United States Toxic Substances EPA-560/5-81 -006
Environmental Protection Washington DC 20460 July 1981
Agency
Toxic Substances
&EPA Airborne Asbestos Levels in
Schools: Design Study
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EPA Report Number 560/5-81-006
July 1981
AIRBORNE ASBESTOS LEVELS IN SCHOOLS: DESIGN STUDY
by
B. Price
Battelle Memorial Institute
Columbus, Ohio 43201
and
D. Watts
E. Logue
T. Hartwell
Research Triangle Institute
Research Triangle Park, North Carolina 27709
EPA Contract Number 68-01-5848
RTI Project Number 1864-23
EPA Task Manager: Cindy Stroup
EPA Project Officer: Joe Carra
Prepared for:
Design and Development Branch
Exposure Evaluation Division
Office of Pesticides and Toxic Substances
U.S. Environmental Protection Agency
Washington, D.C.
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DISCLAIMER
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, subcontrac-
tors, or their employees makes any warranty, expressed of im-
plied, or assumes any legal liability or responsibility for
any third party's use or the results of such use of any in-
formation, 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 sig-
nify that the contents necessarily reflect the joint or sep-
arate views and policies of each sponsoring agency. Mention
of trade names or commercial products does not constitute en-
dorsement or recommendation for use.
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iii
TABLE OF CONTENTS
Page
I. EXECUTIVE SUMMARY 1
II. INTRODUCTION/BACKGROUND 5
III. THE VALIDATION PROBLEM 9
A. Validation: Conceptual Background 9
1. Role of Air Sampling 10
2. Framework 11
B. Validation: Specific 14
1. Description of the Data 18
2. Validation for SA 19
3. Validation for DTA 20
IV. SAMPLE SIZE AND PRECISION 23
A. Sources of Variation 23
B. Sample Size: SA: Controlled Experiment... 25
1. SA Formulation 1 25
2. SA Formulation 1: Summary 34
3. SA Formulation 2 36
4. Comparison: Formulation 1 versus
Formulation 2 45
C. Sample Size: Decision Tree Algorithm
(DTA) : Controlled Experiment 48
D. Sample Size: Measurement Error 51
1. SA Formulation 2 52
2. DTA 53
V. DESCRIPTION OF AVAILABLE SITES IN MONTGOMERY
COUNTY AND NEW YORK CITY 59
A. The Montgomery County Data 59
B. The New York City Data 59
VI. SAMPLE DESIGN 65
A. Introduction 65
B. Study Area 65
C. Overview of the Sample Design 66
D. Construction and Stratification of the
First-Stage Frame 67
E. Allocation and Selection of the First-
Stage Sample 71
F. Construction and Stratification of the
Second-Stage Frame.. 74
G. Allocation and Selection of the Second-
Stage Sample 76
H. Data Collection at Sample Sites 77
I. Design Effect 78
J. Sampling Strategy 82
REFERENCES 84
APPENDIX A: The Noncentral t Distribution
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iv
LIST OF TABLES
Number Title Page
1 Factor Weights for SA Score 13
2 Site Description for DTA 16
3 Required Sample Size (n) for Testing
V V10 = yB vs« Hl: "10 " yB = Ka
(n sites in each group) 28
4 Probability That YIQ <_ YB When Testing
HQ: y1Q = yB versus H^ y1Q - UB = Ka . . . 30
5 Raw Meansurement Units (ng/m ) For The
Alternative Hypothesis When Testing
HQ: y1Q = UB versus H^ y1Q - ufi = Ka . . . 32
6 Effect of Increased Standard Deviation
On Raw Measurement Units (ng/m ) For
The Alternative Hypothesis When Testing
HQ: VIQ - PB versus H.^ y1Q - yfi = Ka . . . 33
7 Summary of Power (%) For Testing
V y10 ~ yB versus =1* ^10 ~ WB = Ka '•• 35
8 Full Factorial Sampling Plan ............ 39
9 Full-Factorial Sampling Plan, Average
Factor Scores ........................... 40
10 Formulation 2 : Full-Factorial Design
HQ: B-j^ = 0; H-j^: B-j^ = Ka/A , n = 32,
Degrees of Freedom = 30, £g = 138.8;
Body of Tables Gives Power of the Test
(%) ..................................... 41
11 Formulation 2: Half-Fraction Design
H0: 6 = 0; H: B = Ka/A' n =
Degrees of Freedom = 14, g = 98.2;
Body of Table Gives Power of the Test
12 Formulation 2: Optimal Design
Hn: B, = 0; H,: Bn = Ka/30 (i.e.,
42
1/9
A=30), 2-s = 49.9 x n ' Body of the
Table Gives Power of the Test (%} ....... 44
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List of Tables (cont'd)
Number Title Page
13 Comparison: Formulation 1 Versus Formu-
lation 2. a = .05, A = 30; Body of
Table Presents Power (%) 47
14 Statistical Test: DTA; HQ: PA = UDA;
H,: UA - PDA = Ka» Body of Table Presents
Power of Test (%) 50
15 DTA: HQ: nA = HDA; EI : t»A - HDA = (l-2p)Ka
Body of the Table Gives Power of Degrada-
tion When Sites Are Misclassified.
Significance Level = .05 56
16 Distribution of Montgomery County Sites .. 60
17 Distribution of New York Sites 62
18 Asbestos Content of New York City Public
Schools 69
19 Distribution of Asbestos-Containing Public
Schools in New York City with Respect to
Asbestos Content, Friability, Condition,
Exposure, and Accessibility 70
20 First-Stage Strata 72
21 Distribution of New York City Public
Schools by Selected Size Categories 73
22 Second-Stage Strata 75
23 The Clustering Effect Corresponding to
Selected Values of p and n, 80
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LIST OF FIGURES
Number
1
2
3
4
Title
Relationship
Decision Tree
Hypothetical
Some Effects
Between Air Levels
Algorithm (DTA) . ,
Air Levels for DTA
and SA
Groups
of an Erroneous Linear
Pag
12
15
17
46
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Vll
ACKNOWLEDGMENTS
The authors gratefully acknowledge the helpful sugges-
tions of Cindy Stroup, Joe Breen and Larry Longanecker of
the U.S. Environmental Protection Agency, Washington, D.C.,
and Steve Williams of the Research Triangle Instititue, Re-
search Triangle Park, North Carolina.
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I. EXECUTIVE SUMMARY
A field study has been proposed to collect data in
schools that are to be used to analyze and validate two as-
bestos exposure assessment algorithms as compared to levels
of airborne asbestos. This field study would involve algo-
rithm scoring (including bulk asbestos sampling) and air
sampling in sites (e.g., classrooms) within selected schools.
The objective of the planning study described in this report
is to establish the characteristics of various alternative
statistical designs (e.g., number and characteristics of
sample sites) for the proposed field study and to recommend
the most appropriate design.
The report is intended to provide EPA with an assess-
ment of precision and completeness that can be expected from
the data collected in the field study.
The approach is to formulate various operational defini-
tions of validation and to develop the statistical character-
istics of specific designs with respect to these definitions.
In order to compare designs, it is necessary to incorporate
assumptions that simplify the validation problem. Although
the problem is simplified, the comparisons of statistical
designs are still informative. The major results of the
investigation follow.
(1) Using information based on data that are currently
available, it has been determined that between 32
and 64 sites are required in the field study in
order to meet most reasonable statistical object-
ives. These sites would be selected in approxi-
mately 16 schools.
(2) The basic 32 sites and 16 schools should be se-
lected according to the two-stage sampling design
with stratification imposed at each stage as des-
cribed in the report. The two-stage sampling pro-
cedure would base selections of sites and schools
on the scores of the factors that make up the as-
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sessment algorithms (e.g., asbestos content, fria-
bility, etc.). The first stage of sampling would
select schools and the second stage would select
the sites within the schools.
(3) Based on currently available asbestos assessment
algorithm data, New York City schools are recom-
mended as the location for the field study if
local cooperation is forthcoming.
(4) It is recommended that the initial 3 or 4 sites
selected for air sampling for the proposed study
have algorithm scores that are at the high and low
ends of the algorithm "scoring scale (for example,
two sites with high asbestos, friability, expo-
sure, accessibility, and bad condition; and two
sites with low asbestos, friability, exposure, ac-
cessibility, and good condition). Air sampling
results from these initial sites would then be
compared (as well as results from background
samples at each site) to determine if clear dif-
ferences are evident. This analysis could be used
to determine if additional air sampling is war-
ranted.
(5) The variabilities found at a fixed site in expos-
ure assessment algorithm scoring and in airborne
asbestos concentration levels are the key factors
in determining sample size. These parameters
should be monitoried in the early stages of the
field study to determine that they have values
that are consistent with the values used to de-
velop the sampling plan. A sample size adjust-
ment upward from 32 to 64 sites may be necessary
if the variability encountered is larger than
expected.
Specific rules for selecting sites and collecting mea-
surements are presented in the body of the report. In addi-
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tion, a description is presented of the population of poten-
tial schools and sites in two school districts (Montgomery
County, Ohio, and New York City) where algorithm data are
currently available.
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II. INTRODUCTION/BACKGROUND
In March 1979, EPA initiated a rulemaking proceeding
under the Toxic Substances Control Act (TSCA) regarding
friable asbestos-containing materials in schools, (44 FR
177790). In September 1979, an Advance Notice of Proposed
Rulemaking (ANPRM) was published (44 FR 54676) that dis-
cussed EPA's plan for rulemaking. EPA's plan includes "(1)
requiring surveys of schools to determine whether they con-
tain friable asbestos-containing materials, requiring that
an exposure assessment be performed for all such materials
identified, and requiring that friable asbestos-containing
materials be marked; (2) requiring corrective actions with
respect to friable asbestos-containing materials for which
the exposure assessment exceeds a level determined by EPA as
presenting an unreasonable risk,' and (3) requiring periodic
reevaluation of the friable asbestos-containing materials to
determine whether the exposure assessment is still valid or
whether additional corrective action is required under the
regulation."
Upon confirmation that friable material is present and
that the material contains at least 1 percent asbestos, an
exposure assessment would be required. A numerical based
inspection scheme (also referred to as algorithm) has been
proposed to serve as the exposure assessment tool. The al-
gorithm value is determined from scores assigned to eight
factors—condition, water damage, exposure, accessibility,
activity, presence of an air plenum, percent asbestos con-
tent, and friability. The factor scores are summarized to
form an exposure number which is compared to a preestablish-
ed numerical scale that indicates what correction or control,
if any, is necessary.
Two forms of summarization have been proposed—Sawyer's
Algorithm (SA) and the Decision Tree Algorithm (DTA). The
usefulness of either of these summarizations in rulemaking
has been an ongoing concern to EPA. Over the past 2 years,
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there have been a number of investigations directed at vali-
dating various algorithms (see Patton et al. 1980; Price and
Townley 1980a; Logue and Hartwell 1981; EPA 1980a). The
experience gained has resulted in the design of a field
study using air sampling to validate SA and DTA (Price et al.
1980b). The recommended field program for collecting data
to validate these algorithms consists of a set of activities
that must be carefully controlled if the resulting data are
to be informative. Care must be taken to insure (1) that
the technical and logistical problems associated with field
air monitoring are satisfactorily solved, and (2) that the
statistical sampling plan provides appropriate levels of
precision to support the types of regulatory decisions that
are being considered.
The research presented in this report addresses the
statistical issue—Item (2) above. Typically, not enough
effort is devoted to understanding the types of precision
statements that are defensible and consistent with a pro-
posed sampling plan. There is often a "gap" between the
precision and completeness of the information expected by
the regulator, and the precision and completeness that can
be validly associated with the data and the statistical
analysis. The consequences of not closing this "gap" are
(1) investments in research that are large, but not large
enough to provide a usable research result, and (2) invest-
ments that should not have been made because there was no
reasonable agreement on the level of precision to be expect-
ed and therefore no chance that the research objective would
be met. The goal of the current planning study is to close
the "expectations gap" in order to provide EPA with an oppor-
tunity to reevaluate the resources required to meet the re-
sources required to meet the objectives of the suggested
validation study.
This report includes an evaluation of alternative sta-
tistical plans that have been suggested for the field study.
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Operational definitions of "validity" are discussed. The
two measurement processes—air sampling and algorithm scor-
ing—are characterized in terms of sources of variability.
The relationship between sample size and precision is de-
veloped for the statistical sampling plans that have been
proposed.
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III. THE VALIDATION PROBLEM
EPA's objective is to select a numerical based visual
inspection scheme to serve as an exposure assessment tool.
One important aspect of the selection process is validation
—that is, documenting that the assessment tool operates as
intended. At this time, there are two proposed tools for
exposure assessment: Sawyer's Algorithm (SA) and the Deci-
sion Tree Algorithm (DTA). Operational definitions of vali-
dity are closely linked to the particular tool that is being
validated. In this section of the report, we present alter-
native definitions of validity that are used later in the
analysis of sample size, statistical design, and precision.
A. Validation; Conceptual Background
The primary concern in assessing exposure to as-
bestos in schools is to be able to initiate corrective
action in those situations that present an unreasonable risk
to the health of persons who use school buildings. The
health effects associated with exposure to asbestos fibers
have been analyzed using epidemiologic and experimental
data. It has been shown that exposure to asbestos via
inhalation increases the risk of numerous diseases including
asbestiosis pleural and peritoneal mesothelioma, and cancers
of the lung and other organs (45 FR 61966, Technical Support
Document for Regulatory Action Against Friable Asbestos-
Containing Materials in School Buildings, USEPA, August
1980). As formulated, validation of the proposed exposure
assessment methods involves two relationships:
(1) the relationship of the risk of adverse health ef-
fects to dose levels of airborne asbestos fibers;
and
(2) the relationship between levels of airborne asbes-
tos fibers and scores on the specific exposure as-
sessment algorithm under consideration.
The scope of the current report is restricted to validation
of the second relationship. Validity of the dose-response
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re lationship is assumed. (See 45 FR 61969 and the associat-
ed list of support documents, 45 FR 61980-61986.)
1. Role of Air Sampling
It has been suggested that exposure assess-
ment should be directly based on measurements of airborne
asbestos concentration levels rather than using a proxy such
as a visual inspection scoring algorithm. Direct measure-
ment would be accomplished by air sampling, and the air
sampling approach has been considered. However, "EPA has
determined that air sampling is an inappropriate test to
determine if an asbestos exposure problem exists in school
buildings" (45 FR 61978). There are at least four objec-
tions. First, air sampling measures the concentration of
airborne asbestos fibers only at the time the sample is
taken. However, airborne concentration levels are the re-
sult of an episodic process more than a continuous process.
Air sampling over relatively short time periods (as would be
necessary in an assessment program) could easily lead to
misclassification of a potentially hazardous site. Second,
the exposure assessment is supposed to identify not only
currently hazardous sites, but it is also supposed to pro-
vide information about,sites that may become hazardous in
the future. Air sampling is capable of identifying a cur-
rent problem only. Third, it is not feasible to use air
sampling in an extensive assessment program because of
seemingly insurmountable technical and logistical problems.
To be effective, air sampling would have to be conducted for
periods of at least one week at sites where normal activity
was in progress. Difficulties of implementation include the
nuisance factor of sampling equipment operating in a school
environment, problems in establishing background asbestos
levels, and assuring the integrity of the samples collected
in an uncontrolled field setting. Finally, the resources
necessary for the collection and analysis of scientifically
valid air samples are not available. Although these diffi-
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culties render air sampling inappropriate for widespread use
as an exposure assessment tool, air sampling may be used as
a basis for validating the proposed assessment methods.
For the implementation of the validation study, it has
been agreed that airborne asbestos data will only be used to
validate the algorithm factors that are indicators of current
asbestos levels. Validation of those factors that are main-
ly predictors of future health risk must be established by
other means. The validation study is designed to collect
air data over time intervals that are long enough to capture
the effects of an episodic process. Air sampling will be
conducted for one week during regular school hours so that
regularly occurring activity is represented. These long
sampling periods are acceptable for the validation study
because the overall scope of sampling is limited. The re-
lated technical and logistical problems are manageable be-
cause the validation•field program is small relative to the
field program that would be necessary if air sampling were
used as the exposure assessment method. Also, other poten-
tial problems such as the nuisance factors associated with
classroom sampling and assuring the integrity of samples may
be costly to solve, but they are manageable because the val-
idation effort is well defined, and appropriate controls can
be imposed.
2. Framework
As mentioned above, our concern is the valid-
ity of the decision rules related to taking a corrective
action or deferring action to a later time. We now describe
the framework underlying the decision rules for both SA and
DTA.
When considering SA, the decision rule is derived from
the relationship between air levels of asbestos and SA
scores. Conceptually, the relationship takes the form shown
in Figure 1. As SA increases, airborne asbestos concentra-
tion increases (see Table 1 which shows how SA is computed).
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Airborne
Asbestos
ng/m
n
(U
-p
(0
s
o
c to
tJ»-H
•H OS
01
c
0 Deferred S
Action
Action
SA
Figure 1. Relationship between air levels and SA scores,
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Table 1. Factor Weights for SA Score
Weighted
Factor Scores
1. Condition (Fl)
No Damage ." • 0
Moderate Damage 2
Severe Damage 5
2. Accessibility (F2)
Not Accessible 0
Rarely Accessible 1
Accessible 3
3. Part of Air Moving System (F3)
No 0
Yes 1
4. Exposure (F4)
Material is not exposed 0
10 percent or less of the material is exposed.. 1
Greater than 10 percent of the material is
exposed 4
5. Water Damage (F5)
No water damage 0
Minor water damage 1
Moderate or major water damage 2
6. Activity or Movement (F6)
None or low activity level 0
Moderate activity level 1
High activity level 2
7. Friability (F7)
Not friable 0
Low friability 1
Moderate friability 2
High friability 3
8. Percentage Asbestos (F8)
Less than or equal to 1 percent 0
Greater than 1 percent and less than or
equal to 50 percent 2
Greater than 50 percent 3
SA SCORE = (F-j^ + F2 + F3 + F4 + FS + Fg) x F_ x Fg
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In the figure, A represents the airborne asbestos level
that defines the cutoff between safe and unsafe levels. A
in turn determines S on the SA scale which represents the
cutoff point defining corrective action versus the deferral
of corrective action. It is assumed that the value of A
has been determined from the dose-response analysis relating
*
airborne asbestos levels to health risk. Validation of SA
is based on the precision associated with the establishment
of the curve in Figure 1.
When considering DTA (Decision Tree Algorithm), the de-
cision rule for action or deferred action arises from the
classification of sites into one of two groups. For conven-
ience, the groups are named Action (Group A) and Deferred
Action (Group DA). A site is classified into Group A or
Group DA according to the tree shown in Figure 2. Equiva-
lently, there are 16 types of sites as shown in Table 2.
Five sites belong to Group A; eleven sites belong to Group
DA. From a conceptual perspective, there is a distribution
of airbone asbestos concentrations corresponding to each
group (see Figure 3). Validation of DTA involves the com-
parison of the characteristics of these distributions.
B. Validation; Specific
In the previous paragraphs, the conceptual frame-
work for a validation analysis was developed. Before enter-
taining any questions concerning sampling design, sample
size, and precision, it is necessary to formulate the vali-
dation problem in precise, quantitative terms. A few al-
ternative formulations will be considered. Each formulation
is specific and therefore each one appears to be limited in
scope. However, it is necessary to focus on limited objec-
tives if the ensuing discussion of sample size and precision
is to be meaningful. The exercise is constructive in that
* As formulated, the validation analysis does not involve
any specific values of A .
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Action « Yes
Is the Material
Normally Exposed (F4)?
Yes (4)
Is the Material Badly
Damaged (Fl)?
No (0,2)
Is the Material
Accessible (F4)?
Yes (3)
Action
.Yes (2,3)
Is the Material Highly
Friable (F7)?
No (0,1).* Deferred
Action
No (0,1)^
Deferred
Action
No (0,1).
Deferred
Action
* The numbers in parentheses are factor weights (see Table 1)
NOTE: The decision tree is applied to sites with more than
1 percent asbestos in bulk sample.
Figure 2. Decision tree algorithm (DTA).
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Table 2. Site Description for DTAe
Site
ID
Factors
Friability |
Condition |
Exposure
Accessibility
Group
2
3
5
8
9
12
14
15
17
20
22
23
26
27
29
32
Low
Low
Low
Low
High
High
High
High
Low
Low
Low
Low
High
High
High
High
Good
Good
Bad
Bad
Good
Good
Bad
Bad
Good
Good
Bad
Bad
Good
Good
Bad
Bad
Low
High
Low
High
Low
High
Low
High
Low
High
Low
High
LOW
High
Low
High
High
Low
Low
High
Low
High
High
Low
Low
High
High
Low
Low
Low
LOW
High
DA
DA
DA
A
DA
A
DA
A
DA
DA
DA
A
DA
DA
DA
A
Decison tree is applied to sites with more than 1 percent asbestos
in bulk sample.
Friability (low) = 0 or 1 (see Table 1 for codes)
Condition (good) = 0 or 2
Exposure (low) = 0 or 1
Accessibility (high) = 3
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AIR
Group DA
Figure 3. Hypothetical air levels for DTA groups,
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it focuses attention on a precise formulation of the valida-
tion problem and allows for sensitivity analysis regarding
sample size requirements.
1. Description of the Data
The first step in establishing a precise for-
mulation of the validation problem is to characterize the
type of data that will be collected. One set of assumptions
about the data will serve each formulation of the validation
problem whether it involves SA or DTA. The characterization
follows.
It is planned that air sampling will be conducted at a
flow rate of 5 liters/min., using 47mm Millipore filters
mounted at a breathing height of 1.5 meters above the floor
(see Price et al. 1980b). The samples will be analyzed by
transmission electron microscopy (TEM). The basic measure-
ment will be obtained by sampling for a full week of 40
hours when school activities are in progress.
Let us denote airborne asbestos concentration by X,
measured in units of nanograms per cubic meter (ng/m ).
For a given site, let XB and Xg denote the background levels
and the concentration levels at the site, respectively.
Background measurements will be taken near the site (e.g.,
outside the school). The exact location for background
sampling will depend on the structure of the building and
the location of potential sources of background asbestos.
We shall use the conservative assumption that X_ and Xg are
statistically independent and that Y_ = ln(X_) and Yg =
ln(X0) are normally distributed with expected value y_ (y0)
222
and common variance OB (ag = a ). The increment due to site
is Z = Y_ - Y_ which is normally distributed with the ex-
22
pected value ne = Uo - yn and variance a = 2a_.
O O~ B O
The variance, a , is a composite of various components
including analytical error, site variation, and sampling in-
2
terval variation. As presented, a , represents total vari-
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ability. The roles of individual components of variability
are discussed later in the report.
2. Validation for SA
We consider two formulations of the valida-
tion problem for SA.
Formulation 1;
Suppose that values of SA equal to 10 and 40 are used
as decision cutoff points. In the discussion below, these
two values are used to state specific statistical hypotheses
to be used for validation. It is clear that the formulation
applies as well to any two selected values of SA.
We expand the notation slightly to be able to indicate
the level of SA at which the air data were collected. Let S
take on the value of SA at the site under consideration.
Consider two statistical hypotheses:
(a) HQ: y1(J = yB
H0: U40
Hypothesis (a) tests whether or not measured asbestos
concentration levels at sites where SA is 10 can be distin-
guished from background levels. Hypothesis (b) tests for
differences in airborne asbestos levels between sites where
SA is 40 and 10. This formulation of a validity test repre-
sents the minimum that would be required of the relationship
between airborne asbestos and SA if decisions are to be
based on scores of 10 and 40.
Formulation 2 ;
Assume that the relationship between Z and SA can be
approximated by the model
Z = BQ + B.^ SA + e
where e has expected value equal to zero and standard devi-
ation, a. Test the hypothesis
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This formulation of the validation test is an attempt to
establish validity at every value of SA and includes hypoth
eses such as H_: y.Q = y... as a special case.
Comparison of these formulations regarding sample size
and precision is found in a later section of this report.
3. Validation For DTA
The probability distribution and the parame-
ters used to characterize the data for validating DTA are
basically the same as used for SA. However, since the
scores on SA are not part of the DTA scheme, a slightly
different notation must be used.
Table 2 shows the 16 possible combinations that define
unique sites based on the four factors that are used in DTA.
Each site is classified into either the action group (Group
A) or the deferred action group (Group DA) (see the right-
hand column of Table 2). Denote the expected value of the
air measurement for a given type of site by y. where i is
the site ID. Then define
Group A
and
^na
Group DA
corresponding to Group A and Group DA, respectively. The
statistical hypotheses are:
H . ,. — ,.
HO.
This formulation of the validity test represents the minimum
information that could be required on the relationship be-
tween airborne asbestos levels and DTA if DTA is to be used
in rulemaking. Acceptance of H, indicates that it has been
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confirmed that airborne asbestos levels are higher at sites
designated for immediate corrective action than at sites
where the decision is to defer corrective action.
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IV. SAMPLE SIZE AND PRECISION
In this section, the relationship between sampling de-
sign, sample size, and precision is analyzed. The discus-
sion is restricted to those designs that are presented in
the Battelle study design report dated November 20, 1980.
Validation has been formulated as a problem in statistical
hypothesis testing (see previous section). In the testing
problem, the sample size required depends on the .size of the
difference between the parameters of interest and the proba-
bility of detecting that difference. Precision enters
through the variability associated with the process that is
*
producing the measurements. Both the magnitude and type of
variation can be important. Variation is summarized in the
parameter 0 for the airborne asbestos process. In general,
as variation increases, the probability of detecting any
fixed difference declines.
As suggested earlier, variation may be a composite of
many components—site, measurement, and model to name a few.
Proper understanding and evaluation of these components are
critical to the discussions of sample size that follow. The
issue of variance components is discussed next. Then sample
size tables are presented and discussed in relationship to
rnative.assumptions about 1
A. Sources of Variation
2
alternative.assumptions about the structure of a .
There are two basic measurement processes to be
considered—air sampling and algorithm scoring. Variation
associated with the data collected from each process arises
from a number of different sources. For air sampling, the
variation may be due to:
(1) Site—Airborne asbestos levels vary because of
systematic differences among sites. There is also
a random component of variation that arises even
when site characteristics are identical.
* In this discussion, precision may be thought of as the
reciprocal of a.
-------
-24-
(2) Measurement error—The laboratory measurement pro-
cedure involving TEM to determine asbestos concen-
tration is subject to error which can be characte-
rized as random.
(3) Time—Collecting weekly samples at a given site
under seemingly constant external conditions may
result in a distribution of values when data from
different weeks are pooled.
For algorithm scoring, the variation may be due to:
(1) Sites—Sites exhibit systematic differences.
(2) Rater—Different raters may score the same site
0
differently. Also, any one rater may not be able
to duplicate his or her score at a fixed site.
This source of variation is similar to measurement
error in laboratory procedures.
(3) Content—One important factor in the algorithm is
based on the percentage of asbestos found in a
bulk sample. Variability in content is the vari-
ability associated with the bulk sampling proto-
col. Included as sources of variation are
(a) location—random variation across locations
where bulk samples are taken; and
(b) measurement error—random variation associat-
ed with the laboratory procedure.
The structure of variation imposed by the sources des-
cribed above is used in the analysis of sample size that
follows. Three cases are considered. The first case is
developed from the most simplifying assumptions about sour-
ces of variation. That case corresponds to performing the
validation study in a controlled laboratory environment.
The second case consists of Case 1 with the inclusion of
measurement error as a source of variation. The final case
includes variation that results from selecting sites accord-
ing to a probabilistic sampling scheme.
-------
-25-
B. Sample Size: SA; Controlled Experiment
This first case (controlled experiment) is an ob-
vious oversimplification of the actual measurement process.
The composition of a is kept as simple as possible. We
assume that the sites included have, in essence, been con-
structed in a laboratory for the validation study. We
assume that algorithm scoring is not subject to error.
Under these assumptions, the validation study is similar to
a controlled experiment conducted in a laboratory setting.
The experimental units are sites which have been selected to
reflect the desired distribution of SA scores.
Although this formulation is an oversimplification of
the actual problem, it serves as a reference case for more
realistic formulations. The fact that the complexities of a
realistic model of variability are not considered allows us
to focus clearly on the sample size issues.
1. SA Formulation 1
In Formulation 1, we consider two statistical
hypotheses:
(a) HQ: y1Q = yfi
and
(b) H0: y40 = y!0
Hl: y40 > y!0 •
Recall that SA scores of 10 and 40 have been designated as
decision points for corrective action. Hypothesis (a) dis-
tinguishes airborne asbestos levels at sites where SA is 10
from background asbestos levels. Hypothesis (b) distinguish-
es between asbestos levels at sites where the SA scores are
10 and 40, respectively. Note that no specific airborne as-
bestos concentration levels are mentioned in the hypotheses.
If specific levels were to be used, they would come from the
dose-response curve that relates "risk" to asbestos dose
level. The proposed validation tests (hypotheses (a) and
-------
-26-
(b)) are only indirectly related to "risk". It is possible
however that u4Q is found to be larger than UIQ, but both
values are well below the acceptable level based on dose-
response studies. On the other hand, H_: y. = y-_ may be
accepted and both values may be large enough to pose a seri-
ous health risk. As mentioned earlier, the validation tests
described in this report deal only with the relationship
between air levels and exposure assessment algorithm scores.
The other validation step linking air levels to health risk
and establishing maximum safe dose levels is treated else-
where. (See Support Document, Asbestos-Containing Materials
in Schools, Health Effects and Magnitude of Exposure, USEPA,
October 1980).
Hypothesis (a). We consider hypothesis (a) first. The
statistical test of null hypothesis, HQ, will be carried out
using the two sample "t" statistic,
t = (n/2)1/2 (Y1Q - YB) / s ,
with 2n-2 degrees of freedom, and s is an estimate of a us-
ing the pooled set of data. It is assumed that the number
of sites of each type (S = 10 and background) is n. In or-
der to calculate the required sample size, the following
quantities must be specified:
(1) Significance level (a)—probability of choosing H,
when HQ is true;
(2) Specific alternative hypothesis—magnitude of the
difference between y,_ and y_ that is considered
to be important; and
(3) power (1 - 6)—probability of choosing a specific
alternative, H,, when it is correct.
In the planning stages of the experiment, it is often
difficult to arrive at one particular set of the three val-
ues to be used in determining the sample size. Because of
this difficulty, it is instructive to select a range of
values and generate a table of corresponding sample sizes.
-------
-27-
In this way, is it possible to develop a perspective on how
significance level, alternative, power, and sample size are
related.
Table 3 has been constructed for this purpose. The
columns correspond to various values of 1 - 3, the power of
the test. The rows (taken in pairs), correspond to various
alternatives. The alternatives are differcences between
y-j. and yB stated in units of o for convenience. For example,
Rows 5 and 6 correspond to the alternative, H,: yin - y_ =
X J.U 15
2o; that is, the difference is two standard deviation units.
The first row in each pair corresponds to a significance lev-
el of a = .05. The second row in each pair is for a = .01.
Note in general that larger sample sizes are required
as (1) the power is increased, and (2) K is decreased
(i.e., the difference between y,Q and yB is decreased). In
Row 6 we see that with a = .01, it takes 10 sites with SA
scores of 10 and 10 background sites to have a 90 percent
chance of finding a difference between y,Q and y_ of two
standard deviation units (y1Q - yB = 2a). Increasing the
power from 90 to 99 percent requires 5 additional sites of
each type. However, holding the power at 90 percent and
changing the alternative to a difference of one standard
deviation unit (Row 4) requires an increase in the sample
size of 19 sites of each type.
If the value of a were known, the determination of re-
quired sample size would be simplified by using the known
value of a to state the alternative in terms of the original
measurement units, ng/m in this case. If the value of a is
not known, it is instructive to find other representations
that can be used to help interpret the alternative hypoth-
eses. Many representations are possible. The one selected
for discussion here is based on a probability. Consider two
measurements Y, _ and Y from one site where the SA score is
10 and a background site, respectively. It is expected that
Y,_ >_ Yg. Based on our assumptions concerning the structure
-------
-28-
Table 3. Required Sample Size (n) For Testing
H
Versus
(n sites in each group)
Standard
Deviation
Units (K)
0.5
1.0
2.0
3.0
4.0
Significance
Level (a)
a = .
a = .
a = .
a = .
a = .
a = .
a = .
a = .
a = .
a = .
05
01
05
01
05
01
05
01
05
01
.50
23
45
7
15
3
6
2
4
2
4
Power
.80
57
80 >
15
23
5
8
4
6
2
5
(1
.90
70
100
19
29
6
10
4
6
3
5
- 6)
.95
90
>100
23
34
7
12
5
7
4
6
.99
>100
>100
35
49
10
15
6
8
4
6
Source: Handbook of Statistical Tables, D.W. Owen,
Addison-Wesley, 1962, Section 2.2.
-------
-29-
of the data, the probability of the "unexpected", namely Y,.
<_ YB/ depends only on K when y1Q - y = Ka. This probabil-
ity may be interpreted as a measure of "overlap" between the
two distributions of airborne asbestos concentration levels.
Table 4 shows values of the probability that Y,_ < Y_
J.U — a
corresponding to different values of K.
Alternative hypotheses of the form H. : y,Q - y_ = Ko
are interpreted in terms of the "overlap" measure as follows,
When K = 2, the overlap is small: .079. A minimal amount of
data should be required to distinguish between the two means
when the overlap of the two distributions are so small.
When K = .5, the overlap is substantial. In this case, a
large amount of data would be required to distinguish be-
tween the means when the overlap of the two distributions is
that large. For the numerical example given above, when a =
.01, it takes 10 sites of each type to have a 90 percent
chance of finding a difference between y,_ and y_, when the
J.U D
overlap is .079. However, for the same significance level
and power, it takes 29 sites of each type to find a diffe-
rence between y1Q and yfi when the overlap is equal to .239.
The analysis of sample size requirements in terms of
multiples of a is instructive, and introducing the "overlap"
measure is helpful, but it would be more informative to
compare these requirements relative to the actual units of
measurement—ng/m . In order to accomplish that goal, it is
necessary to have preliminary estimates of a and y0. Data
D
that are presented in the report "Measurement of Asbestos
Air Pollution Inside Buildings Sprayed With Asbestos" (Se-
bastien 1980) have been used to obtain estimates. In that
report, a distribution of background values is given for
Paris. The report indicates that the lognormal distribution
is appropriate for background values. An estimate of yn =
13
-.755 is given, and it is stated that 99 percent of the val-
ues are less than or equal to 7 ng/m . Solving the equation
yn + 2.33 a = In 7
£5
-------
-30-
Table 4. Probability That YIQ <_ YB When Testing
H
0 y!0
Versus
= Ka
Standard
Deviation
Units (K)
IV
.5
1
2
3
4
.363
.239
.079
.017
.002
Based on Assumptions stated in Section IV.B,
P(Yin < YD) = $(-K//2), where $ (•) is the
J.U D
Standard Normal Distribution Function.
-------
-31-
yields an estimate of o equal to 1.159. Table 5 shows the
numerical values of the alternatives in ng/m corresponding
to values of the multiple, K.
Using the example introduced above, if a = .01, it
takes measurements on 10 sites of each type to have a 90
percent chance of detecting a difference of 8.4 ng/m . For
the same significance level and power, it takes 29 sites of
each type to correctly identify a difference of 2 ng/m (re-
fer to Tables 3 and 5). This example which is based on an
estimate of a from the Paris data (Sebastien 1980) suggests
that the statistical test of HQ: y1Q = yB is very sensitive
to small differences between y,Q and yB. Requiring a total
of only 20 sites to detect a difference of 8.4 ng/m seems
remarkable. The difficulty, if there is any difficulty, may
be that the estimate of o is too small. The estimate may be
too small either because the sample on which it is based is
not truly representative of background concentration levels,
or because the assumption that the value of y is independent
of 0 is incorrect. One plausible alternative assumption is
that there is a value, a , for background measurements and
another value, a, for all other concentration levels. This
assumption adds an element of complication to the statisti-
cal test used for the hypothesis Hn: y,n = y_.; however,
U -LU c
tests for the other hypotheses are unaffected. It is to be
expected that a is larger than o . Therefore, in our sample
size determination planning effort, it is conservative to
consider values of a larger than 1.159 (obtained from the
Sebastien 1980 data).
Table 6 shows examples of how the numerical alternatives
change when a is increased to 2. Referring once again to
the example introduced above, if ct = .01, it takes 10 sites
of each type to have a 90 percent chance of detecting a dif-
ference of 186.1 ng/m between y1Q and yQ. For the same
significance level and power, 29 sites of each type are re-
-------
-32-
Table 5. Raw Measurement Units (ng/m ) For
The Alternative Hypothesis When
Testing Hn: uln = y_, Versus
U J-U D
H, : P,n - ]in = Ko
•J- J. W AJ
Standard
Deviation
Units (K)
.5
1
2
3
4
Raw Measurement
Units a ng/m
.7
2.0
8.4
28.9
94.0
LeK° - lj,
yB + o2/2
Obtained as e Le^u - lj, rounded
to the nearest tenth. Values of y_, = -.755
13
and a = 1.159 were estimated from Sebastien,
1980.
-------
-33-
Table 6. Effect of Increased Standard Deviation
On Raw Measurement Units (ng/m ) For
The Alternative Hypothesis When Testing
H
0:
Versus
= Ka
Standard
Deviation
Units (K)
.5
1
2
3
4
Raw Measurement
a = 1.159
.7
2.0
8.4
28.9
94.0
Units , ng/m a
a = 2.0
6.0
22.2
186.1
1397.6
10349.2
B + ° /2
Obtained as ey
the nearest tenth.
[•": •].
rounded to
-------
-34-
quired to correctly identify a difference of 22.2 ng/m (re-
fer to Tables 3 and 6).
2. SA Formulation 1: Summary
It is clear from the preceding discussion
that the process of arriving at a required sample size in-
volves a number of decisions concerning parameter values.
There is an element of subjectivity associated with each
decision. The most difficult decision is choosing a fixed
alternative. As was presented above, there are various ways
to quantify an alternative. It may be equivalently specified
as (1) a multiple of o, (2) the probability of overlap
(P[Y_ Y_]), or (3) in raw measurement units of nanograms per
O— o
cubic meter, as well as other ways that have not been intro-
duced here. Since there is usually no one quantification
that is easily interpreted in all cases, it is useful to
consider various specifications.
The question remains: "What sample size?" or more fun-
damentally, "How does one select sample size?" The task is
best accomplished by first understanding the tradeoffs among
the relevant parameters listed above. A summary table,
Table 7, has been constructed for this purpose. The table
displays power levels corresponding to a cross classifica-
tion of proposed sample sizes and plausible alternatives.
Four quantifications of each alternative are listed. If we
settle on a significance level of 5 percent (a = .05), it
follows that with 16 or more sites of each type, most rea-
sonable validation objectives will be met. Note that in-
creasing n to either 24 or 32 is not sufficient to have a
test that is sensitive to alternatives corresponding to K =
.5. A sample size of n = 16 may be problematic if the true
values of a is larger than 2. If o is as large as 2.5 (not
included in the table), the alternative corresponding to K
= 2 takes the value 1576.9 ng/m . In this case (a = 2.5),
a value of n approximately equal to 30 would be necessary to
meet the objectives that are satisfied by n = 16 when a = 2.
-------
-35-
Table 7. Summary of Power (%) For Testing H_:
0
B
Versus
= Ka
n
(sites
per
group)
4
8
16
24
32
K (std. dev. units)
P(YB > Y10}
ng/m , 0=1.159
ng/m , o=2 . 0
a = .05
a = .01
a = .05
a = .01
a = .05
a = .01
a = .05
a = .01
a = .05
a = .01
Alternative Hypotheses a
.5
.363
.7
6.0
<50
<50
<50
<50
<50
<50
50
<50
60
<50
1
.239
2.0
22.2
<50
<50
54
<50
83
54
95
82
98
93
2
.079
8.4
186.1
65
<50
96
80
>99
>99
>99
>99
>99
>99
3
.017
28.9
1397.6
90
50
>99
99
>99
>99
>99
>99
>99
>99
4
.002
94.0
10349.2
95
50
>99
>99
>99
>99
>99
>99
>99
>99
Linear Interpolation From Table 3.
-------
-36-
It is clearly essential to have a reliable preliminary esti-
mate of 0 in order to arrive at a judicious choice of n.
Hypothesis (b). Hypothesis (b) states that H_: y4Q =
y1Q is to be tested against H.^: y4Q > y-,Q. The test statis-
tic is similar to the statistic used in Hypothesis (a),
namely,
t = (n/2)1/2 (Y4Q - Y1Q) / s ,
which has a "t" distribution with 2n-2 degrees of freedom.
The characteristics of the test are found in Tables 3 and 7.
In the current formulation of the validation problem, the SA
designations of "background," S = 10 and S = 40 serve only
as labels. Therefore, Hypotheses (a) and (b) have identical
characteristics. The discussion presented for Hypothesis
(a) applies without modification to Hypothesis (b).
3. SA Formulation 2
We assume that the relationship between Z,
(Z = Y_ - Y ), and the SA score (denoted by S) can be ap-
proximated as a linear function
Z = BQ + B-j^S + e
where e is a random quantity having an expected value equal
2
to zero and variance equal to a . Since we continue to
treat the data as though they were generated in a controlled
2
laboratory setting, a is a composite of variation in Z and
variation associated with errors due to linear approximation.
S is assumed to be measured without error.
The hypothesis of interest is
HQ: BI = 0
H-j^: B-L > 0 .
In this formulation, the acceptance of H, means that the
relationship has been validated for all values of the SA
score. As with Formulation 1, the alternative hypothesis
must be made specific in order to arrive at the required
sample size. The alternative considered is similar to those
-------
-37-
formulated earlier. In particular, we shall look at hypoth-
eses such as HQ: u4Q = UIQ. Note that the model implies
that
Therefore, the hypothesis that y4Q = y,Q is equivalent to
306, = 0 or 8, = 0. The alternative, H,: y.Q - y,Q = Ka is
equivalent to H,: 308, = Ka or H,: 3n = Ka/30. It follows
that the hypotheses considered in the previous section cor-
respond to special cases of the more general hypothesis,
HO: B-. = 0. However, it is demonstrated below that the
characteristics—sample size and power—of the statistical
test of HO: 3, = 0 are different from the characteristics of
the statistical test used in Formulation 1. As expected,
there is a reduction in sample size resulting from the as-
sumption that the relationship between Z and SA is linear.
Test Characteristics; HQ; g, = 0
The test of HO is based on a "t" statistic defined as
t =
(s/Eo)
'S'
with n-2 degrees of freedom where:
(1) b, and s are standard regression model estimators
of 3, and a, respectively, and
11/2
[-1
£
-------
-38-
according to a half-fraction of the 2 design. There are
sound arguments for using one or the other of these designs
that are not based on formal quantitative characteristics.
Those arguments appear in the Battelle Study Design report
and will be discussed later in this report. For the pre-
sent, we analyze the characteristics of the design relative
to the validation test of Formulation 2.
Design; 2 Full-Factorial and Half-Fraction. The des-
cription of sites that make up the 2 design is given in
Table 8. The rows that are marked (*) are those that are
designated for the half-fraction design. In order to ana-
lyze the characteristics of these designs, it is necessary
to compute an SA score for each site and then compute the
value of 2^s- Although the design is based on factor scores
that are dichotomous, the actual sites will be scored accord-
ing to the original SA weights shown in Table 1. If the
original weights were used to specify unique sites, there
would be 324 unique combinations. By dichotomizing 5 of the
factors (and ignoring others), each of the 324 sites is asso-
ciated with one of the 32 sites shown in Table 8. Factor
scores for each of those 32 types of sites have been calcu-
lated as averages of the weights that are combined by the
dichotomization process. These average factor scores and
their associated SA scores are presented in Table 9. A
value of 2JS has been computed for each design. For the
full-factorial design, n = 32, i^ = 138.8; for a half-
fraction design, n = 16, 2-»_ = 98.2.
A table showing power of the statistical test of
H_: 8, = 0 has been developed for each design. Tables 10
and 11 have the results for the full-factorial design and
the half-fraction design, respectively. Recall that the
alternative hypotheses are designated to detect differences
in mean values of airborne concentration levels of asbestos
corresponding to different values of SA. We considered
Hl: W40 ~ W10 = Ka wn^cn is equivalent to H,: 6, = K0/30.
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-39-
Table 8. Full-Factorial Sampling Plan
Site
No.
1
*2
*3
4
*5
6
7
*8
*9
10
11
*12
13
*14
*15
16
*17
18
19
*20
21
*22
*23
24
25
*26
*27
28
*29
30
31
*32
Algorithm Factors3
Content
Low
Low
Low
Low
Low
Low
LOW
LOW
Low
Low
Low
Low
Low
Low
Low
Low
High
High
High
High
High
High
High
,. High
High
High
High
High
High
High
High
High
Friability
Low
Low
Low
LOW
Low
Low
Low
Low
High
High
High
High
High
High
High
High
Low
Low
Low
Low
Low
Low
Low
Low
High
High
High
High
High
High
High
High
Condition
Good
Good
Good
Good
Bad
Bad
Bad
Bad
Good
Good
Good
Good
Bad
Bad
Bad
Bad
Good
Good
Good
Good
Bad
Bad
Bad
Bad
Good
Good
Good
Good
Bad
Bad
Bad
Bad
Exposure
Low
Low
High
High
Low
Low
High
High
Low
Low
High
High
Low
Low
High
High
Low
Low
High
High
Low
Low
High
High
Low
Low
High
High
Low
Low
High
High
Accessibility
Low
High
Low
High
Low
High
Low
High
Low
High
Low
High
Low
High
Low
High
Low
High
Low
High
Low
High
Low
High
Low
High
Low
High
Low
High
Low
High
Content (high) >_ 50%
Friability (high) > 1 (see Table 1 for codes)
Condition (Bad) = 5
Exposure (high) = 4
Accessibility (high) = 3
denotes sites designated for the half-fraction design.
-------
-40-
Table 9. Full-Factorial Sampling Plan, Average Factor Scores
Algorithm
Site
No.
1
*2
*3
4
*5
6
7
*8
*9
10
11
*12
13
*14
*15
16
•17
18
19
*20
21
*22
*23
24
25
*26
*27
28
*29
30
31
*32
Con-
tent
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
3
Fria-
bility
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.0
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
Con-
dition
1
1
1
1
5
5
5
5
1
1
1
1
5
5
5
5
1
1
1
1
5
5
5
5
1
1
1
1
5
5
5
5
Expo-
sure
.5
.5
4.0
4.0
.5
.5
4.0
4.0
.5
.5
4.0
4.0
.5
.5
4.0
4.0
.5
.5
4.0
4.0
.5
.5
4.0
4.0
.5
.5
4.0
4.0
.5
.5
4.0
4.0
Factors
Accessi-
bility
.5
3.0
.5
3.0
.5
3.0
.5
3.0
.5
3.0
.5
3.0
.5
3.0
.5
3.0
.5
3.0
.5
3.0
.5
3.0
.5
3.0
.5
3.0
.5
3.0
.5
3.0
.5
3.0
Other3
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
2.5
SA Scoreb
9.0
14.0
16.0
21.0
17.0
22.0
24.0
29.0
22.5
35.0
40.0
52.5
42.5
55.0
60.0
72.5
13.5
21.0
24.0
31.5
25.5
33.0
36.0
43.5
33.75
52.5
60.0
78.75
63.75
82.5
90.0
108.75
a Average of air moving system and activity (see Table 1).
b SA score given in Table 1.
-------
-41-
Table 10. Formulation 2: Full-Factorial Design
HQ: 3-, = 0; H, : g, = Ko/A , n ••
Degrees of Freedom = 30, £s :
Body of Tables Gives Power of
= 32,
= 138.8;
the Test (%)
K
.5
1
2
3
4
Significance
Level
a = .05
a = .01
a = .05
a = .01
a = .05
a = .01
a = .05
a = .01
a = .05
a = .01
5a
**
**
**
**
**
**
**
**
**
**
10
**
**
**
**
**
**
**
**
**
**
A
20
97
85
**
**
**
**
**
**
**
**
30
75
45
99
98
**
**
**
**
**
**
40
52
25
97
85
**
**
**
**
**
**
** Power exceeds 99%.
a Recall A = 5 implies H- : \i ... = y ;
r 0 S+D s
Hl: ys+5 - ys = KCT-
Source: Handbook of Statistical Tables, D.W. Owen,
Addison-Wesley, 1962, Section 2.2.
-------
-42-
Table 11. Formulation 2: Half-Fraction Design
V ei = 0; H:
. : 3, = Ka/A, n
Degrees of Freedom = 14 , 2-»s
Body of Table
Gives Power of
= 16,
= 98.2;
the Test
(%)
K
.5
1
2
3
4
Significance
Level
a = .05
a = .01
a = .05
a = .01
a = .05
a = .01
a = .05
a = .01
a = .05
a = .01
5a 10
** **
** 99
** **
** **
** **
** **
** **
** **
** **
** **
A
20
75
45
**
99
**
**
**
**
**
**
30
45
20
92
72
**
**
**
**
**
**
40
25
13
75
45
**
99
**
**
**
**
** Power exceeds 99%.
a Recall A = 5 implies HQ: yg+5 = ws;
Source: Handbook of Statistical Tables, D.W. Owen,
Addison-Wesley, 1962, Section 2.2.
-------
-43-
A more general statement of the testing problem is
H0:
which is equivalent to
= Ka '
HQ: BI = 0
in Formulation 2. Tables 10 and 11 show powers correspond-
ing to values ofK= .5, 1, 2, 3, 4, and A = 5, 10, 20, 30,
40.
Design; Optimal. If the assumption that the relation-
ship between Z and SA is linear is accepted without question,
then the most efficient use of resources is to choose sites
in a way that maximizes 2L.C. 2-> is maximized by choosing
O O
one-half of the sites at the low end of the SA scale and the
other half at the high end of the SA scale. The resulting
value of 5Z0 is 49.9 n ' . Table 12 shows the power of the
D
statistical test corresponding to sample sizes n - 4, 8, 16,
and 32 where n/2 sites are at each end of the SA scale. The
alternatives considered are H,: B-, = Ka/A . The value of A
has been fixed at 30 in Table 12.
The statistical test based on this design is more pow-
erful than the tests based on either the full-factorial or
the half-fraction designs. By comparing Tables 10, 11, and
12, we see that for any fixed alternative (K, A = 30), the
power values read from Table 12 are at least as large as
comparable values found in Tables 10 and 11. Equivalently,
it is possible to meet power objectives with fewer sites if
the optimal design is used. For example, if the alternative
hypothesis is H.: ii._ - U,Q = .5o, then 32 sites are requir-
ed to reach a power level of 75 percent with the full-
factorial design (Table 10, Row 1). When using the optimal
design, approximately 10 sites would be required for power
to be equal to 75 percent (Table 12, Row 1).
-------
-44-
Table 12. Formulation 2: Optimal Design
HQ: B-L = 0; I
Eg = 49.9 x
Power of the
*1: &1 = Ka/30 (i-e' • A
n1/2 Body
Test (%)
of the Table
= 30) ,
Gives
K
.5
1
2
3
4
Significance
Level
a = .05
a = .01
o = .05
o = .01
a = .05
a = .01
o = .05
a = .01
a = .05
a = .01
4
30
7
30
7
**
**
**
**
**
**
Sample Size
8
65
35
65
35
**
**
**
**
**
**
(n)
16
94
75
94
75
**
**
**
**
**
**
32
99
99
99
99
**
**
**
**
**
**
** Power exceeds 99%.
-------
-45-
The advantage enjoyed by the optimal design is not
without consequence. In effect, an assumption—the linear-
ity of the relationship between Z and SA scores—replaces
the need for additional sites. The savings in sites is
satisfying provided the assumption of linearity is correct.
However, if the optimal design is used, no data will be ob-
tained that are useful for assessing the assumption.
The types of systematic errors that may occur as a re-
sult of an incorrect assumption about linearity are depicted
in Figure 4. Consider for example, the effects of an incor-
rect linearity assumption on the test of v.Q = v-,Q- In Fig-
ure 4a, the true relationship is convex and the assumed re-
lationship is linear. The change in airborne concentration
levels corresponding to a change from 10 to 40 on the SA
scale may be understated if the linear representation were
accepted. If the true relationship is concave (Figure 4b),
the change in airborne levels may be overstated. In both
situations, the discrepancies could cause the outcome of the
validation analysis to be in error.
4. Comparison: Formulation 1 Versus Formulation 2
Table 13 has been prepared to compare the ef-
ficacy of the different designs proposed under Formulation 1
and Formulation 2. A formal analysis of statistical power
indicates that Formulation 2 is preferred and that the opti-
mal design provides for the most efficient use of resources.
However, there are other issues besides statistical power
that must be considered in selecting a design.
As mentioned earlier, Formulation 2 is superior provid-
ed that the assumption of linearity between Z and SA is cor-
rect. But linearity is only an assumption. It is prudent
to implement a design that is efficient (high level of sta-
tistical power) and that also incorporates information that
makes it possiblie to check assumptions. Clearly the "opti-
mal" design of Formulation 2 does not meet this last crite-
rion. The optimality of the "optimal" design rests totally
-------
-46-
Airborne
Asbestos
ng/m
Figure 4a
Airborne
Asbestos
ng/m
"40 * "10 (TRUE)
y40 - y10 (ASSUMED)
SA Score
'40
(TRUE)
SA Score
Figure 4b
Figure 4. Some effects of an erroneous linear assumption.
-------
Table 13. Comparison: Formulation la Versus Formulation 2.
a = .05, A = 30; Body of Table Presents Power (%)
Formulation 2
K
.5
1
2
3
4
ruj.lliu.Le
n=32
50
83
99
**
**
lUJ-UJl J.
n 1 fi
50
54
96
**
**
Full Factorial
n = 32
75
99
**
**
**
Half Fraction
n = 16
45
92
**
**
**
Optimal
n = 32 n = 16
99 94
** **
** **
** **
** **
** Power exceeds 99%.
a Formulation 1 = H-: Pg+A = us versus H, : p - y = Ka
Formulation 2 = H_: 3, = 0 versus H..: B-. = Ko/30
-------
-48-
on the assumption of linearity. The full-factorial and
half-fraction designs include data points that allow the
linearity assumption to be checked. Each of these designs
force the data to be judiciously scattered across the SA
scale. The full-factorial design is most appealing. By
covering all factor combinations, this design provides the
best opportunity to observe any irregularities in the rela-
tionship between Z and SA.
It should be noted that the power of the proposed va-
lidity tests under Formulation 2 can be increased by devi-
ating slightly from the formal full-factorial and half-
fraction designs. Recall that power increases as £S' t*16
spread in the SA scores, increases. If the half-fraction
design were chosen as a base, it could be supplemented with
16 additional sites having SA scores toward the extremes of
the SA scale. That strategy would result in a design that
retained the ability to assess departures from the linearity
assumption and would simultaneously use the linearity assump-
tion to provide a more powerful test than is available using
the full-factorial design. A variety of adjustments to the
basic designs (such as the one described above) are feasible.
The rationale for adjusting the design is often a result of
scientific judgment or practicality required during field
implementation. In most cases, the adjusted designs can be
evaluated using the techniques that led to Tables 10 through
13 so that it is possible to evaluate the information gain
or loss associated with any adjustment.
C. Sample Size: Decision Tree Algorithm(DTA);
Controlled Experiment
The validation test for DTA is based on the com-
parison of two groups of airborne asbestos concentration
levels. The groups are defined in terms of an "action-
deferred action" decision that results from applying the
algorithm. DTA is based on four factors (see Figure 2)
which determine 16 different types of sites. The classifi-
-------
-49-
cation of sites into Group A (action) and Group DA (deferred
action) is shown in Table 2. Five sites (8, 12, 15, 23, 32)
are in Group A; eleven sites are in Group DA (2, 3, 5, 9,
14, 17, 20, 22, 26, 27, 29). The validation test has been
formulated as
Hn = H* = "
0 A
versus
where
Group A
VDA = Z y. / 11 .
UA Group DA 1
Table 14 summarizes the power to sample size relation-
ship associated with the alternative hypothesis formulated
as H,: y. = y + Ka. The first column of the table (label-
ed n = 16, df = 14) corresponds to the basic half-fraction
design which allows for one site of each type in the sample.
The remaining three columns describe the effects of increas-
ing the sample size by replicating the design once, twice,
and three times, respectively.
The replicated designs are preferred not only because
they provide additional power, but because they provide more
basic information than the unreplicated design (Column 1).
If the basic 16 sites are sampled once, it is necessary to
assume that the means for sites in Group A are identical and
equal to y-. It is also necessary to assume that the means
for sites belonging to Group DA are identical and equal to
yna. In addition, the variances for all sites are assumed
JL/r\
to be equal. In this case, the statistical test is based on
Y - Y
A X
sl=-
with 14 degrees of freedom where
-------
Table 14. Statistical Test: DTA; Hn; MA =
\ \J f\
Body of Table Presents Power of Test (%)
Ko
K
.5
1
2
3
4
Significance
Level
a = .05
a = .01
a = .05
a = .01
a = .05
a = .01
a = .05
a = .01
a = .05
a = .01
Number of Replications (m) a
0
n=16.
DF=14D
20
10
55
25
95
85
**
**
**
**
1
n=32
DF=16
35
15
80
50
**
99
**
**
**
**
2
n=48
DF=32
50
25
95
80
**
**
**
**
**
**
3
n=64
DF=48
60
30
98
90
**
**
**
**
**
**
a Number of times the basic 16 sites design is replicated.
b DF = degrees of freedom.
** Power exceeds 99%.
i
?
-------
-51-
YA = Z * / 5, Y = £ Y / 11
Group A UA Group DA 1
and
•'•I
.Group A Group DA
If the design is replicated, there is sufficient data
to estimate the mean for each type of site and to check on
the validity of the assumption that all variances are equal.
In this case, the test becomes
4- -
u —
(YA - YDA)
72
S l 5 + IT,
with 16 m degrees of freedom where m is the number of repli-
cations and where
~ "~* f±. / 5 (m + 1)
Y, . / 11 (m + 1)
and
s2 =
r
£
L Group
V •• •
Y =
E
A Site
E
Group A
Group DA
ttij-Ti)
E
Site
E
Site
Gr
L (Y..-Y. )2 /16m
DA Site 3 J
with Y. being the site mean. Therefore, the argument for
taking at least one replication is not based totally on pow-
er considerations. At least one replication is required to
keep from having to impose unrealistic assumptions on the
validation test. Considering statistical power, it appears
from Table 14 that at least one replication is also required
if the statistical test is to distinguish group differences
as small as one standard deviation (K = 1).
D. Sample Size; Measurement Error
In the previous discussions of sample size and
precision, it was assumed that sites are scored without
error. To score without error means that a group of raters
presented with a given site will produce identical ratings
on all factors. This assumption was made in the previous
-------
-52-
section to simplify the formulation of the statistical prob-
lem so that the presentation of sample size issues would not
be clouded. However, the assumption that exposure scores
are produced without error is not realistic. Previous work
(Battell, 1980; Price et al. 1980) strongly suggests that
different raters score a given site differently. In this
section we formulate the validation problem in a way that
allows for variation due to error in exposure assessment
scoring. For validation of SA, Formulation 2, the linear
model approach is reanalyzed allowing for scoring errors.
For validation of DTA, a model using composite distributions
is presented and analyzed. In both cases, 'the effect of
errors in assessment exposure scoring reduces the power of
the validation test associated with any fixed sample size.
1. SA: Formulation 2
We assume that the relationship between air-
borne asbestos concentration levels and the algorithm score
is linear. The formulation of the model that is used in the
validation test when errors are present in scoring is slight-
ly different than the model when scoring is without error.
The model formulation follows.
Assume that the observed values of Z and SA are given
by
zi - Zi + ei' si ' Si + "i
where e and n are random quantities, uncorrelated, with ex-
2 2
pected values zero and variances a and a , respectively.
The relationship between Z and SA is specified as
Z = BO + IJj/S + e .
The validation test is formulated as before, HQ: 8^ = 0 ver-
sus H.,: 8-, + Ka/A, where a = ag. The test is based on an
estimate of 6-, and the standard error associated with that
estimate. The formulation given above follows Britt and
Luecke 1973. An estimate, BI of ^ is obtained by constrain-
ed maximum likelihood estimation with
-------
-53-
(R B? + 1) o2
Var (B,) =
^•^* *5 ^"^ O O O
where £ = Zi(s; - S) and R = a /a . (Note that when
o i n e
2
R=0, o=0 which means that SA is measured without error;
^ ~ ? — p *
then Var (3,) = a / 2-(S. ~ s) which is the variance of 3,
in the classical regression model.)
/\
In the case when R ^ 0, the variance of 3, under the
alternative hypothesis that B, 7* 0 depends on the actual
value of B, being considered. The dependence of the vari-
ance on the true value of the parameter in question makes
the evaluation of the power of the statistical test diffi-
A.
cult. The variance of B-, when there is error in the algo-
J- y\
rithm score, is larger than the variance of B, when there is
no error. When arguing qualitatively, because of the in-
crease in variance, it is expected that more sites would be
required to maintain the power of the validation test when
errors are present than when algorithm scoring is error
free. We have chosen not to assess the required increase in
*
sites quantitatively. The necessary calculations are out-
lined in Appendix A. In the next section, the effects of
errors in algorithm scoring are analyzed with respect to
validation of DTA. Tables showing the degradation of sta-
tistical power are included for that analysis.
2. DTA
When attempting to validate DTA, errors in
scoring the algorithm factors may lead to the misclassifi-
cation of a site into the action group when deferred action
is correct, or conversely into the deferred action group
when action is the correct decision. If the probability
* This analysis requires direct calculation of probabil-
ities associated with the "noncentral t" distribution. All
other analyses of sample size have been based on published
tables of the noncentral t. It is not within the scope of
this project to develop computer programs to make these cal-
culations.
-------
-54-
that misclassification occurs approaches one-half, it be-
comes impossible to validate DTA. In order to demonstrate
the effects of errors in algorithm scoring (equivalently,
misclassification of sites) , we formulate the validation
problem as a hypothesis testing problem involving mixtures
of distributions. This formulation is presented below fol-
lowed by an analysis of the degradation in statistical power
associated with misclassification of sites.
Formulation . We first introduce the notation that is
necessary to describe the parameters that must be considered
in the validation analysis. Let P(A/DA) be the probability
of designating a site for Group A (action) when it should be
in Group DA, and let P(DA/A) be the probability of designat-
ing a site for Group DA when it correctly belongs to Group
A. For convenience, it is assumed that these probabilities
are equal:
p = P(A/DA) = P(DA/A) .
2 2
Let y- and a (^ an<^ a ^ denote the mean and variance of
the sites that belong to Group A (Group DA) . These symbols
designate the parameter values when there is no misclassifi-
cation, p = 0. If p ?£ 0, then for Group A,
Mean: n = (1 - p) P + P U
DA
22
Variance: O = a + p(l - p)
and for Group DA,
Mean: nDA = P PA + (1 - p)
Variance: a* = a2 + p(l - p) (UA -
As described earlier in this report, validation is equiva-
lent to testing HQ: WA = UDA against H^: PA - UDA = Ko .
However, because of the possibility of misclassification,
the actual statistical test is
H0: nA ~ nDA
versus
H
1
-------
-55-
Note that HQ: nA = nDA is equivalent to HQ: P = U. For
tive,
is equivalent to
the alternative, our interest is in H.^ VA - yDA = Ka which
Hl: nA ~ nDA = (1 ~ 2p) Ka '
This last statement of the alternative partially shows the
effect of misclassification on the validation test. The
alternative of interest, K
-------
Table 15. DTA: H,
= n
DA;
H
I'
" n
DA
= (l-2p)Ko
Body of the Table Gives Power of Degradation When Sites
Are Misclassified. Significance Level = .05
K
.5
1
2
3
4
Probability
of Misclass-
ification (p)
0
.1
.25
.4
0
.1
.25
.4
0
.1
.25
.4
0
.1
.25
.4
0
.1
.25
.4
Number of Replications (m)
0
n = 16
DF = 14
20
20
10
5
55
40
20
10
95
80
35
15
**
90
45
15
**
98
55
15
1
n = 32
DF = 16
35
25
10
5
80
65
25
10
**
95
60
15
**
**
75
20
**
**
80
25
2
n - 48
DF = 32
50
40
20
10
95
80
45
15
**
**
80
25
**
**
90
25
**
**
90
30
3
n = 64
DF = 48
60
40
25
10
98
90
55
15
**
**
85
25
**
**
95
30
**
**
95
40
DF = degrees of freedom.
n = number of sites.
** Power exceeds 99%.
i
Ul
T
-------
-57-
es. For example, when K = I, 32 sites yield a power of 80
percent if algorithm scoring is done without error (p = 0) .
If p = . 1, the power drops to 65 percent. It would require
48 sites to maintain the power of that test at approximately
the 80 percent level when p = .1. Other examples may be ex-
tracted from the table. The important observation is that
variability in algorithm scoring that leads to misclassifi-
cation of sites can have a material impact on the strength
of the validation results. For both the validation study
and the ultimate utilization of the algorithm in the field,
it is essential to implement training and techniques that
will minimize scoring variation.
-------
-58-
-------
-59-
V. DESCRIPTION OF AVAILABLE SITES IN MONTGOMERY COUNTY AND
NEW YORK CITY
In this section a description of algorithm scores in
two school districts where the proposed field study could be
carried out is presented. These two districts were selected
because RTI had current data on Sawyer Algorithm scores for
schools in these districts.
A. The Montgomery County Data
Two hundred schools were inspected by sanitarians
from the Montgomery County, Ohio, Health District. Friable,
asbestos-containing materials were identified in 75 sites in
23 schools. Table 16 presents the distribution of these
original 75 sites in terms of a full-factorial sampling plan
as described in Section IV (Table 8). The sites subject to
some type of abatement activity since the original survey
are identified in the footnotes. Inspection of Table 16 re-
veals that 12 sites have been subject to abatement activity.
Currently, 33 percent (21/63) of the available sites have
low friability and low percent asbestos. About 55 percent
(35/63) of the available sites have high friability and low
percent asbestos. Only 3 percent of available sites (2/63)
have high percent asbestos and low friability, and only 8
percent of the available sites (5/63) have high friability
and high percent asbestos. The available data thus suggest
that approximately 90 percent (56/63) of the possible sites
in Montgomery County have suspect materials with relatively
small amounts of asbestos and that only 7 sites have asbes-
tos levels of greater than 30 percent. This lack of high
asbestos sites certainly argues against the initiation of a
validation study in this area.
B. The New York City Data
The Division of School Buildings of the New York
City Board of Education became aware of a potential asbestos
exposure problem in January of 1977 shortly after six New
Jersey elementary schools were closed. During that year,
-------
-60-
Table 16. Distribution of Montgomery County Sites
Dichotomized Algorithm Variables'3
%
Asbes-
Stratum tos
1
2
3
4
5
6
7
8
9
10
11
12
13
14
IS
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
a 3
b 4
c 1
d 1
e 1
f 2
Low
Low
Low
Low
Low
Low
Low
Low
Low
Low
Low
Low
Low
Low
Low
Low
High
High
High
High
High
High
High
High
High
High
High
High
High
High
High
High
sites fixed
sites fixed
site fixed
site fixed
site fixed
sites fixed
Fri-
abil-
ity
Low
Low
Low
Low
Low
Low
Low
Low
High
High
High
High
High
High
High
High
Low
Low
Low
Low
Low
Low
Low
Low
High
High
High
High
High
High
High
High
Con-
di-
tion
Good
Good
Good
Good
Bad
Bad
Bad
Bad
Good
Good
Good
Good
Bad
Bad
Bad
Bad
Good
Good
Good
Good
Bad
Bad
Bad
Bad
Good
Good
Good
Good
Bad
Bad
Bad
Bad
g
Sites Total Sites
Expo- Access- Before Available
sure ibility Repair After Repair
Low Low
Low High
High Low
High High
Low Low
Low High
High Low
High High
Low Low
Low High
High Low
High High
Low Low
Low High
High Low
High High
Low Low
Low High
High Low
High High
Low Low
Low High
High Low
High High
Low Low
Low High
High Low
High High
Low Low
Low High
High Low
High High
% Asbestos (high)
(see table 1 for
Friability (high)
Condition (bad) -
Exposure (high) -
0
0
18
3
0
0
o.
3a
0
°b
34c
3C
0
°d
4°
0
0
oe
3|
2f
0
0
0
0
0
0
5
0
0
0
0
0
more than
codes)
more than
5
4
21
35
2
5
30%
1
Accessibility (high) - 3
-------
-61-
the specifications of 321 buildings in New York City that
were constructed between 1956 and 1976 were reviewed. The
specification review identified approximately 185 buildings
with asbestos-containing materials. By January 1981, 1411
buildings from approximately 1000 schools were physically
inspected. Bulk samples confirmed the presence of asbestos
materials in 257 buildings. Conflicting laboratory results
were obtained in an additional 5 buildings.
RTI has received survey data that were collected in 266
schools. In the borough-specific RTI files, there are com-
plete algorithm data from 2,265 sites in the 266 schools;
948 sites have incomplete data." The actual percentage of
asbestos estimated from the bulk samples is available for
approximately 10 percent of the 3,213 sites in the RTI files.
Sites within the 1411 school buildings that have asbes-
tos materials, which have not been eliminated by abatement
activities, constitute the major portion of the population
from which validation study sites should be selected. Table
17 presents the distribution of the original, and of current-
ly available, sites in terms of the full-factorial sampling
plan. It should be noted that sites with less than 1 percent
asbestos have not been deleted from Table 17, although it
could be argued that these sites should not be considered
eligible for the validation study. Inspection of Table 17
suggests that 52 percent (639/1226) of the available sites
have relatively low percentages of asbestos and low friabil-
ity, while 10 percent (117/1226) of the available sites have
high percentages of asbestos and high friability. Thirty-
seven (457/1226) percent of the available observations have
low asbestos percentages and high friability, while only 1
percent (13/1226) of available sites have high asbestos per-
centages and low friability. Tables 16 and 17 clearly show
that compared to Montgomery County, New York City has many
more sites available for the proposed validation study.
-------
Table 17. Distribution of New York Sitea*
Dichotomized
Stratum
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
%
Asbes-
tos
Low
Low
Low
Low
Low
Low
Low
LOW
Low
LOW
Low
Low
Low
Low
Low
Low
High
High
High
High
High
High
High
High
High
High
High
High
High
High
High
High
Fria-
bility
Low
Low
Low
Low
Low
Low
Low
Low
High
High
High
High
High
High
High
High
Low
Low
Low
Low
Low
Low
Low
Low
High
High
High
High
High
High
High
High
Algorithm Variables0
Condi-
tion
Good
Good
Good
Good
Bad
Bad
Bad
Bad
Good
Good
Good
Good
Bad
Bad
Bad
Bad
Good
Good
Good
Good
Bad
Bad
Bad
Bad
Good
Good
Good
Good
Bad
Bad
Bad
Bad
Expo-
sure
Low
Low
High
High
Low
Low
High
High
Low
Low
High
High
Low
Low
High
High
Low
Low
High
High
Low
Low
High
High
Low
LOW
High
High
Low
Low
High
High
Access-
ibility
Low
High
Low
High
Low
High
Low
High
Low
High
Low
High
Low
High
Low
High
Low
High
Low
High
Low
High
Low
High
Low
High
Low
High
Low
High
Low
High
Sites
Original
8
1
685
102
1
0
11
27
549
9
192
21
0
0
12
20
0
0
11
2
0
0
0
0
520
0
75
4
0
0
9
6
a
h
Current"
2
1
555
66
0
0
4
11
303
0
120
14
0
0
6
14
0
0
11
2
0
0
0
0
89
0
16
1
0
0
6
5
Total
Presently
Available
639
457
13
117
(k
V
a missing data, 948 sites, sites with
less than 1% asbestos are included.
b after abatement activity.
% Asbestos (high) 50% or more (see
Table 1 for codes)
Friability (high) greater than 1
Condition (bad) - 5
Exposure (high) *> 4
Accessibility (high) - 3
-------
-63-
In the New York City schools, data collection "sites"
were defined by the nature of the asbestos-containing mate^-
rial and the situation where the material was found. Soft
friable material was usually found in boiler rooms, fan
rooms, music rooms, and cafeterias. Sprayed-on fireproofing
material was usually found on structural steel which might
be exposed, or hidden behind a suspended ceiling.
Troweled-on acoustic plaster was usually found in cor-
ridors and auditoriums. The current recommendation to tak-
ing one bulk sample for every 5,000 square feet of homogen-
eous material was not followed strictly. Since a classroom
is approximately 700 square feet in area, the one sample per
5,000 square feet recommendation necessitates one sample for
every seven classrooms. The Division of School Buildings
reports that acoustic plaster (bulk sample) results can
vary from room to room and floor to floor even though the
plaster's appearance is homogeneous. But this variation may
be a function of laboratory error and not where the bulk
sample was obtained.
-------
-64-
-------
-65-
VI. SAMPLE DESIGN
A. Introduction
This chapter presents the probability sample
design proposed to study the relationships of the algorithms
to airbrone asbestos levels in schools. The sample design
will provide valid inferences for the study area discussed
below. Note that a purposive selection of schools and sites
within schools is not proposed because it would not permit
statistically valid inferences to be made for the entire
study area; the results would apply just to those sites
where data were collected. Also, there would be little
assurance that the results were not controlled or biased by
researcher preconceptions or objectives.
It is deemed feasible at this time that the study in-
clude only a geographically restricted area, as opposed to
being a national study. The inferential ability of the in-
formation generated by the study is concomitantly restrict-
ed. It is still thought important, however, that a sample
design be employed to allow conclusions to be drawn at the
level of the study area. The study area is in fact a real-
world situation, rather than some purposively or convenient
collection of classrooms that may well be atypical of any
real setting.
B. Study Area
The proposed study area includes all eligible
sites in New York City public schools. As discussed in Sec-
tion V of this report, reasons for using New York City pub-
lic schools are the availability of data for use in sample
selection and the variety of asbestos-containing sites pre-
sent. Eligible sites within a school include classrooms,
hallways, cafeterias, kitchens, gymnasiums, locker rooms,
libraries, and auditoriums. A protocol for partitioning
hallways into sites remains to be developed. Storage rooms
and offices are not included as eligible sites, as suggested
-------
-66-
in Price et al. (1980b). This suggestion was based on the
suspected lower activity levels in these rooms.
C. Overview of the Sample Design
A two-stage sample design with stratification im-
posed on each stage is proposed. First-stage sampling units
are public schools in New York City. Stratification of the
first-stage frame (i.e., list of all schools) is first pro-
vided by their classification into the following three stra-
ta: (1) asbestos-containing schools (according to prior in-
vestigation) , (2) schools with unknown asbestos content, and
(3) all remaining schools (those believed to have no asbes-
tos, including those that have had asbestos problems correct-
ed) . Using data collected in a survey of New York City pub-
lic schools for the presence of asbestos by the New York
City Asbestos Task Force, the class of asbestos-containing
schools will be further stratified according to asbestos
content, friability, condition, exposure, and accessibility.
This will result in 16 strata in all. At this time it is
thought that the first-stage sample will probably consist of
16 schools. In this case, the first-stage sample of schools
will be allocated equally among the first-stage strata.
Schools will be selected from the first-stage strata with
probability proportional to size measures based on school
enrollment.
The second-stage frame will consist of all eligible
sites in the first-stage sample of schools. By visual in-
spection, a trained rater will rate each of these sites as
to friability, condition, exposure, and accessibility. All
possible low/high combinations of these four factors will
form the second-stage strata. The sample size will be allo-
cated among the second-stage strata proportional to the
stratum totals of numbers of sites, with the restriction
that at least one site be selected from each nonempty second-
stage stratum. Within second-stage strata, sites will be
selected with equal probability and without replacement. To
-------
-67-
facilitate variance estimation, two independent samples (of
equal size) will be selected from the second-stage frame.
Possible total sample sizes are 32 sites, 48 sites, or 64
sites.
At each selected site, bulk samples of friable material
will be collected, according to the guidance given in Lucas
et al. (1980). Air sampling will be conducted as described
in Price et al. (1980b). Prior to air sampling, an exact
protocol for placement of samplers within a site will be
developed. This protocol should provide standardization in
the procedures used across all sites in order to avoid bias-
ing the results of the study. Additionally, at each select-
ed site, both trained and untrained raters will independent-
ly score all algorithm factors.
The sample design outlined above is a statistically
valid design that will give estimates for the study area
that are free from selection bias. Every public school in
the study area has a known positive probability of selec-
tion, and every eligible site in the study area has a known
positive probability of selection. Additionally, the repli-
cation at the second-stage facilitates variance estimation.
This design provides reasonable assurance that virtually all
of the 32 factorial cells (see Section IV) will be filled,
provided there exist such factor level combinations in the
study area. Supplementary selection procedures can be used
in the event that a cell believed to be nonempty is not in
the sample as desired.
D. Construction and Stratification of the First-
Stage Frame
The first-stage frame consists of all public
schools in New York City. A previous survey of New York
City public schools for the presence of asbestos is describ-
ed in Section V of this report. Based on the results of
this survey and records of corrective action taken since
that time, there are currently 69 public shools in New York
-------
-68-
City known to have at least one asbestos-containing site.
(Asbestos is said to be present at a site if the average
asbestos concentration at the site exceeds 1 percent.) The
remaining schools fall into one of the following categories:
(1) asbestos content unknown, (2) asbestos found present but
subsequently removed or corrected, or (3) asbestos thought
to be not present. Table 18 shows the distribution of
schools among these categories.
The first-stage frame will first be stratified into the
following three classes: (1) asbestos-containing schools,
(2) schools with unknown asbestos content, and (3) all re-
maining schools. The class of asbestos-containing schools
will be further stratified according to asbestos content,
friability, condition, exposure, and accessibility.
Table 19 shows the classification of asbestos-contain-
ing schools with respect to all possible low/high combina-
tions of the five factors listed above. A school is placed
in a given category if any surveyed site in that school ex-
hibited the specified combination of factor levels. This
means that it is possible for one school (with more than one
surveyed site) to be in more than one category. For asbestos-
containing schools, it is proposed that strata be constructed
from all possible low/high combinations of the five factors,
with the exception that only one stratum will include all
factor combinations involving high asbestos and low friabil-
ity. (A site having both high asbestos content and low fri-
ability is known to be a fairly rare occurrence (see Table
17)). This yields 25 strata of asbestos-containing schools.
The strata will be filled in ascending order of the number
of schools (not yet placed in strata) exhibiting the corre-
sponding factor level combinations, with the restriction
that each school belong to only one stratum. That is, if a
stratum has only one school, then that school will be in
that stratum and no other. Then if a stratum has two schools,
those two schools will be in that stratum and no other, etc.
-------
-69-
Table 18. Asbestos Content of New York City Public Schools
Asbestos Number of
Content Schools
Asbestos Known Present 69a
Asbestos Content Unknown 48a
Asbestos Found Present but 89a
Later Removed or Corrected
Asbestos Thought Not Present 792
TOTAL 988b
a Data from survey of New York City public schools by the
New York Asbestos Task Force and records of corrective
action.
b 1978 Curriculum Information Directory.
-------
-70-
Table 19. Distribution of Asbestos-Containing Public Schools
in New York City With Respect to Asbestos Content,
Friability, Condition, Exposure, and Accessibility
Asbestos
Content
Low
Low
Low
Low
Low
Low
Low
Low
Low
Low
Low
Low
Low
Low
Low
Low
High
High
High
•High
High
High
High
High
High
High
High
High
High
High
High
High
Missing
Fria-
bility
Low
Low
Low
Low
Low
Low
Low
Low
High
High
High
High
High
High
High
High
Low
Low
Low
Low
Low
Low
Low
Low
High
High
High
High
High
High
High
High
Condition
Good
Good
Good
Good
Bad
Bad
Bad
Bad
Good
Good
Good
Good
Bad
Bad
Bad
Bad
Good
Good
Good
Good
Bad
Bad
Bad
Bad
Good
Good
Good
Good
Bad
Bad
Bad
Bad
Exposure
Low
Low
High
High
Low
Low
High
High
Low
Low
High
High
Low
Low
High
High
Low
Low
High
High
Low
Low
High
High
Low
Low
High
High
Low
Low
High
High
Accessi-
bility
Low
High
Low
High
Low
High
Low
High
Low
High
Low
High
Low
High
Low
High
Low
High
Low
High
Low
High
Low
High
Low
High
Low
High
Low
High
Low
High
Number of
Schools
0
0
26
6
0
0
0
4
6
0
27
4
0
0
4
5
1
0
1
1
0
0
0
0
5
0
7
1
0
0
4
2
1
Data from survey of New York City public schools and records of
corrective action.
A school was placed in one of the above categories if any sur-
veyed site in that school exhibited the combination of factor
levels corresponding to that category. It is possible for one
school to be in more than one category.
a 69 schools in New York City are currently known to have
greater than 1 percent asbestos.
-------
-71-
Table 20 lists the 27 first-stage strata. The 25 stra-
ta of asbestos-containing schools are listed in the order in
which they were filled, using the procedure discussed above.
The number of schools in each stratum is given. Note that
11 strata contain no schools; of course no schools can be
selected from these strata. The fact that these strata are
empty does not mean, however, that no sites having any of
these factor level combinations will appear in the sample.
A site with one of these factor level combinations may exist
as a nonsurveyed site in an asbestos-containing school, or
such a site may exist in a school in stratum 15 or stratum
16. Any such site will have a known positive probability of
appearing in the sample.
E. Allocation and Selection of the First-Stage Sample
At this time it is thought that the first-stage
sample will probably consist of 16 schools. In this case,
the first-stage sample of schools will be allocated equally
among the 16 first-stage strata; i.e., one school will be
selected from each of the first-stage strata.
It is proposed to select schools from the first-stage
strata with probability proportional to size measures based
on school enrollment. Information on the number of eligible
sites per school is not readily available; however, it is
thought that school enrollment will have a fairly strong
positive relationship with the number of sites in the school.
The idea is to have selection probabilities at the first-
stage be such that a self-weighting sample can be obtained,
to the extent possible given the design constraints. The
distribution of New York City public schools by enrollment
categories is shown in Table 21. Note in Table 20 that two
strata, 1 and 11, contain only one school. Those two schools
will be in the first-stage sample with certainty.
-------
-72-
Table 20. First-Stage Strata
Description
Stratum
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
Asbestos
Content
Low
Low
Low
Low
Low
Low
Low
Low
High
High
High
High
High
High
Low
Low
Low
Low
Low
High
High
High
Low
Low
Low
Asbestos
Asbestos
Fria- Condi-
bility tion
Low Good
Low Good
Low Bad
Low Bad
Low Bad
High Good
High Bad
High Bad
High Good
High Bad
High Bad
High Good
Low
High Bad
Low Bad
Low Good
High Good
High Bad
High Bad
High Bad
High Good
High Good
High Good
Low Good
High Good
Content Unknown
Found Present but
Expo-
sure
Low
Low
Low
Low
High
Low
Low
Low
Low
Low
Low
High
— _
High
High
High
High
High
High
High
Low
High
Low
High
High
Accessi-
bility
Low
High
Low
High
Low
High
Low
High
High
Low
High
High
High
High
High
High
High
Low
Low
Low
Low
Low
Low
Low
Later Removed or
Corrected or Asbestos Thought Not
Present
No. of
Schools3
0
0
0
0
0
0
0
0
0
0
0
1
2
2
4
4
3
3
3
4
4
1
5
18
15
48
O O T
K K I
o o j.
a A school can only appear in one stratum.
-------
.-73-
Table 21. Distribution of New York City Public
Schools by Selected Size Categories
School
Enrollment
1 -
51 -
101
201
301
401
501
751
50
100
- 200
- 300
- 400
- 500
- 750
- 1,000
1,001 +
Total
Number of
Public Schools
2
16
30
29
45
54
232
212
378
998
The information in this table was taken from the
1978 Curriculum Information Directory.
-------
-74-
F. Construction and Stratification of the Second-
Stage Frame
The second-stage frame will consist of all eligible
sites in the first-stage sample of schools. A trained rater
will visit each sample school and list all eligible sites.
By visual inspection, the rater will rate each of these
sites as to friability, condition, exposure, and accessibil-
ity. All sites on the second-stage frame will be stratified
into the 16 categories formed from all possible low/high
combinations of friability, condition, exposure, and acces-
sibility. The 16 second-stage strata are listed in Table
22.
In this stratification proposed for the second-stage
frame, the factor of asbestos content is not used. There
are several reasons for this. First, the distribution of
the first-stage sample of schools with respect to asbestos
content has already been controlled to some extent by first-
stage stratification. The use of asbestos content to stra-
tify sites at the second stage presents several difficulties.
Information on asbestos content is not readily available for
all sites; in fact, in some schools bulk sampling was per-
formed at only one site. This situation requires the follow-
ing: (1) development of a method to classify as many sites
as possible as to asbestos content, using the available in-
formation, and (2) application of a two-phase selection pro-
cedure to the sites that cannot be classified. Developing a
classification method, (1), will undoubtedly result in some
misclassifications and the number of these misclassifications
will increase as the classification procedure becomes less
strict. The two-phase selection procedure, (2), will in-
volve bulk sampling at a sample of the sites that cannot be
classified. This bulk sampling and laboratory analysis will
take time and will be costly. Another aspect of the problem
is that the information currently available on asbestos con-
tent is not in terms of percentages but is in categorical
-------
-75-
Table 22. Second-Stage Strata
Stratum
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
Friability
Low
Low
Low
Low
Low
Low
Low
Low
High
High
High
High
High
High
High
High
Condition
Good
Good
Good
Good
Bad
Bad
Bad
Bad
Good
Good
Good
Good
Bad
Bad
Bad
Bad
Exposure
Low
Low
High
High
Low
Low
High
High
Low
Low
High
High
Low
Low
High
High
Accessibility
Low
High
Low
High
Low
High
Low
High
Low
High
Low
High
Low
High
Low
High
-------
-76-
form. The categories used are (1) 1 to 5 percent, (2) 6 to
50 percent, and (3) more than 50 percent. Thus, 50 percent
is used as the low/high asbestos content dividing point. It
is thought more appropriate, on the basis of data reviewed,
that a number in the neighborhood of 30 percent be used as
the dividing point.
Considering all of the factors discussed above, it is
concluded that not using asbestos content in second-stage
stratification will save time and money, while most probably
not substantially altering the distribution of the sample
from that desired. Perhaps the cost savings at this point
can be used to increase the total number of sites in the
sample.
G. Allocation and Selection of the Second-Stage Sample
The total sample size will be allocated among the
second-stage strata proportional to the stratum totals of
number of sites (i.e., if stratum 2 has 10 percent of the
sites, then approximately 10 percent of samples will be from
strata 2), with the restriction that at least one site be
selected from each nonempty second-stage stratum. Within
second-stage strata, sites will be selected with equal prob-
ability and without replacement. It should be noted that,
with the potential control of the distribution of the sec-
ond-stage sample implied by a large number of strata at this
stage, selection of the second-stage sample independently
within first-stage units is not possible given the proposed
allocation. As a result, sampling variances are not esti-
mable unless the design is replicated. Therefore, to facil-
itate variance estimation, it is proposed to select two in-
dependent samples of sites (of equal size) from the second-
stage frame.
As previously mentioned, it is likely that the first-
stage sample size will be 16 schools. Possible total sample
sizes are 32 sites, 48 sites, or 64 sites. (Cost/variance
modelling to determine the optimal average number of sites
-------
-77-
per school was not within the scope of this task.) Suppose
the total sample size is 48 sites. Then two independent
samples of 24 sites will be selected from the second-stage
frame. The 24 sites will be allocated among the second-
stage strata proportional to the stratum totals of number of
sites, but selecting at least one site from each nonempty
stratum. Intuitively speaking, proportional allocation is
in response to the desire to obtain more information where
more of the population of interest exists. For example,
suppose that .001 percent of all sites in the study area
have factor level combination 1, and 30 percent of all sites
have factor level combination 2. Then it is more important,
in terms of recommending future use of an algorithm, to val-
idate the algorithm for factor level combination 2.
H. Data Collection at Sample Sites
At each sample site, algorithm factors will be rat-
ed, bulk samples of friable material will be collected, and
air sampling will be performed. Both trained and untrained
raters will rate the algorithm factors. Collection of bulk
samples of friable material will follow the guidance given
in Lucas et al. (1980), using the sample site as the Sampling
Area. This guidance suggests that 3, 5, or 7 bulk samples
be collected, depending on whether the size of the Sampling
Area is less than 1,000 square feet, between 1,000 and 5,000
square feet, or greater than 5,000 square feet, respectively.
The guidance given in Lucas et al. (1980) concerning labora-
tory analysis should also be followed. Air sampling will be
conducted as described in Price et al. (198Ob). A randomi-
zation procedure for placement of air samplers within sample
sites is not within the scope of this study. However, an
exact protocol for placement of a sampler within a site
should be developed prior to air sampling. This protocol
should provide standardization in the procedures used across
all sites. It is very important that this be done in order
to avoid jeopardizing the validity of the study results.
-------
-78-
I. Design Effect
To insure that survey results are adequately pre-
cise to satisfy user needs, minimum allowable precision is
usually specified, at least for the more pertinent estimates
being sought. The sample variance or precision associated
with parameter estimates depends upon the following charac-
teristics of the sample design:
(a) the sample size to be used, depending upon the
cost contraints associated with the study,
(b) the magnitude of the finite population effect, df,
(c) the magnitude of the stratification effect, d ,
5
(d) the magnitude of the clustering effect, d ,
(e) the magnitude of the unequal weighting effect, d .
The product of the factors mentioned in (b) through (e)
above is referred to as the design effect (DEFF), as discuss-
ed in Kish (1965). The design effect is the ratio of the
variance obtained using the sample design to the variance
that would have been obtained using a simple random sample
of the same size. The sample size (n) divided by the design
effect (DEFF) is referred to as the effective sample size
*
(n ) of the sample design:
*
n = n/DEFF .
The effective sample size is the sample size that would be
required under simple random sampling to obtain the preci-
sion resulting from using the sample design with sample size
n.
In estimating the design effect expected under the sam-
ple design described in this section, the effects of clust-
ering and unequal weighting are considered. The effect of
the finite population correction factor is assumed to be
negligible, as are gains in precision due to stratification.
The clustering effect, d , is the increase in variance of
G
the sample due to the homogeneity of clusters. This effect
can be expressed as
-------
-79-
dc = 1 + p(n2 - 1) ,
where p is the intracluster correlation and n~ is the aver-
age cluster size, or the average number of sample sites per
school. The intracluster correlation p measures the homo-
geneity within clusters in terms of the protion of the total
element variance that is due to cluster membership. Possi-
ble values of n~ for this study are 2, 3, and 4 sites per
school. An appropriate value of p appears not to be known
with any degree of certainty, although values in the range
. 1 < p ^ .3 may be reasonable. The relationship of airborne
asbestos levels in sites within the same school versus be-
tween schools may be affected by several factors. More than
for sites in different schools, sites within the same school
may have friable material with similar asbestos content and
be similar in age and condition. Air circulation patterns
within a school may also affect the relationship of airborne
asbestos levels among sites within the same school. Taking
these factors into account, intracluster correlations of .1,
.2, and .3 are considered. Table 23 shows the clustering
effects corresponding to these values of p and average
cluster sizes of 2, 3, and 4.
The unequal weighting effect is the increase in vari-
ance due to the fact that the sampling weights (inverse se-
lection probabilities) are unequal. The sampling weight for
a site is constructed from (1) a first-stage component,
school selection probability, and (2) a second-stage compon-
ent, site selection probability, conditional on the first-
stage sample of schools. Because this second-stage compon-
ent depends on the first-stage sample of schools, the sam-
pling weights are not known prior to sample selection. It
is thought, however, that the major unequal weighting impact
will occur at the first stage of selection. This is due to
the following reasons: (1) a somewhat proportional alloca-
tion will be used at the second stage of selection, and (2)
if the first-stage sample of schools is allocated equally
-------
-80-
Table 23. The Clustering Effect Corresponding to
Selected Values of p and n~
Average No. of
Sites per School
(52)
2
3
4
Intracluster Correlation
.1
1.1
1.2
1.3
.2
1.2
1.4
1.6
.3
1.3
1.6
1.9
The entries in this table were calculated
according to
Clustering effect d_ = 1 + p (n0 - 1),
C fc
where p is the intracluster correlation, and
n_ is the average number of sites per school,
-------
-81-
among the first-stage strata (see Table 20), then the sec-
ond-stage strata (see Table 22) are not expected to vary
greatly in number of sites (e.g., not over a tenfold diffe-
rence) , although some differences are certainly expected.
Thus, only the first stage of selection will be considered
in the unequal weighting effect calculations. Also, first-
stage strata 15 and 16 (see Table 20) will not be included
in these calculations as it is thought that the variance in
airborne asbestos levels within each of these strata will be
very small compared to that within each of the other 14
strata.
To facilitate calculation of the unequal weighting ef-
fect, it is assumed that schools are selected from first-
stage strata with equal probability. (In fact, schools are
to be selected with probability proportional to size mea-
sures based on enrollment.) Assuming that one school is
selected from each stratum, the unequal weighting effect can
be expressed as
14
d.. = 14
where N. is the number of schools in first-stage stratum i.
Taking the values of N. from Table 20,
dw = 14 x 67S/692
d = 1.985 = 2.
w
Under all of the assumptions stated, it follows that
the design effect is
DEFF = 2 x d ,
c '
where d is the clustering effect given in Table 23. A rea-
G
sonable value of the intracluster correlation is .2. Then
the design effect is expected to be 2x1.2 =2.4 when the
average number of sites per school is 2. This means that
the ratio of the variance under this sample design to the
variance using a simple random sample of the same size is
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expected to be 2.4. When the average number of sites per
school is 3, then the design effect is expected to be 2.8.
The design effect is expected to be 3.2 when there is an
average of 4 sites per school.
J. Sampling Strategy
Measurements of airborne asbestos concentration
levels have been collected in schools and public buildings
by several investigators (see Sebastien 1980, Nicholson 1975
and 1978, Logue 1981, Patton 1980). Unfortunately, correla-
tions of these airborne levels with algorithm scoring has
had limited success to date. This could be due to several
factors including (1) the episodic nature of airborne con-
centrations which may be undetected with air sampling is
carried out over relatively short time periods; (2) the
problem of establishing background asbestos levels; and (3)
the fact that previous studies were not specifically design-
ed to investigate the relationship between airborne asbestos
levels and algorithm scores over a wide range of conditions
(e.g., high asbestos, low friability, good condition, etc.).
The currently proposed field study has been designed so that
several of these previous shortcomings have been taken into
account (i.e., air sampling for one week at each site, back-
ground air levels at each site, and a designed experiment
which covers a wide range of algorithm scores). Hopefully,
this design will lead to a significant relationship between
the various variables.
However, because of the limited success in past studies
and the fact that air sampling is extremely expensive, it
seems reasonable for the proposed study described in this
report to be selective in which sites are initially sampled.
That is, it is suggested that the initial 3 or 4 sites sam-
pled have widely different algorithm scores (for example,
two sites with high asbestos, friability, exposure, access-
ibility, and bad conditions (i.e., high algorithm scores),
and two sites with low asbestos, friability, exposure, acces-
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sibility, and good condition (i.e., low algorithm scores)).
In conjunction with these initial sites, air sampling will
also be done for background levels at each site. These ini-
tial comparisons (i.e., between high and low scoring sites
and background sites) should indicate early in the study if
there are going to be detectable relationships between air
levels and algorithm scores. If significant relationships
are not noted in these early sites, then a reassessment of
the field study should be undertaken to determine if further
sampling is worthwhile, considering the cost of asbestos air
sampling.
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REFERENCES
Britt HI, Luecke RH. May 1973. The estimation of parameters
in nonlinear, implicit models. Technometrics, Vol. 15,
No. 2.
EPA. September 1980a. Conference on the Exposure Assessment
Algorithm. U.S. Environmental Protection Agency. Wash-
ington , D. C.
EPA. October 1980b. Asbestos-containing materials in
schools: health effects and magnitude of exposure.
U.S. Environmental Protection Agency.
Kish L. 1965. Survey Sampling. John Wiley & Sons, Inc.
New York.
Logue EE, Hartwell TD. January 1981. Characteristics of
the asbestos exposure algorithm: empirical distribu-
tions, correlations, and measurement validity. Re-
search Triangle Institute. Research Triangle Park,
North Carolina.
Lucas D, Hartwell TD, Rao AV. December 1980. Asbestos-
containing materials in school buildings: guidance for
asbestos analytical program. Research Triangle Insti-
tute. Prepared for U.S. Environmental Protection
Agency. Washington, D.C. Contract No. EPA 560/13-80-
017A.
Nicholson WJ, et al. 1975. Asbestos-contamination of the
air in public buildings. Contract No. 68-02-1346.
Environmental Prtection Agency, Office of Air and Waste
Management, Office of Air Quality Planning and Standards:
Environmental Sciences Laboratory: Mt. Sinai School of
Medicine of the City University of New York.
Nicholson WJ, et al. 1978. Control of sprayed asbestos
surfaces in school buildings: a feasibility study.
Report to the National Institute of Environmental
Health Sciences. Environmental Sciences Laboratory:
Mt. Sinai School of Medicine of the City University
of New York.
Owen DW. 1962. Handbook of statistical tables. Addison-
Wesley.
Patton JL, et al. March 1980. Draft final report on asbes-
tos in schools. Battelle Columbus Laboratories.
Columbus, Ohio.
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-85-
References (continued)
Price BP, Townley CW. November 1980a. Asbestos algorithm
validation studies. Battelle Columbus Laboratories.
Columbus, Ohio.
Price BP, Melton C, Schmidt E, Townley CW. November
1980b. Airborne asbestos levels in schools: a design
study. Battelle Columbus Laboratories. Columbus,
Ohio. Contract No. 68-01-3858 to U.S. Environmental
Protection Agency, Washington, B.C.
Sebastien P, et al. August 1980. Measurement of asbestos
air pollution inside buildings sprayed with asbestos.
Laboratories d1Etude des Particules Inhalers. Paris,
France.
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APPENDIX A
The Noncentral t Distribution
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APPENDIX A
The Noncentral t-Distribution
The tables of statistical power appearing in this re-
port were developed using graphs of the noncentral t-distri-
bution found in Handbook of Statistical Tables by D.B. Owen,
Addison-Wesley, 1962. The critical parameter that must be
identified in order to use the tables is the noncentrality
parameter of the distribution. The noncentrality parameter,
6, takes different forms depending on the exact null hypoth-
esis and alternative being tested. As a point of reference,
we record the form of the noncentrality parameter used in
each table. Where interpretation is necessary, some addi-
tional discussion is presented.
Table 3; HQ: V10 = VB; Hl: U10 = UB + KG
6 = (n/2)1/2 K
/•
Table 7 ; same as Table 3
Table 10; HQ: BI = 0; Hi: &i = Ka/A
6 = K
1/2
where £g = (E(S± - S ) 2 )
Table 11; same as Table 10
Table 12: same as Table 10 with A = 30.
Table 14: V "A = PDA; Hi: "A = "DA
To determine this probability, the noncentral t-distribution
2 1/2
must be evaluated at 1.64 / (R S-, + 1) for the various
2 *-
values of R B-, . These probabilities are not available from
the tables in Owen (1962).
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Table 15: H0: *A = y; H: * = * + K°
r d-2p)2 i
Ll + K2 p (1 - p)J
p (1 - p)
5 11
The case is slightly more complicated where the hypoth-
eses are H_: 3, = 0 and H, : 8, = Ka/A and it is assumed that
algorithm scoring is not error free. In this case, it is
not possible to obtain power directly from the published
tables found in Owen. The difficulty is that the variance
of the estimator of B, depends on the value of 3-, hypothe-
sized under H, . (Estimation of S, follows Britt and Luecke,
1973) . The variance is given as
Var (B1) = (R B2 + 1) a2 / £ g2
2 2
where R = a /a as defined in the text of the report. The
statistical test of HQ: g, = 0 (significance level set at 5
percent) says to reject H if
> 1.64 .
To calculate the power of the test, we need to compute the
probability that 2
PI
> 1.64
when B, is not zero. Appropriate algebraic manipulation
leads to the following expression for power:
Prob < noncentral t variate > 1.64 / (R B2 + I)1/2/
with
« = Es Sj/ffg (R 82 + 1)1/2.
To determine this probability, the noncentral t-distribution
2 1/2
must be evaluated at 1.64/(R B-, +1) for various values
2
of R B, . These probabilities are not available from the
tables in Owen (1962).
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TECHNICAL REPORT DATA
(Please read Instructions on the reverse before completing)
1. REPORT NO.
EPA 560/5-81-006
3. RECIPIENT'S ACCESSION NO.
4. TITLE AND SUBTITLE
Airborne Asbestos Levels in Schools
Design Study
5. REPORT DATE
July 1981
6. PERFORMING ORGANIZATION CODE
7. AUTHOR(S)
Bert Price, Donna Watts, Everett Logue, Tyler Hartwell
8. PERFORMING ORGANIZATION REPORT NO.
9. PERFORMING ORGANIZATION NAME AND ADDRESS
Research Triangle Institute
Post Office Box 12194
Research Triangle Park, North Carolina 27709
10. PROGRAM ELEMENT NO.
11. CONTRACT/GRANT NO.
12. SPONSORING AGENCY NAME AND ADDRESS
Environmental Protection Agency
Office of Pesticides and toxic Substances
Washington, D.C. 20460
13. TYPE OF REPORT AND PERIOD COVERED
14. SPONSORING AGENCY CODE
15. SUPPLEMENTARY NOTES
16. ABSTRACT
This document describes a proposed field study to collect data in schools that
are to be used to analyze and validate two asbestos exposure assessment algorithms
as compared to levels of airborne asbestos. This field study would involve algorithm
scoring (including bulk asbestos sampling) and air sampling in sites (e.g., classrooms)
within selected schools. The objective of the planning study described in this report
is to establish the characteristics of various alternative statistical designs (e.g.,
number and characteristics of sample sites) for the proposed field study and to recom-
mend the most appropriate design.
The report is intended to provide EPA with an assessment of precision and com-
pleteness that can be expected from the data collected in the field study.
7.
KEY WORDS AND DOCUMENT ANALYSIS
DESCRIPTORS
b.lDENTIFIERS/OPEN ENDED TERMS C. COSATI Held/Group
Airborne Asbestos Levels
Asbestos in Bulk Samples
Asbestos Algorithm Scoring
Asbestos Levels in Schools
Statistical Design
8. DISTRIBUTION STATEMENT
19. SECURITY CLASS (This Report I
21. NO. OF PAGES
95
20. SECURITY CLASS (Thispage)
22. PRICE
EPA Form 2220-1 (Rev. 4-77) PREVIOUS EDITION is OBSOLETE
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