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
-------
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
-------
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-
-------
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-
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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-
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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-
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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-
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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-
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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-
-------
-38-
-------
APPENDIX A
REVISIONS IN SECTION 2.3.1
-39-
-------
-40-
-------
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-
-------
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-
-------
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.
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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-
-------
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-
-------
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.
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