Document ID: EPA-HQ-OW-2003-0002-0091
Agency: epa
Document Type: Supporting & Related Material
Title: 
Posted Date: 2003-03-12T05:00Z

MEMORANDUM
TO:
Chuck
White,
EPA
WAM
FROM:
Raphael
Kuznetsovski,
SAIC
WAM
DATE:
February
27,
1998
REFERENCE:
EPA
Contract
No.
68­
C4­
0046;
Work
Assignment
No.
3­
17
SAIC
Project
No.
01­
0813­
08­
2108­
170
SUBJECT:
Updated
Annotated
Bibliography
Attached
please
find
an
updated
version
of
the
"
Annotated
Bibliography
on
Measurement
Limitations
and
Calibration."
Since
the
last
report
on
12/
19/
97
SAIC
has
added
annotated
reviews
of
four
articles
to
the
calibration
section
of
the
document.
The
new
articles
are

Buonaccorsi,
J.
P.
(
1986)
Design
considerations
for
calibration.
Technometrics,
28,
149­
155.

Naszodi,
L.
J.
(
1978)
Estimation
of
the
bias
in
the
course
of
calibration.
Technometrics,
20,
201­
205.

Spiegelman,
C.
H.
and
Studden,
W.
J.
(
1980)
Design
aspects
of
Scheffe's
calibration
theory
using
linear
splines.
Journal
of
Research
of
the
National
Bureau
of
Standards,
85,
295­
304.

Thomas,
M.
A.
and
Myers,
R.
H.
(
1973)
Optimal
designs
for
the
inverse
regression
method
of
calibration.
Communications
in
Statistics,
2,
419­
433.

Per
your
January
15,
1998
request,
hard
copies
of
these
and
other
articles
on
calibration
included
in
the
annotated
bibliography
were
delivered
to
you
on
January
26,
1998.
Please
provide
us
with
further
technical
directions
and
future
deadlines
on
this
and
related
issues.
1
EPA
Contract
N.
68­
C4­
0046;
WA
3­
17
2
SAIC
Project
No.
01­
0813­
08­
2108­
170
1
Annotated
Bibliography
on
Measurement
Limitations
and
Calibration12
(
Draft
Report)

Updated:
February
27,
1997
Introduction
This
bibliography
summarizes
recent
publications
on
statistical
issues
related
to
the
measurement
limitation
and
calibration.
These
subjects
have
received
extensive
consideration
in
the
literature
over
the
past
few
decades.
The
bibliography
covers
both
theoretical
and
applied
aspects
of
the
field
and
primarily
concentrates
on
material
which
has
been
published
in
the
last
thirty
years
from
peer
reviewed
journals,
books,
and
manuscripts.
Papers
in
languages
other
than
English
are
not
included.
Although
the
coverage
of
the
bibliography
is
not
comprehensive,
it
does
include
core
materials
on
the
subjects
of
interest
and
related
topics.
The
bibliography
includes
41
entries
by
34
authors.
The
bibliography
is
classified
by
topics,
and
within
a
topic,
the
articles
are
organized
alphabetically
by
author.
Annotations
in
the
bibliography
are
intended
only
to
summarize
the
views
and
claims
of
authors
writing
in
relation
to
the
subject
of
concern.
There
is,
in
the
bibliography,
no
intent
to
endorse
any
views
expressed
by
any
author.

Annotated
bibliography
of
method
detection
limits
and
quantification
Bayley,
C.
J.,
Cox,
E.
A.,
and
Springer,
J.
A.
(
1978)
High
pressure
liquid
chromatographic
determination
of
the
intermediates
side
reaction
products
in
FD&
C
Red
No.
2
and
FD&
C
Yellow
No.
5:
Statistical
analysis
of
instrumental
response.
J.
Assoc.
Off.
Anal.
Chem.,
61,
1404­
1414.

The
authors
apply
the
statistical
methods
of
regression
analysis
and
detection
limits
to
calibration
data
obtained
by
high
pressure
liquid
chromatographic
analysis
for
the
intermediates
and
side
reaction
products
of
FD&
C
Yellow
No.
5
and
the
former
FD&
C
Red
No.
2.
The
calibration
line
is
calculated
Annotated
Bibliography
on
Measurement
Limitations
and
Calibration
Updated:
2/
27/
98
2
from
concentrations
and
corresponding
peak
area
response
of
the
calibration
data
by
the
method
of
least
squares.
The
formulas
for
prediction
intervals
are
also
provided.
The
Hubaux
&
Vos
(
1970)
approach
is
applied
by
the
authors
to
calculate
the
limit
of
detection,
.
That
is,
is
defined
as
X
LD
Y
Q
the
response
estimate
calculated
from
the
calibration
line
which
corresponds
to
.
The
authors
X
LD
accent
importance
of
value
since
it
defines
the
response
above
which
quantitation
is
to
be
Y
Q
performed.
is
the
lowest
amount
in
units
of
area
that
can
be
measured
above
the
blank.
Y
Q
Clayton,
C.
A.,
Hines,
J.
W.,
and
Elkins,
P.
D.
(
1987)
Detection
limits
with
specified
assurance
probabilities.
Analytical
Chemistry,
59,
2506­
2514.

Clayton
and
co­
workers
discuss
the
following
topics:
(
1)
calibration
models;
(
2)
detection
limits
for
linear
calibration
with
known
parameters;
(
3)
detection
limits
for
linear
calibration
with
unknown
parameters;
(
4)
estimation
of
detection
limits
and
detection
rates
for
linear
calibration.
First,
the
authors
consider
the
straight­
line
calibration
and
assume
that
.
In
(
2)
they
consider
Y
x



x



>
0
the
concept
of
detectability
within
the
context
of
a
linear
calibration
model
in
which
it
is
assumed
(
a)
that
the
slope
and
intercept
of
the
straight­
line
calibration
are
known
and
(
b)
that
the
variation
of
measurement
errors
(
presumed
homogeneous
over
the
range
of
concentrations
of
interest)
is
known.
They
state
that
the
assumption
of
constant
error
variance
can
often
be
accomplished
by
transforming
the
response
Y.
The
following
detection
rule
is
adopted:
"
Assert
(
i.
e.,
the
analyte
is
present)
if
X>
0
"
where
is
a
threshold
response
value
chosen
so
that
the
rate
of
false
positives
(
in
repeated
Y>
y
p
y
p
applications
of
the
rule
to
blank
samples)
is
fixed
at
a
desired
probability
level,
.
In
(
3)
detection
p
limits
and
detection
rates
are
formally
defined
for
the
case
of
a
linear
calibration
model
for
which
the
model
parameters
and
error
variance
are
unknown
and
must
be
estimated.
Clayton
et
al.
first
pointed
out
that
in
the
case
in
which
is
replaced
by
its
sample­
based
estimate
s,
the
appropriate
distribution

under
the
alternative
hypothesis
(
i.
e.,
)
is
the
non­
central
t­
distribution.
For
this
situation,
in
(
4)
X>
0
tables
and
a
four­
step
procedure
for
generating
detection
limit
estimates
are
provided.
Experiments
that
illustrate
and
validate
the
approach
are
then
described
in
subsequent
sections.

Coleman,
D.,
Auses,
J.,
and
Grams,
N.
(
1997)
Regulations
­
From
an
industry
perspective
or
Relationships
between
detection
limits,
quantitation
limits,
and
significant
digits,
Chemometrics
and
Intelligent
Laboratory
Systems,
37,
71­
80
Coleman
et
al.
interpret
and
extend
the
calibration­
based
definition
of
detection
limit
due
to
Hubaux
and
Vos
(
1970).
For
these
purposes
the
authors
use
a
definition
of
significant
digits
to
determine
relative
measurement
error
(
RME)
and
signal­
to­
noise
ratio
(
SNR).
A
number
expressed
in
scientific
notation
is
said
to
have
w
significant
digits
if
the
value
has
an
error
interval
of
width
.
It
can
be
shown
that
a
number
with
w
significant
digits
(
1/
2)

10

w

1
satisfies
the
following
inequality:
Annotated
Bibliography
on
Measurement
Limitations
and
Calibration
Updated:
2/
27/
98
3
1
2

10

w

RME
<
1
2

10

w

1.

SNR

log
10
signal
noise

log
10
measurement
total
measuremt
err


log
10(
2

RME).
The
authors
then
define
SNR
as
Therefore,
w

SNR
>
w

1.
Further,
Coleman
et
al.
show
that
the
Hubaux­
Vos
detection
limit,
,
is
approximately
DL
H

V
equal
to
the
lowest
concentration
with
RME=
50%,
SNR=
0,
or,
equivalently,
at
least
zero
significant
digits
in
the
measurement.
They
also
define
the
quantitation
limit
based
on
Hubaux­
Vos
(
)
as
QL
H

V
the
lowest
concentration
with
RME=
5%,
or
equivalently,
SNR=
1.
At
this
concentration,
at
least
one
significant
digit
can
be
ensured
in
the
measurement
at
a
specified
level
of
confidence
of
.
If
measurement
error
were
constant,
the
authors
expect
to
have
100(
1




)
%
,
since
RME=
50%
at
,
and
RME=
50%/
10
=
5%
at
.
This
does
not
QL
H

V

10

DL
H

V
DL
H

V
QL
H

V
quite
hold
true
due
to
the
`
flaring'
of
the
prediction
intervals
away
from
the
calibration
line
as
the
concentration
moves
further
from
the
value
of
in
the
calibration
design.
x
Currie,
L.
A.
(
1968)
Limits
for
quantative
detection
and
quantitative
determination,
Analytical
Chemistry,
40,
586­
593
Three
limiting
levels
are
defined
and
discussed
in
this
article:
(
1)
a
"
decision
limit"
at
which
one
may
decide
whether
or
not
the
result
of
an
analysis
indicates
detection,
(
2)
a
"
detection
limit"
at
which
a
given
analytical
procedure
may
be
relied
upon
to
lead
to
detection,
and
(
3)
a
"
determination
limit"
at
which
a
given
procedure
will
be
sufficiently
precise
to
yield
a
satisfactory
quantitative
estimate.
(
Critical
Level):
The
net
signal
level
(
instrument
response)
above
which
an
observed
signal
L
C
may
be
reliably
recognized
as
"
detected".
Mathematically,
the
critical
level
is
given
as
,
L
C

z
 

0
where
is
the
abscissa
of
the
standardized
normal
distribution
corresponding
to
probability
level
z
 
and
represents
the
variance
of
the
blank.
1



2
0
(
Detection
Limit):
The
"
true"
net
signal
level
which
may
be
a
priori
expected
to
lead
to
L
D
detection.
is
defined
so
that
the
probability
distribution
of
possible
outcomes
intersects
such
L
D
L
C
that
the
fraction,
,
will
correspond
to
the
(
correct)
decision,
"
detected".
This
may
be
written
as
1


,
where
is
the
abscissa
of
the
standardized
normal
distribution
corresponding
to
L
D

L
C

z
 

D
z
 
probability
level
and
is
the
variance
of
the
distribution
of
the
(
net)
signal
when
its
true
value
1



2
D
is
equal
to
.
L
D
(
Determination
Limit):
The
level
at
which
the
measurement
precision
will
be
satisfactory
for
L
Q
Annotated
Bibliography
on
Measurement
Limitations
and
Calibration
Updated:
2/
27/
98
4
quantitative
determination.
In
other
words,
a
result
must
be
satisfactory
close
to
the
true
value
(
limiting
mean).
Therefore,
for
the
Determinination
Limit,
,
the
standard
deviation,
,
must
be
L
Q

Q
but
a
small
fraction
of
the
true
value.
The
Determination
Limit
so
defined
may
be
written
as
,
where
is
the
true
value
of
the
net
signal
having
a
standard
deviation,
,
and
is
the
L
Q

k
Q

Q
L
Q

Q
1
k
Q
requisite
relative
standard
deviation.
In
order
to
make
the
significance
of
definitions
clearer,
a
number
of
specific
choices
for
,

,

and
`
s
may
be
helpful.
If
the
risk
of
making
both
kinds
of
mistake
are
set
equal,
i.
e.
,
and,
in



addition,
if
is
approximately
constant,
then
and
.
Making
the
additional

L
C

z

L
D

2L
C
requirement
that
,
we
find
that
.
k
Q

10
L
Q

10

The
principles
are
illustrated
by
examples
of
spectrophotometry
and
radioactivity.

Currie,
L.
A.
(
1988)
Detection:
Overview
of
historical,
societal,
and
technical
issues,
In
Detection
in
Analytical
Chemistry:
Importance,
Theory
and
Practice.
American
Chemical
Society,
Washington,
DC.

This
overview
presents
some
of
the
societal,
historical,
and
broad
conceptual
issues
relating
to
detection
and
chemical
measurements.
Balanced
coverage
was
the
intent,
but
special
attention
has
been
given
to
the
following
new
topics
not
covered
elsewhere:
multiple
decisions
and
probabilistic
pattern
detection,
utilization
of
physical
constraints
(
on
variance),
and
some
effects
of
varying
probability
density
function
as
related
to
experimental
design
and
variation.

The
first
part
of
this
overview
introduces
the
basic
concept
of
(
chemical)
detection,
together
with
applicability
to
selected
societal
problems.
Part
two
comprises
a
brief
historical
review,
highlighting
major
contributions
to
the
concept
and
realization
of
detection
in
chemical
applications.
Part
three
seeks
to
expose
most
of
the
technical
issues
involved
in
deriving
meaningful
detection
decisions
and
detection
limits,
considering
the
overall
Chemical
Measurement
Process:
issues
affecting
the
validity
of
detection
decisions
(
null
hypothesis
testing);
the
type
II
error
and
(

)
detection
for
the
alternative
hypothesis;
multiple
detection
decision
and
Discrimination
Limits
for
chemical
species
and
chemical
patterns.

Currie,
L.
A.
(
1992)
In
pursuit
of
accuracy:
Nomenclature,
assumptions,
and
standards.
Pure
&
Applied
Chemistry,
64,
455­
472.

In
this
paper,
among
other
topics,
Currie
briefly
discusses
issues
related
to
terminology,
assumptions,
and
basic
concepts
of
the
decision
and
detection
limit
theory
used
by
the
scientific
community.
False
positive
and
false
negative
errors
and
homogeneity
of
variance
are
among
these
issues.
Annotated
Bibliography
on
Measurement
Limitations
and
Calibration
Updated:
2/
27/
98
5
Currie,
L.
A.
and
Svehla,
G.
(
1994)
Recommendations
for
the
presentation
of
results
of
chemical
analysis.
Pure
&
Applied
Chemistry,
66,
595­
608.

Among
other
general
terminologies
relating
to
the
precision
and
accuracy
of
experimental
results,
Currie
&
Svehla
provide
definitions
of
the
minimum
significant
signal
(
Critical
Level)
and
minimum
detectable
quantity
(
Detection
Limit).
These
definitions
are
equivalent
to
those
used
by
Currie
in
his
earlier
works.
Expressions
for
calculating
the
estimates
are
also
provided.

Currie,
L.
A.
(
1995)
Nomenclature
in
evaluation
of
analytical
methods
including
detection
and
quantification
capabilities.
Pure
&
Applied
Chemistry,
67,
1699­
1723.

Currie
concentrates
mainly
on
establishing
uniform
terminology,
notation,
and
formulation
for
performance
characteristics
of
the
chemical
measurement
process
(
CMP)
in
concurrence
with
ISO
and
IUPAC
scientists.
The
paper
describes
the
fundamental
quantities
related
to
the
observed
response,
calibration
and
the
evaluation
function.
Precision
and
accuracy
related
to
performance
characteristics
of
a
CMP
are
discussed
at
length.
The
latter
part
of
the
paper
concentrates
on
terminology
and
concepts
underlying
detection
and
quantification
capabilities
in
chemical
metrology.
The
author
also
points
out
the
distinction
between
the
sampled
and
target
population
in
an
interlaboratory
environment.
The
following
terminologies
are
recommended
by
the
author.
The
critical
value,
of
the
L
C
appropriate
chemical
variable
represents
the
detection
decision.
The
minimum
detectable
(
true)
value,
of
a
chemical
variable
is
a
measure
of
the
inherent
detection
capability
of
a
CMP.
The
L
D
minimum
quantifiable
(
true)
value
,
,
measures
the
inherent
quantification
capability
of
a
CMP
and
L
Q
L
is
used
as
the
generic
symbol
for
the
quantity
of
interest.
This
is
replaced
by
when
treating
net
S
analyte
signals
and
when
treating
analyte
concentrations
or
amounts.
x
The
statistical
theory
of
hypothesis
testing
forms
the
basis
for
the
determination
of
detection
and
quantification
capabilities.
To
calculate
,
,
and
,
two
kinds
of
errors
(
Type
I
and
Type
L
C
L
D
L
Q
II)
are
considered.
The
Type
I
error
is
committed
when
a
false
alternative
hypothesis
(
analyte
present)
is
accepted
and
the
Type
II
error
is
committed
when
a
false
null
hypothesis
(
analyte
absent)
is
accepted.
The
probability
of
committing
Type
I
and
Type
II
errors
are
usually
fixed
at
0.05
each
(
default
values).
The
decision
"
detected"
or
"
not
detected"
is
made
by
comparison
of
estimated
quantity
with
the
critical
value
(
).
Currie
points
out
that,
with
the
exception
of
some

L
L
C
nonparametric
techniques,
detection
limits
cannot
be
derived
in
the
absence
of
known
or
assumed
distributions.

Currie,
L.
A.
(
1997)
Annotated
Bibliography
on
Measurement
Limitations
and
Calibration
Updated:
2/
27/
98
6
Detection:
International
update,
and
some
emerging
di­
lemmas
involving
calibration,
the
blank,
and
multiple
detection
decisions,
Chemometrics
and
Intelligent
Laboratory
Systems,
37,
151­
181.

Currie
first
presents
a
brief
resume
of
the
history
of
detection
and
quantification
capabilities
of
the
chemical
measurement
process.
Following
that,
he
reviews
key
aspects
of
detection
and
quantification
concepts
and
their
realization,
together
with
open
questions
that
have
not
been
resolved
and
hence
provide
important
opportunities
for
further
research.
Among
those
questions
are
detection
and
quantification
capabilities
in
the
signal
and
concentration
domains,
respectively;
and
the
link
between
calibration
and
detection
and
quantification
limits,
together
with
the
blank­
intercept
dichotomy.
Also
included
are
special
treatments
and
approximations
involving
the
non­
central
t,
and
the
exact
(
nonnormal
distribution
of
the
estimated
concentration.
The
author
also
introduces
issues
and
approaches
to
multiple
independent
and
multivariate
detection
decisions
and
limits,
and
concludes
with
a
glimpse
at
some
challenges
involving
the
multivariate
blank
and
non­
monotonic
calibration
functions.

Davis,
C.
B.
(
1994)
Environmental
regulatory
statistics.
In
Handbook
of
Statistics,
Vol.
12,
G.
P.
Patil
and
C.
R.
Rao,
(
eds.)

A
few
crucial
issues
and
methodologies
have
been
outlined
in
some
detail,
including
in
particular
`
detection
limit'
definition
problems.
Section
5.3
of
the
paper
contains
an
illuminating
discussion
about
the
nature
of
detection
limits
and
the
statistical
issues
associated
with
it.
In
the
author's
opinion,
one
of
the
more
pressing
areas
where
research
is
needed
involves
the
definitions
of
`
detection
limits'
and
other
limits.
Conceptual
frameworks
by
Currie
(
1968),
Glaser
et
al.(
1981),
Clayton
et
al.(
1987),
and
Gibbons
et
al.(
1991,1992)
are
discussed
and
critically
evaluated.
Suggestions
for
practice
and
for
research
are
also
given.

Garner,
F.
C.
and
Robertson,
G.
L.
(
1988)
Evaluation
of
detection
limit
estimators.
Chemometrics
and
Intelligent
Laboratory
Systems,
3,
53­
59.

Garner
&
Robertson
offer
a
general
discussion
on
definitions
and
methods
developed
for
estimating
the
minimum
detectable
level
of
a
measurement
process.
Their
attention
is
mainly
focused
on
the
decision
limit,
which
is
also
known
from
Currie
(
1968)
as
critical
level,
and
the
detection
limit.
The
authors
discuss
assumptions
and
some
characteristics
of
analytical
methods
that
are
typically
employed
in
deriving
measurement
limitations.
Among
the
assumptions
mentioned
in
the
article
are
assumptions
on
distribution,
independence,
and
homogeneity
of
variance
of
the
instrumental
responses.
The
authors
also
point
out
that
at
the
moment
there
is
a
lack
of
models
developed
for
Annotated
Bibliography
on
Measurement
Limitations
and
Calibration
Updated:
2/
27/
98
7
multivariate
instrumental
responses
and
incorporating
detection
limit
estimators.

Gibbons,
R.
D.,
Jarke,
F.
H.,
and
Stoub,
K.
P.
(
1991)
Detection
limits:
For
linear
calibration
curves
with
increasing
variance
and
multiple
future
detection
decisions.
Waste
Testing
and
Quality
Assurance:
ASTM
STP
1075,
D.
Friedman,
Ed.,
American
Society
for
Testing
and
Materials,
Philadelphia,
3,
337­
390.

In
this
paper,
the
authors
propose
a
detection
limit
estimator
which
is
a
generalization
of
the
method
of
Clayton
et
al.
(
1987).
Clayton
and
co­
workers
have
provided
a
test
for
the
null
hypothesis
that
the
concentration
in
the
solution
is
zero
(
i.
e.,
X=
0)
with
fixed
type
I
and
type
II
error
rates.
The
potential
limitations
of
their
method
are:
(
1)
The
estimator
assumes
a
constant
variance
across
the
calibration
line
and,
(
2)
the
estimator
applies
only
to
the
next
single
detection
decision.
The
authors
of
this
paper
propose
a
new
detection
limit
estimator,
which
allows
variance
of
the
response
signal
ratio
(
i.
e.,
peak
area
of
the
analyte
to
peak
area
of
the
internal
standard)
to
be
proportional
to
the
concentration.
The
new
detection
limit
estimator
can
also
provide
multiple
future
detection
decisions;
with
the
new
detection
limit,
99%
of
all
future
detection
decisions
and
not
just
the
next
signal
detection
can
be
made
with
99%
confidence.
The
detection
limit
for
non­
constant
variance
calibration
designs
can
be
obtained
by
using
the
weighted
least
squares
method.
Calibration
data
for
10
volatile
organic
priority
pollutant
compounds
is
used
by
the
authors
to
illustrate
the
new
detection
limit
estimators.

Gibbons,
R.
D.,
Grams,
N.
E.,
Jarke,
F.
H.,
and
Stoub,
K.
P.
(
1992)
Practical
quantitation
limits.
Chemometrics
and
Intelligent
Laboratory
Systems,
12,
225­
235.

Gibbons
et
al.
develop
an
operational
definition
for
the
practical
quantitation
limits
(
PQL)
and
a
corresponding
statistical
methodology
for
obtaining
facility­
specific
PQL
estimates
and
regions
of
confidence.
The
authors
define
the
PQL
as
the
concentration
at
which
the
instrument
response
signal
is
times
its
standard
deviation.
This
definition
is
similar
to
that
used
by
Currie
(
1968)
who
1
0
0
/

defined
the
determination
limit
(
)
as
the
level
of
concentration
"
at
which
a
given
procedure
will
L
Q
be
sufficiently
precise
to
yield
a
satisfactory
quantitative
estimate."
Gibbons
et
al.
suggest
estimating
the
PQL
of
a
given
compound
in
a
given
laboratory
using
calibration
data
obtained
for
a
series
of
concentrations
in
the
range
0
to
2­
5
times
the
hypothesized
PQL.
Least
squares
regression
techniques
are
used
to
fit
the
calibration
line
for
the
relationship
between
actual
concentration
and
instrument
response.
A
variance
stabilizing
transformation
is
proposed
in
cases
when
the
variability
is
not
constant
throughout
the
range
of
the
calibration
function.
The
PQL
is
then
set
to
a
level
of
concentration
corresponding
to
the
response
value
,
which
is
a
solution
of
the
following
x

 
y

 
equation
Annotated
Bibliography
on
Measurement
Limitations
and
Calibration
Updated:
2/
27/
98
8

100

s(

)
,

where
is
the
predicted
response,
and
is
the
estimated
standard
deviation.
Confidence
limits

s(

)
for
are
obtained
to
incorporate
the
uncertainty
in
the
PQL
estimator
due
to
the
fact
that
estimates
x

 
of
standard
deviation
and
calibration
line
parameters
are
based
on
a
finite
sample
of
observations.
The
proposed
statistical
methods
are
illustrated
using
199
calibration
samples
for
10
volatile
organic
priority
pollutant
compounds.

Gibbons,
R.
D.
(
1994)
Statistical
methods
for
groundwater
monitoring.
John
Wiley
&
Sons,
Inc.

Chapter
5
and
6
of
this
book
present
statistical
methods
for
computing
method
detection
limits
(
MDLs)
and
practical
quantitation
limits
(
PQLs).
In
Chapter
5,
the
historical
literature
on
MDLs
is
reviewed,
and
the
strengths
and
weaknesses
of
various
estimators
are
compared
and
contrasted.
Estimators
based
on
the
entire
calibration
function
and
not
a
single
concentration
are
shown
to
be
the
methods
of
choice.
In
Chapter
6,
the
ideas
developed
in
Chapter
5
are
expanded
to
the
case
of
estimating
the
PQL.
It
is
shown
that
while
both
MDLs
and
PQLs
can
be
estimated
from
similar
analytical
data,
the
estimators
have
little
to
do
with
one
another,
the
former
being
a
test
of
the
null
hypothesis
that
the
concentration
is
zero
and
the
latter
being
a
point
estimate
of
analytical
precision.

Gibbons,
R.
D.
(
1995)
Some
statistical
and
conceptual
issues
in
the
detection
of
low­
level
environmental
pollutants.
Environmental
and
Ecological
Statistics,
2,
125­
167.

Traditional
detection
limit
estimators
are
described
and
critically
evaluated.
The
methods
are
categorized
into
single
concentration
design
versus
calibration
design
methodology.
This
work
demonstrates
that
the
most
critical
source
of
bias
in
estimating
limits
of
detection
is
the
effect
of
nonconstant
variance.
The
solution
proposed
is
to
base
limits
of
detection
on
multiple
concentration
calibration
data
and
either
directly
model
association
between
variability
and
concentration
or
use
a
variance
stabilizing
transformation.
The
various
calibration­
based
methods
are
illustrated
using
real
data
and
experimental
design
issues
for
detection
limit
studies
are
discussed.

Gibbons,
R.
D.,
Coleman,
D.
E.,
and
Maddalone,
R.
F.
(
1997)
An
Alternative
Minimum
Level
definition
for
analytical
quantification.
Environ.
Sci.
Technol.,
31,
2071­
2077.
Annotated
Bibliography
on
Measurement
Limitations
and
Calibration
Updated:
2/
27/
98
9
y
C

K
0.95,0.99
s
0

b
0w

K
0.95,0.99
a
0

b
0w,

L
C

y
C

b
0w
b
1w
.

x
Q

t
b
1w
V(
y
Q),

AML

x
Q

ts
yQ
b
1w
.
The
authors
present
a
new
approach
to
determining
the
quantification
limit
in
analytical
chemistry.
The
proposed
procedure
is
as
follows:
First,
Gibbons
et
al.
model
the
relationship
between
the
standard
deviation
(
),
and
the

concentration
(
)
using
calibration
data.
They
consider
the
following
two
models:
(
1)
exponential,
x
;
and
(
2)
the
Rocke­
Lorenzato
model,
.
Weighted
least
squares

(
x)

a
0
exp(
a
1
x)

(
x)

a
0

a
1
x
2
(
WLS)
method
is
then
used
to
obtain
a
calibration
line
,
I=
1,...,
n,
by
regressing
the

y
i

b
0w

b
1w
x
i
measured
concentration
y
on
the
true
concentration
x.
The
limit
for
measured
concentrations
when
the
true
concentration
is
zero
is
given
by
where
is
the
95%
confidence
99%
coverage
one­
sided
tolerance
limit
factor
for
n
K
0.95,0.99
observations.
The
corresponding
critical
level
is
The
standard
deviation
at
the
,
,
is
computed
using
the
previously
fitted
model
for
.
L
C
s
LC

(
x)
Therefore,
is
the
measured
concentration
at
10
times
the
standard
deviation
at
the
.
y
Q

10s
LC

b
0w
L
C
Finally,
the
AML
is
computed
as
the
99%
prediction
limit
where
and
t
is
the
upper
99th
percentile
of
Student's
t­
distribution.
Alternatively
x
Q

(
y
Q

b
0w)/
b
1w
when
n
is
large
enough
(
n>
25),
the
AML
can
be
approximated
by
The
proposed
procedure
is
illustrated
using
U.
S.
EPA
Method
1638
ICPMS
data
for
cadmium
at
mass
114.

Glaser,
J.
A.,
Foerst,
D.
L.,
McKee,
G.
D.,
Quane,
S.
A.
and
Budde,
W.
L.
(
1981)
Trace
analysis
for
wastewaters.
Environmental
Science
&
Technology,
15,
1426­
1435.
Annotated
Bibliography
on
Measurement
Limitations
and
Calibration
Updated:
2/
27/
98
10
MDL

2.681

S
pooled,
The
purpose
for
developing
a
procedure
to
evaluate
detection
limits
was
to
design
a
methodology
not
limited
by
instrumentation
or
analytical
methodology.
For
pragmatic
reasons,
authors
focused
on
an
operational
definition
of
detection
limit.
The
fundamental
difference
between
the
proposed
approach
to
detection
limits
and
former
efforts
is
the
emphasis
on
the
operational
characteristics
of
the
definition.
The
method
detection
limit
is
defined
as
the
minimum
concentration
of
a
substance
that
can
be
identified,
measured,
and
reported
with
99%
confidence
that
the
analyte
concentration
is
greater
than
zero
and
is
determined
from
replicate
analyses
of
a
sample
of
given
matrix
containing
analyte.
The
single
step
procedure
for
determining
MDL
is
based
on
the
analysis
of
seven
samples
of
the
matrix
containing
analyte.
The
standard
deviation,
,
of
the
seven
replicate
measurements
is
S
C
calculated
and
the
MDL
is
computed
as
,
where
is
the
MDL

t(
n

1,1



.99)
S
C
t(
n

1,1



.99)
student's
t­
value
at
99%
confidence
level
with
degrees
of
freedom.
n

1
An
additional
iterative
procedure
is
presented
to
test
the
reasonableness
of
the
MDL
estimate
and
subsequent
MDL
determinations
on
the
same
matrix.
At
each
iteration
of
the
MDL
calculation,
the
variance
from
the
current
MDL
calculation
and
the
preceding
MDL
calculation
are
compare
by
computing
the
F­
ratio,
which
is
compared
with
the
tabulated
F­
ratio,
.
If
the
computed
F.
95(
6,6)

3.05
F­
ratio
is
less
than
3.05,
then
the
pooled
standard
deviation
is
calculated
using
the
standard
deviation
of
the
current
MDL
determination
and
the
preceding
iteration.
The
MDL
is
then
calculated
as
where
2.681
is
equal
to
and
.
Convergence
of
the
iterative
t(
12,1



.99)
S
pooled

(
S
2
A

S
2
B)/
2
procedure
will
depend
on
the
closeness
of
the
estimated
MDL,
or
the
background
level
of
analyte
present
in
the
matrix,
to
the
calculated
MDL.
Both
theory
and
applications
are
discussed
for
reliable
wastewater
analyses
of
priority
pollutants.

Hubaux,
A.
and
Vos,
G.
(
1970)
Decision
and
detection
limits
for
linear
calibration
curves,
Analytical
Chemistry,
42,
849­
855.

Hubaux
and
Vos
were
the
first
to
apply
the
theory
of
statistical
prediction
to
the
problem
of
detection
limit
estimation.
For
linear
calibration
curves,
two
kinds
of
detection
limits
may
be
connected
to
the
notion
of
confidence
limits
around
the
regression
line.
Here,
the
variance
of
the
error
distribution
of
the
signals
around
their
expectation
is
supposed
to
remain
constant.
On
either
side
of
the
regression
line,
two
confidence
limits
may
be
drawn,
with
an
a
priori
chosen
level
of
an
overall
100(
1




)
%
confidence.
The
lowest
measurable
signal
is
defined
as
the
point
at
which
the
upper
y
C
(
1


)
confidence
limit
intersects
the
y­
axis.
This
definition
corresponds
to
the
definition
of
critical
level
L
C
by
L.
A.
Currie.
The
lowest
content
(
or,
"
detection
limit",
in
Currie's
notation)
is
then
defined
as
x
D
the
point
at
which
one
can
have
%
confidence
that
the
measurable
signal
is
greater
than
.
(
1


)
y
C
Annotated
Bibliography
on
Measurement
Limitations
and
Calibration
Updated:
2/
27/
98
11

(
c)


p(
m)
f(
m|
c)
dm.
It
can
be
obtained
graphically
by
locating
the
abscissa
corresponding
to
on
the
lower
confidence
y
C
limit.
Algebraic
solutions
for
and
are
also
provided.
The
influence
of
the
precision
of
the
y
C
x
D
analytical
method,
the
number
of
standards,
the
range
of
their
contents,
the
various
modes
of
their
repartition,
and
the
replication
of
measurements
on
the
unknown
sample
are
studied
from
a
statistical
point
of
view.
Several
practical
rules
such
as
improving
the
precision
of
the
method,
increasing
the
number
of
standards,
increasing
the
range
of
the
contents
of
these
standards,
optimizing
the
repartition
of
these
standards
within
this
range
are
deduced
to
lower
the
measurement
limits.

Ingle,
J.
D.,
Jr.
(
1974)
Sensitivity
and
limit
of
detection
in
quantitative
spectrometric
methods.
Journal
of
Chemical
Education,
51,
100­
105.

Ingle
discusses
some
the
many
existing
mathematical
definitions
of
the
limit
of
detection
which
are
based
on
the
use
of
the
z­
statistic
or
the
t­
statistic
and
are
concerned
with
errors
of
the
first
type
(
false
positive).
Definitions
based
on
other
test
statistics
or
those
concerned
with
errors
of
the
second
type
(
false
negative)
are
stated
to
be
too
complex
for
usual
routine
procedures
and
not
discussed
in
this
article.

Lambert,
D.,
Peterson,
B.
and
Terpenning,
I.
(
1991)
Nondetects,
detection
limits,
and
the
probability
of
detection,
JASA,
86,
266­
277.

When
chemists
cannot
quantify
the
concentration
in
a
sample,
they
report
a
nondetect.
There
are
two
widely
accepted
procedures
dealing
with
nondetects.
One
is
to
assume
that
all
nondetects
are
smaller
than
all
detects.
Another
is
to
assume
that
nondetects
are
below
a
threshold
such
as
the
detection
limit.
This
article
shows
that
these
assumptions
can
be
wrong.
Furthermore,
new
methods
for
describing
which
measurements
were
detectable
in
a
given
study
are
provided.
Two
new
concepts
are
introduced.
The
probability
of
acceptance
curve
shows
how
likely
a
measurement
of
is
to
be
p(
m)
m
accepted
by
chemists
and
reported
as
a
detect.
The
90th
percentile
of
is
called
the
censoring
p(
m)
limit
and
recommended
as
a
bound
for
nondetects.
The
probability
of
detection
curve
describes
how
likely
a
field
concentration
of
is
to

(
c)
c
give
a
measurement
reported
as
a
detect.
Suppose
that
the
measurements
for
field
samples
with
a
true
concentration
of
a
pollutant
c
have
a
density
and
the
probability
that
a
positive
measurement
passes
the
identification
tests
f(

|
c)
m
is
.
Then
p(
m)
Annotated
Bibliography
on
Measurement
Limitations
and
Calibration
Updated:
2/
27/
98
12
Finally,
the
minimum
reliably
detected
concentration
is
defined
as
the
90th
percentile
of
.

(
c)
To
estimate
,
and
therefore,
the
minimum
reliably
detected
concentration,
the
estimates

(
c)
of
and
are
needed.
Local
logistics
regression,
developed
by
Hastie
&
Tibshirani,
is
used
p(
m)
f(

|
c)
to
estimate
the
probability
of
acceptance
curve,
.
Local
logistics
regression
relaxes
the
linearity
p(
m)
assumption
of
traditional
logistic
regression
by
allowing
the
intercept
and
slope
of
the
regression
function
to
vary
over
the
range
of
measurements.
might
be
estimated,
for
example,
from
f(

|
c)
samples
with
known
concentration.
The
statements
are
illustrated
with
examples
from
the
1988
Love
Canal
study.

Liteanu,
C.
and
Rîc

,
I.
(
1980)
Statistical
Theory
and
Methodology
of
Trace
Analysis.
Halsted,
Chicago.

In
this
work,
general
applications
of
mathematical
statistics
in
chemical
analyses
are
extended
by
the
introduction
of
knowledge
on
stability
of
analytical
systems,
and
also
elements
of
the
informational
treatment
of
analytical
measurements.
In
fact,
specific
statistical
problems
in
trace
analysis,
such
as
limits
of
detection
and
determination,
are
treated
in
details.
For
these
purposes,
the
authors
use
the
statistical
theory
of
signal
detection.
In
the
case
of
detection,
the
analytical
measurement
is
translated
into
a
detection
decision
by
means
of
statistical
decision
rules.
In
determination,
the
result
is
obtained
from
the
measurement
by
use
of
the
calibration
function.
The
authors
adopt
a
statistically
based
concept
of
`
detection
limit'
similar
to
one
introduced
by
Currie
(
1968).
In
particular,
they
make
the
following
statement:
the
detection
limit
is
the
particular
value
of
the
amount
of
component
at
which
the
probability
of
correct
detection
evaluated
with
c
d
respect
to
the
decision
threshold
(
defined
on
the
basis
of
a
probability
of
false
detection)
is
equal
y
k
to
a
preselected
value.
The
authors
then
describe
various
methods
for
estimating
the
detection
limit
and
the
statistical
confidence
interval
for
its
true
value
using
a
calibration
approach.
They
also
address
the
problem
of
evaluation
of
the
detection
limit
when
the
standard
deviation
of
the
signals
varies
with
concentration.
They
introduce
a
frequentometric
principle
for
estimating
the
detection
limit
based
on
interpolating
the
probability
of
correct
detection
in
the
detection
characteristics
given
the
probability
of
false
detections
fixed.
Linking
together
the
theory
and
the
problems
of
analytical
practice,
the
authors
discuss
the
detection
capacity
and
detection
limits
of
the
most
frequently
used
methods
of
trace
analysis.

Oppenheimer,
L.,
Capizzi,
T.
P.,
Weppelman,
R.
M.,
and
Mehta,
H.
(
1983)
Determining
the
lowest
limit
of
reliable
assay
measurement.
Analytical
Chemistry,
55,
638­
643.

The
"
lowest
limit
of
reliable
assay
measurement"
is
usually
used
to
evaluate
the
limitations
of
an
assay
and
make
meaningful
comparisons
between
different
assays.
This
paper
describes
the
derivation
of
Annotated
Bibliography
on
Measurement
Limitations
and
Calibration
Updated:
2/
27/
98
13

2

µ
2e
 
2

(
e
 
2

1).
appropriate
expressions
for
obtaining
various
types
of
assay
limits
under
heterogeneous
variances.
These
results
are
illustrated
by
using
data
from
an
assay
developed
to
monitor
tissue
residues
of
Ivermectin,
an
antiparasitic
agent.
The
paper
also
discusses
related
issues
which
also
affect
assay
limit
determinations,
such
as,
the
method
used
to
select
the
variance
weights,
alternative
models
which
transform
the
variance
structure
of
the
responses,
and
experimental
design
considerations
used
in
obtaining
the
standard
curve.
These
issues
are
also
illustrated
by
using
the
tissue
residue
data.
Dramatic
differences
were
observed
by
the
authors
depending
on
whether
a
valid
weighted
or
an
inappropriate
analysis
without
weights
was
used.

Rocke,
D.
M.
and
Lorenzato,
S.
(
1995)
A
two­
component
model
for
measurement
error
in
analytical
chemistry.
Technometrics,
37,
176­
184.

A
new
model
for
measurement
error
in
analytical
chemistry
is
proposed.
The
model
can
be
written
as
,
where
there
are
two
analytical
error,
and
.
Here
represents
x

µ
e
 




N(
0,

2
 )

N(
0,

2
 )

the
proportional
error
that
is
exhibited
at
(
relatively)
high
concentrations
and
represents
the
additive

error
that
is
shown
primarily
at
small
concentrations.
The
standard
deviation
of
the
model
can
be
written
as
This
formula
illustrates
the
key
feature
of
the
new
model
­
when
is
small,
the
error
is
nearly

constant
in
size;
when
is
large,
the
size
of
the
error
is
roughly
proportional
to
.
The
new
model


allows
the
error
at
large
concentrations
to
be
lognormal,
rather
than
normal.
If
viewed
in
terms
of
the
coefficient
of
variation,
the
picture
is
that
large
concentrations
have
a
constant
CV,
whereas
small
concentrations
have
an
increasing
CV
that
tends
to
infinity
as
the
concentration
approaches
0.
The
computational
approach
used
is
based
on
maximum
likelihood
estimation.
Maximizing
the
likelihood
function
of
the
model
leads
to
estimates
of
the
necessary
parameters
,
and
.
Once

 


estimates
and
have
been
derived,
the
precision
of
any
measured
value
can
be
estimated
from

 

the
formula
for
standard
deviation
by
substituting
estimated
values
for
the
parameters.
Discussion
is
provided
on
the
application
of
the
new
model
to
some
common
issues
including
determination
of
detection
limits.
In
fact,
the
new
model
provides
a
reliable
way
to
estimate
the
precision
of
measurements
that
are
near
the
detection
limits
so
that
they
can
be
used
in
inference
and
regulation.

Singh,
A.
(
1993)
Multivariate
decision
and
detection
limits.
Analytica
Chimica
Acta,
277,
205­
214.
Annotated
Bibliography
on
Measurement
Limitations
and
Calibration
Updated:
2/
27/
98
14

(
Y)

Y,
if
Y

c
any
fixed
constant,
if
Y<
c
L(
d,

)

(
1

p)(
d

)
,
if
d

p(

d),
if
d


Principal
component
analysis
is
used
to
develop
approaches
for
estimating
multivariate
decision
and
detection
limits
for
gas
chromatography­
mass
spectrometry
studies
where
the
instrument
response
is
multivariate
in
nature.
When
the
first
principal
component
explains
most
of
the
variation
contained
in
the
data,
it
may
be
used
to
express
the
multicomponent
instrumental
response
as
a
univariate
composite
signal
representing
all
of
the
monitored
ions
associated
with
the
analyte
of
interest.
Least
squares
regression
is
used
to
obtain
a
calibration
function
for
the
composite
instrument
response
onto
the
analyte
concentration.
When
the
original
response
variables
are
Gaussian,
the
principal
components
are
also
Gaussian
as
a
linear
combinations
of
normally
distributed
random
variables.
When
the
original
response
variables
are
not
Gaussian,
Singh
appeals
to
the
central
limit
theorem
to
justify
the
normality
assumption
for
the
principal
components.
The
first
principal
component
is
used
to
derive
decision
and
detection
limits
by
using
two
approaches;
those
of
Currie
(
1968)
and
of
Hubaux
and
Vos
(
1970).
The
scores
for
the
first
principal
component
are
then
transformed
back
to
the
original
responses
to
give
rise
to
multivariate
decision
and
detection
limits.
Calibration
data
of
2,3,7,8­
tetrachlorodibenzo­
p­
dioxin
are
used
to
numerically
illustrate
the
proposed
approaches.

Spiegelman,
C.
H.
(
1997)
A
discussion
of
issues
raised
by
Lloyd
Currie
and
a
cross
disciplinary
view
of
detection
limits
and
estimating
parameters
that
are
often
at
or
near
zero.
Chemometrics
and
Intelligent
Laboratory
Systems,
37,
183­
188.

Spiegelman
applies
methods
of
statistical
decision
theory
to
the
problem
of
estimating
the
detection
limits.
The
definition
of
the
detection
limit
used
by
the
author
is
taken
from
Currie
(
1988).
Let
be
a
measurement
such
that
.
For
some
constant
,
define
the
critical
value
Y
Y

N(
µ
,1)
c
rule
as
follows

(
Y)

The
author
then
proves
that
given
the
mean
squared
error
loss,
the
detection
limit
estimators

(
Y)
are
inadmissible
from
the
decision
theory
point
of
view.
However,
given
an
asymmetric
family
of
loss
functions
of
the
form
where
,
is
a
uniform
density
on
the
interval
[
0,1],
,
and
is
p(

)



(
0)

(
1


)
q(

)
q(

)


(
0,1)

(
0)
the
Dirac
delta
function
with
point
mass
0,
some
estimates
of
the
form
Annotated
Bibliography
on
Measurement
Limitations
and
Calibration
Updated:
2/
27/
98
15

(
Y)

Y,
if
Y

1

(
Y),
if
c

Y<
1
0,
if
0

Y<
c
S
L

S
2
Y

S
2
B
10
.

L
C

t(
18,

)
S
2
Y

S
2
B
10
,
where
and
is
an
increasing
function,
are
admissible
and
also
Bayes
rules.
0


(
Y)

1

(
Y)

Spiegelman,
C.
H.
and
Tarlow,
P.
(
1997)
A
mock
trial
for
critical
values
(
detection
limits).
STATS:
The
Magazine
for
Students
of
Statistics,
20,
13­
16.

Based
on
Currie's
definition
of
the
critical
value
(
see
Currie
(
1995)),
,
the
authors
provide
an
L
C
example
of
how
this
definition
could
be
used
in
practice.
They
denote
the
means
and
standard
deviations
of
ten
background
and
ten
chemical
sample
measurements
as
,
;
,
¯
B
S
B/
10
¯
Y
S
Y/
10
respectively.
The
measurement
is
and
its
estimated
standard
error
is

L

¯
Y

¯
B
Assuming
homogeneity
of
variance
for
background
and
chemical
sample,
has
18
degrees
of
S
L
freedom.
The
critical
value
(
or
detection
limit)
is
then
defined
as
where
is
the
upper
­
quantile
of
a
t
distribution
with
df
degrees
of
freedom.
The
authors
t(
df,

)

say
that
is
above
the
detection
limit
if
and
only
if
.

Y
¯
Y

¯
B

L
C
The
rest
of
the
article
is
dedicated
to
the
discussion
on
whether
actual
values
or
zeroes
should
be
reported
as
the
result
of
an
experiment
given
that
the
raw
measurements
are
below
the
detection
limit.

Zorn,
M.
E.,
Gibbons,
R.
D.,
and
Sonzogni,
W.
C.
(
1997)
Weighted
least­
squares
approach
to
calculating
limits
of
detection
and
quantification
by
modeling
variability
as
a
function
of
concentration.
Analytical
Chemistry,
69,
3069­
3075.

Zorn
et
al.
use
a
weighted
least­
squares
regression
analysis
of
replicates
spiked
at
a
series
of
Annotated
Bibliography
on
Measurement
Limitations
and
Calibration
Updated:
2/
27/
98
16
concentrations
for
estimating
detection
and
quantification
limits.
In
addition,
models
for
estimating
the
weights
used
in
calculating
weighted
prediction
intervals
and
weighted
tolerance
intervals
are
presented.
To
calculate
detection
and
quantification
limits
using
weighted
prediction
and
tolerance
intervals,
the
weight
at
zero
(
),
the
weight
at
the
limit
of
detection
(
),
and
the
weight
at
limit
w
0
w
LD
of
determination
(
)
must
be
estimated.
In
this
study,
the
weights
have
been
set
equal
to
the
w
LQ
inverse
variance,
.
Therefore,
can
be
estimated
by
modeling
the
standard
deviation
(
or
s
i

2
w
j
variance)
as
a
function
of
concentration
x.
The
following
models
of
standard
deviation
were
s
x
examined:
a
quadratic
model,
;
an
exponential
model,
;
and
a
two­
s
x

a
0

a
1
x

a
2
x
2
s
x

a
0
e
a1x
component
model
proposed
by
Rocke
and
Lorenzato
(
1995),
.
In
presented
examples,
s
x

a
0

a
1
x
2
the
quadratic
model
provides
the
best
overall
fit
to
the
data.

Annotated
bibliography
of
calibration
and
related
regression
techniques.

Berkson,
J.
(
1969)
Estimation
of
a
linear
function
for
a
calibration
line;
consideration
of
a
recent
proposal.
Technometrics,
11,
649­
660.

Berkson
briefly
reviews
the
results
of
the
Krutchkoff
(
1967)
article.
The
remainder
of
the
paper
is
dedicated
to
pointing
out
that
the
inverse
calibration
method
is
inconsistent,
whereas
the
classical
calibration
method
is
consistent.
Since
the
work
of
Krutchkoff
(
1967)
was
based
on
relatively
small
sample
sizes,
Berkson
is
interested
in
how
these
methods
perform
for
large
sample
sizes
(
i.
e.,
).
N


Controlling
the
calibration
range
as
in
Krutchkoff's
article,
the
ratio
of
mean
squared
errors
were
determined
for
both
and
.
For
this,
asymptotic
formulas
were
used
to
determine
N

6
N


the
asymptotic
variance
for
each
of
the
calibration
methods.
For
,
the
ratios
of
the
mean
X

1.2
squares
are
larger
than
unity
for
both
samples
sizes
state
previously.
However,
for
large
samples
(
)
and
,
the
mean
squared
errors
for
the
classical
calibration
approach
are
smaller
than
N


X>
1.2
those
found
using
the
inverse
calibration
approach.
Berkson
notes
that
the
asymptotic
formulas
for
variance
are
accurate
enough
for
and
N

20
states
that
since
the
m.
s.
e.
is
not
smaller
for
the
inverse
estimator
for
all
values
of
,
it
undermines
X
the
basis
of
Krutchkoff's
argument
that
the
inverse
calibration
method
is
superior
to
the
classical
calibration
method.
Also
noted
is
the
fact
that
the
inverse
estimator,
though
inconsistent,
shows
smaller
m.
s.
e.
than
the
classical
estimator
in
a
large
region
of
the
scale
around
the
mean
of
.
An
X
example
using
the
relationship
between
Farenheit
and
Centigrade
with
both
the
calibration
methods
is
used
to
show
how
the
inverse
calibration
method's
inconsistent
results
depends
on
.
The
author
X
completes
his
paper
by
stating
that
the
m.
s.
e.
of
the
classical
calibration
estimator
will
actually
be
smaller
than
the
inverse
calibration
method
estimator
when
(
1)
sufficient
number
of
samples
are
take
Annotated
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Measurement
Limitations
and
Calibration
Updated:
2/
27/
98
17

2(
x)


2(
m

1

n

1

(
x

¯
X
)
2T

1)


2
1
.
to
determine
the
calibration
line
and
(
2)
the
calibration
range
is
restricted
to
[
0,1]
as
in
Krutchkoff's
article.

Buonaccorsi,
J.
P.
(
1986)
Design
considerations
for
calibration.
Technometrics,
28,
149­
155.

Buonaccorsi
investigates
the
effects
of
the
choice
of
design
points
on
calibration
(
X
1,
X
2,

,
X
n)
in
the
presense
of
a
linear
regression
model,
,
where
X's
are
under
the
Y
i


0


1
X
i


i
control
of
experimenter
and
.
It
is
assumed
that
,
belong
to
the

i

iid(
0,

2)
X
i,
i

1,

,
n
calibration
interval
,
.
As
an
approach
to
finding
optimal
design
the
author
considers
[
a,
b]
a<
b
minimization
of
the
asymptotic
variance
Several
criteria
are
considered
by
the
author,
which
include
(
i)
optimality
that
minimizes
V
x
;
(
ii)
AV
optimality
that
minimizes
an
average
asymptotic
variance
with
respect
to
some

2(
x)
prior
distribution
for
;
and
(
iii)
M
optimality
that
minimizes
the
maximum
asymptotic
variance
x
over
certain
values
of
.
If
,
then
an
optimal
design
by
criteria
is
one
so
that
the
x
x

(
a,
b)
V
x
mean
of
two
calibration
points
is
the
prior
"
guess"
of
.
By
AV
criteria,
an
optimim
design
is
an
x
endpoint
design
so
that
the
mean
of
the
prior
distribution
is
.
M
optimal
when
the
(
a

b)/
2
maximum
variance
is
considered
over
an
interval
symmetric
with
respect
to
.
In
summary,
[
a,
b]
Buonaccorsi
suggests
that
if
one
has
a
prior
guess
at
,
say
then
choose
so
that
it
is
x
x
0
[
a,
b]
symmetric
about
and
to
make
as
wide
as
possible.
Then
take
n
1
observations
at
,
n
2
x
0
[
a,
b]
a
observations
at
and
n
3
observations
at
,
so
that
n
1
=
n
2
and
n
1
+
n
2
+
n
3
=
n.
b
(
a

b)/
2
Using
accuracy
curves,
the
probability
that
a
confidence
region
contains
when
the
true
x
0
value
is
,
to
develop
optimality
criteria,
Buonaccorsi
concludes
that
the
equally
weighted
end­
x
point
design
is
optimal
for
extreme
values
of
.
However,
it
is
generally
known
a
priori
that
x
x
will
fall
in
some
finite
interval.
Thus,
Buonaccorsi
suggests
that
a
confidence
region
be
truncated
over
that
finite
range
of
.
x
Eberhardt,
K.
R.
and
Mee,
R.
W.
(
1994)
Constant­
width
calibration
intervals
for
linear
regression.
Journal
of
Quality
Technology,
26,
21­
29
Eberhardt
and
Mee
develop
constant­
width
simultaneous
tolerance
intervals
for
which
the
bound
on
the
nominal
coverage
probabilities
is
exact
under
normality.
Their
method
allows
one
to
control
the
Annotated
Bibliography
on
Measurement
Limitations
and
Calibration
Updated:
2/
27/
98
18

X

Y

a
b
,
(
conditional)
probability
that
a
confidence
interval
captures
the
true
value
of
X
to
be
greater
or
equal
to
any
chosen
value
P,
provided
that
the
calibration
data
set
belongs
to
a
set
G
of
"
good"
outcomes.
The
set
G
is
implicitly
defined
in
the
procedure
when
the
user
chooses
an
appropriate
high
probability
that
the
calibration
data
set
will
belong
to
G.
The
proposed
method
is
a
modification
of
the

procedure
of
Mee,
Eberhardt,
and
Reeve
(
1991).
Constant­
width
intervals
have
a
"
minimax"
appeal
in
that
they
are
narrower
than
the
maximum
width
obtained
by
the
other
calibration
interval
methods
having
this
same
interpretation.

Eisenhart,
C.
(
1939)
The
interpretation
of
certain
regression
methods
and
their
use
in
biological
and
industrial
research.
The
Annals
of
Mathematical
Statistics,
10,
162­
186.

Eisenhart's
work
is
one
of
the
first
publications
in
statistical
literature
on
regression
analysis
discussing
the
problem
of
predicting
X
corresponding
to
a
future
value
of
Y,
which
is
known
as
statistical
calibration
problem.
This
paper
presents
a
review
of
some
of
the
ideas
involved
in
curve
fitting
practices
related
to
the
problem
of
estimating
from
.
The
author
considers
some
of
the
X
Y
instances,
when
it
is
necessary
to
follow
curve
fitting
practices
with
as
the
dependent
variable
and
Y
then
using
the
inverse
of
the
relation
found
to
get
an
estimate
of
.
The
paper
also
points
out
the
X
types
of
problems
to
which
this
method
of
inverse
regression
provides
a
solution
and
the
confidence
interval
nature
of
the
estimates
it
gives.
Eisenhart
briefly
discusses
effects
of
limiting
the
ranges
of
both
independent
and
dependent
variables
in
the
sampling
process.
The
method
of
inverse
regression
is
demonstrated
by
working
out
a
problem
arising
in
the
manufacture
of
cheese,
and
a
problem
concerned
with
the
biological
assay
of
a
hormone
substance.

Krutchkoff,
R.
G.
(
1967)
Classical
and
inverse
regression
methods
of
calibration.
Technometrics,
9,
425­
439.

Krutchkoff
considers
the
classical
and
inverse
methods
of
the
statistical
calibration
problem
and
compares
them
by
Monte
Carlo
simulations.
Suppose
that
the
relationship
between
a
controlled
variable
x
and
a
measured
variable
y
can
be
written
as
,
where
and
obtained
by
the
y




x




ordinary
least
squares
method.
The
classical
estimate
of
an
unknown
X
corresponding
to
a
future
reading
Y
is
where
Annotated
Bibliography
on
Measurement
Limitations
and
Calibration
Updated:
2/
27/
98
19
b


(
x
i

¯
x)
(
y
i

¯
y)

(
x
i

¯
x)
2
and
a

¯
y

b
¯
x
.

X

c

dY
,

d


(
x
i

¯
x)
(
y
i

¯
y)

(
y
i

¯
y)
2
and
c

¯
x

d
¯
y
.
In
the
inverse
approach,
Krutchkoff
considers
the
regression
of
x
on
y
rather
than
y
on
x,
which
leads
to
a
different
estimate
of
the
unknown
X,

where
The
classical
approach
is
then
compared
with
the
inverse
approach
by
Monte
Carlo
simulations.
Krutchkoff
considers
an
interval
[
0,1]
as
the
calibration
range
for
x
and
designs
the
calibration
experiment
so
that
there
are
three
observations
at
each
of
the
end
points
(
x=
0,
x=
1).
The
experiment
is
repeated
10,000
times.
Based
on
the
results
of
the
simulation
study
the
author
concludes
that
the
inverse
estimation
has
a
uniformly
smaller
mean
squared
error
than
the
classical
estimator.

Krutchkoff,
R.
G.
(
1969)
Classical
and
inverse
regression
methods
of
calibration
in
extrapolation.
Technometrics,
11,
605­
608.

The
author,
in
an
earlier
paper,
compared
the
inverse
method
with
the
calibration
method
by
Monte
Carlo
simulations
and
found
that
the
inverse
method
has
a
uniformly
smaller
average
squared
error
in
the
range
of
controlled
variable.
In
this
paper,
the
author
extends
his
previous
results
by
including
extra
points
outside
the
calibration
range.
Author
also
points
out
that
results
quoted
for
X=
5
and
X=
10
in
the
original
paper
were
incorrectly
labeled
and
were
independent
repetitions
of
X=
2
column.
Based
on
the
results
of
the
new
simulation
study,
Krutckoff
demonstrates
that,
for
a
sufficiently
large
number
of
observations,
the
classical
method
gives
a
smaller
average
squared
error
outside
the
range
of
the
calibration,
however
the
conclusions
drawn
in
the
previous
paper
remain
unchanged
for
X
values
within
the
calibration
range.

Lechner,
J.
A.,
Reeve,
C.
P.
and
Spiegelman,
C.
H.
(
1982)
An
implementation
of
the
Scheffé
approach
to
calibration
using
spline
functions,
illustrated
by
a
Annotated
Bibliography
on
Measurement
Limitations
and
Calibration
Updated:
2/
27/
98
20
p

p(
v)


g
1(
v),

0
<
v<

1

g
2(
v),

1
<
v<

2

g
k

1
(
v),

k
<
v<

k

1
,

v


h
0
A(
x)
dx,
pressure­
volume
calibration.
Technometrics,
24,
229­
234.

This
paper
presents
a
Scheffé­
type
calibration
procedure
to
the
pressure­
volume
relationship
for
nuclear­
materials
processing
tanks.
Piecewise
polynomials
(
splines)
are
used
to
fit
this
relationship
for
nuclear­
materials
processing
tanks.
Since
differential
pressure
can
be
simply
and
quickly
measured,
the
relationship
between
volume
and
pressure
is
used
to
estimate
the
volume
indirectly
by
measuring
the
pressure.
The
settings
for
the
calibration
experiment
are
as
follows.
The
nuclearmaterials
tanks
are
considered
to
be
composed
of
segments
for
which
an
idealized
model
could
be
considered
as
a
good
representation.
The
authors
assume
that
a
tank
is
composed
of
distinct
k

1
and
known
regions,
where
the
idealized
relationship
between
the
two
variables
pressure,
and
p
volume,
is
given
by
v
where
is
a
polynomial
function
of
.
At
height
the
volume
in
the
container
is
g
i(
v)
v
h
where
A(
x)
is
the
cross
sectional
area
at
height
.
When
liquid
height
is
,
where
is
the
x
hp

h

g,

density
of
the
homogeneous
liquid
and
is
the
acceleration
due
to
gravity.
In
the
areas
of
the
tank
g
where
is
constant,
the
volume­
pressure
is
a
straight
line.
Hence
A(
x)

p

g
i(
v)


i


i
v,
for

i

1
<
v<

i
,
i

1,......,
k

1.

The
authors
assume
that
each
pressure
measurement
has
a
random
error
associated
with
it
and
the
errors
are
independently
and
normally
distributed,
with
mean
zero
and
constant
variance
The

2.
calibration
experiments
obtain
estimates
of
and
,
and
these
coefficient
estimates
are
then
used

i

i
to
obtain
the
estimates
of
the
volume
of
the
tank,
using
the
inverse
of
the
relationship
given
in
(
1).
The
authors
also
obtain
the
interval
estimates
of
volume
for
measured
values
of
pressure.

Mee,
R.
W.
and
Eberhardt,
K.
R.
(
1996)
A
comparison
of
uncertainty
criteria
for
calibration.
Technometrics,
38,
221­
229.
Annotated
Bibliography
on
Measurement
Limitations
and
Calibration
Updated:
2/
27/
98
21
Mee
and
Eberhardt
briefly
review
the
literature
for
constructing
confidence
intervals
for
multiple­
use
calibration,
in
which
a
single
fitted
regression
line
is
used
repeatedly
to
estimate
many
x's.
In
addition,
they
derive
probability
expressions
for
computing
exact
prediction
intervals
that
enable
the
construction
of
tighter
limits
than
are
currently
available
based
on
that
criterion.
As
an
example,
the
authors
examine
the
performance
of
simultaneous
prediction
intervals,
simultaneous
tolerance
intervals,
and
tolerance
intervals
procedures
using
the
context
of
a
linewidth
data.
The
widths
of
10
reference
lines
were
used
to
calibrate
optical
imaging
systems
measuring
features
in
the
size
range
0.5
to
12
µ
m
on
photomasks
for
integrated
circuits.
The
range
of
the
10
certified
linewidths
on
the
standard
is
0.74
to
10.56
µ
m.
The
authors
simulate
estimates
corresponding
to
10,000
samples
at
x=
0.0,
0.1,
0.2,
...
,
12.9
and
provide
summary
statistics
for
each
procedure.

Naes,
T.
(
1985)
Multivariate
calibration
when
the
error
covariance
matrix
is
structured.
Technometrics,
27,
301­
311.

In
this
paper,
multivariate
calibration
in
linear
models
is
considered.
The
author
assumes
the
error
covariance
matrix
to
have
a
linear
factor
structure.
Parameters
of
the
distribution
of
the
variable
to
be
predicted
(
)
is
estimated
from
the
s
in
the
calibration
set
and
the
distribution
of
is
x
x

x
incorporated
in
the
predictor.
The
approach
to
calibration
in
this
paper
involves
estimating
the
parameters
of
the
best
linear
predictor
in
the
assumed
model.
The
author
compares
his
approach
with
the
ordinary
multiple
linear
regression
approach
to
natural
calibration.
The
predictor
is
tested
on
two
(
meat
and
fish)
data
sets.
The
first
data
set
contains
observations
on
60
samples
of
meat
and
for
each
sample
water
concentration
is
determined
in
a
laboratory
and
the
spectra
are
obtained
on
NIR
spectrophotometer.
The
data
set
is
divided
randomly
into
two
subsets,
called
A
and
B,
each
containing
30
samples.
The
data
set
A
was
first
used
for
calibration,
and
the
constructed
predictor
was
tested
on
the
set
B.
Afterwards
the
two
sets
were
used
in
opposite
order.
Similar
technique
was
used
for
the
second
data
set
(
fish
data).
Computations
on
NIR
data
sets
show
that
the
linear
factor
structure
may
be
adequate
and
may
give
improved
predictions.

Naszódi,
L.
J.
(
1978)
Elimination
of
the
bias
in
the
course
of
calibration.
Technometrics,
20,
201­
205.

Naszódi
considers
the
exact
relation
between
and
to
be
,
,
where
is
the
x
y
y

p
1(
x

¯
x)

p
2
x

H
H
domain
of
.
The
unknown
value
of
an
object
is
estimated
with
the
help
of
a
measured
value
x
x
of
the
object
and
of
the
fitted
calibration
line
by
inserting
the
value
to
the
y
m

y


p
1(
x

¯
x)


p
2
y
m
inversion
of
the
function
,

y
Annotated
Bibliography
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Measurement
Limitations
and
Calibration
Updated:
2/
27/
98
22

x

¯
x

(
y
m

p
2)

p
1
.

1
b

a

b
a
|
T(
x)|
dx

>
min
xi
.

a

x
1

x
2



x
n/
2,
x
n/
2

1

x
n/
2

2

x
n

b
The
author
shows
that
this
point
estimate
is
biased
even
if
is
an
unbiased
estimate
of
.
A
new

y
y
estimate
is
suggested
which
can
be
used
with
a
slight
restriction
for
the
distribution
of
the

x
errors.
The
new
is
practically
unbiased
and
can
easily
be
derived
from
.
Naszódi
then

x

x
considers
a
problem
of
finding
an
optimal
experimental
design
by
minimizing
the
average
of
absolute
value
of
the
bias
of
on
the
calibration
domain
,
i.
e.
x
H

[
a,
b]

If
is
even,
then
the
solution
to
this
problem
is
the
end
point
plan
n
and
the
reached
minimum
is
.
If
instead
of
minimizing
the
average
of
the
absolute

2/(
np
2
1
(
b

a))
value
of
the
bias
one
considers
important
the
optimum
criterion
that
,
one
max
x

H
|
T(
x)|

>
min
xi
obtains
the
same
plan.
Simulations
with
equidistant
and
optimal
calibration
designs
were
performed
to
compare
the
classical
and
inverse
estimates
to
the
new
estimate
from
the
viewpoint
of
bias.
The
simulation
study
shows
that
the
new
estimate
is
more
efficient
then
the
classical
estimate
and
has
the

x

x
advantage
of
consistency
over
the
inverse
estimate
.

x
Ott,
R.
L.
and
Myers,
R.
H.
(
1968)
Optimal
experimental
designs
for
estimating
the
independent
variable
in
regression.
Technometrics,
10,
811­
823.

The
problem
of
optimal
calibration
design
is
studied
under
correct
classification
and
misclassification
of
a
regression
model.
The
design
criterion
considered
by
Ott
and
Myers
is
minimization
of
the
integral
of
over
the
calibration
range
of
x
with
respect
to
design.
It
is
assumed
that
the
E(

x

x)
2
calibration
range
of
interest
is
of
finite
length.
Therefore,
without
loss
of
generality,
the
authors
consider
that
of
.
They
evaluate
the
optimal
calibration
design
for
linear
estimation
when

1

x

1
the
function
is
truly
linear
and
also
when
the
function
is
truly
quadratic.
They
found
that
when
the
function
is
linear,
regardless
of
whether
x
or
y
is
estimated,
that
the
optimal
calibration
is
that
half
Annotated
Bibliography
on
Measurement
Limitations
and
Calibration
Updated:
2/
27/
98
23
of
the
calibration
observations
are
at
the
lower
end
of
the
calibration
range
of
x
and
half
of
the
observations
are
at
the
upper
end
of
the
calibration
range.
If
an
odd
number
of
samples
is
used
for
calibration,
one
observation
should
be
at
the
mid­
point
of
the
range
and
the
other
observations
should
be
evenly
split
between
the
two
end­
points.
In
the
case
where
the
function
is
quadratic
in
nature,
but
a
linear
approximation
is
used
for
calibration,
the
problem
is
more
complex.
Thus,
if
a
quadratic
function
of
the
form
is
approximated
by
,
the
optimal
calibration
y

c
0

c
1
P
1(
x)

c
2
P
2(
x)


y

a

bx
design
is
found
to
be
a
function
of
,
where
and
involves
solving
an
equation
f

|


/
c
2|


2

2

2
n
of
the
fourth
order.
Numerical
solutions
for
various
sets
of
parameters
were
obtained
by
the
authors
in
this
case.

Scheffé,
H.
(
1973)
A
statistical
theory
of
calibration.
The
Annals
of
Statistics,
1,
1­
37.

Scheffé
considers
two
related
quantities
U
and
V
such
that
U
is
relatively
easy
to
measure
and
V
relatively
difficult,
requiring
more
effort
or
expense.
If
a
measurement
u
of
U
is
made
when
V
has
the
value
v,
it
is
assumed
for
v
in
an
interval
that
u
is
a
normal
random
variable
with
mean
[
v
(
1),
v
(
2)]
and
variance
,
where
the
{
}
are
unknown
parameters
and
{
}
are
m(
v,

)


p
j

1

j
g
j(
v)

2

j
g
j(
v)
known
functions.
In
a
calibration
experiment,
measurements
are
made
at
known
n
U
1,
U
2,

,
U
n
values
.
Let
,
.
The
closed
interval
V
1,
V
2,

,
V
n
v
(
1)

min(
V
i,
i

1,

,
n)
v
(
2)

max(
V
i,
i

1,

,
n)
is
called
the
calibration
interval
of
;
it
is
a
known
interval,
assumed
to
be
T

[
v
(
1),
v
(
2)]
V
nondegenerate.
The
author
makes
no
assumption
about
the
"
fine
structure"
of
the
distribution
of
u
i
for
outside
the
calibration
interval
T.
The
estimates
of
the
{
}
are
calculated
from
the
calibration
v
i

j
experiment
by
the
method
of
least
squares.
After
the
calibration
experiment,
measurements
of
U
are
made
at
unknown
values
of
u
i
v
i
V,
for
,
and
it
is
assumed
that
the
random
variables
are
independent.
For
i

1,2,

U
1,
U
2,

,
U
n,
u
1,
u
2,

each
measurement
,
a
lower
limit,
a
point
estimate,
and
an
upper
limit
are
made
for
the
u
i
corresponding
unknown
.
Associated
with
these
intervals
are
two
probabilities:
and
.
By
the
v
i


Scheffé
procedure,
the
probability
is
that
at
least
of
the
intervals
calculated
will
contain
1


1


the
true
value
of
.
v
i
The
procedure
for
finding
the
interval
estimates
is
as
follows.
First,
the
numerical
values
of
the
least­
squares
estimates
{
},
the
estimate
,
and
the
coefficients
{
}
are
calculated,
where

j


2
b
jk
b
jk
is
the
covariance
between
and
divided
by
.
The
calibration
chart
is
constructed
by
plotting

j

k

2
the
calibration
curve
,
the
lower
calibration
curve
,
and
the
upper
u

m(
v,

)
u

m(
v,

)


[
c
1

c
2
S(
v)]
calibration
curve
.
The
function
S(
v)
is
defined
as
for
u

m(
v,

)



[
c
1

c
2
S(
v)]
S(
v)

STD(
m(
v,

)
)/

.
Constants
and
are
functions
of
.
Corresponding
to
v

[
v
(
1),
v
(
2)]
c
1
c
2

,

,
p,
minS(
v),
maxS(
v)
an
observation
,
falling
in
the
calibration
interval
of
U,
a
point
estimate
of
may
be
read
as
the
u
i

v
i
v
i
ordinate
of
the
calibration
curve
at
.
In
this
case
the
confidence
interval
for
is
determined
by
u

u
i
v
i
the
ordinates
read
from
the
lower
and
upper
calibration
curves
at
.
u

u
i
As
an
illustration
Scheffé
considers
how
general
method
applies
in
the
case
of
linear
Annotated
Bibliography
on
Measurement
Limitations
and
Calibration
Updated:
2/
27/
98
24
regression
with
p=
2.
In
this
case
the
author
present
an
explicit
solution
of
the
calibration
curve
equation
without
resorting
to
numerical
or
graphical
methods.
In
fact,
the
upper
and
lower
calibration
curves
are
found
to
be
the
two
branches
of
hyperbolas.
The
upper
and
lower
calibration
curves
will
be
closest
to
the
calibration
line
if
has
its
minimum,
which
is
attained
for
even
n
if
half
1/

(
V
i

¯
V)
2
the
observations
in
the
calibration
experiment
are
taken
at
the
value
and
half
at
.
However,
v
(
1)
v
(
2)

Scheffé
does
not
recommend
using
this
calibration
design,
but
rather
a
more
or
less
uniform
distribution
of
the
within
the
calibration
range
.
V
i
[
v
(
1),
v
(
2)]

Shukla,
G.
K.
(
1972)
On
the
problem
of
calibration.
Technometrics,
14,
547­
553.

Shukla
compares
Krutchkoff's
(
1967)
classical
and
inverse
estimators
using
MSE's
when
the
number
of
calibration
samples
(
n)
is
large
and
the
number
of
replicates
of
the
unknown
quantity
(
n')
is
a
positive
integer.
Shukla
shows
that
the
classical
estimator
is
consistent,
but
the
inverse
estimator
is
not.
He
demonstrated
that
when
n'=
1
and
n>=
8,
the
MSE
of
the
inverse
estimator
is
always
less
than
the
classical
estimator.
When
the
number
of
observations
used
in
calibration
is
small,
the
inverse
estimator
yields
a
smaller
MSE.
However,
the
inverse
estimator
is
unlikely
to
be
advantageous
when
a
large
number
of
observations
are
used
in
calibration
and
n'
is
also
large.
Quite
often,
only
one
observation
can
be
made
on
the
unknown
.
In
that
situation,
the
inverse
estimator
is
preferable
X
0
when
lies
close
to
the
mean
of
the
known
X's.
Shukla
suggests
that
due
to
its
consistency,
the
X
0
classical
estimator
be
used
with
large
sample
sizes
when
there
is
no
prior
information
about
the
unknown
X.
Shukla
mentions
that
the
optimum
calibration
design
for
estimating
parameters
and
prediction
is
the
end
point
design
such
that
a<
X<
b
and
where
half
of
the
observations
are
taken
at
a
and
half
of
the
observations
are
taken
at
b.
He
mentions
that
the
endpoint
design
is
optimum
for
classical
estimation,
but
it
is
not
obvious
if
it
is
optimum
for
the
inverse
estimation.

Spiegelman,
C.
H.
and
Studden,
W.
J.
(
1980)
Design
aspects
of
Scheffe's
calibration
theory
using
linear
splines.
Journal
of
Research
of
the
National
Bureau
of
Standards,
85,
295­
304.

Spiegelman
and
Studden
discuss
the
design
of
the
calibration
experiment
in
the
context
of
Scheffe's
approach
(
1973)
to
the
uncertainties
of
a
calibration
curve
and
in
particular
for
the
case
in
which
the
calibration
curve
is
a
linear
spline.
Their
goal
is
to
minimize
the
measuremnt
error
caused
by
calibration
by
using
splines
to
determine
a
separate
calibration
curve
for
certain
intervals.
As
an
example
the
authors
consider
the
design
problem
for
the
pressure­
volume
tank
calibration
curve
based
on
splines
(
see
Lechner,
Reeve,
and
Spiegelman
(
1982)
for
details.)
The
pressure­
volume
relationship
can
be
represented
as
a
spline
Annotated
Bibliography
on
Measurement
Limitations
and
Calibration
Updated:
2/
27/
98
25
m(
v)


k

1
i

0

i
N
i(
v)
,

N
0(
v)


1

v

1


0
,

0

v
<

1
0,

1

v
N
i(
v)

v


i

1

i


i

1
,

i

1

v
<

i

i

1

v

i

1


i
,

i

v
<

i

1
0,
elsewhere
i

1,2,

,
k
N
k

1
(
v)

v


k

k

1


k
,

k

v
<

k

1
0,
v
>

k

1
where
The
quantities
are
called
"
knots"
and
these
will
occur
wherever
the
cross­
sectional

0
<

1
<

<

k

1
area
changes.
The
design
problem
is
to
see
what
quantative
statements
can
be
made
about
where
the
calibration
points
should
be
chosen
to
minimize
the
width
of
the
confidence
intervals
v
1,
v
2,

,
v
n
.
A
class
of
appropriate
designs
is
given,
which
depend
on
the
location
of
the
knots
and
the
I(
u)
slopes
of
the
segments.
Based
on
these
results,
a
calibration
plan
is
suggested,
which
incorporates
a
two
phase
procedure
for
collecting
observations.
A
numerical
example
is
presented
in
which
the
authors
show
that
in
the
area
where
the
knots
are
concentrated
the
proposed
design
is
superior
to
the
equally
spaced
calibration
design.

Thomas,
M.
A.
and
Myers,
R.
H.
(
1973)
Optimal
designs
for
the
inverse
regression
method
of
calibration.
Communications
in
Statistics
2,
Annotated
Bibliography
on
Measurement
Limitations
and
Calibration
Updated:
2/
27/
98
26
1
n

x
2
i

1/
3
S
Q


(
x
2
i

¯
x
2)

n
10
.
419­
433
Whereas
Ott
and
Myers
look
for
optimal
calibration
designs
using
the
"
classical"
estimator,
Thomas
and
Myers
look
at
optimal
designs
using
the
inverse
estimator
and
an
average
mean
square
error
criteria.
Designs
are
developed
for
a
linear
approximation
when
the
true
model
is
linear
and
when
it
is
quadratic.
Let
first
the
relation
between
and
be
linear
and
given
by
,
where
x
y
y




x


.
They
found
that
the
optimal
designs
of
the
linear
relationship
which
exist
rely
on


N(
0,

2)
n
and
the
unknown
parameters,
and
and
are
not
realistically
usable.


When
evaluating
calibration
designs
when
the
relationship
between
and
is
truly
x
y
quadratic
(
),
optimal
designs
rely
on
and
unknown
parameters,
,
,
.
y


0


1
x


2
x
2


n

1

2

However,
using
methods
of
numerical
integration
and
minimization,
they
found
nearly
optimal
designs
when
the
following
conditions
were
satisfied:

where
is
limited
to
the
calibration
range
of
[­
1,
1]
and
when
x
There
is
no
optimal
two­
point
design
for
a
quadratic
relationship
using
linear
calibration
and
a
three­
point
design
is
optimal
only
when
is
a
multiple
of
19.
Nearly
optimal
designs
were
found
n
for
four­
and
five­
point
designs
when
the
selection
of
the
calibration
points
are
symmetric
about
the
center
and
the
number
of
observations
are
also
symmetric
about
the
center.
Using
the
range
of
as
[­
1,
1],
0
is
the
central
value
(
),
and
refers
to
the
number
of
observations
at
0.
x
x
0
n
0
through
refer
to
the
number
of
observations
at
,
respectively,
such
that
,
n
1
n
4
x
1

x
4
x
1


x
4
,
,
and
.
Thomas
and
Myers
found
nearly
optimal
design
when
is
x
2


x
3
n
1

n
4
n
2

n
3
2
(
n
3

n
4)/
n
greater
than
or
equal
to
0.526.
They
went
on
to
explore
six
different
designs
when
and
n

10
determined
the
number
of
observations
at
each
value
of
and
what
the
optimum
values
of
x
x
were.
In
summary,
since
optimum
calibration
designs
rely
on
unknown
parameters
for
underlying
linear
or
quadratic
relationships,
they
suggest
using
the
near
optimum
designs
found
when
the
relationship
was
truly
quadratic.

Vecchia,
D.
F.,
Iyer,
H.
K.
and
Chapman,
P.
L.
(
1989)
Calibration
with
randomly
changing
standard
curves.
Technometrics,
31,
83­
90.
Annotated
Bibliography
on
Measurement
Limitations
and
Calibration
Updated:
2/
27/
98
27
This
paper
concerns
with
changing
calibration
curves
from
one
time
to
the
next
due
to
various
factors,
such
as
changing
environmental
conditions,
instrument
aging
and
several
other
causes.
The
usual
practice
in
such
situations
is
to
recalibrate
the
measuring
devices
at
frequent
intervals
and
obtain
estimate
of
the
unknown
values
of
the
test
samples
using
only
data
from
the
corresponding
calibration
period.
The
authors
consider
a
random
coefficient
regression
model
for
the
different
calibration
curves
and
combine
the
data
from
all
calibration
periods
to
estimate
the
unknowns.
A
simulation
study
was
performed
to
compare
the
two
inverse
estimators
where
the
calibration
curve
is
a
straight
line
through
the
origin.
The
calibration
design
was
as
follows:
number
of
periods,
T=
4,
8,
36;
number
of
standards,
n=
2,
4;
design
points,
;
slope/
error
ratio,
=
5,
10,
20,
50,
100;
x
1

x
2



x
n

1/
n

/

between/
within
ratio,
=
1.01,
2,
5;
and
unknowns,
=
0.25,
0.50,
0.75,
1.00.
The
authors

2
0
u
n
mention
that
the
parameter
values
chosen
for
the
Monte
Carlo
study
cover
regions
that
correspond
to
many
practical
situations.
Using
the
Pitman
closeness
criteria,
the
authors
show
that
it
is
possible
to
obtain
more
efficient
estimate
of
the
unknown
values
by
this
method
as
compared
to
the
usual
estimator.
The
authors
also
propose
a
large
sample
combined­
periods
confidence
interval
for
the
value
of
an
unknown
test
sample.

Williams,
E.
J.
(
1969)
A
note
on
regression
methods
in
calibration.
Technometrics,
11,
189­
192.

In
this
paper,
a
brief
review
of
the
classical
and
the
inverse
calibration
methods
is
performed
to
set
up
a
response
to
the
questions
raised
by
the
Krutchkoff
(
1967)
article
which
simulates
the
classical
and
the
inverse
calibration
approaches
using
the
calibration
of
[
0,1].
The
author
firsts
examines
the
classical
calibration
method
and
determines
that
the
classical
estimator
has
undefined
expectation
and
infinite
variance,
and
hence
infinite
mean
squared
deviation
(
MSD).
The
inverse
estimator
is
then
examined
and
is
determined
to
have
finite
mean
squared
deviation
for
.
Therefore,
the
inverse
N

4
estimator
is
"
better"
than
the
classical
estimator
in
terms
of
MSD.
However,
the
author
points
out
that
showing
the
MSD
of
an
estimator
is
less
than
infinity
is
not
proving
very
much,
and
thus
brings
into
question
the
use
of
MSD
as
a
criterion
for
goodness
of
fit
of
the
estimators.
To
further
weaken
the
argument
of
Krutchkoff
(
1967),
the
author
uses
Blackwell
(
1947)
to
show
that
no
unbiased
estimator
(
i.
e.
the
classical
estimator)
will
have
finite
variance.
This
determination,
combined
with
the
fact
that
the
inverse
estimator
examined
in
the
Krutchkoff
(
1967)
article
has
constant
variance,
provides
the
author
with
the
determination
that
the
minimum
variance
or
minimum
MSD
criterion
is
not
suitable
for
this
type
of
problem.
Suggested
references
for
using
confidence
interval
are
then
given
as
an
alternative
for
what
is
needed
for
the
calibration
type
problems.

Yao,
Y.
C.,
Vecchia,
D.
F.
and
Iyer,
H.
K.
(
1988)
Linear
Calibration
when
the
coefficient
of
variation
is
constant.
Probability
and
Statistics:
Essays
in
Honor
of
Franklin
A.
Graybill,
297­
309.
Annotated
Bibliography
on
Measurement
Limitations
and
Calibration
Updated:
2/
27/
98
28
The
authors
consider
point
estimation
of
the
unknown
parameters
in
the
calibration
problem.
They
assume
the
calibration
curve
to
be
a
straight
line
through
the
origin
and
the
responses
are
normal
with
variances
proportional
to
the
square
of
their
mean
.
The
problem
is
stated
in
the
following
manner:
Let
be
the
values
of
the
standards
and
the
corresponding
responses.
Let
X
1,
X
2,

,
X
n
Y
1,
Y
2,

,
Y
n
u
be
the
true
unknown
value
of
a
test
sample
and
the
corresponding
response
be
.
The
observations
z
are
mutually
independent
and
normally
distributed
and
the
unknown
is
assumed
to
be
nonzero.
u
Also,
it
is
assumed
that
,
,
where
E[
Y
i]


X
i
Var(
Y
i)


2X
2
i
,
i

1,2.....,
n,
E[
z]


u,
Var(
z)


2u
2
are
unknown.
The
main
objective
is
to
obtain
an
efficient
point
estimator
of
The
authors

2,

,
u
u.
consider
weighted
least
square
estimation,
maximum
likelihood
estimation,
equivariant
estimation
and
inverse
estimation.
They
compare
the
estimators
using
the
Pitman's
closeness
criterion.
Three
different
sample
sizes
were
considered.
In
each
case
a
2­
point
calibration
design
with
equal
number
of
observations
at
each
point
was
used.
Asymptotic
results
indicate
that
the
classical
estimator
(
weighted
least
squares)
is
superior
to
the
maximum
likelihood,
equivariant
and
inverse
estimator
and
the
same
conclusions
can
be
drawn
for
small
samples.
The
classical
estimator
is
preferable
to
the
other
estimators,
if
the
sensitivity
(
)
of
the
calibration
procedure
is
greater
than
or
equal
to
1.4
in



/

absolute
value,
which
is
usually
the
case
in
practice.