United States
Environmental Protection
Agency
Office of Toxic Substances
Office of
Toxic Substances
Washington, DC 20460
EPA 560/13 80 017B
December 1980
Asbestos-Containing Material:
in School Buildings

Guidance for Asbestos
Analytical Programs
Statistical Background Document

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                                       EPA 560/13-80-017B
                                       December 1980
ASBESTOS-CONTAINING MATERIALS IN SCHOOL BUILDINGS
    Guidance for Asbestos Analytical Programs

         Statistical Background Document
                       by

                    D. Lucas
                    A. V. Rao
                   T. Hartwell

           Research Triangle Institute
             Research Triangle Park,
              North Carolina 27709
         EPA Contract Number 68-01-5848
           Task Manager:  Cindy Stroup
           Project Officer:  Joe Carra

          Design and Development Branch
          Exposure Evaluation Division
           Office of Toxic Substances
              Washington,  DC  20460
    OFFICE OF PESTICIDES AND TOXIC SUBSTANCES
      U. S. ENVIRONMENTAL PROTECTION AGENCY
              WASHINGTON, DC  20460

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                   DISCLAIMER NOTICE








     This report was prepared under contract to an agency of



the United States Government.  Neither the United States



Government nor any of its employees, contractors, subcon-



tractors, or their employees makes any warranty, expressed



or implied, or assumes any legal liability or responsibility



for any third party's use or the results of such use of any



information, apparatus, product, or process disclosed in



this report, or represents that its use by such third party



would not infringe on privately owned rights.



     Publication of the data in this document does not



signify that the contents necessarily reflect the joint or



separate views and policies of each sponsoring agency.  Men-



tion of trade names or commercial products does not consti-



tute endorsement or recommendation for use.
                            11

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                          CONTENTS

                                                        Page
  I.  INTRODUCTION  	    1
 II.  SAMPLING FRIABLE MATERIAL  	    3
      A.  Summary of the Sampling Procedure  	    4
      B.  Comparison of the Recommended Sampling
          Procedure with Purposive Sampling  	    6
      C.  Recommended Sample Size 	    8
          1.  Variance of Asbestos Concentration of
              Friable Material 	    8
          2.  Confidence Intervals 	  12
          3.  Power Calculations 	  16
          4.  Conclusions 	  23
III.  LABORATORY QUALITY ASSURANCE 	  27
      A.  General Description of the Quality
          Assurance Program 	  27
      B.  Probability of Agreement Between the
          Results Based on the Analysis of Two
          Parts of a Split Sample 	  29
      C.  Sample Sizes 	  31
      D.  Initial Laboratory Quality Assurance 	  33
      E.  Procedure for Monitoring on an On-Going
          Basis 	  34
      F.  A Central Administrative Structure to
          Monitor Lab Quality 	  35
REFERENCES  	  37
APPENDIX A  	  39

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                    LIST OF TABLES
Number    Title _  _ _____       Page



  1       Summary of Variability Data  ...........        10



  2       90% Confidence  Intervals  for  y, when



          a  =  .0155 ............................        14
           c


  3       90% Confidence  Intervals  for  y, when



          ac =  .04  ..............................        17



  4       Power of  the  Test  H  :  y  =  .01 Versus



          H,:   y =  y,  > .01  for  Selected Values



          of y, , when  a  =  .0155  ................        21
              _L         Ox


  5       Power of  the  Test  H  :  y  =  .01 Versus



          H, :   y =  y ,  > .01  for  Selected Values



          of y, , when  a  =  .04 ..................        22
              -L         v_»


  6       False Positive  and False  Negative



          Rates  .................................        32



  A-l     Power of  the  Test  H  :   y  =  y   =  .01



          Versus H, :   y = y, > .01  for Selected



          Values of MI  ..........................        43



  A-2     Power of  the  Test  HQ:   y  =  yQ =  .01
           Versus H •   y = y^  > .01 for Selected
Values of y,
                                                         44

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I.   INTRODUCTION




     A Guidance Document, Asbestos-Containing Materials in




School Buildings:  Guidance for Asbestos Analytical Programs




(USEPA 1980) presents detailed sampling procedures and labo-




ratory quality assurance measures for bulk samples collected




in school buildings.   This background document is designed




to supplement the Guidance Document by presenting statisti-




cal support for these procedures.  In this document, the




statistical basis for the proposed bulk sampling procedures




is given in Chapter II,  and Chapter III presents the basis




of the proposed quality  assurance methods.

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-2-

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II.  SAMPLING FRIABLE MATERIAL



     The sampling procedure for friable material  in  school



buildings given in Chapter 2 of the Guidance Document  (USEPA



1980) is directed towards meeting the following two  require-



ments:   (1) construction of a probability sample  from which



meaningful conclusions can be drawn concerning the presence



or absence of asbestos and (2) provision of sampling



instructions that can be carried out without undue effort



and with minimum error by school personnel who are untrained



in sampling.



     This chapter briefly summarizes the proposed sampling



methodology and discusses the statistical motivation for its



recommendation.  The advantages of this methodology over



plans involving personal judgment or convenience are given



in Section II.B.  Section II.C presents statistical consid-



erations in determining the necessary sample size to give



the amount of confidence that is of practical significance



for reaching a decision as to the presence or absence of



asbestos.  Data collected in two studies conducted by Bat-



telle are used in the sample size calculations.   These data



are described in (Patton et al.  1980)  and (Rao et al. 1980).



     The variance estimation presented in Section II.C.I is



a revision of that given in an early draft of this document



(Lucas et al.  1980).   The reasons for this revision,  and its



implications,  are discussed in Appendix A.  The  revision



does not change the recommended sampling procedure or the



recommended number of bulk samples to be collected.
                         -3-

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     A.    Summary of the Sampling Procedure

          All school building areas should first be inspect-

ed for friable material.  The total friable material area of

the school building is then partitioned into Sampling Areas

using results of visual inspection, knowledge of the school

building's history, and building records if available.  A

Sampling Area is defined to be a homogeneous area of friable

material — that is, all friable material in a Sampling Area

is of the same type and was applied during the same time pe-

riod.  A decision as to the presence or absence of asbestos

in the friable material is necessary for each Sampling Area.

Asbestos is considered  "present" in the Sampling Area if the

average concentration of asbestos in that area exceeds 1%;

otherwise, asbestos is  considered to be "absent."

     A scale diagram of each  Sampling Area should be prepar-

ed according to the instructions given in Chapter 2 of the

Guidance Document  (USEPA 1980).  Based on area, the number

of bulk samples to be collected  from the Sampling Area is

determined as indicated in the table below.

      If the  size  (square           Then the number  (n)
      feet) of the  Sampling         of bulk samples to
      Area is                       be collected is
      Less  than 1,000                        3
      Between 1,000  &  5,000                  5
      Greater than 5,000                     7

 The required number of sample locations  should be  selected

 from the Sampling Area using the random  number pair proced-

 ure.   In addition to  detailed instructions  on the  use  of  a
                            -4-

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Table of Random Digits, a  Selection  of  Sample Locations



Worksheet is provided in the Guidance Document to aid in



completion of this task.



     The procedure proposed for sampling  is  simple random



sampling; that is, there is one stage of  sampling,  and all



elements of the Sampling Area have equal  probabilities of



selection, where an element is a subdivision  of  friable ma-



terial that is of the size to be collected in a  sampling



container.  Let y be the true average asbestos concentration



in the Sampling Area.  Let X, , X» , ..., X be the  measured



asbestos concentrations in the n collected bulk  samples.



Assuming that the laboratory analysis and quantitation of



asbestos is unbiased, the sample mean




                    n
                 =  £  xi/n
is an unbiased estimator of y .   (Unbiasedness refers to the



fact that in repeated sampling the distribution of X is



centered around y.)  An estimator of the standard deviation



of the measured asbestos concentration is




                     n          „
               3 =      CX. - X)  / (n-l)               (2-2)
and an estimator of the standard deviation of X is
               s- = s/-n .                              (2-3)




A (1 - a) -100% confidence interval for y is given by






     (X " t(n-l), (1-a)  SX'  X + t(n-l), (1-ct)  sx} '      (2~4)
                           -5-

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where t(n_1)^  (i-a) is the  ^"f^ quantile  of  the  t-distri-
                 2"
bution with n-1 degrees of  freedom.   The interpretation  of

the confidence interval  (2-4)  is that in the  long run

(l-a)'100% of  the intervals constructed according to  (2-4)

will contain y, the true  average asbestos  concentration  in

the Sampling Area.

     Chapter 4 of the Guidance Document gives instructions

for computation of the sample  mean and the sample standard

deviation  for  the reported  asbestos concentrations.  A 90%

confidence interval for  the true average asbestos concen-

tration  in the Sampling  Area is formed.  A Statistics  Com-

putations  Worksheet is provided for use in performing  these

calculations.  The following rule  is presented for reaching

a  decision as  to  the  presence  or absence of asbestos  in  the

Sampling Area, where  asbestos  present is taken to mean that

the average  concentration of asbestos exceeds 1%.

      (.1)   If the  entire  confidence interval is below  1%,

           then conclude  asbestos absent;

      (2).   If the  entire  confidence interval is above  1%,

           then conclude  asbestos present;  or

      (.3)   If the  confidence interval contains 1%, then there

           remains uncertainty as  to the presence or absence

           of asbestos.

      B.    Comparison  of  the Recommended  Sampling Procedure
           with Purposive Sampling

           In one  form of purposive sampling the person se-

 lecting sample locations will, during visual inspection of
                           —6 —

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the Sampling Area, classify certain areas of  the  friable



material as "representative" and then collect bulk  samples



from these locations.  Attempting to make inferences  con-



cerning y, the true average percentage of asbestos  in the



Sampling Area/ from data collected in such a  manner will



lead to many difficulties.  First, the judgment of  the per-



son selecting sample locations is a factor and will differ



from person to person, as discussed in (Raj 1968).  The



probability that a given element of the Sampling Area will



be selected is unknown.  In this situation it is not  possi-



ble to determine the distribution of X, the estimator of y.



The sampling error cannot be objectively determined,  and the



accuracy of the resulting estimator cannot be assessed.



     The above comments also apply to selecting sample lo-



cations haphazardly or according to convenience.  In conven-



ience sampling, all locations not deemed convenient by the



person selecting sample locations have probability  zero of



being included in the sample.  Making inferences for the



entire Sampling Area from such a sample cannot be statisti-



cally justified or defended.   Under certain conditions, ei-



ther haphazard or convenience sampling may happen to give



good results.   However, due to the lack of structure of the



methods, there is no way of predicting how frequently good



results will occur.



     In the proposed procedure for sampling friable materi-



al, it is known that each element of the  Sampling Area has
                           -7-

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an equal probability of selection.  An estimate of sampling


variability can be calculated from the data obtained.  Meth-


ods of statistical inference can be used to estimate the


true average percentage of asbestos in the Sampling Area,


give the precision of this estimate, and test hypotheses


concerning the presence or absence of asbestos in the Samp-


ling Area.


     C.   Recommended Sample Size


          1.   Variance of Asbestos Concentration of Fria-
               ble Material


               Determining the  necessary sample size con-


sists of  finding  the minimum number of observations that


will give the amount of precision  in inferences about y, the


true average concentration of asbestos in  the Sampling Area,


that is of practical significance  for reaching a decision.


The desired precision in  inferences about  y will be discuss-


ed in  the following  sections.   Procedures  for sample size


determinations  first require  an estimate of the variability


to be  expected  among the  observations of asbestos concentra-


tion.


     The  variance among observations of  asbestos concentra-


tion  can  be  partitioned  into  the following two components:

                                 2
      (1)  A  variance component  a  attributable to the vari-


           ability in true (not  measured) asbestos concentra-


           tion among elements of the  Sampling Area, where  an


           element is a  subdivision of  friable material  that


           is of the size  to be  collected in a  sampling  con-


           tainer.


                            -8-

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      (2)  A variance component a  attributable  to  the  vari-



          ability in measured asbestos concentration among



          measurements on a bulk sample of friable material.



     To obtain numerical estimates of the above variance



components, data collected in two studies conducted by Bat-



telle are used.  These currently available data are very



limited, and for future study it would be of interest to



have more extensive data.  In the Battelle Duplicate Analy-



sis Study, described in  CRao et al.  1980), asbestos concen-



tration measurements were made on split-samples of friable



material collected at three locations from each of five



buildings.  Eight buildings were included in the Battelle



Bulk Sampling Variability Study,  described in (Patton et al.



1980).  Within each of these eight buildings, bulk samples



were collected at four randomly selected locations within a



5,000 square feet area of homogeneous ceiling material.



Each bulk sample was then split into four parts prior to



laboratory analysis.  For each area included in these two



studies, measurement variability and variability due to



sampling location were estimated using analysis of variance



techniques.  These estimates are displayed in Table 1.



     It can be seen from Table 1 that estimates of a
                                                    c


(square root of the variance component due to location with-



in the Sampling Area)  range from 0 to 15.22%.  The average



a  for the thirteen areas studied is 1.55%.   Note that there



is no reason to assume that a  is constant for various types
                            -9-

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      Table  1.   Summary of Variability Data

Data
Sourcea
(1)
(1)
(1)
(1)
(.1)
(21
(.2)
(2).
(2)
(2)
(2)
(2)
t2)
Sampling
Area ID
Number
1
2
3
4
5
6
7
8
9
10
11
12
13
Number of
Observa-
tions
6
6
6
6
6
16
16
16
16
16
16
16
16
Average
Asbestos
Concentra-
tion, X
.075
.397
.042
.545
.548
.481
.116
.131
.134
.028
.004
.184
.289
Standard Devi- .
ation Estimates
yv
CTc
.0204
.0000
.0000
.0000
.1522
.0000
.0198
.0000
.0000
.0051
.0039
.0000
.0000
/•>
°e
.0204
.0619
.0385
.1866
.0261
.0661
.0725
.0750
.1197
.0360
.0122
.1008
.1350
Data Source (1) is the Battelle Duplicate Analysis Study,
  described in  (Rao et al. 1980).  Data Source  (2) is
  the Battelle Bulk Sampling Variability Study, described
  in (Patton et al. 1980).
^
a  is the square root of  the variance component due to
 C                                       A
  location within the Sampling Area, and a  is the square

  root of the variance component due to laboratory mea-
  surement.
                         -10-

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of friable material.   In  fact,  it  is  suspected that the var-



iability due to location  is different  for  different types of



friable material.  Some friable materials  may  be  fairly het-



erogeneous with respect to asbestos concentration,  while



others may not be.  In the sample  size calculations of  the

                    l*\              XN.

following sections, a  =  .0155 and a  =  .04 are considered.
                     C_x              O
          /\

The value a  = .04 was arbitrarily selected for considera-



tion; it is thought possible that  a  values in the  neighbor-
                                   c


hood of .04 may be encountered in practice.



     In (Patton et al. 1980) it was observed that there ap-



pears to be a strong positive relationship between  asbestos



concentration and measurement variability when asbestos con-



centration is less than approximately 30%.  The following



linear regression model is formulated to describe this  rela-



tionship:




               ae = 30 +  BI y ,



where BQ and g,  are parameters to be estimated.  The least



squares estimates of these parameters are computed  using



those points from Table 1 for which X is less than  30%  (X.



is used to estimate y), yielding the following estimated re-



gression equation:



               ae = .0177 + .45 y,                      (2-5)



with a correlation coefficient of  .90.  The estimate of to-



tal variance among observations of asbestos concentration


 222
a  = a  + a  is then given by
      C    G
                           -11-

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               ~2   "2   "2
               a  = a  + a
                     c    e
               a2 = a2 +  (.0177 +  .45y)2,               (2-6)
                     C^


      "2          2     ~2         2
where a  =  (.0155)  and a  =  (.04)  are considered.
       c                 c


     It should be noted that  there are currently  available



no data that can be used to examine the relationship between



total area  and variability of asbestos concentration within



the Sampling Area.  As mentioned,  the Battelle  Bulk Sampling



Variability Study included areas of 5,000  square  feet.   Some



increase in variability as the area increases might be  ex-



pected due  to increased time  span  of material application,



increased number of batches of material used, or  other  fac-



tors depending upon the application method.  However, it is



not thought likely that such  a relationship  between total



area and variability would be directly proportional.  In



other words, doubling  the total area would not  require  doub-



ling the sample  size to obtain equal estimation precision.



Because of  this,  the rule of  sampling at  one location for



every  5,000 square feet of  friable material, recommended in



 (USEPA  1979),  is  questionable.  The  factor of area, however,



is taken into  account  in  the  sample  size  recommendations pre-



sented  in Section  II.C.4  of this  document.



          2.   Confidence  Intervals



               Let n denote the number of bulk  samples  col-



lected  from a  Sampling Area,  and  let X be the average mea-



sured  asbestos concentration. According  to  the discussion
                            -12-

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of Section II.C.I, assume that the estimate  of  the  standard



deviation is
               s = Ja^ +  (.0177 +  .45y)2,




       2
where a  is the variance component  due to location within



the Sampling Area.  Values of this  component considered  here


     2          222
are a  = (.0155)  and a  = (.04) .  The standard deviation
     c                 c


of X is estimated by S:T = s/Jn, and a  (1-a) -100% confidence



interval for y is given by
     (X   t(n-l)., (1-a) SX'  X"l"t(n-l),  (1-a) SX} '
as discussed in Section II.A.  For selected values of X and



n, 90% confidence intervals for y are displayed in Table 2



(a  = .0155) and Table 3 (a  = .04).
  C                        C


     Consider Table 2, which assumes that cr  = .0155.  If
                                           C


the variance model given in Section II.C.I describes the



true situation for a Sampling Area fairly well, then the



expected 90% confidence interval for y is (.009, .091) when



X = .05, computed from measurements on 5 bulk samples.  In



other words, the true average asbestos concentration is ex-



pected to be between .9% and 9.1%, with 90% confidence.



(The phrase "with 90% confidence" means that in repeated



sampling, 90% of the intervals constructed in this way will



contain y.)   Again, with X = .05, the expected 90% confidence



interval for y.is C.025, .075)  when 10 bulk samples were col-



lected.  The length of this confidence interval is 7.5% -
                           -13-

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Table 2.  90% Confidence Intervals for y, when a  =  .0155
                                                c

V
n\
2
3
5
7

10
15
20
30
50

.005
.060a
.033a
.022a
.019a
a
.016a
.014a
.013a
.ona
.oioa

.010
.069a
.039a
.029a
.025a

.022a
.019a
(.000, .020)
(..002, .018)
(.004, .016)

.015
.078a
.047a
-035a
.031a
a
.028a
(.002, .028)
(.004, .026)
(.006, .024)
(.008, .022)

.020
.087a
.054a
.041a
.037a

(..002, .038)
(.006, .034)
(.008, .032)
(.010, .030)
(.013, .027)


\ X
A
2
3
5
7
10
15
20
30
50

.030
.106a
.068a
.054a
(.004, .056)
(.010, .050)
(.014, .046)
(.017, .043)
(.019, .041)
C. 022, .038)

.040
.125a
.082a
(.003, .077)
(.011, .069)
(.017, .063)
(.022, .058)
(.025, .055)
(..028, .052)
(.031, .049)

.050
.144a
.097a
(.009, .091)
(.018, .082)
(.025, .075)
(.030, .070)
(.033, .067)
(.037, .063)
(.040, .060)

.075
.192a
.133a
(.024, .126)
(.036, .114)
(.044, .106)
(.051, .099)
(.054, .096)
(.058, .092)
(.062, .088)
                                                   continued
                          -14-

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Table 2.  (continued)

Y
n\
2
3
5
7
10
15
20
30
50



(.
(..
c.
(.
(.
(.
c.



038,
053,
063,
071,
075,
080,
085,
.100
.240a
.170a
.162)
.147)
.137)
.129)
.125)
.120)
.115)


C.004,
(.067,
(.086,
(.100,
(.111,
(-117,
(.123,
(..129,
.150
.338a
.296)
.233)
.214)
.200)
.189)
.183)
.177)
.171)


(.017,
(.096,
(.120,
(.137,
(.151,
(.158,
(-166,
(.174,
.200
.437a
.383)
.304)
.280)
.263)
.249)
.242)
.234)
.226)


(.029,
(.125,
(.154,
(.174,
(.190,
(.199,
(..209,
(.219,
.250
.535a
.471)
.375)
.346)
.326)
.310)
.301)
.291)
.281)
a
   90% Upper Confidence Bound.   (.90% Confidence  Interval
     contains zero.)

   The entries in the above table were calculated  according
     to


     (* ~ fc(n-l) , .95 SX' X + t(n-l),  .95  SX* '

     where t,  ,«    g5 is the 95th quantile of the t-distri-

     bution with n-1 degrees of freedom, n is the  number of

     bulk samples collected, and
     s^ -  J(..0155)2 +  (.0177 +  .45X)2  / J~~n~~.


     This variance model is based on data in  (.Patton  et  al.
     1980)  and (Rao et al. 1980) .
                           -15-

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2.5% = 5.0%, as compared to a length of 9.1% -  .9% = 8.2%



for a sample size of 5.  It can be seen from the table that



the length of the confidence interval decreases as the num-



ber of bulk samples collected is increased.  The desired



confidence interval length is a factor in choosing an appro-



priate sample size.



     Table 3 displays  confidence intervals for  p, again us-



ing the variance model of Section II.C.I, but with a  =
                                                    O


.04.  This larger variability due to location leads to



longer confidence intervals, for the same sample size and



the same X.  As mentioned previously, it is thought likely



that the variability due to location is different for dif-



ferent friable materials.  Considering Table 3, the expected



90% confidence interval  for y is  (.017,  .083) when X =



.05, computed from measurements on 10 bulk samples.  When 5



bulk samples were collected and X =  .05, the expected 90%



upper confidence bound is  .089.  The interpretation of a 90%

                  4

upper confidence bound is that in repeated sampling, 90% of



the bounds  constructed in this way will exceed  the true av-



erage asbestos concentration.  A 90% upper confidence bound



is  presented  in Table  2  or  3 whenever the corresponding 90%



confidence  interval  includes 0.



           3.   Power Calculations



                It  is desired to decide whether  or not asbes-



tos is present  in  the  Sampling Area, where  "asbestos present"



is  defined to occur  when the average asbestos concentration
                          -16-

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Table 3.  90% Confidence Intervals for y, when a  = .04

V
-\
2
3
5
7
10
15
20
30
50

.005
.102a
-054a
.036a
.029a
.025a
.021a
.018a
.016a
.013a

.010
.noa
.060a
.041a
.035a
,030a
.026a
.024a
.021a
.018a

.015
.117a
.066a
.047a
.041a
.036a
.031a
.029a
(.000, .030)
(.004, .026)

.020
.125a
.072a
.053a
.046a
.041a
.037a
(.001, .039)
(.005, .035)
(.009, .031)


V
n\
2
3

5
7
10 (.001,
15 (.007,
20 (.010,
30 (.014,
50 (0.18,

.030
.140a
.085a
=a
.065a
.058a
.059)
.053)
.050)
.046)
.042)

.040
.157a
.098a

.077a
(.001, .079)
(.009, .071)
(.016, .064)
(.019, .061)
(.023, .057)
(.027, .053)

.050
.173a
.112a

.089a
(.008, .092)
(.017, .083)
(..024, .076)
(.028, .072)
(.032, .068)
(.037, .063)

.075
.217a
.146a

(.013, .137)
(..027, .123)
(.037. ,113)
(.045, .105)
(.050, .100)
(.055, .095)
(..060, .090)
                                                 continued
                        -17-

-------
Table 3.  (continued)

V
n\
2
3
5
7
10
15
20
30
50



(.
I.
(..
(.
(.
(.
t-



029
045
057
066
071
077
082
.100
.262a
.181a
,.171)
,.155)
,.143)
,.134)
,.129)
,.123)
,.118)



(.060,
(.081,
(.095,
(-.107,
(.114,
(.121,
(.128,
.150
.355a
.252a
.240)
.219)
.205)
.193)
.186)
.179)
.172)


(.006,
(.090,
(.116,
(.133,
(-148,
(-156,
(.164,
(.173,
.200
.450a
.394)
.310)
.284)
.267)
.252)
.244)
.236)
.227)


(.020,
(.120,
(.150,
(.171,
(.188,
(.197,
(.208,
(.218,
.250
.546a
.480)
.380)
.350)
.329)
.312)
.303)
.292)
.282)
    90%  Upper Confidence Bound.  (90% Confidence Interval
      contains zero.)

    The  entries in the above table were calculated according
      to

      CX ~ Hn-l), .95 SX' * + t(n-l), .95 SX) '

      where t,  , v   Q[. is the 95th quantile of the t-distri-
              in— -Lji / . y D
      bution with n-1 degrees of freedom, and


      S£ =  J  (.04)2 +  (.0177 + .45X)2  / J~n  .

      This variance model is based on data in  (Patton et al.
      1980) and (JRao et al. 1980).
                            -18-

-------
exceeds 1%.  Putting this in terms of statistical  hypothesis




testing, it is desired to test the null hypothesis HQ:  y  <_



.01 versus the alternative H,: y > .01, where  y  is the  true



average concentration of asbestos in the Sampling  Area.   A



test can be constructed according to the following decision



rule:



          Reject H  if X > .01 + z(1_a) a





          Accept HQ if X <_ .01 + Z(1_ . a/-\J~n,






where Z,,_ .  is the (1-a)  quantile of the normal distribu-



tion, a is the standard deviation when H  is true,  n is the



number of bulk samples collected, and X is the average of



the measured asbestos concentrations.  (Under the  assumptions


                                   2                2
presented in Section II.C.I,  a =  a  + (.0177+.45y) , where
                                   \*f

 2

a  is the variance component due to location within the Samp-
 O


ling Area.)  The size of the test, denoted by a, is the



probability that the test leads to rejection of the null



hypothesis H : y <_ .01 when in fact the null hypothesis is



true.  Denote by B(y-,)  the probability that the null hypo-



thesis is accepted when actually the alternative y = y,



> .01 is true.  The power of the test under y-, is  defined as



l-g(y,), the probability that the test correctly rejects H



when y = y. > .01.  In other words, the power of the test



under y-, is the probability of correctly concluding that



asbestos is present when the average asbestos concentration



is y , some concentration greater than .01.  The power of



the test is an important consideration when deciding how
                           -19-

-------
many bulk samples to collect.  A  sufficient  number  of  bulk



samples to give good  (i.e., high) power  for  alternatives  of



interest  (.y-^'s) is desired.



     Let  a, the size of  the test, be  .05.  This  means  that



the probability of the test concluding that  asbestos  is pre-



sent when in  fact y <_  .01  is  5%.  For this a and the  vari-



ance model presented in  Section II.C.I,  Tables  4 and  5 give



the power of  the test  for  selected values of n  and  y, . In


                            2         2
Table  4 it is assumed  that a   =  (.0155)  , and in Table 5  it


                 22          2
is assumed that o  =  (..04)  ,  where a  is the variance com-
                 c                   c


ponent due to location within the Sampling Area. Consider-



ing Table 4,  suppose  5 bulk samples  were collected  and



actually  y =  y, =  .05.   Then, if  the variance assumptions



describe  the  situation fairly well,  the  power is expected to



be  .851.  In  other words,  with probability  .851 the test



will  conclude that asbestos is present when  the true  average



asbestos  concentration is  .05.  In this  case the test will



incorrectly  conclude  that  asbestos  is absent with probabil-



ity  .149.  It can  be  observed in  Table  4 that,  for  a fixed



alternative  y-, ,  the power  increases  as  the  sample size n



increases.   Also,  for a  fixed sample size,  the  power in-



creases  as  y, increases; that is, the test  can  better dis-



tinguish the difference  between .01  and  y,  when y,  is far-



ther  from .01.
                            -20-

-------
Table 4.  Power  of  the  Test  H  :  y  = .01 Versus

          H, :  y  = y,  >  .01 for Selected Values of
          y,, when  a  =  .0155

V1
"\
2
3
5
7
10
15
20
30
50

.015
.098
.107
.124
.139
.160
.192
.221
.276
.374

.020
.162
.189
.236
.279
.337
.425
.502
.629
.801

.025
.238
.286
.368
.441
.535
.660
.753
.874
.969

.030
.320
.388
.502
.595
.704
.827
.901
.969
.997

.040
.478
.575
.718
.814
.902
.967
.989
.999
>.999

.050
.610
.717
.851
.922
.971
.995
.999
>.999


.060
.710
.813
.922
.968
.992
.999
>.999




V1
n\
2
3
5
7
10
15
20
30
50

.070
.783
.875
.959
.987
.998
>.999




.080
.835
.915
.977
.994
.999
>.999




.090
.872
.941
.987
.997
>.999





.100
.900
.958
.992
.999
>.999





.125
.941
.980
.998
>.999






.150
.962
.989
.999
>.999






.200
.980
.996
>.999






The entries in the above table were calculated according  to

     Power of the Test H :  y = y  =  .01 versus
= 1 - $ {(..01 - y-L + 1.645 a
                            yo
                                   Afli)  / a  /J~~n}
                                             yl  *

     where a   = J(.0155)2 +  (.0177 +  .45y.)2,
            yi   *                        *-
     and ${z} is the cumulative distribution function  of  a
     standard normal random variable.  The variance model
     is based on data in  (Patton et al. 1980) and  (Rao et al,
     1980).

                           -21-

-------
Table 5.  Power of  the  Test  H  :  y  =  .01  Versus

          H,: y = y   >  .01 for Selected  Values of

          y,, when  a  =.04

V1
n x
2
3
5
7
10
15
20
30
50

.015
.073
.078
.086
.093
.103
.117
.130
.154
.198

.020
.102
.114
.136
.156
.183
.224
.263
.336
.463

.025
.137
.160
.200
.236
.287
.365
.435
.556
.734

.030
.178
.212
.274
.330
.407
.518
.611
.751
.904

.040
.271
.333
.440
.531
.643
.778
.864
.952
.995

.050
.371
.459
.599
.706
.817
.920
.966
.994
>.999

.060
.470
.576
.729
.829
.917
.976
.993
>.999



V1
n\
2
3
5
7
10
15
20
30
50

.070
.561
.675
.824
.906
.965
.993
.999
>.999


.080
.639
.754
.888
.950
.985
.998
>.999



.090
.705
.815
.929
.973
.994
>.999




.100
.758
.861
.955
.986
.998
>.999




.125
.851
.930
.985
.997
>.999





.150
.904
.962
.994
.999
>.999





.200
.954
.986
.999
>.999





 The entries in the above table were calculated according to

      Power of the Test H : y = y  = .01 versus
                                     yo = *01
      = 1 - $
(C-01 - y,  + 1.645 a  /J n)  / a  AHi}
         J-           yQ v       y-L v
      where a   = Jt.04)2 +  (.0177 +  .45vi)  ,

      and ${z} is the cumulative distribution function of a
      standard normal random variable.  The  variance model is
      based on data in  (Patton et al. 1980)  and  (Rao et al.
      1980).
                             -22-

-------
          4.   Conclusions



               The confidence interval calculations  of  Sec-



tion II.C.2 can be used in sample size determination upon



establishing the following:   (1) the confidence interval



length that is of practical significance for estimation and



(2). the maximum risk that can be tolerated that the  confi-



dence interval will not include y.  The confidence intervals



given in Section II.C.2 are 90% confidence intervals; that



is, in the long run 90% of the intervals so constructed will



contain y.



     The power calculations of Section II.C.3 can be  used



upon establishing the following:  (.1) the magnitude  of real



difference that is of practical significance for detection,



(2) the maximum risk that can be tolerated in concluding



asbestos is present Cy > .01)  when actually there is  none



(y <_ .01) and (3)  the maximum risk that can be tolerated in



not detecting presence of asbestos when actually asbestos is



present.  It is very difficult to assess these factors.  In



considering di,  the relationship between asbestos concen-



tration and level of danger to those in the school building



is not presently completely understood.  Additional  factors



that increase exposure risk such as water damage and  acces-



sibility will be considered in making a decision as to the



necessity of corrective action.   In the power calculations



of Section II.C.3, the risk of concluding that asbestos is



present when in fact it is not is set at 5%.  The power
                           -23-

-------
(probability of concluding asbestos is present when it

actually is present) for selected sample sizes and alter-

native y^s is displayed in Tables 4 and 5.  A reasonable

requirement is power of at least 90% for alternatives of

interest.  It is emphasized that a small sample size giving

insufficient power would lead to further questioning whenever

the test "concludes" that no asbestos is present.  Such a

situation is obviously undesirable, especially in light of

the harmful effects undetected asbestos may be having on

school children.  It may be that alternatives y, > .05 or

y1 >  -10 are of interest, and that the sample sizes  (expense

and effort) necessary for sufficiently powerful tests against

alternatives y1 <_ .05 or y, <_ .10 are not  justified by pre-

sently known risks to health.

     An  additional consideration is the variance of asbestos

concentration.  As discussed in Section II.C.I, the variance

used  in  calculations for sample size determination is esti-

mated using presently available data to the extent possible.

This  can be refined when more data become  available.

      Taking into account all the factors discussed above,

the number of recommended sample locations for a Sampling

Area  is  given in the table below.

          If the size  (square      Then the number (n)
          feet) of the Sampling    of samples to be
          Area  is                  collected is
          Less  than  1,000                  3
          Between  1,000  &  5,000            5
          Greater  than 5,000               7
                            -24-

-------
The relationship between area and recommended sample  size  is

in response to the considerations discussed at the end  of

Section II.C.I.  There are no data presently available  that

could be used to examine the relationship between total area

and variability of asbestos concentration within the  area  of

friable material.  The proposed procedure is considered the

minimal procedure (with respect to effort and expense)

capable of producing adequate results for the estimation and

testing problems concerning presence or absence of asbestos

in friable material.

     A comparison is given in the table below of the pro-

posed sample size and the sample size determined by the rule

of sampling at one location for every 5,000 square feet of

friable material, previously recommended in (USEPA 1979).

Size of Sampling    Proposed       Sample Size Based on
Area (Square Feet)   Sample Size    5,000 Square Feet Rule
750
3,000
7,500
20,000
40,000
75,000
125,000
3
5
7
7
7
7
7
1
1
2
4
8
15
25
It can be seen that, for larger areas, the proposed guidance

results in the collection of fewer bulk samples than recom-

mended by earlier EPA guidance.
                           -25-

-------
-26-

-------
III.  LABORATORY QUALITY ASSURANCE

      The laboratory quality assurance procedures given  in

Chapter 3 and Appendix B of the Guidance Document  (USEPA

1980). are designed to ensure reliable results for laboratory

analyses of bulk samples.  In particular, Chapter 3 of the

Guidance Document presents procedures to monitor laboratory

results on an on-going basis while Appendix B is designed to

evaluate the initial performance of an unknown laboratory

(.i.e., a laboratory not choosen from the list given in Ap-

pendix A of the Guidance Document).  In general, the on-

going and initial quality assurance evaluation procedures

are the same except for the number of split-samples analyzed.

     This chapter gives a brief summary of the concepts

underlying the suggested quality assurance procedures and

then presents the statistical bases for the number of split-

samples recommended in practice to carry out the quality

assurance program.

     A.   General Description of the Quality Assurance
          Program

          Results of the analysis of a number of bulk split-

samples form the basis of the proposed laboratory quality

assurance procedures.   For the purpose of this discussion, a

split-sample is defined as the two parts of a sample.   In

general,  certain number of split-samples will be sent to the

laboratory(s) performing the asbestos analysis;  the results

of the analysis of these samples will be analyzed to deter-
                           -27-

-------
mine whether the laboratory(s) is capable of performing  the



analysis to the desired level of accuracy.  This general



procedure applies either initially  (i.e., for an unknown



lab) or for monitoring a laboratory's performance over time.



     Specifically, the laboratory report for each sample



will include whether the asbestos level in that sample is



above a specified level or not  (a yes or a no).  This data



will be analyzed to determine the extent of agreement in the



results of the split-samples.  The  laboratory will be con-



sidered to be performing satisfactorily if the number of



split-sample disagreements is less  than a specified number



referred to in this report as the critical number of dis-



agreements.  This general split-sample procedure is appli-



cable for the following situations:



      (1)  When both parts of  the split-samples are sent  to



          the same laboratory;



      (2)  When the two parts  of the split-sample are sent to



          different laboratories;  (In this case, disagree-



          ment in the results is indicative of difficulties



          in one or both of the laboratories.)



      (3)  When the decision is whether asbestos is present



          in the  sample or  not; and



      (4)  When the decision is whether the level of asbestos



          in the  sample  is  above a  predetermined value or



          not.
                            -28-

-------
For example, suppose the quality assurance decision  is  to  be

based on five split-samples.  Then if the conclusions for

two or more of these five split-samples are different  (be-

tween the two splits) , the laboratory procedure is not  sa-

tisfactory and should be investigated.  Following an inves-

tigation, when the problem is corrected, all samples analy-

zed since the last time the laboratory was determined to be

in a satisfactory state should be re-analyzed.

     The critical number of disagreements allowed for a

laboratory depends on the probability of agreement between

the results of the analysis of the two parts of the split-

sample.  The algebraic expression for this probability is

discussed in the next section.

     B.   Probability of Agreement Between the Results Based
          on the Analysis of Two Parts of a Split Sample

          Let C  denote the probability of a positive result

when the sample is truly positive (has asbestos)  and C  de-

note the probability of a negative result when the sample is

truly negative.   Let P (A)  denote the probability of the

sample being truly positive.   If the two analyses are sta-

tistically independent,  then the probability p of observing

an agreement in the results of the two analyses is given by
p =  Cp  + d-Cp)    P(A) +  cn  + d-Cn)     1-P(A).   (3-1)


If the process of selecting samples is heavily biased to-

wards positive samples, then
                           -29-

-------
               p * cp2 +  (1-cp12-
If C  is very close to 1, i.e., positive samples can be
classified as positive with near certainty then


In this case, in order for p to be equal to 0.90, C  should
be of the order 0.95.
     The above assumptions are used to calculate the number
of split-samples required for the quality assurance program.
These sample sizes are calculated assuming p = 0.90.
     Note, if C  or C  changes with the level of asbestos  in
the sample, then the assumption of a  constant  'p1 implicitly
assumed in arriving at' the quality assurance procedures giv-
en below  is not  strictly valid.   If,  for example, for bor-
der line  samples C  is small and the  process of selecting
samples is biased toward borderline samples, then one would
expect more disagreement between the  results of such split-
samples than in  other situations; p will be smaller than 0.9
in such a case.
      If the two  parts of the split-sample are  analyzed  in
two different  laboratories, the  above discussion is still
appropriate  if C and C   are the  same for both the  labs.   If
they  are  different  for the two labs,  the expression for p
can be  easily  modified to account  for these differences.
                            -30-

-------
     C.   Sample Sizes



          Let N be the number of split samples tested  in  the



laboratory and let d denote the number of samples for  which



there was disagreement.  Further let us assume we would like



to test the null hypothesis that p = 0.90 against the  alter-



native hypothesis that p = 0.70.  For the purposes of  this



discussion, we will define the false positive rate as  the



probability of rejecting p = 0.9 when p is really =0.9 (in



statistical terms this is usually referred to as the signi-



ficance level).  In other words, the false positive rate is



the probability of incorrectly rejecting an acceptable batch



of bulk samples associated with the split-samples.  The



false negative rate is defined here as the probability of



accepting p = 0.9 when p is really = 0.7 (this is usually



referred to as 1-power).   The false negative rate is the



probability of incorrectly accepting an unacceptable batch



of bulk samples.   Table 6 gives the false positive and false



negative rates for several values of the critical number of



split-sample disagreements (d)  for sample size (N) equal to



5, 10, 15, 20 and 25.



     The procedure to be followed in an actual situation is



to reject the null hypothesis if the observed number of dis-



agreements is equal to or exceeds the critical number of



disagreements d for the appropriate sample size.   For ex-



ample, consider the case where N=15 and the critical number



d selected is 5.   If the observed number of disagreements is
                           -31-

-------
Table 6.  False Positive and False Negative Rates
          Corresponding to Various Critical Numbers
          of Split-Sample Disagreements

No. of
Split
Samples
(N)
5


10



15



20




25




Critical
Number of
Split-
Sample
Disagree-
ments (d)
1
2
3
2
3
4
5
3
4
5
6
3
4
5
6
7
4
5
6
7
8
False
Positive
Rate For
p=0.90
(significance
level)
0.4095
0.0815
0.0086
0.2639
0.0702
0.0128
0.0016
0.1841
0.0556
0.0127
0.0022
0.3231
0.1329
0.0432
0.0127
0.0022
0.2364
0.0980
0.0334
0.0095
0.0023
False
Negative
Rate For
p=0.70
(1-power)
0.1681
0.5282
0.8369
0.1498
0.3828
0.6493
0.8497
0.1268
0.2969
0.5155
0.7216
0.0355
0.1071
0.2375
0.5155
0.7216
0.0332
0.0905
0.1935
0.3407
0.5118
                            -32-

-------
5 or more, the procedure will call for rejecting p=0.90  and

accepting p=0.7.

     The decision on sample size N and the critical number

of split-sample disagreements d should be arrived at by  tak-

ing into consideration the cost of testing, the acceptable

level (p) of agreement, the false positive rate, and the

false negative rate for the appropriate alternate hypothe-

sis.

     The usual statistical practice is to choose a critical

number for which (1)  the false positive rate does not exceed

0.05 or  .10 and the false negative rate does not exceed 0.20

and (2)  the false negative rate is the smallest among all

those satisfying criterion 1.

     Using this guideline, if the null hypothesis is p=0.9

and the alternate hypothesis is p=0.7, and the split-sample

size is 25, then choosing 6 for the critical number of

split-sample disagreements is satisfactory.  Similarly, the

choice of 2, 4, 5,  and 6 for samples of size 5, 10,  15, and

20 respectively is  satisfactory.

     Accordingly, the following table provides the suggested

critical number of  split-sample disagreements for different

sample sizes.
                                   Critical Number
                                   of Split-Sample
     No. of Split-Samples          Disagreements
              5                           2
           6 to 8                         3
           9 to 14                         4
          15 to 20                         5
          21 to 25                         6
                           -33-

-------
     D.   Initial Laboratory Quality Assurance



          The quality assurance procedure given in Appendix



B of the Guidance Document recommends a sample of size 25



for initial quality assurance except in the case of a school



or school system where a very small number of samples will



be analyzed  (i.e., less than 25 samples).  With 25 split-



samples the critical number of split-sample disagreements is



6.  That is,  if 6 or more disagreements are noted in an  ana-



lysis  of 25  split samples, the procedure used in the labora-



tory Cs)_ involved is unacceptable.  The laboratory should be



rejected if  there are no satisfactory explanations for the



large  number  of disagreements.



     A minimum sample size of 5 split-samples is recommended



initially.   Notice  from Table 6 that  in this case the prob-



ability of  incorrectly accepting  the  null hypothesis is  high



 (.0.5282).   This suggests that it  would be better for schools



to combine  their efforts for determining the initial quality



of the laboratory Cs) .



     E.   Procedure  for Monitoring on an On-Going Basis



          The recommended  procedure  (for school  systems  with



more than  100 samples) is  to decide  on the  basis of  the  re-



 sults  of  sets of  20  split-samples (see Figure  3.3 of the



 Guidance  Document).   If  5  or more disagreements  are  noted  in



 a set  of  20 split-samples,  the  recommendation  is to  inves-




 tigate the  laboratory procedure.   The false positive rate



 and false negative  rate  for this  procedure  (under  the  assump-



 tions) are respectively  0.0432  and 0.2375  (see Table 6).






                            -34-

-------
     If the null hypothesis is true, then this procedure

will call for checking the laboratory incorrectly  in  1  out
                                               ^
of every 23 split-sample sets of size 20  (1/23 = 0.04); on

the other hand, if the alternative hypothesis is really true

it will call for checking the lab  (correctly!!) in 3  out  of

every 4 sets of split-samples of size 20  (3/4 = 1  -.2375).

     In the case of school systems with a limited number  of

samples to be analyzed (.less than 100) , the procedure recom-

mended is to decide on the basis of results of 5 split-sam-

ples (see Figures 3.1 and 3.2 of the Guidance Document).  If

2 or more disagreements are noted in a set of 5 split-sam-

ples, the recommendation is to investigate the laboratory

procedure.  Note that if the null hypothesis is true, this

procedure will call for checking the laboratory incorrectly

in 1 out of 12 split-samples of size 5 (1/12 = .0815); on

the other hand, if the alternative hypothesis is really true

it will call for checking the lab (correctly)  in 1 out of
                                                  s^
every 2 or 3 sets of split-samples of size 5 (1/2 = 1 -.5282)

     The sampling rate for monitoring on an on-going basis

for school systems with more than 100 samples is 1 in 5

split-samples initially up to 100 samples and then at a re-

duced rate of 1 in every 10 samples thereafter (see Figure

3.3 of the Guidance Document).

     F.   A Central Administrative Structure to Monitor Lab
          Quality

          The quality assurance procedure as is proposed

calls for every school or school system to monitor labora-
                           -35-

-------
tory quality.  This- results in an enormous multiplication of



quality assurance efforts.  The effort in terms of the over-



all number of split-samples required to be analyzed can be



reduced considerably if the quality assurance efforts are



centralized.  As an example, consider a group of six schools



with expected number of samples of 10, 20, 30, 40, 50, and



60.  If each school requires independent initial and on-



going quality assurance procedures, then 151 of the 210 sam-



ples should be  split-samples.  If initial quality assurance



is  not required at all, then 46 of the 210 samples should be



split-samples.   However,  if the group combines their quality



assurance  program, then with initial quality assurance the



group will include 51  split-samples  (compared 151) and with-



out it they will include  only  21  (compared to 46) split-



samples.



     The  increased efficiency  achieved by centralizing the



quality assurance program would result in a  significant sav-



ings in the  overall  cost  of the asbestos analytical program



for a  school  district.  It would  require, however, the com-



mitment or designation of an  individual with overall respon-




sibility  for the program.
                             -36-

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                      REFERENCES
Lucas D, Hartwell T, Rao AV.  Research Triangle Institute,
     1980.  Background Document.  Asbestos-Containing
     Materials in School Buildings:  Guidance for Asbestos
     Analytical Programs.  Draft Report.  Washington, DC:
     Office of Pesticides and To-xic Substances, U.S. Envi-
     ronmental Protection Agency-  Contract no. 68-01-5848.

Patton JL, Price BP, Ogden JS.   Battelle Columbus Labora-
     tories.  1980.  Asbestos in Schools:  Bulk Sampling
     Variability Study.  Draft Report.  Washington, DC:
     Office of Pesticides and Toxic Substances, U.S. En-
     vironmental Protection Agency.  Contract no. 68-01-
     3858.

Raj D. 1968.  Sampling Theory.   New York:  McGraw-Hill Book
     Company -

Rao AV, Myers LE, Lentzen DE, Hartwell TD.   Research Tri-
     angle Institute.  1980.  Analysis of Battelle Bulk
     Asbestos Duplicate Samples.  Draft Report.  Washington,
     DC:  Office of Pesticides  and Toxic Substances, U.S.
     Environmental Protection Agency.   Contract no. 68-01-
     5848.

USEPA. 1979.  U.S. Environmental Protection Agency-  Office
     of Toxic Substances.  Asbestos-Containing Materials in
     School Buildings:   A Guidance Document,  Part 1.

USEPA.  1980.   U.S. Environmental Protection Agency.  Office
     of Toxic Substances.  Asbestos-Containing Materials in
     School Buildings:   Guidance for Asbestos Analytical
     Programs.  Washington, DC:   Office of  Toxic Substances,
     USEPA.  EPA 560/13-80-017A.
                           -37-

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-38-

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         APPENDIX A




REVISIONS IN SECTION 2.3.1
               -39-

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-40-

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A.   REVISIONS IN SECTION II.C.I



     Estimation of the variability to be expected among  the



observations of asbestos concentration is described  in Sec-



tion II.C.I.  The variance estimation differs from that  pre-



sented in an early draft of this document (Lucas et  al.



1980).   The following two changes have been made:  (1) Data



from (Patton et al. 1980) are used, in addition to data  from



(Rao, Myers et al. 1980)  , to estimate the variance compon-



ents.   (2) A variance component attributable to variability



among laboratories is not included in the variance model.



This appendix discusses the reasons for these changes and



their implications.



     In the early draft of this document, data from the Bat-



telle Duplicate Analysis Study, described in (Rao, Myers et



al. 1980), were used to estimate measurement variability


  2                                             2
(a )  and variability due to sampling location (a ).   After
  6                                             C


preparation of the early draft, data from the Battelle Bulk



Sampling Variability Study (Patton et al. 1980)  became



available.  This study, designed to estimate measurement



variability and variability due to sampling location, gives



more extensive information than the Duplicate Analysis



Study.   The results are used in the calculations of Section



II.C.I to give better variance component estimates.



     The variance component model in the early draft includ-



ed a component attributable to variability among laborator-



ies.   This component was estimated using data made available
                           -41-

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to EPA by the Bureau of Mines  (Rao, Hartwell et al. 1980).



For most Sampling Areas, however,  it is thought that all



bulk samples will be sent to the same laboratory  (with the



exception of the quality assurance procedures described in



Chapter III of this document).  When only one laboratory  is



involved, it is not appropriate to include  a variance com-



ponent attributable to variability among laboratories.



Thus, this component was omitted from the model,  and the


                             2      2
revised model includes only a  and a  .
                             c      e


     Tables A-l and A-2 are power  tables from  (Lucas et al.



1980).  The calculations were  based on data from  (Rao,



Hartwell et al. 1980) and  (Rao, Myers et al. 1980), and the



variance model  included a  laboratory variability  term.  The



size  of the test,  defined  in Section  II.C.3, was  set equal



to .10.   (For Tables  4  and  5 in Chapter II, the size was  set



equal to  .05.)  Under the  assumptions of Table A-l (smaller



laboratory variability), power of  .913  is  expected when  3



bulk  samples  are  collected and actually y  = .05.   In other



words, with probability .913 the  test will  correctly conclude



that  asbestos  is  present,  when 3  bulk  samples  are collected



and y =  .05.   Under the assumptions  of  Table A-2  (larger



laboratory variability), power of .914  is  expected when  3



bulk  samples  are  collected and y  = .15.   Power of .952  is



expected when 5 bulk samples  are  collected and y  = .10.



      Tables A-l and A-2 can be compared with Tables  4  and 5



 in Chapter  II of  this document.   Under  the assumptions  of
                            -42-

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Table A-l.  Power of the Test H  :   y  =  y   =  .01 versus

            H, :  y = y, >  .01 for  Selected Values of y-,

             (assuming  smaller laboratory  variability and
            the variance model from (Lucas et  al. 1980))

X
n \
3
5
7
10
12
15
17
20
30
50


X1
3
5
7
10
12
15
.015

.379
.441
.493
.558
.595
.644
.673
.712
.810
.916


.050
.913
.968
.988
.997
.999
>.999
.020

.571
.662
.730
.805
.842
.884
.905
.930
.975
.997 >


.060 .075
.931 .946
.977 .984
.992 .995
.998 .999
.999 >.999
>.999
.025

-682
.776
.839
.901
.928
.995
.967
.979
.995
.999


.100
.991
.999
>.999



.030

.748
.838
.893
.942
.961
.978
.985
.992
.999
>-999







.040

.819
.898
.941
.973
.984
.993
.995
.998
>.999








The above table is taken from  (Lucas et al. 1980).  The
entries were calculated according to

     Power of the test H :  y = y  = .01 versus
                        o        o


     = 1 - $ { (.01 - y, + 1.282 a  //n)/a  //n} ,
                                 yo      yl
     where a   = y±  V.H25 + k2 ,  i = 0, 1,
            yi                 yi

     k   = 1.00 for y± < .05, k   = 0.75 for  .05 <_ y .  <  .10,
     k   = 0.50 for .10 <_ y.  < .20, k   = 0.25 for y.  >_  .20,
      yj_                   !         y^_             i  —

     and ${z> is the cumulative distribution function  of a
     standard normal random variable.
                           -43-

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Table A-2.  Power of  the  Test H :   y = y  = .01 versus
                                o        o


HL: y = •
(assuming
y-L > -01
larger
for Selected Values of y.
laboratory
variance model from (Lucas
variability
et al. 1980) )
and the


nX
3
5
7
10
12
15
17
20
30
50
.015
.284
.313
.337
.369
.388
.414
.430
.452
.518
.621
020
415
464
505
555
584
623
646
678
761
865
.025
.500
.559
.606
.664
.696
.737
.760
.791
.866
.943
.030
.556
.621
.671
.730
.762
.801
.823
.851
.915
.971
040
626
694
745
802
832
867
886
909
956
989


nX
3
5
7
10
12
15
17
20
30
50
.050
.713
.795
.851
.904
.928
.953
.965
.977
.994
>.999 >.
075
774
850
897
940
958
975
982
989
998
999
.100
.891
.952
.978
.993
.997
.999
.999
>.999


.150
.914
.965 >.
.985
.996
.998
.999
>.999



200
994
999








 The above table is taken from  (.Lucas et al. 1980).  The
 entries were calculated according to
      Power of the test H :  y = yQ = .01 versus
      = 1 - $ {(..01 - u1 + 1.282 a  //n)/a  //n},


      where a   = yi  J.1125 + 4k2  , i = 0, 1,
              i                   i

      k   = 1.00 for U:L <  .05, k   = 0.75 for  .05  <_  Vi  <  .10,
       ui                       yi

      k   = 0.50 for .10 <_ yi < .20,  k   = 0.25 for y.^ >_ .20,

      and ${z} is the cumulative distribution  function  of a
      standard normal random variable.

                            -44-

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Table 4, power of .915 is expected when 3 bulk samples are



collected and p = .08.  Power of .922 is expected when 5



bulk samples are collected and p = .06.  From Table 5, for



5 bulk samples and p = .09, power of .929 is expected.  For



7 bulk samples and p = .07, power of .906 is expected.



     It can be seen from comparing these tables that the re-



sults of the power calculations using the revised variance



estimation are not identical to the results in the early



draft.  These differences are not thought to be appreciable



in light of the qualifying assumptions made Ce.g., unbiased



laboratory measurement, variance estimation based on limited



data) and the uncertainty as to alternatives (i.e.,  p,)  of



interest.  The revised variance estimation does not merit



any change in the recommended number of bulk samples to



collect.
                              -45-

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                         REFERENCES
                         (Appendix A)
Lucas D, Hartwell T, Rao AV.  Research Triangle Institute.
     1980.  Background Document.  Asbestos-Containing Ma-
     terials in School Buildings:  Guidance for Asbestos
     Analytical Programs.  Draft Report.  Washington, DC:
     Office of Pesticides and Toxic Substances, U.S. En-
     vironmental Protection Agency-  Contract no. 68-01-
     5848.

Patton JL, Price BP, Ogden JS.  Battelle Columbus Labora-
     tories. 1980.  Asbestos in Schools:  Bulk Sampling
     Variability Study.  Draft Report.  Washington, DC:
     Office of Pesticides and Toxic Substances, U.S. En-
     vironmental Protection Agency.  Contract no. 68^01-
     3858.

Rao AV, Hartwell TD, Myers LE, Lentzen DE, Breen J, Campbell
     W, Breeden CH, Gustafson NF.  Research Triangle Insti-
     tute.  1980.  Analysis of the Data from the Roundrobin
     Evaluation of Miscroscopic Procedures for Identifica-
     tion  and Quantification of Asbestos in Sprayed Con-
     struction Materials.  Washington, DC:  Office of Pest-
     icides and Toxic  Substances, U.S. Environmental Pro-
     tection Agency.   Contract no. 68-01-5848.

Rao AV, Myers LE, Lentzen DE, Hartwell TD.  Research Tri-
     angle Institute.  1980.  Analysis of Battelle Bulk
     Asbestos Duplicate  Samples.  Draft Report.  Washington,
     DC:   Office of Pesticides and Toxic Substances, U.S.
     Environmental  Protection Agency.  Contract no. 68-01-
     5848.
                               -46-

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