RESEARCH   TRIANGLE
             RTI/2757/05-01F
INSTITUTE
                               March  1986
                      DEMONSTRATION OF A TECHNIQUE FOR ESTIMATING DETECTION
                          LIMITS WITH SPECIFIED ASSURANCE PROBABILITIES
                                            Authors:
                                        C. Andrew Clayton
                                          John W. Hines
                                        Tyler D. Hartwell
                                   Research Triangle Institute
                          Research Triangle Park, North Carolina 27709
                                               and
                                  Peter M. Burrows, Consultant
                                       Post Office Box 105
                                  Cletnson, South Carolina 29631
                                  Contract Number:  68-01-6826
                                   Task Manager:  .John Warren
                                  Project Officer:  John Warren
                                     Statistical Policy Branch
                             Chemicals  and  Statistical Policy Division
                                Office  of Standards and Regulations
                               U.S.  Environmental Protection Agency
                                        401 M  Street, SW
                                     Washington, D.C. 20460

                       RESEARCH T R I A N G L E P A R K ,  N ORTH C AR 0 LI N A 27709-2 1 94

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RTI/2757/05-01F                                             March 1986
         DEMONSTRATION OF A TECHNIQUE FOR ESTIMATING DETECTION
             LIMITS WITH SPECIFIED ASSURANCE PROBABILITIES
                               Authors:

                           C. Andrew Clayton
                             John W. Hines
                           Tyler D. Hartwell
                      Research Triangle Institute
             Research Triangle Park, North Carolina 27709
                                  and
                     Peter M. Burrows, Consultant
                          Post Office Box 105
                     Clemson, South Carolina 29631
                     Contract Number:  68-01-6826

                      Task Manager:  John Warren
                     Project Officer:  John Warren
                       Statistical Policy Branch
               Chemicals and Statistical Policy Division
                  Office of Standards and Regulations
                 U.S. Environmental Protection Agency
                           401 M Street, SW
                        Washington, D.C. 20460
Submitted by:                           Approved by:
Tyler D. Hartwell                   w"R>  E' Mason
Delivery Order Leader               (r   Project Director

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TABLE OF CONTENTS
1.
INTRODUCTION. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.
CAL IBRA TrON ..............................................
2.1
2.2
2.3
Calibration Models...... .............................
Estimation of the Calibration Model.................
Calibration Designs .................................
3.
DETECTION LIMITS WITH SPECIFIED ASSURANCE PROBABILITIES ..
3.1 Linear Calibration with Known Parameters ............
3.2 Linear Calibration with Unknown Parameters ..........
3.3 Estimation of Detection Limits for Linear
Calibration. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . . .. . .
3.4
EPA Procedures for Determining a "Method

Detection Limit" ....................................
4.
AN EXPERIMENTAL EVALUATION OF THE RECOMMENDED APPROACH ...

4.1 Experimental Method .................................

4.2 Phase I Des ign ......................................

4.3. Phase I Data........................................

4.4 Determination of the Calibration Function ...........
4.5 Estimation of Detection Limits ......................

4.6 Phase I I Des ign .....................................

4. 7 Phase II Data.......................................

4.8 Phase II Detectability ..............................
5.
CONCLUSIONS AND RECOMMENDATIONS ..........................
REFERENCES. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Detectability and Detection Limits
SAS Algorithms for Determining Parameters
Associated with the Noncentral t-Distribution
Definition and Procedure for the Determination
of the Method Detection Limit
Extraction and Analysis of Selected Organics
in Sediments by Ultrasonication
APPENDIX A:
APPENDIX B:
APPENDIX C:
APPENDIX D:
Page
1
3
3
5
7
14
15
19
22
26
29
29
30
32
39
63
73
81
81
96
97

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Number
2-1
2-2
4-1
4-2
4-3
4-4
4-5
4-6
4-7
4-8
4-9
4-10
4-11
4-12
4-13
4-14
4-15
4-16
LIST OF TABLES
Title
Nine Design Patterns With Four Standard Concen-
trations ...........................................
Properties of Designs in Table 2-1 Under Linear
Calibration. . . . . . .. . ... ... . ... . . . . . . . .. .. . . . . .. . . ..
Actual Analyte Concentrations (ppm) Used in
Phase I of the Experiment..........................
Listing of Calibration Data for 2-Chloronaphthalene
Listing of Calibration Data for Dimethylphthalate ..
Listing of Calibration Data for Hexachlorobenzene ..
Listing of Calibration Data for Anthracene .........
Listing of Calibration Data for Phenanthrene
.......
Listing of Calibration Data for Fluoranthene
.......
Means and Standard Deviations of Adjusted Peak
Areast by Analyte and Concentration Level..........
Means and Standard Deviations of Transformed
Responses (square root of adjusted peak areas)t
by Analyte and Concentration Level.................
Calibration Design Parameterst by Analyte
..........
Calibration Model Parameter Estimatest by Analyte ..
Threshold Response Values y , By Analyte, for
p=0.01 and 0.05, and for r=~, 2t and 3 .............
Detection Limit Estimates for Selected p and q

Values, For r=1 ....................................
Detection Limit Estimates for Selected p and q

Values, For r=2 ....................................
Detection Limit Estimates for Selected p and q

Values, For r=3 ................................... 0
Estimated Phase I Detection Rates Associated
With Each Phase II Concentration Levelt For r=1 ....
Page
12
13
31
33
34
35
36
37
38
46
48
64
64
66
74
75
76
78

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LIST OF TABLES (continued)
Number
4-17
4-18
4-19
4-20
4-21
4-22
4-23
4-24
4-25
4-26
4-27
4-28
Title
Estimated Phase I Detection Rates Associated
With Each Phase II Concentration Level, For r=2 ....
Estimated Phase I Detection Rates Associated
With Each Phase II Concentration Level, For r=3 ....
Listing of Validation Data for 2-Chloronaphthalene .
Listing of Validation Data for Dimethylphthalate ...
Listing of Validation Data for Hexachlorobenzene ...
Listing of Validation Data for Anthracene ..........
Listing of Validation Data for Phenanthrene
........
Listing of Validation Data for Fluoranthene
. . . . . . . .
Comparison of Phase I and Phase II Detection
Rate Estimates, by Analyte, For r=1 ................
Comparison of Phase I and Phase II Detection
Rate Estimates, by Analyte, For r=2 ................
Comparison of Phase I and Phase II Detection
Rate Estimates, by Analyte, For r=3 ................
Comparison of Phase I and Phase II Detection
Rate Estimates, Based on Aggregating the Phase
II Results Over Five Analytes ......................
Page
79
80
82
83
84
85
86
87
89
91
92
94

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Number
3-1
3-2
3-3
4-1
4-2
4-3
4-4
4-5
4-6
4-7
4-8
4-9
4-10
4-11
4-12
4-13
LIST OF FIGURES
Title
Illustration of How Detection Probability, 1-q(x),

Varies With Concentration x When Using the Rule

'Assert X > 0 Whenever Y > y , .....................
p


Power Curves For Design 4 With Linear Calibration

N = 32, P = 0.05...................................
Power Curves For Design 4 With Linear Calibration

N = 32, p = 0.01...................................
Calibration Data:
y* Versus x*
Analyte = 2-Chloronaphthalene ......................
Calibration Data: y* Versus X*
Analyte = Dimethylphthalate ........................
Calibration Data:
y* Versus X*
Analyte = Hexachlorobenzene ........................
Calibration Data:
y* Versus X*
Analyte = Anthracene...............................
Calibration Data:
y* Versus X*
Analyte = Phenanthrene .............................
Calibration Data:
y* Versus X*
Analyte = Fluoranthene .............................
Calibration Data: Y Versus X*
Analyte = 2-Chloronaphthalene ......................
Calibration Data: Y Versus X*
Analyte = Dimethylphthalate ........................
Calibration Data: Y Versus X*
Analyte = Hexachlorobenzene ........................
Calibration Data: Y Versus X*
Analyte = Anthracene...............................
Calibration Data:
Y Versus X*
Analyte = Phenanthrene.............................
Calibration Data:
Y Versus X*
Analyte = Fluoranthene .............................
Calibration Data: Y Versus X
Analyte = 2-Chloronaphthalene
......................
Page
16
23
24
40
41
42
43
44
45
50
51
52
53
54
55
57

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LIST OF FIGURES (continued)
Number
4-14
4-15
4-16
4-17
4-18
4-19
4-20
4-21
4-22
4-23
4-24
4-25
Title
Calibration Data: Y Versus X
Analyte = Dimethylphthalate ........................
Calibration Data: Y Versus X
Analyte = Hexachlorobenzene ........................
Calibration Data: Y Versus X

Analyte = Anthracene...............................
Calibration Data:
Y Versus X
Analyte = Phenanthrene.............................
Calibration Data: Y Versus X
Analyte = Fluoranthene .............................
Estimated Power Curves
Analyte = 2-Chloronaphthalene,
p=O .01 ..............
Estimated Power Curves
Analyte = Dimethylphthalate,
p=O. 01 ................
Estimated Power Curves
Analyte = Hexachlorobenzene,
p=O. 0 1 ................
Estimated Power Curves
Analyte = Anthracene, p=O.Ol
...................... .
Estimated Power Curves
Analyte = Phenanthrene,
p=O. 0 1 .....................
Estimated Power Curves
Analyte = Fluoranthene,
p=O. 0 1 .....................
Representative Chromatograms For Analysis of
Fortified Sediment Samples, (a) Phase II Sample,
Dimethylphthalate (DMP) Fortified at 0.173 ppm;
(b) Phase I Sample, Dimethylphthalate (DMP)
Fortified at 0.217 ppm .............................
Page
58
59
60
61
62
67
68
69
70
71
72
88

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1.
INTRODUCTION
Procedures for determining critical concentrations below which
chemical instrumentation or chemical methods are considered "unreliable"
-- in terms of their capability to indicate the presence or absence of a
given substance in some environmental medium -- have received a great
deal of attention within the United States' Environmental Protection
Agency (EPA). This concern has been manifested in the Agency's require-
ment that concentrations below the "detection limit" be clearly identi-
fied in data bases or in reported data. Unfortunately, different
authors and researchers have used different terminology to refer to such
limiting concentrations and have adopted different definitions of
"detection limits." In many cases, procedures for determining such
limits have been established and implemented with little conceptual or
statistical justification.
The purposes of this report are (1) to describe an approach for
defining and estimating detection limits, and (2) to demonstrate the use
of this approach through an actual example relevant to EPA activities.
Errors inherent to chemical measurement of analyte responses -- a
result of instrumental noise, chemical interferences, operator incon-
sistencies, etc. -- require that some type of "detection rule" be
formulated for asserting that a given analyte either is or is not
present in a given source matrix. For example, the rule might take the
form: "declare analyte A to be present if the magnitude of a single
response measurement exceeds some prescribed value". Such a "detection
rule" is employed in EPA's current procedure for defining/estimating the
"Method Detection Limit", for example. Based upon the use of any such
rule, we may claim that the analyte of interest is present in the given
matrix when in fact it is not (a Type I error, or false positive state-
ment). On the other hand, we may conclude that the analyte is absent
when it is in fact present (a Type II error, or false negative state-
ment).
As shown in Chapter 3, EPA's current approach for defining/estimat-
ing the "Method Detection Limit" tends to emphasize achieving protection
against false positives and tends to ignore achieving any specific level
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of protection against false negatives. A preferred approach would
permit protection against both types of errors at prescribed levels that
are consistent with the objectives and needs of the specific EPA study
or program. Such an approach is described in Chapter 3.
The approach advocated herein is based upon the notion that all of
necessary parameters can be estimated using standard calibration proce-
dures.
(The calibration is carried out, however, over a range of "low"
concentrations that is much narrower than the range that would normally
apply to most calibration situations.) Chapter 2 therefore describes
the calibration process. One major function of Chapter 2 is to intro-
duce some of the terminology and notation that is used subsequently in
Chapters 3 and 4. A second major function is to describe some of the
important features and properties of calibration designs in relation to
analyte detection.
Chapter 3 then defines the detection limit, describes procedures
for its estimation, and compares the approach with traditional methods.
Chapter 4 describes a two-phased experiment designed to illustrate
application of the recommended approach (Phase I) and to validate its
performance empirically (Phase II).
Conclusions and recommendations are given in Chapter 5.
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2.
CALIBRATION
Consider a specific environmental medium, exposure source, or
exposed population that may contain a specific analyte of interest. For
a given medium (water, air, soil, sediment, human blood, etc.), assume
that a well defined protocol exists for preparing and processing samples
and taking observations.
Let X denote the true, but unknown, concentra-
tion of the analyte and assume that the method produces an indirect
measure, Y, that is related to X in some known (or assumed but testable)
manner. The relationship between Y and X (i.e., the calibration curve)
is assumed to be a monotonic function of concentration that depends on
unknown parameters.
These parameters are estimated during a calibration
phase by fortifying "uncontaminated" media with known amounts of the
analyte(s), processing these samples according to the established
protocol, measuring the responses, and then fitting the calibration
curve that "relates Y to X.
Estimated concentrations in subsequent
samples are then determined by inverse interpolation of the estimated
calibration curve (standard curve).
When some of the concentrations that may be estimated in this
manner are "low", there is an immediate interest in whether the analyte
is present (X > 0) or absent (X = 0).
In such a case, it is desirable
to be able to assert, with high probability of being correct, that the
analyte either is or is not present.
Therefore there is need for some
rule, based on the magnitude of one or more observations, that provides
the mechanism for making such assertions. Regardless of the actual rule
adopted, such assertions may be incorrect in either of two ways:
Type I Error (false positive): assert that X > 0 when in fact X = 0
Type II Error (false negative): assert that X = 0 when in fact X > o.
2.1
Calibration Models
Assume, for further specificity, that the underlying monotonic
relationship between Y and X (i.e., ignoring unobservable, uncontrol-
lable errors in observations) is such that Y increases as X increases,
and that the expected background value of Y is an unknown constant, a.
The calibration model therefore has the form
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Y = a + f(x,6) + E
x
(2-1)
where
Y is an observed response when X = x,
x
a is an unknown parameter or vector of parameters,

f(x,6) is a monotonic increasing function of x that satisfies
f(0,6) = 0, and
E is the observational error for the particular determina-
tion.
In order to provide a consistent context for the various approaches
considered herein, we will assume that the observational errors, E, from
different determinations (at the same or different x values):
(i)
(ii)
(iii)
(iv)
are independent,
are normally distributed,
have zero mean, and
have constant variance, a2, which does not depend on
concentration.
Clearly, there are cases where the distributional assumptions (ii)
and/or (iv) may not be satisfied; in many such cases, a transformation
of observations may be applied so that the assumptions will be tenable
on the transformed scale.
The most straightforward example of model (2-1) is the straight-
line calibration curve:
Y = a + ax + E
x
In this case, 6 represents the slope of the line.
(2-2)
For (2-2) to be
useful, the slope must not be zero. When dealing with model (2-2), we
will assume (without loss of generality) that a > o.
It should be noted that the model parameters of either (2-1) or
(2-2) must remain constant over all sources (matrices) to which the
analytical protocol is applied. Moreover, the distributional properties
of the errors must not be dependent upon the particular matrices for
which the calibration results are to be extrapolated. (Such assumptions
are essentially required in any calibration situation involving extrapo-
lation to other matrices.)
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2.2
Estimation of the Calibration Model
In most situations, the form of the function f is unknown and the
values of the parameters a, a, and 02 are unknown.
The validity of
distributional assumptions concerning  may also be unknown. Generally,
estimation of parameters and testing for adequacy of the chosen func-
tional form and for validity of the variance homogeneity assumption are
Step 1.
carried out as follows:
Step 2.
Step 3.
Step 4.
Step 5.
A set of n "uncontaminated" sample a1iquots are fortified at m
different standard concentrations xl' x2' ..., xm and nj
determinations are made at x., where
J
n =
m
E
j =1
n. .
J
The resulting responses, Yi (i=1,2,...,n), are examined (e.g.,

via plots against the x.) and sample means y. and variances s~
. J J J
within each standard concentration level are computed.

The sample variances s~ are used to examine the reasonableness
J
of the variance homogeneity assumption. Tests such as those
of Bartlett or Levene can be used (see Snedecor and Cochran,
1980).
Transformations are considered for stabilizing the
variances, if necessary, and steps 2 and 3 are repeated.
Based on steps 2 and 3, various functional forms (f) are
considered and estimates of the model parameters are obtained
using a least squares criterion (equivalent to maximum likeli-
hood estimation by virtue of assumptions (i) - (iv)).
If the
functional form is a linear combination of the unknown param-
eters, then ordinary linear least squares procedures are used;
otherwise, nonlinear least squares procedures would be needed
to estimate the parameters.
Adequacy and lack of fit of candidate functional forms are
evaluated, and a satisfactory model is selected.
When using a
completely random design (see Section 2.3), for instance, the
standard 1ack-of-fit test is conducted by computing
A ~ ~
F = [(n-p) 02 - (n-m) 02] / [(m-p) 02] , m > p
(2-3)
where nand m are as defined previously; p is the number of
model parameters (e.g., p = 2 for model [2-2]);
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~
02 =
m
E
j=l
(n. - l)s~ / (n-m) = pooled within-standard~
J J variance;
(2-4)
and
A
02 is the residual mean square for the particular model
n A
E (y. - y.)2 / (n-p) , where y. is the fitted value
i=l ~ ~ f. th ~.th
or e ~ observation.

The statistic F is compared with the tabulated critical values for

the F distribution with (m-p) and (n-m) degrees of freedom. Lack

of fit of the particular model form is indicated if the F statistic
=
(2-5)
exceeds the tabulated value.
Since the alternative hypothesis for
the test in (2-3) is not explicit, this test is broadly sensitive
to any type of lack of fit, though not highly sensitive to any
particular type. Rather than using (2-3), it is possible (even
preferable) to examine model adequacy with respect to more specific
alternatives (e.g., comparing fits of linear versus quadratic
models).
As an example of step 4 above, assume that model (2-2), along with
assumptions (i) through (iv) , has been judged to be adequate. In this
case, estimates of the model parameters (denoted by A) are given by
A
a = y - ax
and
(2-6)
a - Q / Q
xy xx
where y and x are the means (over all n observations) of the observed

responses and of the standard concentrations, respectively, and
(2-7)
m
~y = E n. (Xj - x) (y. - y), and
j=l J J
m
Qxx = En. (x. - x) 2 .
j=l J J

Standard errors of the parameter estimates
1 X2 ~
s.e.[a] = 0 [-+ -]
n Qxx
are given by
(2-8)
and
A 1
s.e.[a] = 0 Q-~
xx
(2-9)
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Under this model and related assumptions, the residual variance (with
(n-2) degrees of freedom) provides the estimate
A

cr2 = [Qyy - Q~y / Qxx] / (n-2)
(2-10)
where
n
Q = L (y. - y)2 .
yy i=l ~
Assume that an observation, Y,.is obtained for some subsequent
sample having unknown concentration; this concentration would generally
be estimated by solving the calibration function for x. For model
(2-2), for instance, this concentration would be estimated as
x = (Y - a) / 6 .
(2-11)
This estimate is not unbiased, and has infinite variance.
It is,
nevertheless, based upon the statistics y, y, and 6, which are suffi-
cient statistics for the involved parameters. (If cr2 is known and 6>0,
then an unbiased estimator for the unknown concentration that is also
based on sufficient statistics can be obtained [see Williams, 1969]).
Assume now that r subsequent determinations are made from the same
source matrix with unknown concentration X = x.
Let Y denote the mean
of the r observed responses. Then (Y - a - 6x) is distributed normally

with mean zero and variance cr2w2, where
x

w2 = -1- + -1- + (X-X)2 .
x r n Q
xx
(2-12)
Similarly, (Y - a) is distributed normally with mean 6x and variance
cr2W2.
o
Calibration Designs
It is important to distinguish between ana1yte detection and
ana1yte concentration estimation when considering calibration designs.
Several considerations are common to both contexts, but others are
2.3
specific to ana1yte detection and result in different criteria for
optimal design. Complete elaboration of these matters is beyond the
scope of this report which concentrates on detectabi1ity and detection
limits; but the effects of variations in calibration design on detecta-
bi1ity and its estimation cannot be ignored and we discuss them briefly
in this section.
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An initial consideration is a direct consequence of natural or
imposed groupings of sample materials and their processing. This leads
to the classical experimental design features of "blocking" and randomi-
zation (Cochran and Cox, 1957). It is common for groups or sets of
samples to be collected, prepared, processed and analyzed in an orga-
nized fashion. Especially common is an imposed time structure, such as
groups of samples that are preparedand/or processed together on the
same day. Organization with spatial structure is another possibility.
In this circumstance, it is often beneficial to arrange that such
structure is incorporated in the calibration design. And when the
magnitude of extraneous variations (attributable to levels of such
structures) is important, it is essential to arrange that the calibra-
tion samples (known standards) and the unknown samples (subjects for
detection inferences) are processed in an overall designed manner.
As
an example of such considerations, we cite Burrows et a1. (1984) who
employ lattice square designs to achieve two-way control over extraneous
variations when calibrating antigen concentrations by enzyme-linked
immunosorbent assay.
Choice of experimental design, and its consequences, are specific
to context.
We make no attempt to generalize here; indeed, we consider
only the simplest choice (completely randomized designs (CRDs)) in
subsequent chapters.
Nevertheless, with appropriate modifications for
parameter estimation and for degrees of freedom attached to the estimate
cr, the formulations and methods in Chapter 3 continue to apply.
A major consideration is choice of the actual standard analyte
concentrations x., and their replication counts n., to include in the
J J
calibration design. The remainder of this section attempts to summarize
the consequences of three interrelated and, unfortunately, antagonistic
criteria for this choice:
(i)
(ii)
designs useful for exploring the calibration model,
designs yielding sensitive detection (minimum detection
limits) ,
designs yielding precise estimates of detectability and
detection limits.
( iii)
Prior information, such as might be available from preliminary evalua-
tions of the chemical method and instrumentation, is invaluable.
For
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example, if the form of f(x,8) is well-known or strongly suspected
a priori, the choice of standards can be tailored to tha~ specific form
using criteria (ii) and (iii); otherwise, it is necessary to deploy the
standards in a fashion (such as evenly spaced and equally replicated
points) that conflicts with criteria (ii) and (iii) but enables a
determination of that form for f(x,8) considered adequate, as discussed
in Section 2.2.
Similarly, if it is known or suspected that the vari-
ance homogeneity assumption is tenable a priori, then fewer replications
at each standard are necessary (implying that a larger number of distinct
standards might be included). Finally, prior information, often quite
informal, at low concentrations x (as opposed to calibrations against
concentrations far beyond the region where detection is uncertain)
enables choice of a range of standards in the neighborhood of the
probable or speculated detection limit.
Henceforth it is assumed that the form of f(x,8), with homogeneous
error variance, has been determined and, for simplicity, that it yields
the straight line model of (2-2).
Hubaux and Vos (1970) discuss several
variations in choices of x. and n. in this context and their general
J J
conclusions apply here also. But one of their recommendations is not
applicable: They suggest arranging that x coincide with a prior guess
for the detection limit because of an erroneous formulation of the
latter which does not conform to our definition in Chapter 3.
It is convenient to characterize the calibration design (standards
xl' x2' ..., xm and their replication counts n1' n2' n3' ..., nm) by
four summary features that are open to choice:
(a)
overall sample size:
m
n = L n. ,
. J
J
(d)
location: L = min(x.) ,
j J

span: S = max(x.)-L ,
j J

pattern of relative spacings (free of location and span) and

proportionate replication (free of overall size n):

u. = (x.-L)!S and r. = n.!n, so that the u.'s are free to be
J J J J J
deployed over the interval [0,1] inclusive, and L r. = 1.
j J
(b)
(c)
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Thus the important characteristics x, 0 and w
~x 0
of Section 2.2 are
given by
u = E r.u.
j J J
Qxx = nQS2, Q = E rj(uj-u)2,
x = L + Su,
var(6/a) = 1/nQS2
(2-13)
W2 = 1 + 1 [1+(u+L/S)2/Q],
o r n .
A
var(a/a) = W2 - l/r.
o
(2-14)
Recall that r is the number of independent observations providing Y for
a sample source of unknown concentration, and that degrees of freedom v
are fixed at (n-2).
Criterion (ii), designs yielding sensitive detection, is met when
w is minimized (this maximizes detection rates at all concentrations
o
x)O and so minimizes detection limits as described in Chapter 3).
Observe first that w is minimized with respect to location, regardless
o
of span or pattern, by choosing L=zero (that is, always include zero or
blank as one of the standards), and that this choice does not affect Q
or var (6). Then W2 reduces to
o
W2
o
= 1 + 1 [1+u2/Q] .
r n
(2-15)
-
Observe that w is minimized with respect to pattern by choosing u=zero,
o
that is, by placing all n observations at x=zero. This choice conflicts
immediately with criterion (iii), because now S=zero and B is not
estimable, so that detection rates and detection limits (as defined in
Chapter 3), although optimized, are not estimable (since they depend
upon B).
Precision of estimates of detection rates and of detection
limits, criterion (iii), improves as n increases and as the precision of
the estimate B improves (which, with n fixed, occurs as Q and S increase).
But this conflicts with criterion (ii), because Q is maximized when n/2
observations are placed at u=O and n/2 at u=l fixing u=0.5 (instead of
close to zero); fortunately, S can be increased without effect on u.
Thus, taking criteria (ii) and (iii) together, it is necessary to
compromise between the desirability of small U and large Q. This is
achieved by concentrating more of the u.'s closer to zero than to 1.0,
. J
or by increasing the proportionate replications, r., of those u. 's close
J J
to zero at the expense of those closer to 1.0.
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Some examples of these considerations are given in Table 2-1 which
contains nine design patterns with four concentrations each. The
consequences of these manipulations of u1' u2' u3 and u4' and r1' r2'
r3' and r4 are shown in Table 2-2 which contains n, and
SA = standard deviation of (a/a),
A
SB = standard deviation of (S8/a),
WO r = w at (2-12) for r = 1, 2, and 3.
- 0
Further discussion of design strategies is postponed until Section 3.2
where the concept of manipulating calibration design to achieve speci-
fied detectability is introduced.
(2-16)
(2-17)
-11-

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TABLE 2-1. NINE DESIGN PATTERNS WITH FOUR STANDARD CONCENTRATIONS.
DESIGN CONCENTRATION UI (PROPORTIONATE REPLICATION RI)
NO. U1 (R1 ) U2 (R2). U3 (R3) U4 (R4)
1 0.000 (.25) 0.333 (.25) 0.667 (.25) 1.000 (.25)
2 0.000 (.40) 0.333 (.20) 0.667 (.20) 1.000 (.20)
3 0.000 (.40) 0.333 (.30) . 0.667 (.20) 1.000 (.10)
4 0.000 (.25) 0.200 (.25) 0.800 (.25) 1 .000 (.25)
5 0.000 (.40) 0.200 (.20) 0.800 (.20) 1 .000 (.20)
6 0.000 (.40) 0.200 (.30) 0.800 (.20) 1 .000 (. 10)
7 0.000 (.25) 0.250 (.25) 0.500 (.25) 1.000 (.25)
8 0.000 (.40) 0.250 (.20) 0.500 (.20) 1.000 (.20)
9 0.000 (.40) 0.250 (.30) 0.500 (.20) 1.000 (.10)
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TABLE 2-2. PROPERTIES OF DESIGNS IN TABLE 2-1 UNDER LINEAR CALI BRAT I ON 
 DESIGN N SA SB  WO_1 WO_2 WO_3
   8 0.591532 0.948493 1.16186 0.921906 0.826585
   12 0.482984 0.774442 1.11053 0.856314 0.752733
   16 0.418276 0.670686 1.08395 0.821556 0.712943
   20 0.374118 0.599880 1.06769 0.799977 0.687966
   24 0.341521 0.547613 1.05671 0.785262 0.670798
   28 0.316187 0.506991 . 1.04880 0.774580 0.658261
   32 0.295766 0.474247 1.04282 0.766471 0.648699
   36 0.278851 0.447124 1.03815 0.760104 0.641164
   40 0.264541 0.424179 1.03440 0.754972 0.635071
  2 10 0.453708 0.813369 1 .09811 0.840150 0.734292
   15 0.370451 0.66411 3 1.06641 0.798269 0.685979
   20 0.320820 0.575139 1.05020 0.776483 0.660499
   25 0.286950 0.514420 1.04036 0.763112 0.644728
   30 0.261949 0.469599 1.03374 0.754067 0.633996
   35 0.242517 0.434764 1.02899 0.747539 0.626217
   40 0.226854 0.406685 1.02541 0.742605 0.620319
  3 10 0.447146 0.948493 1.09542 0.836624 0.730256
   20 0.316180 0.670686 1.04879 0.774577 0.658258
   30 0.258160 0.547613 1.03279 0.752759 0.632440
   40 0.223573 0.474247 1.02469 0.741610 0.619127
  4 8 0.555719 0.857493 1 . 14404 0.899346 0.801347
   12 0.453743 0.700140 1.09813 0.840168 0.734313
   16 0.392953 0.606339 1.07444 0.808957 0.698387
   20 0.351468 0.542326 1.05997 0.789639 0.675916
   24 0.320844 0.495074 1.05021 0.776493 0.660511
   28 0.297044 0.458349 1.04319 0.766965 0.649283
   32 0.277859 0.428746 1.03789 0.759741 0.640733
   36 0.261968 0.404226 1.03374 0.754074 0.634004
   40 0.248525 0.383482 1.03042 0.749510 0.628568
  5 10 0.436931 0.753778 1.09129 0.831209 0.724046
   15 0.356753 0.615457 1.06173 0.792006 0.678680
   20 0.308957 0.533002 1.04664 0.771657 0.654819
   25 0.276340 0.476731 1.03748 0.759186 0.640076
   30 0.252262 0.435194 1.03133 0.750757 0.630055
   35 0.233550 0.402911 1.02691 0.744678 0.622799
   40 0.218466 0.376889 1.02359 0.740086 0.617301
  6 10 0.417635 0.852493 1.06371 0.821230 0.712567
   20 0.295312 0.602804 1.04269 0.766296 0.648493
   30 0.241121 0.492187 1.02866 0.747087 0.625678
   40 0.208817 0.426246 1.02157 0.737295 0.613953
  7 8 0.547723 0.956183 1.14018 0.894427 0.795822
   12 0.447214 0.780720 1.09545 0.836660 0.730297
   16 0.387298 0.676123 1.07238 0.806226 0.695222
   20 0.346410 0.604743 1.05830 0.787401 0.673300
   24 0.316228 0.552052 1.04881 0.774597 0.658281
   28 0.292770 0.511101 1.04198 0.765320 0.647339
   32 0.273861 0.478091 1.03682 0.758288 0.639010
   36 0.258199 0.450749 1.03280 0.752773 0.632456
   40 0.244949 0.427618 1.02956 0.748331 0.627163
  8 10 0.433013 0.845154 1.08972 0.829156 0.721688
   15 0.353553 0.690066 1.06066 0.790569 0.677003
   20 0.306186 0.597614 1.04583 0.770552 0.653516
   25 0.273861 0.534522 1.03682 0.758288 0.639010
   30 0.250000 0.487950 1.03078 0.750000 0.629153
   35 0.231455 0.451754 1.02644 0.744024 0.622017
   40 0.216506 0.422577 1.02317 0.739510 0.616610
  9 10 0.425685 1.03626 1.08683 0.825353 0.717315
   20 0.301005 0.73274 1.04432 0.768508 0.651105
   30 0.245770 0.59828 1.02976 0.748600 0.627484
   40 0.212843 0.51813 1.02240 0.738446 0.615334
-13-

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3.
DETECTION LIMITS WITH SPECIFIED ASSURANCE PROBABILITIES
Within EPA, calibrations of ana1yte responses versus standard
concentrations Xj (j = 1, 2, ..., m) are usually performed for the
purpose of estimating ana1yte concentrations, as opposed to the purpose
of detecting ana1yte presence/absence. The former purpose implies that
the range covered by the x. should encompass all of the concentrations
J
occurring in matrices for which the analytical protocol and calibration
results apply. The results of Section 2.3, on the other hand, indicate
that calibration designs directed towards the latter purpose would cover
a narrower range of concentrations -- namely, that range over which
detectability was considered uncertain. The definitions and methods of
this chapter presume that a calibration of this latter type will be/has
been conducted. (This does not imply that an overall calibration design
aimed at both purposes should not be considered; on the contrary,
efficient use of resources dictates that such a composite calibration
design be utilized whenever both of the aforementioned purposes are
deemed relevant.)
Section 3.1 below provides the basic concepts associated with
developing detection limits that achieve specified assurance probabili-
ties (i.e., specified rates of protection against both Type I and Type
II errors). In that section, the simple straight-line calibration model
is assumed and its parameters are assumed to be known. The latter
assumption, though obviously impractical, is temporary: it allows many
of the technical details relating to estimation to be avoided. For
example, the methods described in Section 2.2 are unnecessary.
In Section 3.2, the straight-line calibration model is still
assumed but the definitions and methods are developed for the case in
which this relationship between Y and X must be estimated.
Detection limits, as formulated in Section 3.2, are functions of
unknown parameters.
Hence, Section 3.3 deals with the estimation of
detection limits and associated parameters for the linear calibration
case.
A more general calibration framework, along with additional
details concerning estimation of the calibration model, detection
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limits, etc., can be found in Burrows (1985) which is included as
Appendix A.
The chapter concludes with Section 3.4, which furnishes a brief
review of EPA's current approach for defining/estimating the "Method
Detection Limit."
3.1
Linear Calibration With Known Parameters
We assume temporarily that the calibration model (2-1) relating Y
and X is known (i.e., both the model form and the parameters
a, a, and 02 are known) and, for simplicity, we assume that the model is
of the form (2-2). As in Chapter 2, we presume the observed responses Y
at any given concentration x are distributed normally, and assume that r
independent determinations will be made from each source matrix. Hence
the mean of the r responses, Y, has mean (a + Bx) and variance 02/r.
With B > 0, it is reasonable to declare that the analyte of interest
is present if Y is "sufficiently large". We formalize this notion by
adopting the following detection rule:

'Assert X > 0 whenever Y > y "
p
where y is a threshold response value
p
(3-1)
chosen so that the rate of false
positives, in repeated applications of the rule to (blank)
samples, is fixed at some desired (low) level p.
Let Z denote a random variable having a standard normal distribu-
tion (mean = 0, variance = 1), and let z denote the upper lOOp percent-
p
age point of this distribution,
i.e. ,
With the
choose y
p
Pr[Z > z ] = p.
p
assumptions given above, the rule (3-1) is specified if we

such that the probability of making a Type I error is
p =
Pr[false positive] = Pr[Y > y Ix=O]
p
- y - a y - a
Pr [ Y - a > p !X=O] = Pr [Z > P ].
o/fr o/Ir o/Ir

Thus we can satisfy the condition of having a Type I error rate equal to
=
p by choosing
y = a + oz / Ir .
p p
The value y shown in Figure 3-1a,
p
response threshold value.
(3-2)
for instance, illustrates such a
Note that the shaded region under the normal
-15-

-------
y
yp
ex
o
x
Figure 3-la.
Xl x2
Observed Response vs. Concentration, With Y Distributions.
p~
o
l-q(x)
1
-----------------
x
0.5
o
Xl x2
Detection Probability vs. Concentration.
Figure 3-lb.
Figure 3-1.
Illustration of How Detection Probability, l-q(x), Varies With

- ,
Concentration X When Using the Rule 'Assert X > 0 Whenever Y > Y
P
-16-

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distribution pictured at X=O is equal to the specified value p. (This
distribution represents the distribution of values that would be expect-
ed if a large number of Y values were independently determined for a
source matrix having X=O.)
Now suppose that r determinations are made on a single source
matrix having some unknown, but positive, concentration x.
We then
compute the probability that the mean of the r determinations, Y, will
exceed y , the threshold response value defined by (3-1) and (3-2).
p
Since Y is normally distributed with mean (a + ax) and variance a2/r,
the detection .probabi1ity at x is given by
Y - (a + ax)
Pr(Y > y Ix = x] = Pr [
p
a +
az /IT - (a + ax)
p
>
a/IT
a/IT
= Pr (Z > z
p
ax
a/IT
1 - Pr (Z < z
- p
ax
a/IT
] = 1 - q(x) .
(3-3)
Note the following:
(i)
The function q(x), defined by (3-3), depends not only on the true
concentration x but also on the values p, r, and a/a (as well as
the distributional assumptions).
The function q(x) represents the rate of false negatives that
(ii)
would be expected in repeated applications of the rule (3-1) --
i.e., the probability of erroneously concluding that the ana1yte
was absent (X = 0) when it was not (X > 0).
The detection probability, (1 - q(x)], is an increasing function
(iii)
of x bounded as follows:
1 - q(O) = P
and
1 - q(~) = 1.
Figure 3-1b illustrates the function (1 - q(x)].
This curve is
simply the power curve associated with the test of the null hypothesis
that the ana1yte concentration is zero, and derives directly from the
classical Neyman-Pearson theory. For a given x, the point on the curve
in Figure 3-1b represents the area above the point y (in Figure 3-1a)
p
under the normal distribution centered at a + ax. The shaded regions in
Figure 3-1a, for instance, illustrate these areas for three such x
values:
-17-

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x = 0, X = xl' and X = x2.
Of particular interest is the concentration level xl' in Figure 3-1,
because when X = xl' we have
yP = a + aXl
or
Xl = (Yp - a) / a .
Frequently, researchers have adopted xl
tion limit". It is clear from the above
as a definition of the "detec-
discussion that using y =
- p
(a + aXl) as the detection threshold for Y does in fact achieve protec-
tion against Type I errors at the specified rate p. However, it is also
clear from Figure 3-1 that if a particular source matrix had ana1yte
concentration actually equal to xl' then we would expect to assert X = 
(i.e., a false negative) 50 percent of the time. Although a number of
authors (for example, Currie, 1968) have pointed out this problem with
such traditional "detection limits", their use has continued to be
widespread.
--....
Here, we prefer to adopt the notion that a detection limit should
be an ana1yte concentration that is almost assured of detection.
That
is, in addition to fixing the Type I error rate p at some low value, we
also want to specify the Type II error rate q at a realistic level.
Given the relationship between [l-q] and x (for fixed p), as depicted in
Figure 3-1b, it is clear that specification of q is equivalent to
specifying a concentration level xi as a detection limit. For example,
we might report the value x2 in Figure 3-1 as a detection limit that
achieves a satisfactory value for q given p. Clearly, such a formula-
tion of detection limits is inseparable from the detection rule.
In the case of a known relationship between X and Y, as assumed in
this section, the detection rule is given by (3-1); for this situation,
we state
the following definition:
if xi = xi(P,q) is the lowest analyte concentration
for which the rate of detection is at least (l-q)
when using the rule (3-1), then xi is the detection
limit with assurance probability (l-q) for that rule;
alternatively, the rate of false negatives is at most
q for all ana1yte concentrations exceeding xi.
-18-

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With appropriate modification of the detection rule, the above defini-
tion also applies more generally. Values of x~(p,q) for .fixed p, q and
r are determined from (3-3) as
( ) ( z ) ~
x~ p,q = zp + q air
(3-4)
3.2
Linear Calibration With Unknown Parameters
In this section, we again assume the straight-line calibration
model (2-2) but in contrast with the previous section, we now assume
that a, a, and 02 are unknown and must be estimated. The least squares
methods of Section 2.2 are adopted.
In addition to the notation developed previously, we adopt the
following:
S
v
is a random variable that is independent of the random variable Z
(see Section 3.1) and is such that VS2 is distributed as chi-square
v
with v degrees of freedom.
T (0) is a random variable having a noncentra1 t-distribution with v
v
degrees of freedom and noncentra1ity parameter o.
follows Student's t-distribution with v degrees of
Note that T (0)
v
freedom, and
that T (0) = (Z + o)/S .
v v
t represents the upper lOOp percentage point of Student's t-distribu-
v,p
tion with v degrees of freedom, i.e., Pr[T (0) > t ] = p.
v v,p
In most cases, we will suppress the v subscript appearing in the above

to simplify the notation. We will assume that the residual variance OL

(eq. (2-5), with p = 2) is used to provide an estimate of 02; hence

v = (n-2).
Proceeding in the manner of Section 3.1, we first specify a detec-
tion rule that will provide the desired level of protection against
false positives. In the rule (3-1), the threshold concentration y was
p
a function of 0 and of a, both of which were assumed to be known. In
this section, they must be estimated, so that the threshold level
becomes a random variable, which we will denote as y .
p
We then adopt
the detection rule:
'Assert X > 0 whenever Y > y ,
p
where yp is chosen so that the rate of false positives [in repeated
applications of (3-5)] will be fixed at the desired level p. The
prescribed level p will be achieved if we choose
(3-5)
-19-

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y = a + w at ,
pop
where w is defined by (2-12), since this choice of y gives
o p
(3-6)
"

Pr[false positive] = Pr[Y > y Ix=O]
p
=
Pr [ Y - a > t Ix=o ]
p
w 0
o
=
(since Y - a = C1W Z and a = oS)
o
Z
Pr [ S > tp ]

Pr [T(O) > t ] = p.
p
Use of (3-5) and (3-6) corresponds to using a t-statistic to test
=
the hypothesis that the true analyte concentration is zero versus
alternatives that it is greater than zero. The power of this test (or
alternatively, the rate of false negatives) is obtained from the appro-
priate noncentral t-distribution (rather than the normal distribution,
as was the case in Section 3.1).
As in the previous section, we now need to determine the detection
probability for any given x.

t:,. = x6
w a
o
First, we define the following parameter:
(3-7)
Detection rates when X=x, using the rule (3-5), are given by
"
Pr[detectionIX=x] = Pr[Y > y Ix=x]
p
"
= Pr[Y > a + w ot Ix=x]
o p
[from (3-6)]
= Pr [ Y - a > t Ix=x]
p
w a
o
= Pr [
6x + aw Z
o
w as
o
> t ]
P
""
(since Y - a = 6x"~w Z and a = as)
o
= Pr [ t:,. ; Z > tp ]

= Pr [ T(t:,.) > t ].
P
From (3-7) and (3-8), it is
(3-8)
clear that (with 6, 0, and w fixed) we can
o
achieve a specified detection rate (l-q) at x by choosing a value of t:,.
appropriately; this value is a function of p and q (and v) and hence is
-20-

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denoted as ~(p,q).
concentration level
Using (3-7) and this choice of ~ yields an analyte
Xt(p,q) = wo~(p,q)(0/6) (3-9)
such that the detection probability will be (l-q) when we use the rule
(3-5). Thus Xt(p,q) is the detection limit associated with (3-5) that
achieves an assurance probability equal to (l~q).
Note the following:
i)
The detection limit Xt(p,q) is a parameter that characterizes the
methods employed to produce the observed responses; it is not an
estimate based on a particular set of observations.
Those aspects of the method that influence detectability are
ii)
evident in (3-9); they can be separated into three factors:
(0/6), determined by sensitivity of the protocol
leading to observations Y;
~, determined by v and specification of p and q,
the rates of false positives and false
negatives that can be tolerated; and
w , determined by r and characteristics of the
o
calibration design (n, X, Q ).
xx
iii) As n+~, w +1/1:[, ~+(z +z ), and xn(p,q)+(z +z )0/61r, exactly as
o p q ~ p q
given at (3-4) in the previous section.
Apart from the unknown factor (0/6), which is analyte- and measurement-
specific and which must be estimated (see Section 3.3), all of the
components of (3-9) can be determined by
a) specifying a calibration design,
b) specifying a value for p, and
c) specifying a value for q.
In practice, it would be desirable to specify several values of q, and
determine the corresponding values of
* 6
Xt = Xt-o- = wo~(p,q)
(3-10)
which are values of Xt expressed as multiples of (0/6) instead of in
original concentration units.
As an example, suppose we choose design number 4 from Table 2-1 as
the calibration design and assume that eight replicates will be utilized
at each of the four standard concentrations (see Table 2-2). If we also
~\
fix rand p and let vary over the interval (p,l), then values of x!
~
-21-

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can be determined for each such q.
This produces curves such as those
shown in Figure 3-2 and Figure 3-3. Figure 3-2 shows the case p ; 0.05
while Figure 3-3 shows the case p ; 0.01. The three curves in each
figure correspond to r ; 1, 2, and 3. Once an estimate of cr/B is
obtained from the calibration, the horizontal axes of these figures can
be scaled in original concentration units to provide appropriate esti-
mated detection curves (henceforth referred to as power curves).
Values of 6(p,q) for generating the x~ values in these figures were
determined using Gaussian quadrature algorithms developed by Peter M.
1/
Burrows; the SA~ Macros are given in Appendix B. Values of 6(p,q) for
special cases p;q set at 0.05, 0.01 and 0.001 are given in Table 1 of
Appendix A.
Figures 3-2 and 3-3 illustrate an important feature of the concepts
and methods above:
Since such curves can be developed prior to actual
experimentation, we can assess the impact of using alternative calibra-
tion designs (as well as alternative p, q, or r values) as a part of the
overall experimental plan. Moreover, after estimates of the calibration
model parameters become available, we can determine threshold response
value(s) and estimate detection limit(s) for calibration designs and ~
values other than those actually used.
This permits the strategy of
modifying designs for subsequent calibrations so as to produce a detec-
tion limit (with the same p and q) that, in expectation, differs from
that for the initial calibration design. It also permits detection
limits from two different calibrations to be compared in a realistic
fashion (e.g., did laboratory A report lower limits of detection than
laboratory B because of a more sensitive protocol, or because of a
different calibration design, or because of different choices for p, q,
or r?).
3.3
Estimation of Detection Limits for Linear Calibration
As noted in the previous section, if we fix p, q, and the calibra-
tion design, then all quantities needed for determining Xt(p,q) accord-
ing to (3-9) are known except for the factor o/B. Given an estimate of
o/B, we can therefore provide an estimate of the detection limit having
1/
SAS is the registered trademark of SAS Institute, Cary, N.C.
-22-

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FIGURE 3-2
 1 .0
D
E 0.9
T 
E 0.8
C 
T 0.7
I 
0 0.6
N 
 0.5
P 
R 0.4
,0 
~B 0.3
,
A 
B 0.2
I 
L 0. 1
I
T 0.0
Y 
 0.0
POWER CURVES FOR DESIGN 4 WITH LINEAR CALIBRATION
N=32 P=0.05
.".......". ",...--.....
~.. --
.. .,.-
./" ,,""
, "
/ "
, "
/ "
, /
/ //
, /
/ /
, /
/ //
, /
/ /
, /
/ //
, /
/ /
, /
/ //
, /
//
, /
/,,/
,,,
~"
0.5 1 . 0 1 . 5 2.0 2.5 3.0 3.5 4.0 4.5 . 5.0
    WO*DELTA     
 LEGEND: R 1 ----... 2  ---- 3  

-------
FIGURE 3-3
 1 .0
D
E 0.9
T 
E 0.8
C 
T 0.7
I 
0 0.6
N 
 0.5
P 
R 0.4
,0 
~B 0.3
I
A 
B 0.2
I 
L 0. 1
I
T 0.0
Y 
 0.0
POWER CURVES FOR DESIGN 4 WITH LINEAR CALIBRATION
N=32 P=0.01
......-- - -- ~ -- .... ...
, --
,""" " "
/ ",,"
, ""
/ /
, ,/
/ ,/'
, /
/ /
~ /
/ //
, /
/ /
, /
/ //
, /
/ /
, /
/ /
, //
/ /
, /
/ //
, /
//
, /
./ ",,/
,,,,,
~"
'-
0.5 1 .0 1 .5 2.0 2.5 3.0 3.5 4.0 4.5 "5.0
    WO*DELTA     
 LEGEND: R 1 ------ 2  -....--- 3  

-------
specified assurance probability for the detection rule (3-5). Alterna-
tively, if an estimate of 8/a is used in (3-7) to provide an estimated
'"

noncentrality parameter ~ (for fixed p, x, and calibration design), then
an estimate of the detection rate for that x can be determined.
Estimation of 8/a.
An intuitive point estimate of 8/a is 8/a,
where 8 and a are determined from the calibration results, as described
in Chapter 2.
"'-1 . -1
a 1S not a
This estimate is not unbiased, since the expectation of
"'-1 -1
Rather, the expectation of a is M a , where v is the
v
degrees of freedom for a, and
(v/2)~2V-2{[(v-3)/2]!}2/TI~(v-2)!
if v is odd,
M
v
=
(v/2)~TI~(v-2)!/2v-2{[(v-2)/2]!}2
if v is even.
An approximation
for M , suitable
v
+ 3
4(v-l.042) .
for v > 8,is given by Owen (1968):
M :::::: 1
v
Thus the
(minimum variance)
A '"
(8/a) = 8/M a .
v
unbiased estimator for 8/a is
(3-11)
An interval estimate for 8/a with coverage probability (1-y) can be
obtained by finding values 0 and 0+ such that
determine
Pr[T(o_) < 0] = 1-y/2

BQ~ /~.
xx
o and 0 .
+
-~ -~
(o_Qxx' o+Qxx).
and
Pr[T(o+) < 0] = y/2,
(3-12)
where  =
The SAS algorithms of Appendix B can be used to
An interval estimate for ala is then given by
(3-13)
Estimation of a/8. An intuitive point estimate for a/B is a/B; it
'"

has infinite mean square error (like the estimate X in eq. (2-11)). An
interval estimate for a/8 having coverage probability (1-y) is obtained
by simply inverting the inverval end-points of (3-13):
~ ~
(Qxx/o+, Qxx/o_),
(3-14 )
where 0+ and 0 are determined via (3-12). This is an approximation
that neglects the probability of negative estimates B.
-25-

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Estimation of Xt(p,q). Interval estimation of the detection limit
defined by (3-9) is accomplished by multiplying the interval end-points
of (3-14) by w ~(p,q) to produce the interval
o
!::
(w ~(p,q)Q2 10+,
o xx
~
w ~(p,q)Q 10) .
o xx-
(3-15)
This furnishes an interval estimate for Xt(p,q) with coverage probabil-
ity equal to (1-y) when 0- and 0+ are determined according t~ ~3-12).
A point estimate for Xt(p,q) is . obtained by substituting alS, for
alS in (3-9).
Estimation of q(x). Specification of p and x (or several values of
A

and substitution of S/M a for Sia in (3-7) yields a point estimate
v .
A point estimate for the false negative rate q(x) is then
x)
,..
~ of ~.
obtained as
"
q(x) = Pr[T(~) < t ]
p
(3-16)
"

where ~ = xS/w M a.
o v
bility equal to (1-y) is obtained as follows:

1. Determine c and c via (3-12).
+

Compute ~ = xo Iw Q~ and
- 0 xx

~+ = xo Iw Q~
+ 0 xx
An interval estimate for q(x) with coverage proba-
2.
(Interval (~ ,~ ) provides an interval estimate for ~ having
- +
coverage probability equal to (1-y.
3.
and q+ = Pr[T(~ ) < t ].
- p
estimate for q(x) with coverage
(3-17)
Determine q = Pr[T(~ ) < t ]
- + P
Interval (q-, q+) provides an interval

probability (1-y).

The algorithms in Appendix B can be used to determine q(x) in

(3-16) and q- and q+ in (3-17).
3.4 EPA Procedures for Determining a "Method Detection Limit"
Appendix C provides the EPA guidelines for determining a "Method
Detection Limit", which is defined as "the minimum concentration of a
substance that can be measured and reported with 99% confidence that the
true value, corresponding to a single measurement, is above zero".
first note the following:
We
-26-

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1.
The procedures for determining the "Method Detection Limit"
presume that a calibration has been performed, since the
calculations are made in terms of concentration units. The
calibration is not necessarily confined to a range of "low"
2.
concentrations, however.
The procedures involve an (implicit) assumption that variances
3.
of "observed" concentrations are homogeneous, at least over a
limited range in the vicinity of zero. For a straight-line
calibration model, this implies that errors in the response
scale (Y) are also expected to exhibit homogeneous variances.
The procedures assume that "observed" concentrations are
normally distributed (since criteria are based upon t-statis-
tics). For the linear calibration model, this implies that
errors in the response scale are also normally distributed.
Three pertinent comments concerning the above points deserve mention:
a)
For curvilinear calibration relationships, it is not clear why
one would expect the "observed" concentrations, obtained via
inverse interpolation of the estimated calibration model, to
exhibit normality or homogeneous variances.
It is more
realistic, in our opinion, to consider that response measure-
ments (on some scale) will exhibit these properties.
b)
The assumptions required for the methods of Sections 3.2 and
3.3, at least for the linear calibration case, are the same as
c)
those inherent to the EPA procedure.
In most cases, the level of effort involved in generating the
data needed for applying the methods of Sections 3.2 - 3.3 is
not unrealistic in comparison to that required for the EPA
method (including calibration).
Finally, we note that the EPA procedure suffers from the problem
associated with "traditional detection limits" that was mentioned in
Section 3.1.
In particular, assume that a linear calibration model like
(2-2) holds; let s denote the standard deviation of the k replicate
x
"observed" concentrations involved in the EPA procedure, and let

tk-1,O.Ol denote the Student's t-va1ue used in that procedure (in

Appendix D, sand k are denoted as Sand n, respectively). Let x
x m
denote the estimated "Method Detection Limit" defined as
-27-

-------
x = t s
m k-1,0.01 x
If a andB are assumed known, we can rewrite this limit in the response

scale as
where s = Bs
y x
values that led to the k "observed" concentrations.

For a given source matrix with zero concentration, let Y denote the
Ym = a + BXm = a + Btk-1,0.01 Sx

= a + tk-1,0.01 Sy
is the estimated standard deviation of the k response
observed response.
Then
Pr[Y > ; \X=O] = pr[a + aZ > y ]
m m
= Pr[aZ > tk-1,0.01 aSk-1]
= Pr[Tk-1(0) > tk-1,0.Ol] = 0.01
Next assume that a given source matrix has
concentration exactly
equal to x .
Am
Since the observed response, Y, is now given by
Y = a + Bx + aZ, we obtain
m A A
Pr[Y > y Ix=x ] = pr[a + Bx + aZ > a + Bx ] = Pr[aZ > 0] = 0.50.
m m m m
Thus, while providing the specified level of protection against

false positives, the EPA procedure affords only 50 percent protection

against false negatives when the true concentration is equal to the

(estimated) "Method Detection Limit". The method also fails to recog-
nize the effects that the calibration design and calibration model
parameter estimates have on reported "detection limits", since estima-
tion of the "Method Detection Limit" is treated separately from the
calibration (which, in some cases, may involve an instrument rather than
a method calibration) and since the calibration function is treated as
though it is known (e.g., estimates of a and B are treated as known
quantities).
-28-

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4.
AN EXPERIMENTAL EVALUATION OF THE RECOMMENDED APPROACH
A two-phased experiment was conducted to demonstrate and validate
the methods described in Chapter 3.
The first phase -- calibration --
involved generation of the calibration results and their statistical
analysis: estimation of the model parameters and estimation of detec-
tion limits with specified assurance probabilities based upon the
estimated power curve associated with the calibration design and chemi-
cal method. The statistical analysis involved in the first phase
illustrates the types of statistical procedures that are needed in
applications of the recommended approach. The second phase of the
experiment -- validation -- involved generation of results at selected
concentrations from which empirical estimates of detection rates could
be derived and then compared to the theoretically derived estimates from
the first phase.
Experimental Method.
4.1
A single sediment matrix was used as the source matrix for all
determinations in the experiment. This sediment, collected from Uni-
versity Lake (Chapel Hill, North Carolina) was chosen since this source
is regarded as an uncontaminated reservoir and the sediment is unlikely
to contain any of the ana1ytes of interest. A single large batch
(approximately 1 kg) of the University Lake sediment was collected on
December 6, 1984 for use in the experiment. After removing trash and
homogenizing the sediment, 30 g aliquots were removed for analysis as
required. When not in use, all sediment and sediment samples were
stored in the dark at 40 C.
Standard solutions containing known concentrations of eight ex-

tractable analytes (in acetone) were prepared for fortifying the sedi-

ment samples. The eight chosen analytes, seven of which are EPA prior-

ity pollutants, were as follows:

2-chloronaphthalene
2,6-dinitrotoluene
dimethy1phtha1ate
hexachlorobenzene
anthracene
phenanthrene
aldrin
fluoranthene
-29-

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Anthracene was included since it is a common choice for an internal
standard.
Sediment a1iquots were processed according to the specific chemical
protocol described in Appendix D.. This protocol utilized an ultrasonic
extraction technique, which was shown in a previous study (EPA Contract
No. 68-03-3119) to be the "best" available extraction method for sedi-
ments. Except for this extraction method, all other components of the
chemical methodology followed procedures outlined in EPA Test Method
625-B, as applicable. Each sample extract was fortified with pentade-
cane at a fixed, known concentration (.265 ppm, relative to the original
sediment sample) immediately prior to the GC (gas chromatography)
injection; this compound thus served as the internal standard for the
method.
4.2
Phase I Design.
A single concentrated stock solution containing the eight ana1ytes
was first prepared. For Phase I of the experiment, the standard forti-
fying solutions were prepared by serial dilution of this stock solution
to target concentrations of 0.2, 0.8, and 1.0 mg/kg sediment (i.e.,
ppm).
4-1.
Actual fortification levels for each ana1yte are shown in Table
The target levels were chosen on the basis of several preliminary
runs, which were conducted primarily for instrument setup. These levels
appeared satisfactory for all ana1ytes except 2,6-dinitroto1uene, for
which they were much too low.
Based upon these same runs, it was also
apparent that a background interference with aldrin existed. Hence,
these two ana1ytes were dropped trom further consideration, although
they remained in the standard solutions at the indicated concentrations.
Although the problem with 2,6-dinitrotoluene could perhaps have been
remedied by using higher concentrations, this was not done since it
would have required reformulating the initial stock solution.
The Phase I design involved four sets of samples. Each set con-
sisted of eight sediment samples and a control sample that were prepared
and processed together (e.g., during extraction). Sediment samples were
prepared by fortification of individual 30 g aliquots of sediment with
appropriate levels of ana1ytes (Table 4-1). Control samples were
prepared in a similar manner without sediment.
All control samples were
-30-

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Table 4-1.
Actual Analyte Concentrations (ppm) Used in
Phase I of the Experiment
Ana1yte
Target Concentration (ppm)
0.2 0.8 1.0
2-chloronaphtha1ene
2,6-dinitroto1uene
dimethyl phthalate
hexach1orobenzene
anthracene
phenanthrene
aldrin
f1uoranthene
0.215
0.205
0.217
0.200
0.217
0.210
0.203
0.200
0.858
0.818
0.867
0.802
0.870
0.840
0.815
0.802
1.07
1.02
1.08
1.00
1.07
1.05
1.02
1.00
-31-

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fortified with the intermediate level of ana1ytes (Table 4-1).
GC
analyses for all samples in a set were performed on the same day. The
table below indicates the design (each symbol "X" denotes an observa-
tion):
Sediment
Target Concen-  Set Number  Number of
tration (ppm) 1 2 3 4 Observations
0.0 (blank) XX XX* XX XX 8*
0.2 XX XX XX XX 8
0.8 XX XX XX XX 8
1.0 XX XX XX XX 8
0.8 X X X X 4
Type of
Sample
Control
(* one of these samples was not used)
In addition to the runs indicated above, a single GC check sample was
analyzed along with each sample set. Check samples consisted of stand-
ard solutions of ana1ytes at a level corresponding to ana1yte concen-
trations in sediment extracts fortified at the intermediate level (Table
4-1) .
Since all sample preparations and chemical analyses were performed
as close together in time as possible, "set s" were ignored in the
statistical treatment of the results (i.e., the design was regarded as a
CRD). Hence the design shown above corresponds to that designated as
design number 4 in Table 2-1. If a linear calibration model can be
assumed, we would therefore expect to achieve the properties shown for
design number 4 with n=32 in Table 2-2. Figures 3-2 and 3-3 are also
applicable to this design and model.
4.3
Phase I Data
Tables 4-2 through 4-7 provide listings of Phase I results for the
various ana1ytes (2,6-dinitroto1uene and aldrin are excluded due to the
aforementioned problems). These tables indicate the sample type; the
fortification level (X*) in ppm; the set, replicate (REP), and run
numbers; and observed responses (peak-area counts) from the GC analysis
for both the particular ana1yte and the internal standard (pentadecane).
In addition, the tables contain the adjusted peak area, Y*, which is
obtained as the ratio of ana1yte peak area to internal-standard peak
-32-

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TABLE 4:-2. LISTING OF CALIBRATION DATA FOR 2-CHLORONAPHTHALENE
       GC AREAS FOR:  
SAMPLE   SET .  RUN -----------------  
TYPE X* X NO. REP NO. ANALYTE INT. STD. Y* Y
------------------------------------------------------------------
GC_CHK 0.858 0.6625 1  1 10 29001 18941 1. 5311 1.2374
   2  1 20 31126 20420 1.5243 1.2346
   3  1 '30 28778 18882 1. 5241 1. 2345
   4  1 40 26221 17198 1.5247 1.2348
CONTRL 0.858 0.6625 1  1 11 27549 19503 1. 4126 1. 1885
   2  1 21 25760 18239 1.4124 1. 1884
   3  1 31 21511 17927 1. 1999 1.0954
   4  1 41 21204 14656 1.4468 1. 2028
UL SED 0.000 0.0000 1  1 12 3477 23652 O. 1470 0.3834
   1  2 13 3532 25660 O. 1376 0.3710
   2  2 29 1693 28403 0.0596 0.2441
   3  1 34 1381 24573 0.0562 0.2371
   3  2 38 2204 23558 0.0936 0.3059
   4  1 45 1191 22142 0.0538 0.2319
   4  2 44 2091 20175 O. 1036 0.3219
UL SED 0.215 0.2450 1  1 15 10139 25075. 0.4043 0.6359
   1  2 14 9306 25610 0.3634 0.6028
   2  1 22 8990 24944 0.3604 0.6003
   2  2 24 7881 24073 0.3274 O. 5722
   3  1 37 6164 22436 0.2747 O. 5242
   3  2 32 6384 27393 0.2331 0.4828
   4  1 43 7050 24045 0.2932 O. 5415
   4  2 46 5209 20816 0.2502 O. 5002
UL SED 0.858 0.6625 1  1 19 28628 26397 1. 0845 1. 0414
   1  2 18 25194 25244 0.9980 0.9990
   2  1 27 18480 21681 0.8524 0.9232
   2  2 28 24426 28036 0.8712 0.9334
   3  1 39 21226 23939 0.8867 0.9416
   3  2 35 22435 25198 0.8903 0.9436
   4  1 48 20760 24985 0.8309 0.9115
   4  2 47 20991 23978 0.8754 0.9356
UL SED 1.070 O. 7654 1  1 16 33112 30710 1.0782 1.0384
   1  2 17 30837 22250 1. 3859 1. 1773
   2  1 23 32592 24641 1.3227 1. 1 501
   2  2 26 27094 23086 1. 1 736 1.0833
   3  1 33 . 28910 25820 1. 1197 1.0581
   3' 2 36 33094 27314 1. 2116 1. 1007
   4  1 42 27492 21783 1. 2621 1. 1234
   4  2 49 26072 22356 1. 1662 1.0799
-33-

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TABLE 4-3. LISTING OF  CALIBRATION DATA FOR DIMETHYLPHTHALATE
       GC AREAS FOR:  
SAMPLE   SET   RUN -----------------  
TYPE X* X NO.  REP NO. ANALYTE INT. STD. Y* Y
------------------------------------------------------------------
GC_CHK 0.867 0.6671  1 1 10 14944 18941 O. 7890 0.8882
    2 1 20 11284 20420 0.5526 O. 7434
    3 1 30 10658 18882 0.5645 O. 7513
    4 1 40 9981 17198 O. 5804 O. 7618
CONTRL 0.867 0.6611  1 1 11 14709 19503 O. 7542 0.8684
    2 1 21 10963 18239 O. 6011 O. 7753
    3 1 31 11815 17927 0.6591 0.8118
    4 1 41 10190 14656 0.6953 0.8338
UL SED 0.000 0.0000  1 1 12 1912 23652 0.0808 0.2843
    1 2 13 9199 25660 0.3585 O. 5987
   2 2 29 2468 28403 0.0869 0.2948
   3 1 34 4040 24573 O. 1644 0.4055
    3 2 38 2220 23558 0.0942 0.3070
    4 1 45 1635 22142 0.0738 0.2717
    4 2 44 1632 20175 0.0809 0.2844
UL SED 0.217 0.2468  1 1 15 3803 25075 O. 1517 0.3894
    1 2 14 4790 25610 O. 1870 0.4325
   2 1 22 3492 24944 O. 1400 0.3742
   2 2 24 3193 24073 O. 1326 0.3642
   3 1 37 2568 22436 0.1145 0.3383
   3 2 32 4832 27393 0.1764 0.4200
   4 1 43 4573 24045 O. 1902 0.4361
   4 2 46 2974 20816 O. 1429 0.3780
UL SED 0.867 0.6671  1 1 19 9636 26397 0.3650 0.6042
    1 2 18 9009 25244 0.3569 O. 5974
   2 1 27 6302 21681 0.2907 O. 5391
   2 2 28 13558 28036 0.4836 0.6954
   3 1 39 9664 23939 0.4037 0.6354
   3 2 35 12059 25198 0.4786 0.6918
   4 1 48 13998 24985 O. 5603 O. 1485
   4 2 47 9757 23978 0.4069 0.6379
UL SED 1.080 O. 7701  1 1 16 14440 30710 0.4702 0.6857
    1 2 17 12537 22250 O. 5635 O. 7506
   2 1 23 14028 24641 O. 5693 O. 7545
   2 2 26 11553 23086 O. 5004 O. 7074
   3 1 33 15105 25820 0.5850 O. 7649
   3 2 36 15119 27314 O. 5535 O. 7440
   4 1 42 13799 21783 0.6335 O. 7959
   4 2 49 11798 22356 0.5277 O. 7265
-34-

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TABLE 4-4. .LISTING OF CALIBRATION DATA FOR HEXACHLOROBENZENE
      GC AREAS FOR:  
SAMPLE   SET  RUN -----------------  
TYPE X* X NO. REP NO. ANALYTE INT. STD. Y* Y
------------------------------------------------------------------
GC_CHK 0.802 0.6335 1 1 10 9599 18941 O. 5068 0.7119
   2 1 20 9968 20420 0.4881 0.6987
   3 1 30 9304 18882 0.4927 O. 7020
   4 1 40 8496 17198 0.4940 O. 7029
CONTRL 0.802 0.6335 1 1 11 9143 19503 0.4688 0.6847
   2 1 21 9122 18239 O. 5001 O. 7072
   3 1 31 8490 17927 0.4736 0.6882
   4 1 41 7389 14656 O. 5042 O. 7100
UL SED 0.000 0.0000 1 1 12 1090 23652 0.0461 0.2147
   1 2 13 1282 25660 0.0500 0.2235
   2 2 29 1744 28403 0.0614 0.2478
   3 1 34 1038 24573 0.0422 0.2055
   3 2 38 448 23558 0.0190 O. 1379
   4 1 45 604 22142 0.0273 O. 1652
   4 2 44 235 20175 0.0116 O. 1079
UL SED 0.200 0.2315 1 1 15 3280 25075 O. 1308 0.3617
   1 2 14 2781 25610 O. 1086 0.3295
   2 1 22 3274 24944 O. 1313 0.3623
   2 2 24 3068 24073 O. 1274 0.3570
   3 1 37 2762 22436 O. 1231 0.3509
   3 2 32 2592 27393 0.0946 0.3076
   4 1 43 2676 24045 O. 1113 0.3336
   4 2 46 2462 20816 0.1183 0.3439
UL SED 0.802 0.6335 1 1 19 11401 26397 0.4319 0.6572
   1 2 18 10302 25244 0.4081 0.6388
   2 1 27 8525 21681 0.3932 0.6271
   2 2 28 10722 28036 0.3824 0.6184
   3 1 39 9074 23939 0.3790 0.6157
   3 2 35 9616 25198 0.3816 0.6178
   4 1 48- 9733 24985 0.3896 0.6241
   4 2 47 9495 23978 0.3960 O. 6293
UL SED 1.000 0.7326 1 1 16 14415 30710 0.4694 0.6851
   1 2 17 12612 22250 O. 5668 O. 7529
   2 1 23 14877 24641 0.6037 O. 7770
   2 2 26 13644 23086 O. 5910 O. 7688
   3 1 33 13249 25820 0.5131 O. 7163
   3 2 36 14541 27314 0.5324 O. 7296
   4 1 42 11472 21783 0.5266 O. 7257
   4 2 49 11685 22356 O. 5227 0.7230
-35-

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TABLE 4-5. LISTING OF CALIBRATION DATA FOR ANTHRACENE 
      GC AREAS FOR:  
SAMPLE   SET  RUN -----------------  
TYPE X* X NO. REP NO. ANALYTE INT. STD. Y* Y
------------------------------------------------------------------
GC_CHK 0.870 0.6687 1 1 10 32015 18941 1.6902 1. 3001
   2 1 20 35261 20420 1. 7268 1.3141
   3 1 30 32812 18882 1. 7377 1. 3182
   4 1 40 29719 17198 1. 7280 1. 3146
CONTRL 0.870 0.6687 1 1 11 33959 19503 1. 7412 1. 31 96
   2 1 21 32472 18239 1.7804 1. 3343
   3 1 31 30676 17927 1. 7112 1. 3081
   4 1 41 26382 14656 1. 8001 1. 341 7
UL SED 0.000 0.0000 1 1 12 740 23652 0.0313 0.1769
   1 2 13 1580 25660 0.0616 0.2481
   2 2 29 997 28403 0.0351 O. 1874
   3 1 34 811 24573 0.0330 0.1817
   3 2 38 1092 23558 0.0464 0.2153
   4 1 45 982 22142 0.0444 0.2106
   4 2 44 796 20175 0.0395 O. 1986
UL SED 0.217 0.2468 1 1 15 9023 25075 0.3598 O. 5999
   1 2 14 7948 25610 0.3103 O. 5571
   2 1 22 9371 24944 0.3757 0.6129
   2 2 24 7815 24073 0.3246 O. 5698
   3 1 37 7717 22436 0.3440 O. 5865
   3 2 32 7751 27393 0.2830 0.5319
   4 1 43 8109 24045 0.3372 O. 5807
   4 2 46 6926 20816 0.3327 O. 5768
UL SED 0.870 0.6687 1 1 19 37040 26397 1.4032 1. 1846
   1 2 18 32371 25244 1. 2823 1. 1324
   2 1 27 25715 21681 1. 1861 1. 0891
   2 2 28 31503 28036 1. 1237 1. 0600
   3 1 39 28626 23939 1. 1958 1. 0935
   3 2 35 29883 25198 1. 1859 1.0890
   4 1 48 30437 24985 1.2182 1. 1037
   4 2 47 29597 23978 1. 2343 1. 1110
UL SED 1.070 O. 7654 1 1 16 48641 30710 1. 5839 1. 2585
   1 2 17 38841 22250 1. 7457 1. 3212
   2 1 23 48621 24641 1. 9732 1.4047
   2 2 26 36312 23086 1.5729 1.2542
   3 1 33 42071 25820 1.6294 1. 2765
   3 2 36 47124 27314 1. 7253 1. 3135
   4 1 42 37929 21783 1. 7412 1. 3196
   4 2 49 37667 22356 1. 6849 1.2980
-36-

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TABLE 4-6. LISTING OF CALIBRATION DATA FOR PHENANTHRENE 
      GC AREAS FOR:  
SAMPLE   SET  RUN -----------------  
TYPE X* X NO. REP NO. ANALYTE INT. STD. Y* Y
------------------------------------------------------------------
GC_CHK 0.840 0.6533 1 1 10 29108 18941 1. 5368 1.2397
   2 1 20 32214 20420 1. 5776 1. 2560
   3 1 30 30030 18882 1.5904 1. 2611
   4 1 40 27423 17198 1.5945 1. 2628
CONTRL 0.840 0.6533 1 1 11 32030 19503 1. 6423 1. 281 5
   2 1 21 31459 18239 1. 7248 1. 3133
   3 1 31 30365 17927 1.6938 1. 3015
   4 1 41 25711 14656 1.7543 1.3245
UL SED 0.000 0.0000 1 1 12 1472 23652 0.0622 0.2495
   1 2 13 1999 25660 0.0779 0.2791
   2 2 29 1352 28403 0.0476 0.2182
   3 1 34 1904 24573 0.0775 0.2784
   3 2 38 1272 23558 0.0540 0.2324
   4 1 45 1139 22142 0.0514 0.2268
   4 2 44 1592 20175 0.0789 0.2809
UL SED 0.210 0.2405 1 1 15 9918 25075 0.3955 0.6289
   1 2 14 8603 25610 0.3359 O. 5796
   2 1 22 10349 24944 0.4149 0.6441
   2 2 24 8963 24073 0.3723 O. .6102
   3 1 37 8301 22436 0.3700 0.6083
   3 2 32 8740 27393 0.3191 O. 5649
   4 1 43 8842 24045 0.3677 0.6064
   4 2 46 6904 20816 0.3317 O. 5759
UL SED 0.840 0.6533 1 1 19 40871 26397 1. 5483 1.2443
   1 2 18 36950 25244 1.4637 1. 2098
   2 1 27 29973 21681 1. 3825 1. 1758
   2 2 28 35652 28036 1. 271 7 1. 1277
   3 1 39 33499 23939 1.3993 1. 1829
   3 2 35 34803 25198 1. 3812 1. 1752
   4 1 48 34852 24985 1. 3949 1. 1811
   4 2 47 33846 23978 1.4115 1. 1881
UL SED 1.050 O. 7562 1 1 16 53361 30710 1. 7376 1.3182
   1 2 17 42845 22250 1. 9256 1.3877
   2 1 23 52966 24641 2. 1495 1. 4661
   2 2 26 45976 23086 1. 9915 1. 4112
   3 1 33 48360 25820 1.8730 1. 3686
   3 2 36 52988 27314 1.9400 1. 3928
   4 1 42 41945 21783 1. 9256 1.3877
   4 2 49 43104 22356 1. 9281 1.3886
-37-

-------
TABLE 4-7. LISTING OF CALIBRATION DATA FOR FLUORANTHENE 
      GC AREAS FOR:  
SAMPLE   SET  RUN -----------------  
TYPE X* X NO. REP NO. ANALYTE INT. STD. Y* Y
------------------------------------------------------------------
GC_CHK 0.802 0.6335 1 1 10 27177 18941 1.4348 1. 1978
   2 1 20 30911 20420 1. 5138 1. 2303
   3 1 30 28982 18882 1.5349 1.2389
   4 1 40 26794 17198 1.5580 1.2482
CONTRL 0.802 0.6335 1 1 11 30307 19503 1.5540 1.2466
   2 1 21 29592 18239 1.6225 1.2738
   3 1 31 29262 17927 1.6323 1. 2776
   4 1 41 24250 14656 1.6546 1.2863
UL SED 0.000 0.0000 1 1 12 1533 23652 0.0648 0.2546
   1 2 13 1748 25660 0.0681 0.2610
   2 2 29 1222 28403 0.0430 0.2074
   3 1 34 1554 24573 0.0632 0.2515
   3 2 38 1008 23558 0.0428 0.2069
   4 1 45 1440 22142 0.0650 0.2550
   4 2 44 1256 20175 0.0623 0.2495
UL SED 0.200 0.2315 1 1 15 9209 25075 0.3673 0.6060
   1 2 14 7991 25610 0.3120 O. 5586
   --2 1 22 9206 24944 0.3691 0.6075
   2 2 24 7334 24073 0.3047 O. 5520
   3 1 37 7500 22436 0.3343 O. 5782
   3 2 32 7340 27393 0.2680 0.5176
   4 1 43 9963 24045 0.4143 0.6437
   4 2 46 8778 20816 0.4217 0.6494
UL SED 0.802 0.6335 1 1 19 36563 26397 1.3851 1. 1769
   1 2 18 33091 25244 1. 31 08 1. 1449
   2 1 27 24972 21681 1. 1518 1.0732
   2 2 28 34499 28036 1.2305 1. 1093
   3 1 39 28485 23939 1. 1899 1. 0908
   3 2 35 29774 25198 1. 1816 1.0870
   4 1 48 31888 24985 1. 2763 1. 1297
   4 2 47 31998 23978 1.3345 1. 1552
UL SED 1.000 O. 7326 1 1 16 41284 30710 1. 3443 1. 1 594
   1 2 17 38348 22250 1.7235 1. 3128
   2 1 23 51414 24641 2.0865 1. 4445
   2 2 26 42443 23086 1.8385 1. 3559
   3 1 33 44229 25820 1. 7130 1. 3088
   3 2 36 45825 27314 1.6777 1. 2953
   4 1 42 40518 21783 1. 8601 1.3638
   4 2 49 37264 22356 1.6668 1. 2911
-38-

-------
area.
Finally, the tables contain values of two additional quantities
that are used in subsequent sections:
x = I x* + .1 - ~ and
y = ~ .
4.4
Determination pf the Calibration Function.
As indicated in Section 2.2, several steps are involved in deter-
mining a suitable form for the calibration model and estimating its
parameters. We first plotted the observed ana1yte responses (adjusted
peak areas, designated as y* in Tables 4-2 through 4-7) versus concen-
tration (designated as X* in these tables). These plots are shown in
Figures 4-1 through 4-6 for the various ana1ytes and suggest the follow-
ing:
(1)
At least one outlier appears to be present for dimethy1phtha-
late (Figure 4-2) -- y* = .3585 for X* = O. This is listed as
run number 13 in Table 4-3. All subsequent plots and calcula-
tions exclude this observation. Hence n=30 for this ana1yte
and n=31 for all others.
(2)
With the possible exception of dimethy1phtha1ate, variation in
observed responses tends to increase with increasing concen-
(3)
tration level.
Although y* tends to increase with X*, the relationship is not
a simple straight line for most ana1ytes.
This curvi1inearity, and the tendency for the response variation to
increase with increasing concentration level, are further exhibited,
respectively, by the means and standard deviations shown in Table 4-8.
Tests for homogeneous response variation across the concentration levels
yielded the following results:
-39-

-------
FIGURE 4-1
CALIBRATION DATA: Y* VERSUS X*
ANALYTE=2-CHLORONAPHTHALENE
     *
     *
  1 .25   *
 A    =
 D    *
 ~   * *
 U 1.00  * 
 S    
 T   I 
 E   
 D 0.75   
 P    
 E    
~ A 0.50   
0    
I K    
  *  
 A  i  
 R 0.25  
 E    
 A    
  0.00   
o
o
o
o
o
o
o
fa
fa
o
t
.

o
.

1
.

2
.

3
.

4
.

5
.

6
.

7
.

8
.

9
.

fa
.

t
CONCENTRATION CPPM)

-------
FIGURE 4-2.
CALfBRATION DATA: Y* VERSUS X*
ANALYTE=DIMETHYLPHTHALATE
  0.7   
     *
 A 0.6   -
 D   *
 cJ    *
 U 0.5  . *
 S   *
 T    
 E 0.4  * 
 D   * 
 P 0.3  * 
 E    
 A    
I K 0.2 i  
~  
......    
I   !  
 A   
 R   
 E    
 A    
o
o
fa
o
fa
fa
fa
o
o
fa
1
1
.

o
.

t
.

2
.

3
.

4
. -

5
.

6
.

7
.

8
.

9
.

fa
.

1
CONCENTRATION CPPM)

-------
FIGURE 4-3.
CALIBRATION DATA: Y* VERSUS X*
ANALYTE=HEXACHLOROBENZENE
0.7
 A 0.6  i
 D   *
 J   i
 U 0.5 
 S   *
 T  * 
 E 0.4 I 
 D  
 P 0.:3  
 E   
 A   
I K 0.2  
.c-  
N    
I    
 A  I 
 R 0. 1 
 'E   
 A   
  0.0  
o
o
o
o
o
o
o
o
o
o
t
1
.

o
.

2
.

3
.

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.

5
.

6
.

7
.

8
.

9
.

o
.

1
CONCENTRATION CPPM)

-------
FIGURE 4-4.
CALIBRATION DATA: Y* VERSUS X*
ANALYTE=ANTHRACENE
  2.0  *
 A   I
 D  
 J  
 U 1.6  
 S  * 
 T  , 
 E  
 D  * 
  1 .0  
 P   
 E   
 A   
 K   
I  0.5  
.t:'-   
w A   
I I 
 R 
 E   
 A   
  0.0  
o
o
o
o
o
o
o
o
o
o
.

o
.

1
.

2
.

:3
.

4
.

5
.

6
.

7
.

8
.

9
.

o
.

1
CONCENTRATION CPPM)

-------
FIGURE 4-5.
CALIBRATION DATA: Y* VERSUS X*
ANALYTE=PHENANTHRENE
    *
  2.0  i
 A  
 D   
 ~   *
 U  * 
 S 1.5 . 
 T  
 E  * 
 D   
 P 1.0  
 E   
I A   
~   
~ K   
I   
 A 0.5  
 R I 
 E 
 A   
o
o
o
o
o
o
o
o
o
o
1
1
.

o
.

1
.

2
.

3
.

4
.

5
.

6
.

7
.

8
.

9
.

o
.

t
CONCENTRATION CPPM)

-------
FIGURE 4-6.
CALIBRATION DATA: Y. VERSUS X*
ANALYTE=FLUORANTHENE
   *
  2.0 
 A  *
 D 
 ~  *
 U
s 1.5 
 T I *
 E
 D 
 P 1.0 
 E  
I A  
~  
V1 K  
I  
 A 0.5 
 R i
 E
 A  
  0.0 
o
o
o
o
o
o
o
o
o
o
1
.

o
.

1
.

2
.

3
.

4
.

5
.

6
.

7
.

8
.

9
.

o
.

1
CONCENTRATION CPPM)

-------
Table 4.8. Means and Standard Deviations of Adjusted Peak Areast by
 Analyte and Concentration Level  
  Standard Cone.  Adjusted Peak Area (Y*)
Analyte i.. X*(ppm) n. Mean Std. Deviation
-1.
2-chloronaphthalene 1 0.000 7 0.093 0.039
  2 0.215 8 0.313 0.060
  3 0.858 8 0.911 0.086
  4 1.070 8 1.215 0.103
dimethylphthalate 1 0.000 6 0.097 0.034
  2 0.217 8 0.154 0.027
  3 0.867 8 0.418 0.086
  4 1.080 8 0.550 0.051
hexachlorobenzene 1 0.000 7 0.037 0.018
  2 0.200 8 0.118 0.013
  3 0.802 8 0.395 0.018
  4 1.000 8 0.541 0.044
anthracene 1 0.000 7 0.042 0.010
  2 0.217 8 0.333 0.029
  3 0.870 8 1. 229 0.084
  4 1.070 8 1. 707 0.127
phenanthrene 1 0.000 7 0.064 0.014
  2 0.210 8 0.363 0.033
  3 0.840 8 1. 407 0.078
  4 1.050 8 1. 934 0.115
fluoranthene 1 0.000 7 0.058 0.011
  2 0.200 8 0.349 0.054
  3 0.802 8 1. 258 0.082
  4 1.000 8 1.739 0.211
-46-

-------
Analyte
Chi-square
value for
Bartlett's Test
F-value for
Levene's Test
2-chloronaphthalene
dimethylphthalate
hexachlorobenzene
anthracene
phenanthrene
fluoranthene
5.77
9.47*
12.62**
29.21**
22.53**
34.66**
1.97
3.39*
5.05**
3.82*
1. 76
4.01*
* = exceeds critical value at .05 level.
** = exceeds critical value at .01 level.
We considered several transformations of the observed responses in
an attempt to remedy the variance heterogeneity (see, for example,
Kempthorne (1952) and Snedecor and Cochran (1980)).
Three of the
candidate transformations were
R,n (y* + C),
.; y* + C,
and
y*/(x* + C)
where C denotes some constant (various values of C were considered).
Although the performance of the various transformations varied across
the analytes, the square root transformation, with C=O, appeared to
provide reasonably stable variances over the concentration levels.
therefore adopted the transformation
We
Y = IY*;
where y* is the adjusted peak area. Values of both y* and Yare included
in Tables 4.2 through 4.7. Table 4.9 contains the means and standard
deviations of the transformed variate (Y), by analyte and concentration
level.
On this transformed scale, results of the variance homogeneity
tests are as follows:
-47-

-------
Table 4.9. Means and Standard Deviations of Transformed Responses
 (Square Root of Adjusted Peak Areas), by Analyte and
 Concentration Level   
  Standard Cone.  Y = { Adjusted Peak Area
Analyte -L X*(ppm) D. Mean Std. Deviation
J
2-chloronaphthalene 1 0.000 7 0.299 0.064
  2 0.215 8 0.557 0.054
  3 0.858 8 0.954 0.044
  4 1.070 8 1.101 0.047
dimethylphthalate 1 0.000 6 0.308 0.049
  2 0.217 8 0.392 0.035
  3 0.867 8 0.644 0.066
  4 1.080 8 0.741 0.034
hexachlorobenzene 1 0.000 7 0.186 0.050
  2 0.200 8 0.343 0.019
  3 0.802 8 0.629 0.014
  4 1.000 8 0.735 0.030
anthracene 1 0.000 7 0.203 0.025
  2 0.217 8 0.577 0.025
  3 0.870 8 1.108 0.037
  4 1.070 8 1.306 0.048
phenanthrene 1 0.000 7 0.252 0.027
  2 0.210 8 0.602 0.027
  3 0.840 8 1.186 0.033
  4 1.050 8 1. 390 0.041
fluoranthene 1 0.000 7 0.241 0.023
  2 0.200 8 0.589 0.046
  3 0.802 8 1.121 0.037
  4 1.000 8 1.316 0.081
-48-

-------
Ana1yte
Chi-square
value for
Bartlett's Test
F-va1ue for
Levene's Test
2-ch1oronaphtha1ene
dimethyl phthalate
hexach1orobenzene
anthracene
phenanthrene
f1uoranthene
1.03
3.99
11 .58**
3.98
1.56
9.60*
0.88
1.32
6.56**
0.83
0.05
1.46
* = exceeds critical value at .05 level.
** = exceeds critical value at .01 level.
With the possible exception of hexach1orobenzene, the transformed scale
is clearly preferable to the original scale in terms of variance stabi1-
ity over concentration levels.
Even for hexach1orobenzene, there does
not appear to be a systematic trend in the variation of responses with
concentration level (see Table 4.9).
We then prepared plots of Y = ~ versus concentration (X*) , as
shown in Figures 4-7 through 4-12. In an effort to determine a simple
(i.e., one or two parameters) relationship between Y and X*, we consid-
ered a number of monotonic transformations of X*.
these was the following formulation:
The most promising of
Y = A + B v X* + C + 
(C 2. 0) .
(4-1)
This model was fitted by nonlinear least squares for each ana1yte
separately.
Examination of the parameter estimates A, B, and C, which are
highly intercorre1ated, revealed that the residual sum of squares was
very insensitive to the C value. Based on these results, a value of C =
0.1 appeared to be a reasonable choice for all ana1ytes. Using this
value of C, the model (4-1) becomes a two-parameter model that can be
written in the form
Y = A + BX' + 
(4-2)
where X' = v X* + 0.1.
This can be rewritten in the form of model
(2-2), which requires f(O,B) = zero, by choosing
x = (X ' - ro:T ) = (.,I X* + O. 1 - ro:T )
(4-3)
-49-

-------
FIGURE 4-7.
CALIBRATION DATA: Y VERSUS X*
ANALYTE=2-CHLORONAPHTHALENE
Y  
1 .25  
 * I
1 .00 * 
 I 
i
0.75
I
VI
o
I
0.50
0.25
o
o
o
o
o
0-
o
o
o
o
1
I
.

o
.

1
.

2
.

3
.

4
.

5
.

6
.

7
.

8
.

9
.

o
.

I
CONCENTRATION 
-------
FIGURE 4-8.
CALIBRATION DATA: Y VERSUS X*
ANALYTE=DIMETHYLPHTHALATE
Y
0.8
0.6
 *
* I
* *
* 
* 
* 
0.7
I
\J1
f-'
I
-
I
*
o
o
o
o
o
o
o
o
o
o
I
I
.

o
.

I
.

2
.

3
.

4
.

5
.

6
.

7
.

8
.

9
.

o
.

I
CONCENTRATION 
-------
FIGURE 4-9.
Y
0.8
0.7
0.6
0.5
0.4
I
I.J1
N
I
0.3
0.2
0. 1
CALIBRATION DATA: Y VERSUS X*
ANALYTE=HEXACHLOROBENZENE
t
-
*
i
8
*
o
o
o
o
o
o
o
o
o
o
1
1
.

o
.

1
.

2
.

3
.

4
.

5
.

6
.

7
.

8
.

9
.

1
.

o
CONCENTRATION 
-------
FIGURE 4-10.
CALIBRATION DATA: Y VERSUS X*
ANALYTE=ANTHRACENE
0.75
Y  
  *
1 .25  I
 * 
 I 
1 .00  
I
V1
W
I
I
o
o
o
o
o
o
o
o
o
o
1
1
.

o
.

I'
.

2
.

3
.

4
.

5
.

6
.

7
.

8
.

9
.

o
.

1
CONCENTRATION 
-------
FIGURE 4-11.
Y
I .5
I .2.
0.9
 0.6 I
I  
lJ1  
~  
I  
 0.3 
0.0
CALIBRATION DATA: Y VERSUS X*
,ANALYTE=PHENANTHRENE
!
*
*
*
o
o
o
.

o
.

I
.

2
o
o
o
o
o
.

3
.

4
.

6
.

7
.

5
CONCENTRATION 
-------
FIGURE 4-12.
CALIBRATION DATA: Y VERSUS X*
ANALYTE=FLUORANTHENE
Y  
1 .5  
  *
  ;
1 .2 I *
0.9  
 0.6 i
I 
\JI  
\JI  
I  
 0.3 
0.0
o
o
o
o
o
o
o
o
o
o
1
1
.

o
.

1
.

2
.

3
.

4
.

5
.

6
.

7
.

8
.

9
.

o
.

1
CONCENTRATION (PPM)

-------
and letting 6 = B and a = A + B ~.
are included in Tables 4.2 through 4.7.
Values of x defined by (4-3)
The selected model form is thus
identical to the linear calibration model (2-2):

Y = a + 6x + 
x
where a and 6 are the unknown model parameters, and where
(4-4)
Y = square root {observed peak area adjusted for the peak area of
x the internal standard},
x = , X* + 0.1 - ~ where X* is the standard concentration,

 = observational error, which is assumed (on the transformed
scale) to be normally distributed with zero mean and
variance 0'2.
Plots of the transformed responses, Y , versus the transformed concen-
x
trations, x, are shown in Figures 4-13 through 4-18.
These figures also show (1) the fitted model (solid line), based
on least squares estimates a and 6 (eq. (2-6) and (2-7)), and (2) bands
constructed from interval estimates expected to contain 99% of indivi-
dual (future) observations (dashed lines):
a + 6x ! t 005 w 0'
v,. x
where
tv,.005 is the upper 0.5 percentage point of Student's t-distribu-

tion with v degrees of freedom (v=29 for most ana1ytes),
w is defined by (2-12) with r=1, and
x
0' is the estimate of 0' derived from the residual variance.
We examined model adequacy in two ways. First, we conducted an
overall lack of fit test, as described under Step 5 in Section 2.2. The
"
variance 0'2, the residual variance 0'2, and
table below shows the pooled
the F-statistic (eq. 2-3) for
anthracene
phenanthrene
f1uoranthene
testing lack of fit: 
  ,. 
0'2 0'2 F-Statistic
.0027180 .0027966 1.42
.0022934 .0027582 3.84*
.0009418 .0010524 2.70
.0012537 .0018670 8.09**
.0010790 .0013905 5.19*
.0027212 .0029900 2.43
Ana1yte
2-ch10ronaphtha1ene
dimethy1phtha1ate
hexach1orobenzene
-56-

-------
y
1 .2
1 .0
0.8
I
V1
'-I
I
0.6
0.4
0.2
FIGURE 4-13.
CALIBRATION DATA: Y VERSUS X
ANALYTE=2-CHLORONAPHTHALENE
"
"
"
"
,
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,
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,
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"
,
0.0
0. 1
0.2
0.3
0.4
0.5
0.6
0.8
0.7
x

-------
Y
0.8
0.7
0.6
0.5
I
U1
00
I
0.4
0.3
0.2
"
"
"
"
"
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"
"
,
FIGURE 4-14.
0.0
CALIBRATION DATA: Y VERSUS X
ANALYTE=DIMETHYLPHTHALATE
*
*
III
"
"
"
"
* "
"
"
"
"
"
"
"
"
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"
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;
*
0. I
0.2
0.3
0.4
0.5
0.6
0.7
0.8
x

-------
FIGURE 4-15.
CALIBRATION DATA: Y VERSUS X
ANALYTE=HEXACHLOROBENZENE
Y
0.8
0.5
"
"
"
"
"
"
"
"
"
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0.7
0.6
0.4
I
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I
0.3
0.2
0. I
0.0
0.0
0. I
0.2
0.3
0.4
0.5
0.6
0.7
0.8
x

-------
Y
I .50
I .25
I .00
0.75
I
(j'\
o
I
0.50
0.25
0.00
FIGURE 4-16.
CALIBRATION DATA: Y VERSUS X
ANALYTE=ANTHRACENE
,.
--*,'
"
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"
"
"
"
,
0.0
0. I
0.2
0.3
0.4
0.5
0.6
0.7
0.8
x

-------
FIGURE 4-17.
CALIBRATION DATA: Y VERSUS X
ANALYTE=PHENANTHRENE
Y
1 .50
1 .00
* .."
"
"
"
"
"
,
?Is",

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0.2
0.3
0.4
0.5
0.6
0.7
0.8
x

-------
Y
I .50
I .25
I .00
0.75
I
0'\
N
I
0.50
0.25
0.00
FIGURE 4-18.
CALIBRATION DATA: Y VERSUS X
ANALYTE=FLUORANTHENE
"
"
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,""*
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0.7
0.8

-------
The F-statistics, based on 2 and 27 degrees of freedom (2 and 26 for
dimethy1phtha1ate), provide evidence of lack of fit for anthracene, and
perhaps for dimethy1phtha1ate and phenanthrene.
We also tested the straight line model against the specific alter-
native of a quadratic model. With the exception of dimethy1phtha1ate
(for which some erratic behavior has been noted previously), there was
no evidence that the inclusion of a quadratic term was necessary.
We adopted the model form (4-4) for all ana1ytes; on the trans-
formed scales, all of the methods described for the linear calibration
in Chapters 2 and 3 are applicable therefore.
Table 4-10 contains the relevant calibration design characteris-
tics, and Table 4-11 contains estimates of the parameters.
used in these tables is defined in Chapter 2.
The notation
4.5
Estimation of Detection Limits
After estimating the parameters for each ana1yte, we proceeded in
the manner described in Section 3.2.
This involved:
(1)
(2)
(3)
(4)
specification of a detection rule;
construction of the threshold response value y ;
p
estimation of Bfa (Section 3.3); and
estimation of detection limits with specified false positive
and false negative rates.
These four steps are addressed in the subsections below.
Specification of a detection rule. In accordance with the formula-
tion in Chapter 3, the analyte is to be declared detected whenever the
mean of r replicate response observations from the same source matrix
exceeds a response threshold value, which is a function of the chosen
rate p for false positives. In practice, we would therefore choose a
particular value of p and a single value r as a part of the detection
protocol. For purposes of this report we considered two p values (p =
0.01 and p = 0.05) and three r values (r = 1, 2, and 3) in order to
provide a more comprehensive description of detectability. Values of t
p
corresponding to the two chosen p values are shown below (values from
Table 1 of Appendix A):
-63-

-------
Table 4-10. Calibration Design Parameters, By Analyte 
    w Values For 
    0  
  - 0 r=l r=2 r=3
Analyte n x .xx
-     
2-chloronaphthalene 31 .43174 2.90072 1.04715 0.77235 0.65563
Dimethylphthalate 30 .44906 2.74193 1.05208 0.77902 0.66349
Hexachlorobenzene 31 .41228 2.66357 1.04694 0.77206 0.65529
Anthracene 31 .43378 2.91816 1.04725 0.77249 0.65580
Phenanthrene 31 .42581 2.83083 1. 04 705 0.77221 0.65547
Fluoranthene 31 .41228 2.66357 1.04694 0.77206 0.65529
Table 4-11. Calibration Model Parameter Estimates, By Analyte 
     Standard Standard
     Error Error
Analyte n a B a of a of B
-     
2-Chloronaphthalene 31 0.300676 1.02173 0.052883 0.016429 0.03105
Dimethylphthalate 30 0.279346 0.57002 0.052519 0.017170 0.03172
Hexachlorobenzene 31 0.178793 0.73651 0.032441 0.010055 0.01988
Anthracene 31 0.212840 1.39402 0.043208 0.013439 0.02529
Phenanthrene 31 0.247232 1.47915 0.037289 0.011572 0.02216
Fluoranthene 31 0.245677 1. 43041 0.054681 0.016949 0.03350
-64-

-------
'I
False Positive Rate
Degrees of Freedom
v = n-2 = 29*
p = 0.01
2.46202
2.46714
p = 0.05
1. 69913
1. 70113
v = n-2 = 28
* Applicable to all ana1ytes except dimethy1-
phthalate.
Values of w corresponding to r = I, 2, and 3, which also depend on
o
the calibration design, were given previously in Table 4-10.
Determination of the threshold response value.
Threshold response
values for each p and r combination, given the calibration design, were
determined via eq. (3-6) and are given in Table 4-12. These values of
y apply to the transformed scale associated with model (4-4).
p
Estimation of ala.
We next used the results of Table 4-11 to
produce point estimates of ala and of ala.
Section 3.3, are given below:
These estimates, derived in
   ,. ,.
Ana1yte M aiM 0' ala
v v
2-ch1oronaphtha1ene 1.02683 18.81575 0.051758
dimethyl phthalate  1.02782 10.55982 0.092135
hexach10robenzene 1.02683 22.10985 0.044047
anthracene 1.02683 31. 42001 0.030995
phenanthrene 1.02683 38.63073 0.025210
f1uoranthene 1.02683 25.47567 0.038228
Estimation of detection limits.
Point estimates of x~(p,q) were
determined for each ana1yte as
Xt(p,q) = wo6(p,q)ala
(4-5)
where q was allowed to vary over the range (p,l). For each such q, the
6 value satisfying (3-8) was determined using the algorithms of Appendix
B. Estimated power (i.e., detection) curves, such as those shown in
Figures 4-19 through 4-24, were produced by this means.
The curves
shown in these figures correspond to p = 0.01; each figure contains a
curve for each of r = 1, 2, or 3. A common horizontal scale (i.e., the
transformed concentration scale) is used for all six figures in order to
-65-

-------
Table 4-12. Threshold Response Values y , By Analyte, 
 for p=O.Ol and 0.05, and fo 
   r=l, 2, and 3    
 r=l r=2 r=3
Analyte p=O.Ol p=0.05 p=O.Ol p=0.05 p=O.Ol p=0.05
2-Chloronaphthalene 0.43701 0.39477 0.40123 0.37008 0.38604 0.35959
Dimethylphthalate 0.41567 0.37334 0.38029 0.34895 0.36531 0.33862
Hexachlorobenzene 0.26241 0.23650 0.24046 0.22135 0.23113 0.21491
Anthracene 0.32425 0.28973 0.29502 0.26955 0.28260 0.26099
Phenanthrene 0.34336 0.31357 0.31813 0.29616 0.30741 0.28876
Fluoranthene 0.38662 0.34295 0.34962 0.31741 0.33390 0.30656
-66-

-------
FIGURE 4-19.
ESTIMATED POWER CURVES
ANALYTE=2-CHLORONAPHTHALENE
P=0.01
  1 .0      - -- 
      /... .,. 
 D      '" "," 
 E 0.9     / ~ 
     ./  
     I  /  
 T    / 1  
 E 0.8  I 1   
  /  I   
 C   I   
   I    
 T 0.7  /  I    
  J     
 I   I I     
 0 0.6  / J      
 N   I J      
  / /      
  0.6      
  I I      
 P  / J      
 R 0.4 I I      
 0  / J       
  I J       
 B 0.3 1/       
~ A  I I       
~  0.2 /1       
I B       
 I  I 1       
  /./1       
 L 0. 1 "'./       
 I  ~./       
 T 0.0        
 Y         
  0.0 0. 1    0.2 0.3
        X 
   LEGEND:  R 1 ----- 2
0.4
0.6
----- 3

-------
FIGURE 4-20.
ESTIMATED POWER CURVES
ANALYTE=DIMETHYLPHTHALATE
P=0.01
 1 .0 
D  
E 0.9 
T  
E 0.8 
C  
T 0.7 
I  
0 0.6 
N  
 0.5 
P  
R 0.4 
0  
8 0.3 
6-. A   

-------
D
E
T
E
C
T
I
o
N

P
R
o
B
6-A
'F B
I
L
I
T
Y
FIGURE 4-21.
1 .0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0. 1
0.0
ESTIMATED POWER CURVES
ANALYTE=HEXACHLOROBENZENE P=0.01
....- --
" .",
/ '"
J /
/ /
J I
/ /
I I
/ I
I I
1 I
I I
/ /
I I
/ I
I I
1/
I I
/,
I I
/'
I I
III
JI
L/
v
0.0
0. 1
0.2
0.3
x
LEGEND: R
----- 2
1
0.4
0.5
..---- 3

-------
FIGURE 4-22.
ESTIMATED POWER CURVES
ANALYTE=ANTHRACENE P=0.01
  1 .0 
 D  
 E 0.9 
 T  
 E 0.8 
 C  
 T 0.7 
 I  
 0 0.6 
 N  
  0.5 
 P  
 R 0.4 
 0  
 B 0.3 
~ A  
0 B 0.2 
I 
 I  
 L 0. 1 
 I  
 T 0.0 
 Y  
  0.0 0. 1
0.2  0.3  0.4 0.5
X    
LEGEND: R 1 ----- 2 ----- 3 

-------
FIGURE 4-23.
ESTIMATED POWER CURVES
ANALYTE=PHENANTHRENE P=0.01
 1 .0 
D  
E 0.9 
T  
E 0.8 
C  
T 0.7 
I  
0 0.6 
N  
 0.6 
P  
R 0.4 
0  
B 1a.3 
~A  
rB 0.2 
I  
L 0. 1 
I  
T 1a.0 
Y  
 0.0 0. 1
0.2  0.3  0.4 0.5
X    
LEGEND: R 1 ----- 2 ------- 3 

-------
FIGURE 4-24.
ESTIMATED POWER CURVES
ANALYTE=FLUORANTHENE P=0.01
 1 .0     , ",,-    
D      ,/'/     
     ~ "     
E 0.9   / /     
  1 I     
T    / /     
E 0.8   I I       
C   / I       
T 0.7  I I       
I   1 /       
0 0.6  I I       
 1 I         
N   I I         
 0.5  1 I         
P   I I         
R 0.4  1/         
0   I I         
 / I         
B  I I         
~ A  /'         
N  1 I          
16  1/          
I           
 II          
L 0. t 1/          
I  1.'./          
T            
Y            
 0.0  0. 1 0.2 0.3 0.4 0.5
      X   
   LEGEND: R 1 ----- 2 ----. 3 

-------
demonstrate the different detectabililty characteristics of the analytes.
For example, for a fixed q, dimethylphthalate (Figure 4-20) will clearly
have a larger detection limit (less sensitive detection) than any of the
other analytes.
We note that while the horizontal scale used in Figures 4-19
through 4-24 is given in terms of x, a similar set of curves could just
as readily have been produced in the original concentration units (ppm)
A

by determining Xt values according to (4-5) and then using the relation-
ship
x~ = Xt(Xt + 210:1)
(4-6)
= Xt(Xt + 0.632456)
Tables 4-13, 4-14, and 4-15 contain estimated detection limits for
r = 1, 2, and 3, respectively; point estimates x(p,q) are given for two
p values (p = 0.01 and 0.05) and two q values (q = 0.01 and q = 0.05).
These tables also provide the corresponding interval estimates for
Xt(p,q), as determined via (3-15) with coverage probabilities (1-y) =
0.95 or 0.99. Again, all of the results are given in terms of the
transformed scale associated with model (4-4), but could be translated
to ppm through eq. (4-6). For example, for the first line of Table 4-13
(analyte = 2-chloronaphthalene, r=1, p=0.01, q=0.05), we obtain:
x~ = .194 ppm,
Interval for x~ =
Interval for x~ =
(0.145, 0.288) with (1-y) = 0.95, and
(0.134, 0.332) with (1-y) = 0.99.
4.6 Phase II Design
The first phase of the experiment --
mechanism for estimating detection limits
calibration -- provides the
with specified assurance
probabilities. Phase II of the experiment was designed to demonstrate
and validate the Phase I formulation. Phase I results were generated
during December 1984, while Phase II was conducted during February 1985.
Experimental methods and the source matrix for both phases were ident-
ical (see Section 4.1).
The design for Phase II involved fortifying aliquots of the Uni-
versity Lake sediment at three distinct concentration values (x!' x~,
and x~) for each analyte and conducting twelve independent "blind"
-73-

-------
TABLE 4-13. DETECTION LIMIT ESTIMATES FOR SELECTED P AND ~ VALUES, FOR r=1
     DETECTION LIMIT ESTIMATES 
     ---------------------------------------------
     POINT 957. CONF. LIMITS 99r.' CONF. LIMITS
ANALYTE P ~ ESTIMATE LOW HIGH LOW HIGH
----------------------------------------------------------------------------
2-CHLORONAPHTHALENE 0.01 0.05 0.22601 O. 17907 0.30684 O. 16721 0.34108
   0.01 0.26491 0.20990 0.35965 O. 19599 0.39979
  0.05 0.05 O. 18264 O. 14472 0.24796 O. 13513 0.27564
   0.01 0.22051 0.17472 0.29938 O. 16314 0.33279
DIMETHYLPHTHALATE 0.01 0.05 0.40495 0.31616 0.56550 0.29427 0.63748
   0.01 0.47467 0.37060 0.66286 0.34493 O. 74723
  0.05 0.05 0.32695 0.25526 0.45657 0.23758 O. 51468
   0.01 0.39474 0.30819 O. 55124 0.28685 0.62141
HEXACHLOROBENZENE 0.01 0.05 O. 19230 O. 15253 0.26053 O. 14245 0.28929
   0.01 0.22539 0.17878 0.30537 O. 16697 0.33909
  0.05 0.05 O. 15540 O. 12326 0.21054 O. 11512 0.23379
   0.01 O. 18762 O. 14882 0.25420 O. 13899 0.28226
ANTHRACENE 0.01 0.05 O. 13536 O. 10760 O. 18261 O. 10054 0.20232
   0.01 O. 15866 O. 12612 0.21404 0.11784 0.23714
  0.05 0.05 O. 10939 0.08696 O. 14757 0.08125 O. 16350
   0.01 O. 13207 O. 10499 O. 17817 0.09809 O. 19740
PHENANTHRENE 0.01 0.05 O. 11007 0.08755 O. 14833 0.08181 O. 16424
   0.01 O. 12902 O. 10262 O. 17386 0.09589 O. 19251
  0.05 0.05 0.08895 0.07075 0.11987 0.06611 O. 13273
   0.01 O. 10739 0.08542 O. 14472 0.07982 O. 16025
FLUORANTHENE 0.01 0.05 O. 16689 O. 13251 0.22568 O. 12378 0.25034
   0.01 O. 19562 O. 15531 0.26452 O. 14508 0.29343
  0.05 0.05 O. 13487 O. 10708 O. 18238 O. 10003 0.20231
   0.01 O. 16283 O. 12929 0.22019 O. 12077 0.24426
-74-

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TABLE 4-14. DETECTION LIMIT ESTIMATES FOR SELECTED P AND q VALUES, FOR r=2
     DETECTION LIMIT ESTIMATES 
     ---------------------------------------------
     POINT 957. CONF. LIMITS 997. CONF. LIMITS
ANALYTE P q ESTIMATE LOW HIGH LOW HIGH
----------------------------------------------------------------------------
'2-CHLORONAPHTHALENE 0.010.05 O. 16670 O. 13208 0.22631 O. 12333 0.25157
   0.01 O. 19539 O. 15481 0.26527 O. 14455 0.29487
  0.05 0.05 O. 13471 O. 10674 O. 18289 0.09966 0.20330
   0.01 O. 16264 O. 12887 0.22081 O. 12033 0.24546
DIMETHYLPHTHALATE 0.01 0.05 0.29985 0.23411 0.41873 0.21789 O. 47203
   0.01 0.35147 0.27441 0.49082 0.25540 0.55329
  0.05 0.05 0.24209 O. 18901 0.33807 O. 17592 0.38110
   0.01 0.29229 0.22820 0.40817 0.21240 O. 46013
HEXACHLOROBENZENE 0.01 0.05 O. 14181 O. 11248 O. 19213 O. 10505 0.21334
   0.01 O. 16622 O. 13184 0.22520 O. 12313 0.25006
  0.05 0.05 0.11460 0.09090 O. 15526 0.08489 O. 17241
   0.01 O. 13836 O. 10975 O. 18746 O. 10250 0.20815
ANTHRACENE 0.01 0.05 0.09984 0.07937 O. 13470 0.07416 O. 14924
   0.01 0.11703 0.09303 O. 15788 0.08692 0.17492
  0.05 0.05 0.08069 0.06414 O. 10885 0.05993 O. 12060
   0.01 0.09742 0.07744 O. 13142 0.07236 O. 14561
PHENANTHRENE 0.01 0.05 0.08118 0.06457 O. 10939 0.06034 O. 12113
   0.01 0.09515 0.07568 O. 12822 0.07072 O. 14198
  0.05 0.05 0.06560 0.05218 0.08840 0.04876 0.09789
   0.01 0.07920 0.06300 O. 10673 0.05887 0.11819
FLUORANTHENE 0.01 0.05 O. 12307 0.09772 O. 16643 0.09128 O. 18461
   0.01 O. 14426 0.11454 O. 19507 O. 10699 0.21639
  0.05 0.05 0.09946 0.07897 O. 13449 0.07376 O. 14919
   0.01 O. 12008 0.09534 O. 16238 0.08906 O. 18013
-75-

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TABLE 4-15. DETECTION LIMIT ESTIMATES FOR SELECTED P AND ~ VALUES, FOR r=3
    DETECTION LIMIT ESTIMATES 
    ---------------------------------------------
    POINT 957. CONF. LIMITS 997. CONF. LIMITS
ANALYTE P ~ ESTIMATE LOW HIGH LOW HIGH
----------------------------------------------------------------------------
~-CHLORONAPHTHALENE 0.01 0.05 O. 14151 O. 11212 O. 19211 O. 10469 0.21356
  0.01 O. 16586 O. 13142 0.22518 O. 12271 0.25031
  0.05 0.05 0.11435 0.09061 O. 15525 0.08460 O. 17258
  0.01 O. 13807 O. 10940 O. 18744 O. 10215 0.20836
DIMETHYLPHTHALATE 0.010.05 0.25538 O. 19939 0.35663 O. 18558 O. 40202
  0.01 0.29935 0.23371 0.41802 0.21752 O. 47123
  0.05 0.05 0.20619 O. 16098 0.28793 O. 14983 0.32458
  0.01 0.24894 O. 19436 0.34763 O. 18090 0.39188
HEXACHLOROBENZENE 0.01 0.05 O. 12036 0.09547 O. 16307 0.08916 O. 18107
  0.01 O. 14108 O. 11190 0.19114 O. 10451 0.21224
  0.05 0.05 0.09727 0.07715 0.13178 0.07205 O. 14633
  0.01 0.11744 0.09315 O. 15911 0.08699 O. 17667
ANTHRACENE 0.01 0.05 0.08476 0.06738 0.11435 0.06296 O. 12669
  0.01 0.09935 0.07898 O. 13403 0.07379 O. 14850
  0.05 0.05 0.06850 0.05445 0.09241 0.05088 O. 10238
  0.01 0.08270 0.06574 0.11157 0.06143 O. 12361
PHENANTHRENE 0.01 0.05 0.06891 0.05481 0.09286 0.05122 O. 10282
  0.01 0.08077 0.06424 O. 10884 0.06003 O. 12052
  0.05 0.05 0.05568 0.04429 0.07504 0.04139 0.08309
  0.01 0.06723 0.05348 0.09060 0.04997 O. 10032
FLUORANTHENE 0.01 0.05 O. 10446 0.08294 O. 14125 0.07747 O. 15669
  0.01 O. 12244 0.09721 O. 16557 0.09081 O. 18366
  0.05 0.05 0.08442 0.06702 O. 11415 0.06261 O. 12663
  0.01 O. 10192 0.08092 O. 13782 0.07559 O. 15288
-76-

-------
replicate response determinations at each concentration. The lowest
concentration value was chosen initially so that, on the basis of Phase
I results and the theoretical formulation of detectabi1ity, approxi-
mately 10 percent of the observationsAat xr would be expected to be
detected (i.e., exceed the threshold y ) when p=0.01 and r=1. For
p
2-~hloronaphthalene, for instance, Figure 4-19 indicates that choosing
x ~ .06, or x* ~ .042 ppm, will yield a detection rate approximately
equal to 0.10. The highest concentration value, x~, was selected
initially in a similar manner to yield an approximate detection rate of
0.90 when p=0.05 and r=1. The middle concentration value was chosen to
be the mean o~ the highest and lowest concentrations.
We then examined the initially-specified concentrations and adjust-
ed them slightly so that a single stock solution of all analytes could
be diluted serially. This resulted in choosing a Phase II design that
consisted of three equally-spaced concentration levels (apart from
rounding); for each analyte, these are given in Table 4-16 in the
column labeled X* (ppm).
Table 4-16 also shows, by analyte, the following quantities: w
o
values, selected p values (p=0.01 and p=0.05), associated t values, and
p
the corresponding threshold response values yp (denoted Yp). For each
analyte and for each Phase II concentration level, this table also
provides:
X, the transformed concentration level;
DELTA, the estimated noncentrality parameter 6 associated with X
(computed via eq. (3-7) with B/o replaced with its estimate
from eq. (3-11));
the estimated detection rate, 1-q(x), corresponding to X (computed
via eq. (3-16)); and
interval estimates for 1-q(x), as determined by eq. (3-17) with
(1-y) = 0.95 and 0.99.
All of the estimated quantities shown in this table depend solely on the
Phase I results. Tables 4-17 and 4-18 provide comparable results for
r=2 and r=3, respectively.
The purposes of Phase II were (1) to generate empirical estimates
of the detection rates at each of the selected concentration levels, and
-77-

-------
 TABLE 4-16. 'ESTIMATED PHASE I DETECTION RATES ASSOCIATED WITH EACH        
  PHASE II CONCENTRATION LEVEL, FOR r=1           
             ESTIMATE CONFIDENCE INTERVALS FOR l-q
 ANALYTE   Wo  P Tp Yp X* X DELTA OF 1-11. 95 X LIMITS 99 7. LIMITS
 ------------------------------------------------------------------------------------------------------------------------------------
 2-CHLORONAPHTHALENE 1. 04715 0.01000 2.46202 0.43701 0.043 0.06193 1. 11272 O. 10299 0.06385 0.17086 0.05437 O. 19676
          0.086 0.11505 2.06727 0.36126 0.20187 0.58992 O. 16242 0.65846
          O. 129 O. 16231 2.91651 0.67482 0.41172 0.89866 0.33213 0.93685
      0.05000 1. 69913 0.39477 0.043 0.06193 1. 11272 0.28840 0.205300.40651 O. 18264 0.44566
          0.086 0.11505 2.06727 0.64573 0.453070.83420 0.39314 0.87624
          0.129 O. 16231 2.91651 0.88542 0.69465 0.97955 0.61521 0.98945
 DIMETHYLPHTHALATE  1.05208 0.01000 2.46714 0.41567 0.087 0.11621 1. 16638 O. 11211 0.06571 O. 19374 0.05471 0.22498
          0.173 0.20627 2.07031 0.36107 O. 19070 0.60524 O. 14940 0.67709
          0.260 0.28377 2.84824 0.64933 0.37038 0.89232 0.28914 0.93366
      0.05000 1. 70113 0.37334 0.087 0.11621 1. 16638 0.30620 0.20982 0.44174 O. 18363 0.48600
          0.173 0.20627 2.07031 0.64618 O. 43724 0.84467 0.37228 0.88716
          0.260 0.28377 2.84824 O. 87150 0.65558 0.97789 O. 56733 0.98881
 HEXACHLOROBENZENE  1.04694 0.01000 2.46202 0.26241 0.036 0.05255 1.10987 O. 10251 0.06380 O. 16960 0.05442 O. 19520
          0.072 0.09850 2.08021 0.36589 0.20539 0.59505 O. 16554 0.66342
          0.108 O. 13984 2.95328 0.68731 O. 42378 0.90581 0.34304 0.94207
      0.05000 1. 69913 0.23650 0.036 0.05255 1. 10987 0.28745 0.20519 0.40453 O. 18275 0.44338
          0.072 0.09850 2.08021 0.65042 0.45812 0.83755 0.39812 0.87906
I          0.108 O. 13984 2.95328 0.89222 O. 70565 0.98156 0.62685 0.99063
....         
00           
I                  
 ANTHRACENE   1.04725 0.01000 2.46202 0.32425 0.024 0.03591 1. 07734 0.09707 0.06131 O. 15894 0.05261 O. 18257
          0.047 0.06718 2.01550 0.34296 O. 19405 0.56173 0.15733 0.62909
          0.071 0.09729 2.91904 0.67568 0.41763 0.89712 0.33940 0.93538
      0.05000 1.69913 0.28973 0.024 0.03591 1.07734 0.27672 0.19936 0.38753 0.17828 O. 42454
          0.047 0.06718 2.01550 0.62676 0.44169 0.81516 0.38491 0.85894
          0.071 0.09729 2.91904 0.88590 O. 70007 0.97911 0.62299 0.98911
 PHENANTHRENE  1. 04705 0.01000 2.46202 0.34336 0.019 0.02874 1. 06021 0.09430 0.05993 O. 15370 0.05154 0.17640
          0.038 0.05526 2.03866 0.35110 0.19917 0.57235 O. 16157 0.63984
          0.058 0.08126 2.99824 O. 70229 0.44112 0.91315 0.35983 0.94721
      0.05000 1.69913 0.31357 0.019 0.02874 1.06021 0.27115 O. 19607 0.37899 O. 1"7562 0.41511
          0.038 0.05526 2.03866 0.63529 0.44916 0.82245 0.39176 0.86540
          0.058 0.08126 2.99824 0.90013 O. 72104 0.98355 0.64427 0.99176
 FLUORANTHENE  1. 04694 0.01000 2.46202 0.38662 0.030 0.04433 1.07864 0.09729 0.06121 O. 15972 0.05243 O. 18357
          0.060 O. 08377 2.03848 0.35104 O. 19770 O. 57403 O. 15982 0.64192
          0.090 O. 11966 2.91181 0.67321 0.41366 0.89631 0.33520 0.93488
      0.05000 1.69913 O. :34295 0.030 0.04433 1.07864 0.27715 O. 19910 0.38879 0.17784 0.42606
          0.060 0.08377 2.03848 0.63522 0.44703 0.82360 0.38895 0.86663
          0.090 O. 11966 2.91181 0.88453 0.69643 0.97887 0.61850 0.98899

-------
 TABLE 4-17. ESTIMATED PHASE I DETECTION RATES ASSOCIATED WITH EACH         
  PHASE II CONCENTRATION LEVEL, FOR r=2            
             ESTIMATE CONFIDENCE INTERVALS FOR 1-11.
 ANALVTE   Wo  p Tp Vp X* X DELTA OF 1-11. 95 7. LIMITS 99 7. LIMITS
 ------------------------------------------------------------------------------------------------------------------------------------
 2-CHLORONAPHTHALENE O. 77235 0.01000 2.46202 0.40123 0.043 0.06193 1.50862 O. 18746 O. 10791 0.32207 0.08900 0.37079
          0.086 0.11505 2.80281 0.63498 0.38018 0.87159 0.30583 0.91604
          O. 129 O. 16231 3.95421 0.92504 0.69980 0.99463 O. 59412 0.99811
      0.05000 1.69913 0.37008 0.043 0.06193 1.50862 0.43191 0.29791 0.60423 0.26034 0.65533
          0.086 O. 11505 2.80281 0.86250 0.66463 0.97134 0.58605 0.98431
          0.129 O. 16231 3.95421 0.98661 0.89883 0.99960 0.83694 0.99990
 DIMETHYLPHTHALATE O. 77902 0.01000 2.46714 0.38029 0.087 0.11621 1.57521 0.20429 0.11084 0.36228 0.08904 0.41852
          0.173 0.20627 2. 79598 0.63077 0.35666 0.87975 0.27815 0.92415
          0.260 0.28377 3.84658 0.90840 0.64331 0.99351 O. 52538 0.99775
      0.05000 1. 70113 0.34895 0.087 0.11621 1. 57521 0.45707 0.30382 0.64741 0.26070 0.70154
          0.173 0.20627 2. 79598 0.86050 0.64167 0.97411 0.55421 0.98651
          0.260 0.28377 3.84658 0.98242 0.86797 0.99950 O. 78965 0.99987
 HEXACHLOROBENZENE O. 77206 0.01000 2.46202 0.24046 0.036 0.05255 1.50502 O. 18654 O. 10785 0.31976 0.08912 0.36803
          0.072 0.09850 2.82083 0.64141 0.38674 0.87542 0.31197 0.91893
          O. 108 O. 13984 4.00475 0.93161 O. 71436 0.99544 0.60989 0.99844
      0.05000 1. 69913 0.22135 0.036 0.05255 1.50502 0.43053 0.29778 0.60169 0.26059 0.65257
I          0.072 0.09850 2.82083 0.86633 0.67103 0.97256 O. 59300 0.98506
.....          O. 108 O. 13984 4.00475 0.98821 0.90630 0.99968 0.84706 0.99992
'"         
I                   
 ANTHRACENE   O. 77249 0.01000 2. 46202 0.29502 0.024 0.03591 1.46054 0.17538 O. 10281 0.29903 0.08551 0.34424
          0.047 0.06718 2. 73238 0.60952 0.36570 0.84993 0.29590 0.89829
          0.071 0.09729 3.95730 0.92546 O. 70682 0.99444 0.60447 0.99801
      0.05000 1. 69913 0.26955 0.024 0.03591 1.46054 0.41353 0.288050.57826 0.25307 0.62811
          0.047 0.06718 2. 73238 0.84682 0.65023 0.96411 O. 57463 0.97944
          0.071 0.09729 3.95730 0.98672 0.90247 0.99959 0.84362 0.99989
 PHENANTHRENE O. 77221 0.01000 2.46202 0.31813 0.019 0.02874 1.43755 O. 16978 O. 10006 0.28893 0.08342 0.33269
          0.038 0.05526 2. 76424 0.62110 0.37528 0.85843 0.30423 0.90510
          0.058 0.08126 4.06535 0.93888 O. 73423 0.99619 0.63312 O. 99873
      0.05000 1. 69913 0.29616 0.019 0.02874 1.43755 0.40482 0.28265 O. 56648 0.24869 0.61580
          0.038 0.05526 2. 76424 0.85405 0.65980 0.96701 O. 58423 0.98136
          0.058 0.08126 4.06535 0.98991 0.91611 0.99975 0.86138 0.99994
 FLUORANTHENE O. 77206 0.01000 2. 46202 0.34962 0.030 0.04433 1. 46267 0.17590 O. 10264 0.30071 0.08519 0.34632
          0.060 0.08377 2. 76424 0.62110 0.37265 0.85979 0.30090 O. 90642
          0.090 O. 11966 3.94850 0.92428 O. 70231 0.99436 O. 59870 0.99798
      0.05000 1. 69913 0.31741 0.030 0.04433 1.46267 0.41435 0.28772 0.58019 0.25241 0.63029
          0.060 o. 08377 2. 76424 0.85405 0.65719 0.96747 0.58041 0.98173
          0.090 O. 11966 3.94850 0.98642 0.90014 0.99958 0.83991 0.99989

-------
 TABLE 4-18. ESTIMATED PHASE I DETECTION RATES ASSOCIATED WITH EACH        
  PHASE II CONCENTRATION LEVEL, FOR r=3           
             ESTIMATE CONFIDENCE INTERVALS,FOR 1-~
 ANALYTE   Wo  P Tp Yp X* X DELTA OF 1-~ 95 X LIMITS 99 X LIMITS
 ------------------------------------------------------------------------------------------------------------------------------------
 2-CHLORONAPHTHALENE 0.65563 0.01000 2.46202 0.38604 0.043 0.06193 1. 77719 0.26389 O. 14823 0.44809 O. 12046 O. 51002
          0.086 0.11505 3.30177 O. 79380 0.52164 0.95979 0.42685 0.97916
          0.129 O. 16231 4.65813 0.98250 0.84852 0.99968 0.75609 0.99993
      0.05000 1. 69913 0.35959 0.043 0.06193 1. 77719 O. 53612 0.36991 O. 72708 0.32137 O. 77743
          0.086 O. 11505 3.30177 0.94276 O. 78624 0.99432 O. 70841 0.99765
          O. 129 O. 16231 4.65813 0.99814 0.96362 0.99999 0.92637 1.00000
 DIMETHVLPHTHALATE 0.66349 0.01000 2.46714 0.36531 0.087 O. 11621 1.84952 0.28586 O. 15152 0.49634 O. 11976 0.56473
          0.173 0.20627 3.28287 O. 78703 0.48930 0.96258 0.38740 0.98152
          0.260 0.28377 4.51643 0.97535 O. 79824 0.99955 0.68404 0.99991
      0.05000 1. 70113 0.33862 0.087 0.11621 1.84952 0.56345 0.37581 0.76745 0.32045 0.81783
          0.173 0.20627 3.28287 0.94028 O. 76188 0.99491 0.67232 0.99803
          0.260 0.28377 4.51643 0.99710 0.94491 0.99998 0.89094 1.00000
 HEXACHLOROBENZENE 0.65529 0.01000 2.46202 0.23113 0.036 0.05255 1.77320 0.26265 O. 14818 0.44522 O. 12067 O. 50672
          0.072 0.09850 3.32348 O. 79964 0.52984 0.96169 0.43504 0.98029
          0.108 O. 13984 4.71835 0.98483 0.86003 0.99975 O. 77107 0.99995
      0.05000 1.69913 0.21491 0.036 0.05255 1.77320 O. 53457 0.36982 O. 72460 0.32175 O. 77489
          0.072 0.09850 3.32348 0.94515 O. 79233 0.99468 0.71570 0.99782
I          0.108 O. 13984 4. 71835 0.99846 0.96755 0.99999 0.93308 1.00000
ex>         
0                  
I                  
 ANTHRACENE   0.65580 0.01000 2.46202 0.28260 0.024 0.03591 1.72042 0.24652 O. 14071 0.41781 0.11532 O. 47655
          0.047 0.06718 3.21857 O. 77052 O. 50362 0.94876 0.41366 0.97196
          0.071 0.09729 4.66144 0.98264 0.85400 0.99966 O. 76581 0.99993
      0.05000 1. 69913 0.26099 0.024 0.03591 1.72042 0.51406 0.35720 O. 70024 0.31188 O. 75096
          0.047 0.06718 3.21857 0.93283 O. 77250 0.99209 0.69644 0.99651
          0.071 0.09729 4.66144 0.99816 0.96551 0.99999 0.93075 1.00000
 PHENANTHRENE 0.65547 0.01000 2.46202 0.30741 0.019 0.02874 1.69358 0.23853 O. 13671 0.40450 0.11230 0.46188
          0.038 0.05526 3.25655 O. 78133 0.51565 0.95325 0.42479 0.97484
          0.058 0.08126 4.78939 0.98723 0.87498 0.99981 O. 79219 0.99996
      0.05000 1. 69913 0.28876 0.019 0.02874 1. 69358 0.50361 0.35031 0.68794 0.30623 O. 73881
          0.038 0.05526 3.25655 0.93752 0.78172 0.99302 O. 70657 0.99698
          0.058 0.08126 4. 78939 0.99877 0.97242 0.99999 0.94210 1.00000
 FLUORANTHENE O. 65529 0.01000 2.46202 0.33390 0.030 0.04433 1. 72331 0.24739 O. 14053 0.42021 0.11490 0.47939
          0.060 0.08377 3.25680 O. 78140 O. 51247 0.95398 0.42046 0.97540
          0.090 0.11966 4.65209 0.98225 0.85062 0.99966 O. 76058 0.99993
      0.05000 1.69913 0.30656 0.030 0.04433 1. 72331 0.51518 0.35688 O. 70242 O. 31111 O. 75327
          0.060 0.08377 3.25680 0.93755 O. 77930 0.99317 O. 70265 0.99707
          0.090 O. 11966 4.65209 0.99811 0.96435 0.99999 0.92840 1.00000

-------
(2) to compare these with the estimated detection rates derived from
Phase I that appear in Tables 4-16 through 4-18.
4.7
Phase II Data
Results from this validation phase of the experiment are given in
Tables 4-19 through 4-24. Each table contains the results for a parti-
cular analyte. The notation is identical to that used previously for
the Phase I data.
Some problems with dimethylphtha1ate were encountered again (see
Table 4-20). The GC peak areas for this ana1yte, which had been small
even in Phase I, tended to be indistinct for virtually all of the Phase
II runs.
For four of the runs, no peaks were discernab1e at all.
Figure 4-25 shows a pair of chromatograms illustrating the problem with
dimethylphthalate. The first chromatogram (Figure 4-25a) shows a
typical Phase II run for which the dimethy1phthalate fortification level
was 0.173 ppm. The second chromatogram (Figure 4-25b) illustrates a
typical Phase I run with a 0.217 ppm fortification level for this
analyte. The lower peak response for dimethy1phthalate in Phase II
resulted from a deterioration of chromatographic performance for this
ana1yte following repeated injections of sediment extract. This pro-
duced wider and correspondingly shorter peaks which were more difficult
to distinguish from matrix background at the low concentration levels
used in the study.
4.8
Phase II Detectability
Each of the individual Phase II response values Y was compared to
A
the predetermined threshold response values (i.e., the y values corre-
p
sponding to r=l in Table 4-12). An observation was considered to yield
a "detection" if Y > Y .
P
analyte and concentration

observations (equal to 12
The count of detections occurring for each
level was then divided by the number of
in most cases) to yield an empirical estimate
of the detection rate associated with that x* and p value (p=O.Ol or
0.05).
The results are summarized in Table 4-25.
This table shows the false-positive rate p and the associated yp
value for each ana1yte. In addition, for each concentration level (X*),
it provides:
-81-

-------
 TABLE 4-19 . LISTI"G OF \ALIOATICN DATA FOP 2-CHLORO~APHTHALENE
          GC AREAS FOR:  
SA P'P lE     SE T   RU" -----------------  
TYP E X*  X "0. REF NO. ANALYTE IttT.STO. y* Y
 ------------------------------------------------------------------
GC_CHK 0.086 C. 1150 1 1 C   3979 1<15t:8 0.2033 0.4509
      1 1 C   34~5 11265 0.1713 C.4235
      1 1 C   26 ~2 144<15 C.1830 C.4277
      2 1 C   2987 1~244 C.19~9 C.4427
      3 1 C   2941 15830 C.1858 (.4310
      3 1 C   3418 184<18 C.18Gl C.4349
      4 1 C   26 C6 145C7 C.17G6 0.4238
CO t\T RL 0.086 O. 1150 1 1 q   3016 22948 G.1314 C.3625
      2 1 2   2248 18803 0.11<16 C.3458
      3 1 5   21(0 16186 ().12G7 (.3602
      4 1 4   24e3 173G4 G.1428 C.3778
lL SED 0.043 0.0619 1 1 7   2749 25053 C.I0G7 C.3313
      1 2 10   29 ~5 25022 0.11el C. 3437
      1 3 3   1830 25475 C.0718 C.2680
      2 1 5   22 C2 23748 (.0927 C. 3045
      2 2 8   3172 22759 C.13<14 C.3733
      2 3 4   2242 24107 C.0930 C.3050
      3 1 3   2542 21019 C.12C9 C.3418
      3 2 e   2630 25130 C.I047 C.3235
      3 '3 4   2188 22372 C.0978 C.3127
      4 1 2   2358 238<14 C.0987 C.3141
      4 2 3   2756 2341t0 C.1176 C. 3429
      4 '3 1   27<14 11986 0.1553 C.3941
UL SED 0.086 0.1150 1 1 8   3417 24412 C.13G6 C.3737
      1 2 f:   35t:3 25188 C.1415 C.3761
      1 '3 4   2972 21t126 0.1232 C . 3510
      2 1 <1   3518 22026 C.1624 C.1t030
      2 2 lC   1t6C2 244<14 0.187<1 C.4335
      2 3 1   44Gl 2511t2 C.1786 C. 4226
      3 1 1   3858 21t2fJ9 C.15<10 (.3987
      3 2 10   2234 1<1750 C.1131 C.3363
      3 3 t:   3006 23035 C.13C5 C.3612
      4 1 6   3284 1<1824 C.1657 (.4070
      It 2 <1   2279 It:828 0.1354 C.3680
      It 3 5   36e6 2C5<12 C. 1 7<10 C.1t231
lL S ED O. 129 0.1623 1 1 5   1t970 25046 C.1984 C. 4455
      1 2 2   52q2 268<12 C.19t:8 C.4436
      1 3 1   4944 25277 G.1956 C.4423
      2 1 7   4758 22335 0.2130 C.4616
      2 2 6   t:524 24962 C.2611t C.5112
      2 3 3   1t941 26404 C.1871 C.4326
      3 1 2   42C2 1<1308 C.2176 C.4665
      3 2 <1   3742 212<12 C.1757 C.41<12
      3 3 7   43G6 21132 C.2080 C.4561
      4 1 7   3407 21958 C.15~2 C.3939
      4 2 8   2485 180Ql C.1374 C.3706
      4 3 10   1820 134 <18 C.1348 0.3612
           -82-    

-------
 TABLE 4- 20 . LISTING OF ~ALIDATICN D~TA FOR CI~ETHYLPH1~AL'TE
          GC A RE AS F CR:  
SAP'PLE     SET   RUN -----------------  
TVP E xt  X ~O. RE F NO. AN~LVTE  I ~ T . S TD . V* V
 ------------------------------------------------------------------
GC_C HK 0.173 0.20t) 1  1 G   q20 1'15t8 C.0470 C.2168
      1  1 C   1320 1<;265 C.0685 G.2618
      1  1 C   q30 144<;5 C.0642 C.2533
      2  1 C   776 1~Z44 0.05Cq C. 2256
      3  1 0   825 1 ~ 8 ~O C.C521 C.2283
      3  1 0   832 184G8 C.0450 C.2121
      4  1 O'   1226 145C7 C.0845 C.2qC7
CO~TF
-------
 TABLE It- 21 . LISTING OF V AL I IJ AT I ON DATA FOR HE~ACHLOR08ENIENE
          GC ARE AS FCR:  
SAP'PLE     SE T   RUN -----------------  
TYPE x*  X "0. RE F NO. ANALYTE I"T.S1D. v* y
 --------------- --------------------------.-------------- ---------
GC_CHK 0.072 C. C9 85 1  1 C   787 1<1568 C.OltC2 0.2005
      1  1 C   929 19265 C.0482 C.2196
      1  1 C   642 144G5 C.C443 0.21C5
      2  1 0   7t7 1~2ltlt 0.05C3 C.2243
      :3  1 0   803 115830 0.C5C7 C.2252
      3  1 0   978 U!498 C.0529 0.2299
      4  1 (j   786 llt507 0.C542 C.2328
CO" T RL 0.072 C. C9 85 1  1 G   1413 22948 C.0616 C.2481
      2  1 2   922 1880) C.OItGO C.2214
      3  1 5   7 15 It  16186 C.01tt6 C.2158
      4  1 4   870 17394 C.05CO C.2236
lJL SED 0.036 0.0526 1  1 7   1859 25053 C.07"2 0.2724
      1. 2 10   1730 25022 C.06G1 C.2629
      1  3 3   ]002 2 51t 75 0.03<13 C.1983
      2  1 c;   770 237lt8 C.0321t C.1801
       J  
      2  2 8   1322 22759 C.0581 C.2410
      2  3 It   2246 24107 C.0932 C.30152
      3  1 3   I1t4 21019 0.0554 C.2353
      3  2 8   1078 25130 C.C429 C.2071
      3  "1 It   8~7 22372 C.0383 C.1957
       ..  
      4  1 2   10C5 23894 0.0421 C. 20 51
      4  2 3   826 234ltO 0.0352 C.1877
      4  3 1   1035 17986 0.0575 C.2399
UL SED 0.072 c. 0985 1  1 e   1395 2 41t 72 C.0570 C.2388
      1  2 6   1438 25188 0.0571 C.2389
      1  3 4   1638 24126 C.0679 0.26C6
      2  1 <1   1695 22026 C.0770 C.2774
      2  2 10   2613 24494 C.10t7 C. 3266
      2  3 1   1870 2511t2 C.07lt4 0.2727
      3  1 1   1214 24269 0.05CO C.2237
      3  2 1C   878 1<1750 C.0445 0.21C8
      3  "1 6   1536 23035 C.0667 C.2582
      It  1 6   1432 19824 C.0722 C.2688
      4  2 9   887 16828 C.0527 C.2296
      4  3 5   1521t 2C592 C.0740 0.2720
UL SED 0.106 0.1398 1  1 s;   1914 25046 0.07t4 C.2764
 ..  
      1  2 2   1899 268<:12 0.01C6 C.2657
      1  3 1   ]6C8 25277 C.0636 C.2522
      2  1 7   1555 22335 C.06<:16 C.2639
      2  2 t:   2070 24962 C.0829 C.2880
      2  3 3   1537 2tJltC4 C.C582 C.21t13
      3  1 2   1969 1'1308 0.1020 C.3193
      3  2 <1   16tl 212'12 C.0780 C.2793
      3  ':I 7   1754 21132 C.C830 0.2881
      4  1 7   1732 21958 C.0789 C.2809
      It  2 8   1032 180<11 C.0570 C.2388
      4  3 10   659 134 <18 0.0488 C.2210
           -84-    

-------
 TABLE Lf- 22 . LISTINC OF VALl[) AT I ON OAT A FOR ANTHRACENE 
          CC ARE AS FCR:  
SA f'P LE     SET   RU~ -----------------  
TYPE X.  'X "O. RE F ~O. ANALYTE I"T.STO. y* Y
 ------------------------------------------------------------------
CC_CHK 0.OLf7 0.0672 1  1 C   1577 1'1568 0.080t: C.2839
      1  1 C   1629 1 '1265 0.0846 C.2908
      1  1 C   1105 lLf4G5 0.0762 C.2761
      2  1 C   1326 15244 C.0870 C.29Lf9
      3  1 0   13'17 15830 C.0883 C. 29 71
      3  1 0   1738 184 '18 C.09"0 0.3065
      4  1 0   13'12 14507 .G.09l:0 C.30'18
CO" T RL 0.OLf7 0.C672 1  1 '1   2677 229Lf8 C.1H;7 C.3Lf15
      2  1 2   1650 18803 0.0878 C.2962
      3  1 s;   lLf73 16186 0.0910 C. 3017
       .J  
      Lf  1 Lf   1567 173 '14 0.0901 0.3001
lL SED 0.02Lf 0.035'1 1  1 7   2482 25053 0.09 
-------
 TABLE It- 2 3 . LISTIt\G OF '4ALIDATICN OAT A FOR PHENANTHRENE 
          GC AREAS FOR:  
SA "P LE     SET   RUN -----------------  
TYP E x*  X NO. RE F NO. ANAL 'HE U. T. S TO . Vt y
 ------------------------------------------------------------------
GC_CHK 0.038 O.CSS) 1  1 C 1829 1<1568 0.0935 0.3057
      1  ] C 1887 1<12tJ5 C.0979 C.3130
      1  1 0 13tJ5 144<15 C.09lt2 0.3009
      2  1 C 1558 1~244 C.I022 C.31(17
      3  ] C 1677 15830 C.1059 C.3255
      :3  1 0 2004 184 <18 G.I083 C.32
-------
 TABLE 4- 24 . LISTIt\G OF ~ALIDATIDN DATA FOR FLUORA"THENE 
          GC AREAS FOR:  
SA~PlE     SET   RUN -----------------  
TYP E x*  )( "0. RE F NO. ANALYTE I"T.5TD. y* Y
 ------------------------------------------------------------------
GC_C~K 0.060 C. C8 38 1 1 0  1988 1<1568 C.I016 C.3187
      1 1 0  2316 19265 C. 1202 C.3467
      1 1 C  1702 144<15 C.1174 C. 3427
      2 1 C  1<128 15244 0.1265 C. 3556
      3 1 C  2102 15830 0.1328 0.3644
      3 1 C  2680 18498 0.1449 C.3806
      4 1 0  2068 14507 0.1426 C.3776
CO"TRL 0.060 C. C8 38 1 1 <1  4281 22948 0.1866 C.4319
      2 1 2  2613 18803 C. 13 <10 C.3728
      3 1 5  22 C6 16186 0.13t:3 C.3692
      4 1 4  2265 17394 0.13C2 O. 3609
lL SED 0.030 0.0443 1 1 7  2386 25053 C.C952 C.3086
      1 2 lC  2343 25022 C.0936 C. 3060
      1 3 3  2082 25475 C.0817 C.2859
      2 1 5  1849 23748 (J.0779 C.2790
      2 2 8  2225 22759 C.0978 C.3127
      2 3 4  2476 24107 C.I027 0.3205
      3 1 3  2312 21019 O.l1CO C.3317
      3 2 8  2136 25130 C.C850 C.2915
      3 3 4  1968 22372 C.0880 C.2966
      4 1 2  2218 23894 0.0928 C.3047
      4 2 3  2396 23440 C.I022 C.3197
      4 3 1  1680 17986 C.0934 C. 3056
lL SED 0.060 C. 0838 1 1 8  3383 24472 C.1382 C.3718
      1 2 6  3430 25188 0.1362 C. 3690
      1 3 4  3054 24126 0.1266 C.3558
      2 1 9  2990 22026 0.1357 C.3684
      2 2 lC  3284 24494 C.1341 0.3662
      2 3 1  3408 25142 0.1356 C.3682
      3 1 1  3128 24269 0.1289 0.3590
      :3 2 lC  3032. 1 (17 50 C.1535 0.3918
      3 '] 6  3709 23035 0.1610 C. 4013
      4 1 t  3124 19824 C.1576 C.3970
      4 2 9  1341 16828 C.07<17 C.2823
      4 3 5  3614 2 C5 92 0.1755 C.4189
UL SED 0.0 <10 0.1197 1 1 5  3430 250t.6 C.1369 C.3701
      1 2 2  4183 268<12 C.1555 0.3944
      1 3 1  43C7 25277 0.17C4 C.4128
      2 1 7  3981 22335 0.1782 C. 4222
      2 2 6  4892 24962 0.1960 C.4427
      2 3 3  4847 26404 0.1836 0.4285
      3 1 2  3687 11308 C.1910 C.4370
      3 2 9  3695 21292 0.1735 0.4166
      3 3 7  4675 211 32 0.2212 0.4703
      4 1 7  3904 21958 0.1778 0.4217
      4 2 8  4734 18091 C. 26 17 
-------
a)
\..000
00 0
~
I 0
M I-
WWW
(J) --.J(j) Q)

-------
TABLE 4-25. COMPARISON OF PHASE I AND PHASE II DETECTION RATE ESTIMATES,
 BY ANALYTE, FOR 1'=1       
        ESTIMATE NO. NO. PROPORTN
ANALYTE P Yp X* OF 1-q, DETECTED CASES DETECTED
--------------------------------------------------------------------------------
2-CHLORONAPHTHALENE 0.01 0.43701 0.043 O. 10299 0 12 0.00000
      0.086 0.36126 0 12 0.00000
      0.129 0.67482 7 12 0.58333
  0.05 0.39477 0.043 0.28840 0 12 0.00000
      0.086 0.64573 6 12 0.50000
      O. 129 0.88542 9 12 O. 75000
DIMETHYLPHTHALATE 0.01 0.41567 0.087 0.11211 0 9 0.00000
      0.173 0.36107 0 12 0.00000
      0.260 0.64933 0 11 0.00000
  0.05 0.37334 0.087 0.30620 0 9 0.00000
      0.173 0.64618 0 12 0.00000
      0.260 0.87150 0 11 0.00000
HEXACHLOROBENZENE 0.01 0.26241 0.036 O. 10251 3 12 0.25000
      0.072 0.36589 5 12 0.41667
      0.108 0.68731 8 12 0.66667
  0.05 0.23650 0.036 0.28745 5 12 O. 41667
      0.072 0.65042 9 12 O. 75000
      O. 108 0.89222 11 12 0.91667
ANTHRACENE 0.01 0.32425 0.024 0.09707 0 12 0.00000
      0.047 0.34296 3 12 0.25000
      0.071 0.67568 12 12 1.00000
  0.05 0.28973 0.024 0.27672 8 12 0.66667
      0.047 0.62676 12 12 1. 00000
      0.071 0.88590 12 12 1.00000
PHENANTHRENE 0.01 0.34336 0.019 0.09430 1 12 0.08333
      0.038 0.35110 3 12 0.25000
      0.058 O. 70229 8 12 0.66667
  0.05 0.31357 0.019 0.27115 4 12 0.33333
      0.038 0.63529 3 12 0.25000
      0.058 0.90013 11 12 0.91667
FLUORANTHENE 0.01 0.38662 0.030 0.09729 0 12 0.00000
      0.060 0.35104 4 12 0.33333
      0.090 0.67321 11 12 0.91667
  0.05 0.34295 0.030 0.27715 0 12 0.00000
      0.060 0.63522 11 12 0.91667
      0.090 0.88453 12 12 1.00000
-89-

-------
the number of Phase II observations; and
the proportion of cases [(ii) i (iii)] for which a detection
was noted (i.e., the empirical estimate of 1-q(x.
Table 4-26 shows a set of similar results for r=2. In this case,
(i)
(ii)
(iii)
(iv)
the Phase I point estimate of the detection rate;
the number of Phase II observations y for which Y > Y .
p'
the twelve observed responses for each concentration level were paired
together to form six Y values. The observations comprising each pair
were chosen by sequential grouping of the first two runs, then the next
two runs, etc. (The order of the runs is defined by set number and by
run number within set number, as given in Tables 4-19 through 4-24).
A similar sequential grouping of the 12 individual responses was
used to form a mean of triplicate observations (i.e., r=3).
Four such
means were therefore available for each concentration level and each
analyte.
Table 4-27 summarizes these results for the case r=3.
Results given in Tables 4-25, 4-26, and 4-27 indicate the follow-
ing:
1.
No detections were counted for dimethyl phthalate -- attributed
to the aforementioned problem with this analyte.
For the remaining analytes, the observed proportion of cases
detected is, with several exceptions, in general agreement
with the estimated detection rates from Phase I; the Phase II
2.
3.
proportions are quite variable, however, due to the small
sample sizes (especially when r=3).
With one exception (anthracene, p=0.05, r=l), the agreement
between Phase I and Phase II rates tended to be better for the
high concentration level than for the two lower levels.
Because of the deliberate manner in which the Phase II fortifica-
tion levels were chosen, estimated detection rates from Phase I for the
lowest Phase II concentration level were essentially the same for all
"
analytes (e.g., 1-q(x) ~ 0.10 when p=O.Ol and r=1). Similarly, the
chosen fortification levels for Phase II guaranteed that this property
(i.e., approximate constancy of the estimated, Phase I detection rates
over analytes) would apply also to the other fortification levels, to
other choices of p, and to other choices of r.
The approximate Phase I
-90-

-------
TABLE 4-26. COMPARISON OF PHASE I AND PHASE II DETECTION RATE ESTIMATES,
 BY ANALYTE, FOR r=2        
        ESTIMATE NO. NO. PROPORTN
ANALYTE P Yp X* OF 1-q, DETECTED CASES DETECTED
--------------------------------------------------------------------------------
2-CHLORONAPHTHALENE 0.01 0.40123 0.043 O. 18746 0 6 0.00000
      0.086 0.63498 1 6 O. 16667
      0.129 0.92504 5 6 0.83333
  0.05 0.37008 0.043 0.43191 0 6 0.00000
      0.086 0.86250 5 6 0.83333
      O. 129 0.98661 5 6 0.83333
DIMETHYLPHTHALATE 0.01 0.38029 0.087 0.20429 0 4 0.00000
      O. 173 0.63077 0 6 0.00000
      0.260 0.90840 0 5 0.00000
  0.05 0.34895 0.087 0.45707 0 4 0.00000
      0.173 0.86050 0 6 0.00000
      0.260 0.98242 0 5 0.00000
HEXACHLOROBENZENE 0.01 0.24046 0.036 O. 18654 1 6 O. 16667
      0.072 0.64141 6 6 1.00000
      0.108 0.93161 5 6 0.83333
  0.05 0.22135 0.036 0.43053 3 6 0.50000
      0.072 0.86633 6 6 1.00000
      O. 108 0.98821 6 6 1.00000
ANTHRACENE 0.01 0.29502 0.024 0.17538 2 6 0.33333
      0.047 0.60952 6 6 1.00000
      0.071 0.92546 6 6 1. 00000
  0.05 0.26955 0.024 0.41353 5 6 0.83333
      0.047 0.84682 6 6 1. 00000
      0.071 0.98672 6 6 1.00000
PHENANTHRENE 0.01 0.31813 0.019 O. 16978 2 6 0.33333
      0.038 0.62110 2 6 0.33333
      0.058 0.93888 5 6 0.83333
  0.05 0.29616 0.019 0.40482 2 6 0.33333
      0.038 0.85405 2 6 0.33333
      0.058 0.98991 6 6 1.00000
FLUORANTHENE 0.01 0.34962 0.030 0.17590 0 6 0.00000
      0.060 0.62110 5 6 0.83333
      0.090 0.92428 6 6 1.00000
  0.05 0.31741 0.030 0.41435 0 6 0.00000
      0.060 0.85405 6 6 1.00000
      0.090 0.98642 6 6 1. 00000
-91-

-------
TABLE 4-27. COMPARISON OF PHASE I AND PHASE II DETECTION RATE ESTIMATES,
 BY ANALYTE, FOR 1'=3       
        ESTIMATE NO. NO. PROPORTN
ANALYTE P Yp X* OF 1-Ct DETECTED CASES DETECTED
--------------------------------------------------------------------------------
2-CHLORONAPHTHALENE 0.01 0.38604 0.043 0.26389 0 4 0.00000
       0.086 0.79380 2 4 O. 50000
       0.129 0.98250 3 4 O. 75000
  0.05 0.35959 0.043 O. 53612 0 4 0.00000
       0.086 0.94276 4 4 1. 00000
       0.129 0.99814 4 4 1.00000
DIMETHYLPHTHALATE 0.01 0.36531 0.087 0.28586 0 3 0.00000
       O. 173 O. 78703 0 4 0.00000
       0.260 0.97535 0 3 0.00000
  0.05 0.33862 0.087 0.56345 0 3 0.00000
       0.173 0.94028 0 4 0.00000
      0.260 0.99710 0 3 0.00000
HEXACHLOROBENZENE 0.01 0.23113 0.036 0.26265 2 4 O. 50000
      0.072 O. 79964 3 4 O. 75000
       0.108 0.98483 4 4 1.00000
  0.05 0.21491 0.036 O. 53457 2 4 O. 50000
       0.072 0.94515 4 4 1.00000
       O. 108 0.99846 4 4 1.00000
ANTHRACENE 0.01 0.28260 0.024 0.24652 2 4 O. 50000
       0.047 O. 77052 4 4 1. 00000
       0.071 0.98264 4 4 1.00000
  0.05 0.26099 0.024 0.51406 4 4 1.00000
       0.047 0.93283 4 4 1.00000
       0.071 0.99816 4 4 1. 00000
PHENANTHRENE 0.01 0.30741 0.019 0.23853 1 4 0.25000
       0.038 O. 78133 1 4 0.25000
       0.058 0.98723 4 4 1.00000
  0.05 0.28876 0.019 0.50361 1 4 0.25000
      0.038 0.93752 1 4 0.25000
      0.058 0.99877 4 4 1.00000
FLUORANTHENE 0.01 0.33390 0.030 0.24739 0 4 0.00000
      0.060 0.78140 4 4 1.00000
      0.090 0.98225 4 4 1.00000
  0.05 0.30656 0.030 O. 51518 2 4 O. 50000
       0.060 0.93755 4 4 1.00000
       0.090 0.99811 4 4 1.00000
-92-

-------
estimated detection rates at each Phase II concentration level are shown
below:       
   Phase II Concentration Level 
 r p Low Middle High 
 -    
 1 0.01 0.10 0.35 0.68 
  0.05 0.28 0.64 0.89 
 2 0.01 0.18 0.63 0.93 
  0.05 0.42 0.86 0.99 
 3 0.01 0.25 0.79 0.98 
  0.05 0.52 0.94 0.99+ 
This design property allowed us to pool the Phase II results over
analytes in a useful fashion -- namely, by aggregating the number of
detections over analytes. Through such an aggregation we were able to
achieve a larger effective sample size and thereby reduce the overall
variation in the empirically derived estimates of detection rates. The
results of this aggregation over five ana1ytes (dimethylphtha1ate was
excluded) are given in Table 4-28.
The results of this table indicate a close agreement between the
Phase I predictions of detection rates and the observed rates of detec-
tion derived from Phase II. If the 60 cases for r=1, for instance, are
treated as 60 independent binomial trials (though this is not strictly
true), then binomial confidence limits for the true proportion detected
can be constructed.
Similar interval estimates for r=2 (30 trials) and
r=3 (20 trials) can be developed. In all of the eighteen cases listed
in Table 4-28, the point estimate for the detection rate from Phase I
falls within the corresponding 95% confidence interval for the Phase II
proportion, indicating no detectable departure of predicted from observ-
ed rates.
An alternative way of comparing the Phase I and Phase II results is
to compare KI and KII' where
KI is the expected number of detections derived from Phase I
predictions, and
KII is the observed number of detections in Phase II.
Values of KI and KII are shown in the table below:
-93-

-------
  Table 4-28. Comparison of Phase I and Phase II Detection Rate Estimates, Based on Aggregating
     the Phase II Results Over Five Analytes   
     Approximate  Phase II Re-sults   Approximate
     Phase I  Number   Deviation
   Phase II Detection Number of of Proportion in Phase I and
 r p Concentration Rate Est. Detections Cases Detected Phase II Rates
 -        
 1 0.01 low 0.10 4 60 0.0667 -0.03
   medium 0.35 15 60 0.2500 -0.10
   high 0.68 46 60 0.7667 0.08
 1 0.05 low  0.28 17 60 0.2833 0.00
   medium 0.64 41 60 0.6833 0.04
   high 0.89 55 60 0.9167 0.03
 2 0.01 low 0.18 5 30 0.1667 -0.01
   medium 0.63 20 30 0.6667 0.04
   high 0.93 27 30 0.9000 -0.03
I 2 0.05 low 0.42 10 30 0.3333 -0.09
\0
.I:-   medium 0.86 25 30 0.8333 -0.02
I  
   high 0.99 29 30 0.9667 -0.02
 3 0.01 low 0.25 5 20 0.2500 0.00
   medium 0.79 14 20 0.7000 -0.09
   high 0.98 19 20 0.9500 -0.03
 3 0.05 low 0.52 9 20 0.4500 -0.07
   medium 0.94 17 20 0.8500 -0.09
   high 0.99+ 20 20 1.0000 0.00
     -~---- _. -     

-------
     Phase II Concentration Level 
     low medium high 
   No. KI  KII KI KII KI Kn
r p Cases 
-         
1 0.01 60 6.0  4 21.0 15 40.8 46
1 0.05 60 16.8  17 38.4 41 53.4 55
2 0.01 30 5.4  5 18.9 20 27.9 27
2 0.05 30 12.6  10 25.8 25 29.7 29
3 0.01 20 5.0  5 15.8 14 19.6 19
3 0.05 20 10.4  9 18.8 17 19.9+ 20
Among the six combinations of r and p shown above, the worst agreement
occurs in the case r=1, p=0.01. Even for this case, however, the 
magnitudes of the differcences between KI and KII are within the range
that would be expected to occur simply by chance.    
-95-

-------
5.
CONCLUSIONS AND RECOMMENDATIONS
The current approach utilized by EPA for defining/estimating a
"Method Detection Limit" tends to emphasize achieving protection against
reaching false positive conclusions -- i. e., erroneously claiming
presence of an analyte in an environmental or biological medium.
At
least in certain situations,
the consequences of reaching a false
negative conclusion -- i.e., erroneously concluding that the analyte is
absent when it is in fact present -- can be equally important.
Yet the
EPA approach makes no attempt to furnish adequate protection against
this second type of error.
The methods described in Chapter 3 and Appendix A, on the other
hand, provide a conceptually and statistically rigorous technique for
estimating detection limits with specified assurance probabilities
against both types of errors. The methods, which utilize standard
calibration techniques and involve assumptions common to many calibra-
tion situations, are founded on the classical Neyman-Pearson criterion
for performing tests of hypotheses. For a given detection rule involv-
ing a specified false-positive rate p, this method defines a detection
limit Xt(p,q) that provides protection against false negatives with
probability q. Because of the power properties of the associated
hypothesis test, this method (under normality assumptions) is guaranteed
to produce the lowest such limit that satisfies the specified rates p
and q.
The Phase I experiment described in Chapter 4 illustrates an
application of the method using chemical methods similar to those
employed in many EPA studies. The second phase of the experiment
provides an empirical demonstration that the method does, in fact,
perform in accordance with theory in the EPA laboratory environment.
In order to produce and report reliable estimates of detection
limits therefore, EPA should consider the formulation and methods
described in Chapter 3/Appendix A as a basis for defining and estimating
such limits. Characteristics of the detection protocol (the calibration
design, p, q, and r), should not be fixed arbitrarily, but should be
adapted to meet the needs of specific studies and programs.
-96-

-------
REFERENCES
Burrows PM; Scott SWj Barnett OWj McLaughlin MR. (1984). "Use of
Experimental Designs With Quantitative ELISA." Journal of
Virolo~ical Methods, Vol. 8, pp. 207-216.
Burrows PM. (1985). "Detectability and Detection Limits." (paper
submitted for publication, included as Appendix A of this report).
Cochran WGj Cox GM. (1957). Experimental Designs. New York: John
Wiley and Sons.
Currie LA.
(1968) .
"Limits for Qualitative Detection and Quantitative
Analytical Chemistry, Vol. 40, pp. 586-593.
Determination."
Hubaux Aj Vos G.
(1970).
"Decision and Detection Limits for Linear
Calibration Curves." Analytical Chemistry, Vol. 42, pp. 849-855.
Kempthorne O. (1952) . The Design and Anal}'sis of Experiments. New
York: John Wiley and Sons, Inc.
Owen, DB. (1968). "A Survey of Properties and Applications of the
Noncentral t-Districution." Technometrics, Vol. 10, pp. 445-478.
Snedecor GWj Cochran WG. (1980). Statistical Method~ (Seventh Edition).
Ames, Iowa: Iowa State University Press.
Williams EJ. (1969) . "A Note on Regression Methods in Calibration."
Technometr1cs, Vol. 11, pp. 189-192.
-97-

-------
APPENDIX A
Detectability and Detection Limits
[This appendix is a paper prepared by Peter M. Burrows and submitted to
Technometrics]

-------
DETECTABILITY AND DETECTION LIMITS
Peter M. Burrows
Experimental Statistics Unit
Clemson University
Clemson, South Carolina 29631
Detectability of an analyte is considered when detection is based on
a normally distributed measure calibrated against analyte concentrations.
Detection rules are formulated as t-tests.
Estimation of detection rates,
at specified concentrations, requires estimation of the corresponding power
functions (noncentral t-distribution functions).
Estimation of detection
limits with specified assurance probability requires estimation of the
corresponding noncentrality parameters.
Formulations for situations involving
statutory limits, with assured alert rates, are developed in a similar manner.
KEY WORDS: calibration; inverse calibration; detection; noncentral t-distribution.
Technical Contribution No. 2413 from the South Carolina Agricultural Experiment
Station, Clemson University.

-------
1
1.
INTRODUCTION
A substance ~, usually referred to as the analyte or determinand, is
present in unknown concentration X that varies from one sample preparation
to another, including the possibility X = zero.
In practice ~ may be an
element, a compound or a macromolecule (including biologically functional
particles).
There exists a well defined measurement method that produces
observations Y informativ,~ about X.
In most cases Y is obtained from some
convenient indirect measure of X, often not expressed in the same units as
X, and is subject to random observational error resulting from sequential
accumulation of uncontrollable errors at each step of the preparation and
measurement methods.
Measurement of physical, chemical or biological
properties of ~ itself, or of consequences of the presence or activity of
~, are typical sources of observations Y.
Theories of calibration and reverse calibration address the general
problem of estimating X for a particular sample given one or more determinations
of Y on that sample.
But when ~ is present at low concentrations, in the
neighborhood of X = zero, interest shifts to whether or not it is reasonable
to declare that ~ is detectable given that observations Yare subject to
error as just described.
Instead of providing point and interval estimates
for X, the problem is to provide a reliable rubric for asserting that X > zero.
Assertions 'X > zero' are logically equivalent to declarations 'A is detected',
regardless of their veracity or of the actual value of X when X > zero.
A detection protocol is defined by specifying the measurement method,
performing at least a minimal calibration of Y against X, and adopting a

-------
2
detection rule.
With the assumption that the calibration function (standard
curve) is monotonic, a detection rule with known properties in repeated
applications is provided by the Neyman-Pearson apparatus for constructing
tests of the hypothesis X = zero against alternatives X > zero, with ~pecified
protection against frequent false positives (erroneous assertions that X > zero).
The resulting rule always takes the form of referring Y to a specified action
A
value y, also known as the detection threshold: if Y is expected to increase

with increasing X, then assert X > zero whenever Y > y,~but if Y is expected to
decrease with increasing X (as happens when Y results from some property of a
substance in competition with ~), then assert X > zero whenever Y <:y.
If
several determinations of Yare obtained for a single source then their average
Y is substituted for Y in these rules and threshold y is modified accordingly.
Detection rates in repeated applications of the defined detection protocol
are given by the power function of this detection rule.
There are other criteria
that lead to rules of this same general type (see Voller, Bidwell and Bartlett
(1977) for examples), but superior power properties of the Neyman-Pearson
criterion ensure that among a11 such rules it attains the highest detection
rates at all concentrations X > zero, and therefore the lowest detection limits
as defined below.
Previous discussions of detectability, especially among analytical chemists,
have concentrated on detection limits (see Liteanu and Rica (1980) for a review
and bibliography).
Definition of detection limits with specified assurance
probability, and estimation of detection rates and detection limits, are all

-------
3
applications of noncentral t-distributions as shown in ~3.
For this purpose
the following notations are adopted:
Z : a normally distributed random variable with zero mean and unit
variance,
S : independent of Z, vS2 is distributed as chi-square on v degrees
v v
of freedom,
T (0) : follows the noncentral t-distribution with noncentrality parameter
v
o and v degrees of freedom; Tv(O) = (z+o)LSv'

t : the upper lOOp percentage point of Student's t-distribution with
v,p
v degrees of freedom; Pr [T (0) > t] = p.
v v.p
When there is no risk of confusion, v is suppressed in the notations S, T(o)
and t .
':.t;" p
......;/
A comprehensive review of properties of T(o) is given by Owen (1968).


Availability of arithmetic routines that produce Pr[T(o)
-------
4
Some general comments on the usefulness of detection limits, a~d conventional
specification of false positive rates and false negative rates, are included
in the discussion in ~7.

-------
5
2.
CALIBRATION
A model of observations Y is necessary to formulate the detection protocol
and its behavior in repeated applications.
Here it is assumed that when X = x,
known or unknown,
(2.1)
Y = a+ f(x,~) + e ,
13 = (13bf3.z,...,13 )
m
where e is an additive observational error distributed normally with mean
zero and variance 02 constant for all x in the range of X of interest.
Errors
in r repeated determinations of Y, obtained for a single source with X = x,

are assumed independent so that their average Y = [a+f(x,~)+zo/Ir].
Two features of the function f(x,~) are assu~ed throughout.
First,
f(o,~) = zero, so that parameter a is the expected background value of Y,
attributable to such phenomena as leakage and nonspecific reactions. In the
case of a laboratory instrument, a is the expected reading for a 'blank' sample.
Second, f(x,~) is assumed to be a monotonic function of x over all x > zero,


which imposes conditions (usually inequalities) on its parameters ~.
Monotonicity is necessary to avoid possible ambiguities in inverse calibration.
Here f(x,~) is assumed to be monotonic increasing in x so that detection rules
of .the type 'assert. X > zero whenever Y > y' are appropriate.
Modifications
required when f(x,~) is monotonic decreasing in x are obvious throughout.
Calibration is assumed to be based on standard preparations of A with
known concentrations xl,x2,.",x including at least (m+l) distinct standards.
. n
The model for observation Yi' obtained at standard xi' is that at (2.1) with
pairwise independence of corresponding observational errors e..
1
Let ex and
8
denote the resulting least squares estimates of parameters a and ~. and let 02
denote the estimate of 02.
Then the following distributional properties are

-------
6
assumed throughout:
(2.2)
(~,~] follows a multivariate normal distribution with mean (a,~] and
variance-covariance matrix a2v,
(2.3)
(2.4)
a = as
v'
Y and a are independent of each other and of (&,~].
In many cases degrees of freedom v = (n-m-l), but v is left general here to
allow for other possibilities including incorporation of experimental design
features in the calibration exercise, pooling of information from several
independent calibrations, and use of variation among the r replicated observations
about their average Y.
Matrix V is (m+l) x (m+l) with known elements that are
functions of the n standard concentrations and of the form assumed for f(x,~).
If f(x,~) is linear in its parameters Bl,Bi,...,Bm' which includes cases

of polynomial standard curves treated by Scheffe (1973), then properties (2.2),
(2.3) and (2.4) follow from the observational error structure assumed for Y and
Yi.
If f(x,~) is nonlinear in its parameters ~, so that nonlinear least squares
estimation is necessary, then these distributional properties (and those below)
a~e only large sample approximations.
Two immediate consequences of (2.2) and
(2.4) are stated now for use later.
Given any x,
(2.5)
f(x,a) = f(x,B)+Zau ,
- - x
1l- = var(f(x,S)/a]
x -
and when X = x for the source of Y,
(2.6)
(Y-a) = f (x,~) + Zaw ,
Wl = var(a/a) + l/r .
Both u and ware readily obtained from V.
x

-------
7
Let eS = f(x,6)/ou .
x - x
A point estimate for eS is g = f(x,S)/;u ,
x x - x
but when f(x,~) is linear in its parameters 6 the minimum variance unbiassed

estimate is usually 6 /M where E(l/;) = M /0; values of M are given in the
x v v v
second column of Table 1. An interval estimate for eS is obtained by combining
x

(2.5) and (2.3) to yield 6 = (Z+CS )/5 = T(eS). Let~y- (>O)and y+ (>O)be such
x x x !
- + - A + ".
that (y +y ) = Y < 1, and locate d  15') and d (> cS ) such that
x x x x

Pr[T(d:"')>6' ] = y Pr[T(d+)< 6] = y+.
x x x x
(2.7)
- +
Then the interval (d ,d ) covers eS with probability (l-y).
x x x
The choice
y
+
= y = y/2 is obviously convenient, but to obtain an interval that is
A A
symmetric about cS , or the interval that is narrowest given cS and y, requires
x x
+ - +
manipulation of y and y subject to (y +y ) = y.

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8
3.
DETECTION RULES, RATES AND LIMITS I
A A
Let y ,= (a+wcrt ) and tJ. '= f(x,~)/crw.
p p x
Then at concentrations X = x,
rates of detection when using the rule 'assert X > zero whenever Y > Y , are
p
given by
(3.1)
1-q (x) = PrE (y-~) > wat I X = x]
p
= Pr[(Z+tJ. ) > St ]
x p
=
pr[T(tJ. ) > t ]
x p
because (y-~) = wcr(Z+tJ. ) from (2.6) and a, = crS at (2.3).
x
Threshold y
p
has been
chosen so that rates of false positives under the rule 'y > y , are fixed at p:
p
tJ. = zero and
o
1-q(O) = Pr[T(O) > t ] = P .
P
Rates of false negatives at concentrations X = x are given by q(x).
(3.2)
The definition of detection limits adopted here is based on the concept
favored by Currie (1968) who reviewed several inconsistent alternatives: the
detection limit should be an analyte concentration that is almost assured of
detection.
Such a limit is inseparable from the adopted detection rule, so that
a formal definition is ,stated as follows:
if Xt = x1(P,q) is the lowest analyte concentration for which


the rate of detection is at least (l-q) when using the rule
'declare ~ detected whenever Y > y p " then x t is the detection

limit with assurance probability (1-q) for that rule;
alternatively, the rate of false negatives is at most q for all
analyte concentrations exceeding Xt.

-------
9
Thus Xt corresponds to just one point on the curve of false negative rates

q(x) in relation to analyte concentrations x, namely, that point selected by
specification of q.
Let 6 = 6(v,p,q) satisfy Pr[T(6)
-------
10
Then, since Pr[T(o)< t ] is a monotonic decreasing function of 0, the interval
p
(3.6)
{ q- (x) ,q+(x)} :: (Pr[T(D+) < t ],Pr[T(D-) < t ]}
x p x p
covers q(x) with probability (1-y).
An interval estimate for Xt is then
obtained by inverse interpolations from the specified q through curves q-(x)
and q+(x) to (Xt,xi) as illustrated in Figure 1: q~(x1) = q = q+(xi).

last construction ignores the possibility, and its probability, that although
This
f(x,~) is assumed to be monotonic increasing in x, the standard curve fitted
by least squares may not obey this condition.
The consequences of this
possibility are discussed in more detail in the next section.

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11
4.
LINEAR CALIBRATION
Considerable simplification results when a straight line relationship,
f(x,~) = ax, is adopted for the standard curve.
In particular, interval
estimates for Xt are obtained directly ~ithout recourse to inverse interpolation
as in Figure 1.
For this special case with ~ = 1, the elements of V 'are given by
var(a/a) = (n-1 + x2/Q)

where nx = LXi and Q = I(xi-x)2.
i u = x/~ and
x
so that 0 = Q~8/a is not a function of x and its subscript x can be omitted
x
throughout this section: 0 = Q~8/a and 6 = Q~B/a. Note that (8/a) is the
cov(~/a,B/a) = -x/Q ,
var(B/a) = 1/Q
Then, for (2.5) and (2.6),
, -1 -1 -2 ~
wi = (r + n + x /Q)
sensitivity ratio referred to as Gaddum's constant.
Threshold y and
p
noncentrality parameter 6
x
define detection rates for the rule 'assert x > zero
whenever Y > Y , as at (3.1):
p
(4.1)
1-q(x) = Pr[T(6 ) >t ] ,
x p
6
x
= ax/aw .
Hubaux and Vos (1970) offer a curious development of Xt' endorsed by Currie,
I
. ~ .~! ." "
Filliben and DeVoe (1972) and by Liteanu and Rica !(1980), in terms of inverse
interpolations in the space of Y and x, rather than via a graph of q(x) against
x.
With the present terminology and notation, their graphical formulation
can be stated algebraically as follows:
Step (i). ,The rule 'assert X > zero whenever Y > y' with Y = (&+wGt ),
P , P P
achieves level-p protection against false positives as at (3.2).

Step (ii). Let:w = [r-1+n-1+(x-x)2/Q]~ and consider any point (x,y), x not
x
a random variable, on the curve y = (a+8x-wat ).
x q
Then Xt(.,q) = x for the

-------
l:l
. ~
rule' assert X > zero whenever Y . > y', because (Y-~-Bx) = Zaw when X = x and
x
the detection rate when X =\x is
Pr[(Y-~-ax) > -w at I X= x] = PdT(o) > -t ] = (l-q) .
x q q
The qualification that x not be a random variable is required because no
accounting of any stochastic variation in x has been made when formulating
this detection rate.
Since x; x~(.,q) for this rule, any concentration x
can be a detection limit with assurance probability (l-q).
To each different
x there corresponds a different rate of false positives determined by
threshold y.
Step (iii).
Let x be the solution x to y = y :
p
Bx =. a(wt +w..t ) .
P x q
Then Xt(p,q) = x because the rule in step (ii),
(4.2)
with detection rate (l-q)
..
when x = x, has been made to coincide with the rule in step (i) with false
positive rate p.
Hubaux and Vos (1970) do not provide an explicit solution to (4.2) which
is quadratic ~n x.
The typographical inconvenience is reduced by the
following sequence of notations.
Let x = (y -a)/S. x is the analyte
p p , p
concentration obtained by inverse interpolation from y through the fitted
p
calibration line.
Though it has no useful meaning, a popular convention is
to regard x as a detection limit, often in the mistaken belief that concentrations
p
are detectable at rate (l-p), as in the following quotation from
..
X = x
P
Maggio
(1980) :

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13
"If one makes the generally reasonable assumption
that the system noise, the overall variability in the
measurement of enzyme activity, may be described by a
Gaussian distribution, then the minimum detectable
concentration may be conveniently expressed in terms
of the standard deviation. For example, we may define
the minimum detectable concentration as the interpolated
analyte concentration which lies two standard deviations
above the zero value on the dose-response curve. This
concentration may be distinguished from zero with greater
than 95% confidence."
Let (1+R ) = (6/t )2; in practice R is a positive quantity because otherwise
q q q
"
the hypothesis B = zero is not rejected by the level-q t-test which refers 0
to t .
v,q
Indeed, R > zero is the condition for existence of sensible yet
q
nontrivial interval estimates, with coverage probability (1-2q), for inverse
calibration of X given Y using the formulation of Bowker and Lieberman (1959)
which is a special case of Fieller's method (1954); a general (non-Bayesian)
remedy that produces useable intervals for X given Y, even when g < t , is
q
given by Scheffe (1970).
Let x* =
X +(x -x)/R ; then roots of (4.2) are given by
p P q .
" " "2 "2 ~
x = x*+ [x* -x (l-R /R )]
- P P q
and when q = p, x = 2x* is the required solution.

This formulation, Xt = ~, is seductive but obviously fallacious because
if valid this method would always be available to circumvent noncentral
t-distributions when considering power of t-tests.
Hubaux and Vos (1970)
"
regard x first as a definition of
random variable, as an estimate.
Xt.~\d later,. after noting_that;' is a.

The fallacy arises because x = x violates
the restriction that x not be a random variable in step (ii).
The rule in

-------
14
step (ii) is of s~me interest however, and is used in ~6.
Also as n";' 00 ,
B" 13, a + a and t + Z , the upper 100ppercentage point of Z.
p P
limit, (4.2) reduces to
Thus in the
and x does coincide with
Bx = (z + z )a/lr
p q.
the definition of xl(P,q)

13 and a are known perfectly.
for the case where a,
In contrast with xp artd x, the exact formulation of parameter xl(P,q) is

obtained by specifying q(xl) = q in (4.1):
(4.3)
xl = w/}.a/B
where /}. = /}.(v,p,q) as at (3.3); /}.~(z +z )
P q
Estimation of detectability parameters follows from the general recipe
as v + 00 .
in ~3.
Point estimate q(x), for any x, is given by (3.4) with ~
x
= 8x/aw
and interval [q-(x),q+(x)] is given by (3.6).
For this last purpose, note
that d- and d+ (de~ined as at (2.7) but with subscript x ommitted again because


these are not functions of x) provide an interval estimate for 0 = Q~(B/a),
and that
(D-,D+) = (xd-/wQ~,xd+/wQ~) .
x x
Then the interval estimate for xl' defined by q-(xi) = q = q+(xi) at the end
of ~3, is obtained directly as
(4.4)
x- = wMQ~/d+)
i.
xi = w/}'(~/d-) .
Thus in comparison with xi. = w/}.(a/B), the interval (xi,xi) is constructed as


if (cP/d+,Q~/d-) is an interval estimate for (a/B) with coverage probability


(1-y), whereas (d-/Q~,d+/Q~) is defined to cover (B/a) with probability (1-y).
This situation is similar to that discussed by Koopmans, Owen and Rosenblatt
(1964) .
An interval estimate for 0 is available.
An interval estimate for

-------
15
A -.. 1/& is requfred. -1s it satisfactory to invert the endpoints of the
interval for 01
A A
The inituitive point estimate for A is A = 1/0.
Its mean square error
is infinite for the same reason that this is true of inverse calibrated
estimates X = (Y-a)/S, as noted by Williams (1969).
A
Estimate A is distributed
as W(A) = 1/T(0).
It is assumed now that the possibility of negative estimates
A
B has negligible probability (so that the fitted line, a+Sx, is monotonic
increasing in x > zero as is assumed for the line a+Bx; actually, the
A
occurrence of B < zero would almost certainly lead to a discarding of those
particular calibration results and probably to a questioning of the measurement
method also).
A
Then A > zero and, analogous to (2.7), an interval estimate
(1-,1+) for A is constructed by locating 1-  ~) and 1+ (> i) such that
y * = Pr [W(1-) > ~]
y* = Pr[W(R.+)< ~]
-
Pr[T(l/R.-)< 6] -Pr[T(l/R.-)< zero]
-
Pr[T(l/R.+)> 6}f-Pr[T(l/R.-+j < zero]
where (y;+rt) = y < 1.
Unless the ratio B/a, together with the calibration
design features Q and v, are illsuited for sensitive detection, R.+ is so small
that Pr[T(1/1+)< zero] = pr[Z< -1/1+] is negligible; these are precisely the
circumstances under which Pr[a< zero] is negligible as assumed above.
Then,
in comparison with (2.7), R.- = 1/d+ with y; = y+ and 1+ =, 1!d- with y: = y-,

which justifies the use of (Q~/d+,~/d-) as an interval estimate for (a/B)
when Pr [8 < ze ro] is negligible~

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16
5.
AN EXAMPLE OF DETECTABILITY
Liteanu and Rica (1980;Table 7.1) present an example of calibration
results in which analyte~ is tungsten and coricentrations X refer to tungsten
content in steel samples (scaled as ppm here).
The measure Y was obtained
by emission spectrometry with results recorded as coded digits over the range
110 to 280.
A total of n = 84 observations Y. are listed, twelve each at
~
seven standards x = zero,10,60,120,480,770 and 1050 ppm.
A linear calibration
function, f(x,~) = 8x, With constant variance 02 is a satisfactory representation
over this range of concentrations.
The following summary statistics are
obtained:
n = 84,
x = 355.714,
Q~ = 3563.433,
a = 113.022,
,.
8 = 0.153888,
o = 2.39472 on 82 degrees of freedom.
Thus g = (Q~a/~) = 228.992 and the interval (d-,d+) obtained as at (2.7), with
y-
=
y+ = 0.{)25, is (193..927,263.990).
A false positive rate of p = 0.01 is
specified and, with v = 82, t
P
For a detection limit Xt(p,q)
,.
= 2.37269 is required for threshold y .
p
with assurance probability (l-q) = 0.99,
the required 6 = 6(82,0.01,0.01) = 4.73164 (linear interpolation in Table 1
yields 4.73171).
The following listing contains w, Xt' ~t = (w6a/B), and
interval estimate (Xl' x!) obtained from (d-,d+) with (l-y) = 0.95, for

r = 1,2 or 3 observations yielding Y in the rule 'declare ~ detected if
Y > y , :
p

-------
17
   r=1 r=2 r=3
 w: 1.010876 0.722405 0.505989
Xt (ppm) . 4.7831 (aNn 3.4182(a/B) 2.8200(a/B)
.
xR. (ppm) : 74.4 53.2 43.9
interval (ppm) : (64.6,87.9) (46.1,62.8) (38.1,51.8)
A more complete characterization of detectability, under the same
detection rule, is given in Figure 2 which contains curves of detection

rates for r = 1, 2 or 3, constructed from point estimates [1-q(x)] at
(3.4) with ~ = (0.06426)x/w.
x
An indication of the reliability of these
estimated curves is given in Figure 3 which contains interval estimates
from (3.6), with y = 0.05, D- = (0.05384)x and D+ = (0.07324)x, for the
x x
case r = 1 only.

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18
6.
STATUTORY LIMITS
Another form of limiting concentration is that imposed by statute
or contractual obligation.
Let x denote that ana1yte concentration
s
specified as the statutory limit.
Of the two possibilities that in which
x ~ x corresponds to compliance, and X > x to noncompliance, will be
s s
considered here.
Numerous examples can be found in the Acts and Guidelines
governing clean air, clean water and potentially hazardous compounds in
foodstuffs.
Appropriate modifications for the reverse situation,
compliance if X ~ x , are obvious throughout.
s
to assertions of noncompliance which constitute 'alerts'.
The term detection now refers
Implicit in the notion of a statutory limit, if not explicitly stated
in the statu~e or contract, is the requirement that noncompliance be assured
of detection for all X > x .
s
Unless Xt (p ,q) . xs' a detection rule meeting
this requirement with assurance (l-q) is necessary.
Analogous to u ,wand
x
o of ~2, and ~ of ~3, let
x x
.;.
s,x
c:: vad [f (x ,6) -f (x, B) ]f cr }
s - -
w2 = var{[Y-~-f(x ,B)]/cr} ,
s s -
o
s,x
= [f(x ,B)~f(x,B)]/cru
s - . - s,X
~ = u 0 /w ,
s,X s,X s,X s
and let y = [~+f(x ,B)-w at ].
q s - s q
Then, with the distributional assumptions
in ~2, the rule 'assert noncompliance if Y > y , yields alert rates p(x) at
q
concentrations X = x given by
(6.1)
p (x) = pr{ [Y-~-f (x ,6)] > -w ~t I X = x}
s - s q
= Pr[(Z-~ ) >-St ] = Pr[T(~ )
-------
19
because [Y-~-f (x ,B)] = [f (x, 8)-f (x' ,8)] + Zaw when X = x, and
s - - s - S

A
PdT(-eS) > -t] :: PdT(eS) < d. Threshold y has been chosen so that the
q
alert rate at the statutory limit is
p(x ) = Pr [T(o) < t ] = (l-q)
s q
because 6 = zero when xi = x .
s,X . s
Moreover, since f(x,~) is monotonic ~
increasing in x, 6 < zero for all x > x and alert rate p(x) exceeds
s,x s
(l-q) for all x
> x .
s
The price for this specified level of protection
against failure to detect noncompliance is 'false alerts' at rate p(x) when
x < x given by (6.1).. Frequent false alerts, a consequence of insensitivity
s
of the .detection protocol in relation to stringency of the statute and the
specified level (l-q), may be intolerable.
Estimation in this context is limited to interest in p(x) and follows
the pattern established in fi2 and fi3.
Point estimates of p(x), for any x,
... A
can be obtained by substitution of 8 and a for 8 and a in 6 :
s,x
(6.2)
eS
s,X
A A
= [f (x ,8)-f,(x,~) ]j~u
s - . - s,X
,
~ = u 6 /w
s,X s,X S,x s
A
and p (x) = Pr [T ( 6 ) < t ] .
s,x q
- +
developing an interval estimate (d ,d ) for
s,x s,x
Interval estimates for p(x) are obtained by
eS analogous to that for
s,x
eS at (2.7):
x
(6.3) eS
s,X
= T(eS )
s,X
Pr [T(d- ) >6 ] = y
S,x s,x
Pr [T(d+ ) <6 ] = y+
s,X s,X
and then converting (d- d+ ) to an interval for 6 :
s,x' s,x S,x

-------
20
(6.4)
(D- D+) = (u d- /w,u d+ /w)
s,x' s,x s,x s,x s S,x S,x s
Then the interval
(6.5)
- + +] [ - ]
{p (x),p (x)} = {Pr[T(D , ) < t ,Pr T(D ) < t }
s.x q s,x q "
with probability (l-y).
covers p(x)
Under linear calibration, as in ~4, u
/ s.x
w = [r-1.+n-l+(x -x)2/Q]~ so that 6= Q~a/a
s s " "S,x "
- + - +
and interval (d ,d ) in ~4 supplies (d ,d )
s,x S,x
= (x -x)/Q~ and
s
= 6, ~ = (x -x)a/aw ,6" = 6,
S,x s s s.x
at (6.3) for all x. Then in (6.4),
[D- ",D+ ] = [(x -x)d- /w Q\ (x -x)d+ /w Q~]
S,x s,x s s s s
ready for (6.5).

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21
7. DISCUSSION
Detection limits are valuable to a researcher as an indication of
quality of existing methods and of the need for refinement of those methods.
They provide another criterion for comparisons among alternative methods.
They are also a useful feature to report in communications to other researchers
when describing measurement techniques and the meaning of measurements.
In
the context of statutory limits, characterization of detectabi1ity is an
essential ingredient when evaluating methods:
a method with detection limit
exceeding the statutory limit is obviously inadequate.
The general formulations
in ~3 and ~5 provide definitions of parameters, and practical procedures for
their estimation, that are required to characterize detectabi1ity.
One
option, not elaborated here but implicit throughout, is the possibility of
evaluating the effects of different calibration designs on detectabi1ity using
existing estimates ~, ~ and ~.
Then it is possible to match a detection
protocol to a particular criterion of detectabi1ity.
There remains the important matter of choosing rates p and q when
evaluating and reporting detectability.
There is a long tradition of conservatism,
indeed scepticism, inherited from the philosophy of scientific method, when
reporting 'positive' results (especially in technical journals).
It operates
by emphasizing repeatability and protecting against false positives, in the
general sense, by adopting Type I error levels set at p = 0.05 or 0.01.
This
mode of inference is more behavoria1 than inductive and these levels are quite
arbitrary.
Perhaps the best argument in their defense is the empirical
justification offered by Bross (1971): they do appear to have controlled the

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22
rate of misinformation in communications among researchers at a conventional
low level.
To Bross they are linguistic conventions in such communications,
conventions that have evolved and survived because they are useful.
In the context of analyte detection p and q have natural frequency
interpretations (rates) in repeated applications, so that part of the
philosophical dispute is avoided.
If rates of. false positives in this
context are to be set by convention also, then it is advisable to incorporate
this precedent and adopt the same levels of protection in the same communications.
Consequences of such a convention should not be overlooked however.
Arbitrary
designation of p implies that all benefits of improved sensitivity and
calibration design, and of increases in sample sizes nand r, are then devoted
to higher detection rates at all concentrations x > zero, and therefore
lower detection limits at specified assurance level (l-q), all at the expense
of an arbitrarily fixed rate of false positives, p.
There will be combinations
of protocol sensitivity, calibration designs and sample sizes, for which p
can be set below these conventional levels without serious losses in
detectability.
When describing detection limits to other researchers there is no such
precedent or convention for choice of assurance probability (1-q).
A
convenient convention for analyte detection would be p = kq with k a
preassigned constant (see Lehmann (1958) for constructions of this type in a
more general setting).
The choice k = 1 is assumed for values of 6 given in
Table 1.
A similar table can be prepared for other choices: k = 5, for example,
yields (1-q) = 0.998 when p = 0.01 and (1-q) = 0.99 when p = 0.05 .
Another
suggestion, also convenient, is to borrow a convention from toxicology and
report estimates of the 'median detectable concentration' (MDC or Xso) along

-------
23
with p : MDC = x (p,0.5) of course.
t
But convenience is not a dominant
criterion and it is preferable to report estimates of detection limits for
several choices of q, or of detection rates at several analyte concentrations
(as in Figures 2 and 3) for each p.
These suggestions are also appropriate
in some other areas of technical communications where characterizations of
detectability are valuable in demonstrating technical competence when selling
services or competing for contracts that contain detection specifications.
Such characterizations can be included in expert testimony, in legislative
hearings and before Administrative Law Courts for example, whether defending
or disputing the quality and suitability of a particular detection protocol.
The convention q = p, when reporting detection limit Xt(p,q),


corresponds to a reversal of the 'equal probability' test criterion (Arrow, 1960).
Instead of a preassigned departure (X = Xt)' from the null condition (X = zero),

at which the probability of a Type II error (false negative) is to be equal
to the probability of a Type I error (false positive) because at that
departure the costs of these different errors are equal, this convention defines
Xt as that departure at which q = p with p preassigned.


among researchers there are no habitual formulations of costs of false positives
In communications
or false negatives, which explains in part why conventional levels of protection
are considered so useful: they provide a convenient refuge.
But this is not
the case in the context of statutory limits where costs can be characterized
at least on a relative basis.
A false alert, for example, incurs costs that
are not functions of x < x and can be estimated precisely.
s
Appeal to
traditional linguistic conventions is inappropriate in such a context.
In

-------
24
cases where detection protocol and assurance level (l-q) are to be stipulated
in a contract, or in guidelines attendant to a statute, there will be
interested parties with conflicti~g demands regarding q, especially if protocol
sensitivity results in levels p(x) that promise to be costly in false alerts.
In the arena of public policy, where social costs are invoked, this conflict
occurs among increasingly antagonistic camps with their vehement public
advocates; the likelihood of rational resolution of q, by negotiation and
compromise, diminishes accordingly.
A two point decision space (compliance
versus noncompliance) is too simplistic for detection rules in these circumstances.

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25
8.
REFERENCES
ARROW, K. J. (1963), "Decision Theory and Choice of a Level of Significance
for the t-Test," in Contributions to Probability and Statistics, ed. I. Olkin,
Stanford: Stanford University Press.
BOWKER, A. H., and LIEBERMAN, G. J. (1959), Engineering Statistics, Englewood,
N.J.: Prentice-Hall.
BROSS, I. D. J. (1971), "Critical Levels, Statistical Language, and Scientific
Inference," in Foundations of Statistical Inference, ed. V. P. Godambe and
D. A. Sprott, Toronto: Holt, Rinehart and Winston.
BURROWS, P. M. (1985), "Random Foldings of the Noncentral t-Distribution,"
(submitted to Annals of Statistics).
CURRIE, L. A. (1968), "Limits for Qualitative Detection and Quantitative
Determination," Analytical Chemistry, 40, 586-593.
CURRIE, L. A., FILLIBEN, J. J., and DEVOE, J. R. (1972), "Statistical and
Mathematical Methods in Analytical Chemistry," Analytical Chemistry,
44, 497-512.
FIELLER, E. C. (1954), "Some Problems in Interval Estimation," Journal of the
--
Royal Statistical Society, Series B, 16, 175-185.
HUBAUX, A., and VOS, G. (1970), "Decision and Detection Limits for Linear
Calibration Curves," Analytical Chemistry, 42, 849-855.
KOOPMANS, L. H., OWEN, D. B., and ROSENBLATT, J. 1. (1964), "Confidence
Intervals for the Coefficient of Variations for the Normal and Log Normal
Distributions," Biometrika, 51, 25-32.
LEHMANN, E. (1958), "Significance level and power," Annals of Mathematical
Statistics, 29, 1167-1176.

-------
26
'" v
LITEANU, C., and RICA, I. (1980), Statistical Theory and Methodology of
.. Trace Analysis, Chichester: E!lis Horwood Ltd.
MAGGIO, E. T. (1980), "Enzymes as Iunnunochemical Labels," in Enzyme-
Iunnunoassay, ed. E. T. Maggio, Boca Raton, Florida: CRC Press, Inc.
OWEN, D. B. (1968),. "A Survey of Properties. and Applications of the
Noncentral t-Distribution," Technometrics, 10, 445-478.
.-
SCHEFFE, H. (1970), "Multiple Testing versus Multiple Estimation.
Improper
Confidence Sets.
Estimation of Directions and Ratios," Annals of
Mathematical Statistics, 41, 1-29.
.-
SCHEFFE, H. (1973), "A Statistical Theory of Calibration," Annals of
Statistics, 1, 1-37.
VOLLER, A., BIDWELL, D. E., and BARTLETT, A. (1977), The Enzyme Linked
Immunosorbent Assay (ELISA), Guernsey: Flowline Publications.
WILLIAMS, E. J. (1969), "A Note on Regression Methods in Calibration,"
Technometrics, 11, 189-192.

-------
Table 1. Values My,tp and ~ for specified v and p=q=0.05, 0.01 and
0.001 (see footnotes).
v Mv t.os ' t.Ol t.OOl ~.os ~.Ol ~.OOl
 5 1.18942 2.01505  3.36493 5.89343 3.86994 6.68320 12.60124
 6 1.15124 1.94318  3.14267 5.20763 3.75160 6.2126~ 10~8881~
 7 1.12587 1.89458  2.99795 4.78529 3.67302 5.91456..9.86151
 8 1.10778 1.85955  2.89646 4.50079 3.61713 5.71003 9.18600
 9 1 . 09424 1.83311  2.82144 4.29681 3.57538 5.56152 8.71150
10 1.08372 1.81246  2.76317 4.14370 3.54304 5.44903 8.36169
11 1.07532 1.79588  2.71808 4.02470 3.51725 5.36100 8.09409
12 1.06844 1.78229  2.68100 3.92963 3.49622 5.29030' 7.88328
13 1 .06272 1.17093  2.65031 3.85196 3.47872 5.23231 7.71324
14 1.05766' 1.76131  2.62449 3.76739 3.46396 5.16390 7.57337
15 1.05373 1.75305  2.60248 3.73263 3.45133 5.14291 7.45640
16 1.05014 1.74566  2.56349 3.66615 3.44041 5.10776 7.35722
17 ,..04"700 1.73961  2.56693 3.64577 3.43087 5.01727 7.27212
16 1 .Ollll23 1.73l106  2.55236 3.61048 3.42245 5.05060 7.19830
19 1.04176 1.72913  2.53946 3.57940 3.41499 5.02706 7.13372
20 1.03956 1.72472  2.52796 3.55161 3.40633 5.00615 7.07674
21 1.03758 1.72074  2.51765 3.52715 3.40232 4.98744 7.02611
22 1.03579 1.71714  2.50632 3.50ll99 3.39690 4.97059 6.98084
23 1.03416 1.71387  2.49987 3.48496 3.39198 4.95537 6.94012
24 1.03267 1.71088  2.49216 3.46678 3.38749 4.94153 6.90332
.25 1.03130 1.70814  2.46511 3.45019 3.38338 4.92890 6.86989
26 1.03005 1.70562  2.47863 3.43500 3.37960 4.91732 6.83939
27 1.02889 1.70329  2.47266 3.42103 3.37611 4.90667 6.81144
28 1 .02782 1.70113  2.46714 3.40816 3.37288 4.89684 6.78577
29 1.02683 1.69913  2.46202 3.39624 3.36989 4.88174 6.76208
30 1.02590 1.69726  2.45726 3.38518 3.36710 4.87930 .6.74017
32 1.02423 1.69389  2.44868 3.36531 3.36207 4.86411 '6.70094
34 1.02276 1.69092  2.44115 3.34793 3.35765 4.85082 6.66681
36 1.02145 1.6.8830  2.43449 3.33262 3.35374 4.83910 6.63686
38 1.02029 1.68595  2.42857 3.31903 3.35025 4.82870: 6.61039
40 1.01925 1 .68385  2.42326 3.30688 3.34'713 4.81939 6.58660
42 1.01831 1.68195  2.41847 3.29595 3.34431 4.81101 6.56566
44 1.01746 1.68023  2.41413 3.28607 3.34176 4.80343 6.54660
46 1.01668 1.67866  2.41019 3.21710 3.33944 4.79655 6.52934
48  1.01597 1 .61722  ~.40658 3.26691 3.33730 4.79027 6.51363
50  1.01532 1.67591  2.40327 3.26141 3.33536 4.78451 6.49927
52  1.01472 1.67469  2.40022 3.25451 3.33356 4.77921 6.48609
54  1.01416.., 1.67356  2.39741 3~24815 3.33189 4.77433 '.6.47396
56  1.01365" 1.67252  2.39480 3.24226 3.33035 '4.76981' 6.46275
58  1.01317 1.67155  2.39238 3.23680 3.32892 . 4.76561c:~.45237
60  1.01272 1 .67065  2.39012 3.23171 3.32759 4.76170' 6.44272
65  1.01173 1.66864  2.38510 3.22041 3.32462 4.75302 6.42134
70  1.01088 1.66691  2.38081 3.21079 3.32207 4.74562 6.40318
75  1..01014 1.66543  2.37710 3.20249 3.31989 4.73923 6.38757
80  1.00950 1.66412  2.37387 3.19526 3 . 3 1796 4.73367 6.37400
85  1.00893 1.66298  2.37102 3.18890 3.31628 4.72817 6.36210
90  1.00843 1.66196  2.36850 3.18327 3.31478 4.72444 6.35157
95  1 .00798 1.66105  2.36624 3.17825 3. 31344 4.72057 6.34221
100  1.00758 1.66023  2.36422 3.17374 3.31224 4.71711 6.33380
IHF  1.00000 1 .64485  2.32635 3.09023 3.28971 4.65270 6.18046
Column 1: v = degrees of freedom for estimate ~2.
A
Column 2: M such that expectation of 1/0 is M /0.
v v
Columns 3,4,5: values
t such that PdT (0) > t ] = p (Students's t).
P p
~ such that Pr[T(~) 
-------
(Captions for Figures 1,2,3)
Figure 1.
Figure 2.
Figure 3.
- +
Constructiqn of interval es~imate (Xt,Xt)' for the detection

limit Xt(p,q), using inverse interpolation.
Estimated detection rates for the tungsten content
example with r = 1,2, or 3 and p = 0.01 .
Interval estimates (y = 0.05) of detection rates for
the tungsten content example with r = 1 and p = 0.01 .

-------
 o
 o
 -
- 
:...: 
"-J 
w U')
I- r'
a: 
a: 
w 
> 
....... 0
I- U')
a: 
C) 
w 
z 
w U')
N
en 
~ 
a: 
lL. 
 q
 o
 o
~l'
x+
R.
ANALYTE CONCENTRATION

-------
-

811)
"
W
J-
a:
a:
o
ZII)
o
.-.
I--
U
W
1--11)
WN
o
F~6 . 2.
o
o
-
o
o
20
110
60 .
(PPM)
TUNGSTEN CONTENT
80

-------
'"""

~ .....
..... u~.
r-
W
t-
a:
a::
z ~-
o
to---4
t-
U
W
t- In.
WN
o
o
o
-
o
o
h~.5
I' I
I I
20
.
110
.
60
TUNGSTEN CONTENT (PPM)
80

-------
APPENDIX B
SAS Algorithms for Determining Parameters Associated
With the Noncentral t-Distribution
[Author:
Peter M. Burrows]

-------
2
116
117
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
71
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
S A S
OS SAS 82.4
LOG
MVS/XA JOB EXST1001 STEP G
MACRO NCTP;
*
* ASSUMES T(V,DELTA) FOLLOWS NON-CENTRAL T DISTRIBUTION
* ON V DEGREES OF FREEDOM, NON-CENTRALITY PARAMETER DELTA:
*
* LOAD
*
*
*
*
*
* CONDITIONS: DELTA> -6 IF T > ZERO (OTHERWISE Q = 1)
* DELTA < 6 IF T < ZERO (OTHERWISE Q = 0)
*
* PETER M. BURROWS/EXST/CE .NCT1/MAY 4,1985
*
PPI=0.398942280401432;
IF PT=O THEN DO; PQ=PROBNORM(-PD); PQD=-PPI*EXP(-0.5*PD*PD);
PP=0.5; GO TO Pl; END;
PP=l-PROBT(PT,PV);
PNI=l; IF PT < 0 THEN PNI=-l;
PT=PN I *PT; PD=PN I *PD;
PA=-6; IF PO < 6 THEN PA=-PD; PB=6; PC=PB-PA;
PCV= PV / ( PT*PT);
PQ=O; PQD=O;
ARRAY PU (PI) PU1-PU8;
ARRAY PZ PZ1-PZ16;
ARRAY PW PW1-PW16;
PZl =0.0483076656877383;
PZ2 =0.1444719615827965;
PZ3 =0.2392873622521371;
PZ4 =0.3318686022821276;
PZ5 =0.4213512761306353;
PZ6 =0.5068999089322294;
PZ7 =0.5877157572407623;
PZ8 =0.6630442669302152;
PZ9 =0.7321821187402897;
PZ10=0.7944837959679424;
PZll=0.8493676137325699;
PZ12=0.8963211557660521;
PZ13=0.9349060759377397;
PZ14=0.9647622555875064;
PZ15=0.9856115115452683;
PZ16=0.9972638618494816;
DO OVER PZ;
PF=O; PFD=O;
PU1=(7*PA+PB-PC*PZ)/8; PU2=(7*PA+PB+PC*PZ)/8;
PU3=PU1+PC/4; PU4=PU2+PC/4;
PU5=PU3+PC/4; PU6=PU4+PC/4;
PU7=PU5+PC/4; PU8=PU6+PC/4;
DO OVER PU; PCH=PCV*((PU+PD)**2);
PE=PPI*(EXP(-0.5*PU*PU))*(PROBCHI(PCH,PV));
PF=PF+PE; PFD=PFD+PU*PE; END;
PQ=PQ+PW*PF; PQD=PQD+PW*PFD; END; PQD=-PC*PQD/8;
IF PNI=l THEN PQ=1-PC*PQ/8;
ELSE DO; PQ=PC*PQ/8; PD=PNI*PD; PT=PNI*PT; END;
: V IN PV, DELTA IN PO, T IN PT

RETURNS: Q(DELTA,T,V) = PR( T(V,DELTA) < T ) IN PQ
DERIVATIVE Q'(DELTA,T,V) IN PQD
PR( T(V,ZERO) > T ) IN PP (STUDENT'S T)
PWl =0.0965400885147278;
PW2 =0.0956387200792749;
PW3 =0.0938443990808046;
PW4 =0.0911738786957639;
PW5 =0.0876520930044038;
PW6 =0.0833119242269468;
PW7 =0.0781938957870703;
PW8 =0.0723457941088485;
PW9 =0.0658222227763618;
PW10=0.0586840934785355;
PWll=0.0509980592623762;
PW12=0.0428358980222267;
PW13=0.0342738629130214;
PW14=0.0253920653092621;
PW15=0.0162743947309057;
PW16=0.0070186100094701;
PROC S 1

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3

102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
200
S A S
OS SAS 82.4
PROC S 1
LOG
MVS/XA JOB EXST100l STEP G
*
* CLEAN UP (DROP ALL BUT PV PD PT PQ PQD PP)
*
DROP PA PB PC PCH PCV PE PF PFD PI PNI PU1-PU8 PW1-PW16 PZ1-PZ16;
Pl: DROP PPI;
%
MACRO NCTD;
*
* ASSUMES T(V,DELTA) FOLLOWS NON-CENTRAL T DISTRIBUTION
* ON V DEGREES OF FREEDOM, NON-CENTRALITY PARAMETER DELTA:
*
* LOAD
*
* RETURNS:
*
* USES MACRO NCTP
*
* PETER M. BURROWS/EXST/CE .NCT2/MAY 4,1985
*
DD1=DT-PROBIT(DQ)*SQRT(1+DT*DT/(2*DV)); PV=DV; PT=DT;
Dl: PD=DD1; NCTP;
DD=DD1+(DQ-PQ)/PQD;
IF ABS(DD-DD1) < 0.000001 THEN GO TO D2;
DD1=DD; GO TO Dl;
D2: DROP DD1;
%
*
* 5/04/85
*
* EXAMPLE ONE:
*
* GET PR( T(DF,DELTA) < T ) & PR( T(DF,O) > T ) GIVEN DF, DELTA, T
*
V IN DV, T IN DT, Q IN DQ
DELTA IN DD SATISFYING PR( T(V,DELTA, < T ) = Q
DATA ONE; TITLE 'EXAMPLE ONE'; INPUT DF DELTA T;
PV=DF; PD=DELTA; PT=T;
NCTP; PROBD=PQ; PROBO=PP;
CARDS;
NOTE: DATA SET WORK. ONE HAS 6 OBSERVATIONS AND 11 VARIABLES. 510 OBS/TRK.
NOTE: THE DATA STATEMENT USED 0.41 SECONDS AND lOOK.
207
PROC SORT; BY DF PROBD;
NOTE: 4 CYLINDERS DYNAMICALLY ALLOCATED ON DISK FOR EACH OF 3 SORT WORK DATA SETS.
NOTE: DATA SET WORK. ONE HAS 6 OBSERVATIONS AND 11 VARIABLES. 510 OBS/TRK.
NOTE: THE PROCEDURE SORT USED 0.38 SECONDS AND 200K.
208
209
210
211
212
213
PROC PRINT; BY DF; ID DF; VAR DELTA T PROBD PROBO;
*
* EXAMPLE TWO
*
* GET DELTA SUCH THAT PR( T(DF,DELTA) < T ) = Q GIVEN DF, T, Q
*
NOTE: THE PROCEDURE PRINT USED 0.11 SECONDS AND 132K AND PRINTED PAGE 1.

-------
4
214
215
216
299
OS SAS 82.4
MVS/XA JOB EXST1001 STEP G
S A S
LOG
DATA TWO; TITLE 'EXAMPLE TWO'; INPUT DF T Q;
DV=DF; DT=T; DQ=Q;
NCTD' DELTA=DD'
CARDS; ,
NOTE: DATA SET WORK. TWO HAS 18 OBSERVATIONS AND 14 VARIABLES. 4040BS/TRK.
NOTE: THE DATA STATEMENT USED 1.52 SECONDS AND 108K.
318
PROC SORT; BY DF Q;
NOTE: DATA SET WORK. TWO HAS 18 OBSERVATIONS AND 14 VARIABLES. 4040BS/TRK.
NOTE: THE PROCEDURE SORT USED 0.25 SECONDS AND 200K.
319
PROC PRINT; BY DF; ID DF; VAR T Q DELTA;
NOTE: THE PROCEDURE PRINT USED 0.11 SECONDS AND 132K AND PRINTED PAGE 2.
NOTE: SAS USED 200K MEMORY.

NOTE: SAS INSTITUTE INC.
SAS CIRCLE
PO BOX 8000
CARY, N.C. 27511-8000
PROC S1

-------
EXAMPLE ONE    
OF DELTA T PROBD PROBO
8 9.18600 4.50079 0.0010000 0.0010000
 5.71003 2.89646 0.0100000 0.0100000
 3.61713 1.85955 0.0499999 0.0499998
48 6.51363 3.26891 0.0010000 0.0010000
 4.79027 2.40658 0.0100000 0.0100000
 3.33730 1 .67722 0.0500004 0.0500004

-------
EXAMPLE TWO   
DF T Q DELTA
8 4.50079 0.0010000 9.18600
 2.89646 0.0100000 5.71003
 1 .85955 0.0500000 3.61713
16 3.68615 0.0010000 7.35722
 2.58349 0.0100000 5.10776
 1.74588 0.0500000 3.44041
24 3.46678 0.0010000 6.90332
 2.49216 0.0100000 4.94153
 1.71088 0.0500000 3.38749
32 3.36531 0.0010000 6.70094
 2.44868 0.0100000 4.86411
 1.69389 0.0500000 3.36207
40 3.30688 0.0010000 6.58680
 2.42326 0.0100000 4.81939
 1.68385 0.0500000 3.34713
48 3.26891 0.0010000 6.51363
 2.40658 0.0100000 4.79027
 1.67722 0.0500000 3.33730

-------
APPENDIX C
Definition and Procedure for the Determination
of the Method Detection Limit
[EPA-Revision 1.11]

-------
Definition
The.method detection limit (MOL) is defined as the minimum concentration of a
substance that can be measured and reported with 99% confidence that the
analyte concentration is greater than zero and is determined from analysis of
a sample in a given matrix containing analyte.
Scope and Application
This procedure is designed for applicability to a wide variety of sample
types ranging from reagent (blank) water containing analyte to wastewater
containing analyte. The MOL for an analytical procedure may vary as a
function of sample type. The procedure requires a complete. specific and
well defined analytical method.
It is essential that all sample processing
steps of the analytical m~thod be included in the determination of the method
detection llmit.
The MOL obtained by this procedure is used to judge the significance of a
single measurement of a future sample.
The MOL procedure was designed for applicablity to a broad variety of
physical and chemical methods.
To accomplish this. the procedure was made
device- or instrument-independent.
Procedure
1.
Make an estimate of the detection limit using one of the following:
-858-

-------
(a) The concentration value that corresponds to an instrument
signal/noise in the range of 2.5 to 5.
(b) The concentration equivalent of three times the standard
deviation of replicate instrumental measurements of the analyte
in reagent water.
(c) That region of the standard curve where there is a significant
change in sensitivity, i.e., a break in the slope of the
standard curve.
(d)
Instrumental limitations.
It ts recognized that the experience of the analyst is important to
this process.
However, the analyst must include the above consider-
ations in the initial estimate of the detection limit.
2. Prepare reagent (blank) water that is as free of ana lyte as pos-
 si ble. Reagent or interference free water is defined as a water
 salJ1) 1 e in which analyte and interferent concentrations are not
detected at the method detection limit of each analyte of interest.
Interferences are defined as systematic errors in the measured
analytical signal of an established procedure caused by the presence
of interfering species (interferent).
The interferent concentration
is presupposed to be normally distributed in representative samples
-
of a given matrix.
- 859-

-------
3.
(a)
If the MDL is to be determined in reagent (blank) water. pre-
pare a laboratory standard (analyte in reagent water) at a
concentration which is at least equal to or in the same con-
centration range as the estimated method detection limit.
(Recommend between 1 and 5 times the estimated method detection
limit.) Proceed to Step 4.
(b)
If the MDL is ~o be determined in another sample matrix. ana-
lyze the sample.
If the measured level of the analyte is in
the recommended range of one to five times the estimated
detection limit. proceed to Step 4.
If the measured level of analyte is less than the estimated
detection limit, add a known amount of analyte to bring the
level of analyte between one and five times the estimated
detection limit.
If the measured level of analyte is greater than five times the
estimated detection limit. there are two options.
(1) Obtain another sample with a lower level of analyte in the
same matrix if possible.
(2) The sample may be used as is for determining the method
detection limit if the analyte level does not exceed 10
-860-

-------
times the MOL of the analyte in reagent water. The vari-
ance of the analytical method changes as the analyte
concentration increases from the MOL, hence the MOL deter-
mined under these circumstances may not truly reflect
method variance at lower analyte concentrations.
4.
(a) Take a minimum of seven aliquots of the sample to be used to
calculate the method detection limit and process each through
the entire ana1ytical method.
Make all computations according
to the defined method with final results in the method report-
ing units.
If a blank measurement is required to calculate the
measured level of analyte, obtain a separate blank measurement
for each sample aliquot analyzed.
The average blank measure-
ment is subtracted from the respective sample measurements.
(b)
It may be economically and technically desirable to evaluate
the estimated method detection limit before proceeding with
4a. This will:
(1) prevent repeating this entire procedure
when the costs of analyses are high and (2) insure that the
procedure is being conducted at the correct concentration.
It
is Quite possible that an inflated MOL will be calculated from
data obtained at many times the real MOL even though the level
of analyte is less than five times the calculated method
detection limit. To insure that the estimate of the method
detection limit is a good estimate, it is necessary to deter-
mine that a lower concentration of analyte will not result in a
-861-

-------
. '
significantly lower method detection limit.
Take two aliquots
of the sample to be used to calculate the method d~tection
limit and process each through the entire method, including
blank measurements as described above in 4a.
Evaluate these
d at a:
(1)
If these measurements indicate the sample is in desirable
range fo~determination of the MOL, take five additional
aliquots and proceed.
Use all seven measurements for
calculation of the MOL.
(2 )
If these measurements indicate the sample is not in
correct range, reestimate the MOL, obtain new sample as .in
3 and repeat either 4a or 4b.
5.
Calculate the variance (S2) and standard deviation (S) of the
replicate measurements, as follows:
S2 =
1
[~ X.2
. 1 1
1=
( n
- r
i=l
Xi) / J
n - 1
s = (S2)1/2
where:
Xi; i=l to n,
=
ar~ the analytical results in the final
method reporting units obtained from the
n sample aliquots and r refers to the sum
of the X values from i=1 to n.
- 862-

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where: LCL and UCL are the lower and upper 95% confidence limits
respectively based on seven aliquots.
7.
Optional iterative ptocedure to verify the reasonableness of the
estimate of the MOL and subsequent MOL determinations.
a. )
If this is the initial attempt to compute MOL based on the
estimate of MOL fo~mulated in Step 1. take the MOL as calculated
in Step 6. spike in the matrix at the calculated MOL and proceed
through the procedure starting with Step 4.
b.) If this is the second or later iteration of the MOL calcula-
tion. use S2 from the current MOL calculation and S2 from the
previous MOL calculation to compute the F-ratio. The F-ratio is
calculated by substituting the larger S2into the numerator S~ and
the others into the denominator S~. The computed F-ratio is

then compared with the F-ratio found in the table which is 3.05 as
fo llows:
if
S2
A
52
B
<
3.05.
then compute the pooled standard deviation by the following
equat; on:
1/2
Spoo led = [6S~ + 6S~J
12
- 864-

-------
if
~A

T
B
>
3.05, respike at the most recent calculated
MOL and process the samples through the procedure starting with
Step 4.
If the most recent calculated MOL does not permit
qualitative identification when samples are spiked at that level,
report the MOL as a concentration between the current and previous
MOL which permits qualitative identification.
c.) Use the Spooled as calculated in 7b to compute the final MOL
according to the following equation:
MOL = 2.681
(Spooled)
where 2.681 is equal to t(12, l-a = .99).
d.) The 95~ confidence limits for MOL derived in 7c are computed
according to the following equations derived from precentiles of
the chi squared over degrees of freedom distribution.
LCL = 0.72 MOL
UCL = 1. 65 MOL
where LCL and UCL are the lower and upper 95% confidence limits
respectively based on 14 aliquots.
- 865-

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Table of Students' t Values at the 99 Percent Confidence Level
Number of
Repl icates
Degrees of Freedom
(n-1)
t (n-1, .99 )
7
8
9
10
11
16
21
26
31
61
00
6
7
8
9
10
10
20
25
30
60
00
3.143
2.998
2.896
2.821
2.764
2.602
2.528
2.485
2.457
2 . 390
2 .326
Reporting
The analytical method used must be specifically identified by number or
title and the MOL for each analyte expressed in the appropriate method
reporting units.
If the analytical method permits options which affect the
method detection limit, these conditions must be specified with the MOL
value.
The sample matrix used to determine the MOL must also be identified
with MOL value.
Report the mean analyte level with the MOL and indicate if
the MOL procedure was iterated.
If a laboratory standard or a sample that
contained a known amount analyte was used for this determination, also
report the mean recovery.
If the level of analyte in the sample was below the determined MOL or does
not exceed 10 times the MOL of the analyte in reagent water, do not report a
value for the MOL.
-866-

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APPENDIX D
Extraction and Analysis of Selected Organics
in Sediments by Ultrasonication

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EXTRACTION AND ANALYSIS OF SELECTED ORGANICS IN SEDIMENTS
BY ULTRASONICATION
1.
Scope and Application
1.1 Thia method was originally designed for the determination of
extractable priority pollutants in sediments by GC/MS (source: W.
Loy, Environmental Research Laboratory, USEPA, Athens, GA) with a
detection limit of 5 ppm.
1.2 For purposes of this experiment, the method was implemented as
modified for capillary GC/FID in the evaluation of extraction
aethods under EPA Contract No. 68-03-3119 (Evaluation of
Analytical Test Procedures for the Measurement of Organic
Pollutants in Sediment) .
1.3 No cleanup procedure waa used in the application of the method to
this experiaent; ~owever, if required, the procedures given in EPA
Test Method 625-S are appropriate.
Summary of the Method
2.1 A 30g portion of sediment is mixed with anhydrous sodium sulfate
and immediately extracted with methylene chloride/acetone (1:1>
using a 300W ultrasonic probe. The extract is filtered, and
concentrated by K-D and nitrogen blowdown. The extract is
analyzed for selected organics bX the internal standard, capillary
GC/FID procedure.
Apparatus and Materials
3.1 Disposable pipets, 1 aL
3.2 Sonic cell disruptor, Heat Systems - Ultrasonics, Inc. Model 375C
or equivalent (375 watt with pulsing capability and a 3/4" high-
gain probe).
Screwcap ointment Jars, 16 oz
Buchner funnels
Filter paper (Whatman No. 41)
Kuderna-Danish (K-D) apparatus
3.6.1 Concentrator tube - 10 aL, graduated (Kontes K-570040-1029
or equivalent).
3.6.2 Evaporative flask - 500 mL (Kontes K570001-0500 or
equivalent).
3.6.3 Snyder column - three-ball macro (Kontes K-503000-0121 or
equivalent).
3.6.4 Modified Snyder column - (Kontes K-569251 or equivalent).
3.7 Boiling chips - extracted, approximately 10/40 aesh
3.8 Nitrogen blowdown apparatus
3.9 Steam bath or equivalent
3.10 Gas chromato~raph - analytical capillary GC/FID meeting the 2
requirements outlined in EPA Teat Method 625-S (Section 5.2)
3.11 Capillary GC column - 15 . X 0.25 .m ID wall coated open tubular
capillary column coated with SE-30 or equivalent with a 0.25 urn
film thickness. The column should provide at least 25,000
effective theoretical plates, aeasured at C13'
2.
3.
3.3
3.4
3.5
3.6
1

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4.
ReaQents
4.1 Sodium sulfate - Baker anhydrous powder.
4.2 Methylene chloride - pesticide quality, distilled in g18ss.
4.3 Acetone - pesticide quality, distilled in glass.
4.4 Standard cOMpounds
4.4.1 Dissolve 0.060 g of individual reference materials for each
analyte in pesticide quality acetone and dilute to
volume in a 10 .L glass stoppered volumetric flask.
Individual aliquots are combined and diluted to the
appropriate volume for use. Store all solutions in a
refrigerator and protect from light.
4.4.2 Check frequently for signs of degradation or solvent
evaporation (e.g., color or volume changes,
precipitation).
5.
Calibration
5.1 For this experiment, no separate instrument calibration was used.
Where required, calibratio~ is performed according to EPA Test
Method 625-S (Section 7.0) using triplicate standards at three
concentrations corresponding to 20, 50, and 100% of the sediment
fortification level.
Quality Control
6.1 In this experiment, quality control was provided by statistical
procedures. Separate analytical 2uality control procedures as
outlined in EPA Test Method 625-5 were not applied. Where
required, appropriate quality contr~l procedures are outlined in
EPA Test Method 625-5 (Section 8.0) .
Sample Collection. Preservation. and HandlinQ
7.1 Grab samples aust be collected in g18ss containers. Conventional
sampling practices should be followed except that the bottle must
not be pre-rinsed with sample before collection. Composite
samples should be collected in accordance with the requirements of
the program. Automatic sampling equipment must be free of
potential sources of contamination.
7.2 The samples must be maintained at 40C in the dark from the time of
collection until extraction.
Sample Extraction
8.1 Decant and discard the water layer over the sediment. Mix samples
thoroughly and discard foreign obJects such as sticks, leaves, and
rocks using a 1/4" screen.
8.2 Weigh 30 g of sample into a 16 0% Jar and add 65 g of anhydrous
sodium sulfate. Mix well and add more sodium sulfate as necessary
to achieve a sandy texture. Immediatelv add surrogates (if used)
and the extraction solvent (see step 8.3).
8.3 Add 100 mL of 1:1 methylene chloride/acetone to the
sediment/sodium sulfate mixture.
8.4 Place the sonic probe ca 1 ca below the surface of the solvent but
above the sediment layer.
8.5 Sonicate for 3 min at full power with a 50% pulse.
8.6 Decant the solvent into a Buchner fu~nel and suction filter.
Repeat 8.3-8.6 twice more.
8.7 Pour the entire sample into a Buchner funnel and rinse with 1:1
6.
7.
8.
2

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- .
9.
aethylene chloride/acetone.
8.8 Concentrate the extract to 5-7 aL uaing the K-D apparat~s and
subsequently to ca 1 aL by N2 blowdown for the final
concentration. .
NOTE: Discontinue evaporation if the concentrated extract becomes
excesaively viscous or if precipitation occurs. Redilute with the
appropriate solvent, if necessary, and record final volume.
Sample Extract Analysis
9.1 Capillary GC/MS analysis of sample2extracts is outlined in EPA
Test Method 625-S (Section 12.1.1) .
9.2 Capillary GC/FID analysis of sample extracts is described below.
9.2.1 Instrumentation: Varian Model 3700 GC equiped with FID
detector and Grob-type splitless/split capillary
inJector.
9.2.2 Capillary Column: D8-5 (SE-54) WCOT, fused silica column
(15m X .25 mm ID), .25 um film thickness.
9.2.3 Conditions: InJector temperature, 2400C
o
Detector temperature, 260 C
o
Column temperature, 70 C for 3 ain
70-2200C at 50/min
Splitless time, 60 sec
InJection volume, 1.5 uL
Carrier gas, helium at 0.6 aL/min (coluan)
Septum purge, 2.2 mLlmin
InJector split, 76.5 aL/ain
Detector gases, air at 287 aL/ain
hydrogen at 47 aL/min
Makeup gas (detector), helium at 35 aL/min
A representative capillary GC/FID chromatogram of the
analytes included in this study is shown in Figure 1.
9.2.4
10.
References
1. Hines, J. W. Evaluation of Analytical Test Procedures for the
Measurement of Organic Pollutants in Sediment. EPA Contract No.
3119, Draft Final Report, U.S. Environmental Protection Agency,
Cincinnati, Ohio, June, 1985.
68-03-
2. Test Method 625-S: Protocol for the Analysis of Extractable Organic
Priority Pollutants in Industrial and Municipal Wastewater Treatment
Sludge. Environmental Monitoring and Support Laboratory. U.S. EPA,
Cincinnati, Ohio, Deceaber 10, 1982.
3

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