.8
203.5
148.8
191.6
Std. Dev.,
Ib/scf
40.7
22.2
33.8
210.7
108.3
3.0
23.3
26.4
              The eight  pairs of values obtained are shown  in Table B.5.
              As before, we fit a regression line through the origin to
              these points and try to determine the adequacy of the fit.
              For the collaborator-block data, the value of r2 is found to
              be 0.5343.  It is apparent from Figure E.2 that the degree of
              fit is not as strong as for the run data.

                   The value of r is 0.7309, which is, however, still above the
              critical  value in the table, at the 0.05 level.  Thus, we can still
              say that there is evidence of a proportional relationship. This,
              coupled with the results of the previous study, gives credence to
              the use  of this as a model for the data.
             250
             200
             150
             100
                        Collaborator-Block Standard Deviation
                                  Ib/scf X 107
                                   Collaborator- Block Mean, Ib/scf X 10'
                       FIGURE B.2. INTRALABORATORY COLLABORATOR-BLOCK PLOT

B.5  Unbiased  Estimation of Standard Deviation Components

     In Appendix B.4, an investigation into the correlation between the mean and standard deviation for the collabora-
tor-block data revealed that there was an empirical basis for accepting the model for the within-lab standard deviation of
for the data.  To estimate this standard deviation, we use the relationship

                                                Sj/ - Cx,f
                                                    25

-------
where Cis a constant, representing the proportionality. As previously discussed, Sy is a biased estimator for the true
standard deviation, a. The correction factor for removing the bias is dependent on the sample size n. and is given by
Ziegler^  ' as
                                                         n
                                                      r ~
                                                   2    \2
                                            <*„ =   —
where F represents the standard gamma function. Thus, we can say that
or
                                              a = anE(sjj)
                                               = 06.

so that in obtaining an unbiased estimate of 0, we can obtain an unbiased estimate of a as well.  Thus, we define an esti-
mator for a, a, where
      From Appendices B.3 and B.4, we determine that a suitable model for the run data is given by

                                                Ob = fo6

where aj, = \/o\  + a2 is the between-lab standard deviation. Empirically, we have

                                              sjk = Cbxik

and Sjk is a biased estimator for aj,.  Thus, forp collaborators,

                                              E(aps/k) = a

and we have
                                                    26

-------
      Obtaining an estimate of (3^, we have a new estimator, a/,, of a^ given by
and substituting our estimates of a\, and a, we have
so that the laboratory bias standard deviation may be estimated as a percentage of the mean as well.

 B.6  Weighted Coefficient of Variation Estimates

      The technique used for obtaining estimates of the coefficients of variation of interest is to use a linear combi-
 nation of the individual beta values obtained. The linear combination used will be of the form
                                                 7=1

 where j3y is the/th coefficient of variation estimate, k is the total number of estimates, and Wj is a weight applied to
 the/th estimate.

      As previously discussed, the individual estimate of )3 is obtained as
 for a sample of size n. This estimator is shown in B.5 to be unbiased for the true coefficient of variation. However,
 since we are dealing with small samples to obtain our individual estimates, weighting is more desirable in that it pro-
 vides for more contribution  from those values derived from larger samples.  There is more variability in the beta
 values obtained from the smaller samples, as can be seen by inspecting the variance of the estimator.  We have that
                                      Var(j3) = Var
(¥)
                                             = «,2
                                                    2n
for normally distributed samples,* ' and true coefficient of variation,/3.  Rewriting this expression, we have
                                                    27

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and all terms are constant except for a2 and n.  Thus, the magnitude of the variance changes with respect to the
factor a2/n. Now, since an decreases as n increases, the factor a2/« must decrease as n increases, and the variance
is reduced.

     The weights, Wj, are determined according to the technique used in weighted least squares analysis' % which
gives a  minimum variance estimate of the parameter. The individual weight, w/, is computed as the inverse of the
variance of the estimate, |3/, and then standardized. Weights are said to be standardized when
                                              1
                                                k
To standardize, the weights are divided by the average of the inverse variances for all the estimates. Thus, we can
write
                                                w, = —
where
                                              «,-=-
                                                  Var(ft)
and
     Now, from the above expressions we can determine u/, u and w,- for the beta estimates. For any estimate,

                                                1
for sample size «/, and
                                         Ui =
                                             Var(ft)
                                          1 A    i
                                        = 1^-5-
                                          7, •<--'  ,,,2
                                                          +2(32)
                                                    2j32
                                                   28

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Thus, the /th weight, w,, is
                                    w, = —
n,
<&,
2
L/32(l + 202)_
2
|32(1 + 202)
E

«/
4
     The estimated coefficient of variation is
                                                   £«"L-f
                                                   /=] an,   x
                                        _;=!««,_

B.7  Estimation of Precision Components

     In Appendices B.3 and B.4, the relationships are established for the within-laboratory standard deviation, a,
and the between-laboratory standard deviation, a/,,
                                               a = (36
and
                                                 29

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where 6 is the true mean of the determinations. In Appendix B.5, it is shown that for the laboratory bias standard
deviation, ai, the above expressions imply
      In Appendix B.6, the technique for obtaining an estimate of a coefficient of variation as a linear combination of
the individual values is discussed.  The estimator is of the form
                                                */=!

where fy is the estimated beta from the/th sample, Wj is the weight given to that estimate, and k is the number of esti-
mates obtained. For the run data, this becomes
                                                   12
                                              = -V
                                                11
        t-J
        /=!
The factor 11 is used since one run had no usable determinations.  In Table B.6, the individual coefficients of variation
and their corresponding weights are given. Using these in the above equation, we obtain
          TABLE B.6. RUN BETA
            ESTIMATES AND
                WEIGHTS
Run
1
2
3
4
5
6
7
8
9*
10
11
12
Beta Hat
0.7114
0.1928
0.4494
0.6078
0.3647
0.3484
0.3353
0.6427
0.0000
0.2613
0.1940
0.2532
Weight
0.565
1.045
1.507
1.045
1.045
0.565
1.045
1.045
0.000
1.045
1.045
1.045
*No usable determinations
in this run.
   TABLE B.7. COLLABORATOR-BLOCK
     BETA ESTIMATES AND WEIGHTS
Block
1



2



Collaborator
Lab 101
Lab 102
Lab 103
Lab 104
Lab 101
Lab 102
Lab 103
Lab 104
Beta Hat
0.1763
0.1182
0.2394
1.1398
0.4131
0.0183
0.1644
0.1493
Weight
1.310
0.698
1.310
0.377
1.310
0.377
1.611
1.007
                          fe = (0.387)

and consequently

                         ab=\

                            = (0.387)6

This estimate has 4 — 1=3 degrees of freedom for comparison of
four laboratories.

      Similarly, for the within-laboratory coefficient of variation
we have
for the 8 collaborator-block combinations.  The individual estimates
and their weights are shown in Table B.7. Substituting, we have

                          0 = (0.253)

and an estimated standard deviation of
                                                                     = (0.253)5.

                                           This estimate has 32  - 8 = 24 degrees of freedom, for the 32 determinations
                                           divided into 8 collaborator-block combinations.
                                                 30

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      Using these values in the equation for the laboratory bias coefficient of variation, we estimate
                                       fa = ( V(0.387)2 - (0.253)2]




                                          = \/67086




                                          = 0.293,




and thus, the estimated laboratory bias standard deviation is
                                                 = (0.293)6.
                                                    31

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                                        LIST OF REFERENCES
 1.    Environmental Protection Agency, "Standards of Performance for New Stationary Sources," Federal Register,
      Vol. 36, No. 247, December 23, 1971, pp 24876-24893.

2.    Dixon, W. J. and Massey, F. J., Jr., Introduction to Statistical Analysis, 3rd Edition. McGraw-Hill, New York,
      1969.

3.    Hamil, Henry F. and Camann, David E., "Collaborative Study of Method for the Determination of Particulate
      Emissions from Stationary Sources (Portland Cement Plants)," Southwest Research Institute report for
      Environmental Protection Agency, in preparation.

4.    Hamil, Henry F. and Camann, David E., "Collaborative Study of Method for the Determination of Nitrogen
      Oxide Emissions from Stationary Sources," Southwest Research Institute report for Environmental Protection
      Agency, October 5, 1973.

5.    Hamil, Henry F. and Thomas, Richard E., "Collaborative Study of Method for the Determination of Particulate
      Matter Emissions from Stationary Sources (Fossil-Fuel Fired Steam Generators)," Southwest Research
      Institute report for Environmental Protection Agency, in preparation.

6.    Searle, S. R., Linear Models. Wiley, New York, 1971.

7.    Ziegler, R. K., "Estimators of Coefficients of Variations Using k Samples," Technometrics, Vol. 15, No. 2,
      May, 1973, pp 409414.

8.    Cramer, H., Mathematical Methods of Statistics, Princeton University Press, New Jersey, 1946.
                                                 33

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