187 / 274
observed from his research that the estimate of
(
r
R
, i.e.,
the ratio of the sample repeatability standard deviation to the
sample reproducibility standard deviation
s
s
r
R
, for most
accepted methods ranged from 1/2 to 2/3 (i.e., 0.500 to 0.667).
Because for any
R
,%
p
is at a maximum when
(
= 0.5,
relative to the
p
obtained when
(
= 0.667, we recommend
using Horwitz’s lowest observation limit of
s
s
r
R
= 0.5 as a
consensus value for
(
.
Example 1
In this example, we assume that a Study Director has no
knowledge of
R
,%and
(
but would like to know the largest
RSD
R
,% that might be confidently obtained in a collaborative
study on a given material having a specified
concentration (
C
). Given the above, we will start by using the
"Horwitz equation," if analytically applicable, to predict a
consensus value for the population percent relative
reproducibility standard deviation as follows:
R C
R
PRSD C
,
.
,% ,% 2
0 1505
(using for
C
a known spike or
a consensus level of analyte) to provide a consensus value for
R
,% . Assume that the spike level or consensus value for the
concentration is
C
= 5.1147
)
10
–5
. Substituting the value for
C
in
R C
R
PRSD C
,
.
.
,% ,%
.
)
2
2 5147 10
0 1505
5 0 1505
,
we obtained
R C
,
,% = 8.8398. For use in calculations later,
R C
,
,% will be converted to a decimal,
i.e.,
R C
R C
,
,
,% .
.
100
88398
100
0088398.
Next, we assume that we want a 95% upper limit for future
sample
RSD
R
,% values (
0.95
) obtained from a collaborative
study employing
L
= 8 laboratories each analyzing duplicates
(
n
= 2). We assume further a consensus value of
(
= 0.5. Upon
substituting the special case values
L
= 8,
n
= 2,
(
= 0.5, and
z
0.95
= 1.645 (the standard normal deviate for
p
= 0.95) into
p
z p
n
n L
n n
n L
z p R
n
l
l
l
l
l
(
(
4
2
2
2
2
2
2
2 2
n
nL
R n n
nL
R
1
2
2
2
(
(
l
l
R
l
2 2
2
1 2
z p n n
nL
l
(
/
we obtained an easier-to-use formula for computing
0.95
,
given the above special case values as follows:
0 95
2
2
1 1645 005566 009293
1 029597
.
.
.
.
.
R
R
R
Substituting
R,C
= 0.88398 for
R
in the previous general
formula and performing the indicated mathematical
operations, we obtained
0.95
= 0.12321 or
0.95
,% = 12.321.
This is the 95% upper limit for sample
RSD
R
, % arising from a
population whose true mean percent relative reproducibility
standard deviation is
RC
,% = 8.84.
Provided in the following is an easier-to-use formula for
computing a 99% upper limit (
0.99
) for future sample
RSD
R
,% values obtained from collaborative studies
employing
L
= 8 laboratories each analyzing duplicates
(
n
= 2). Here, we substituted the special case values
L
= 8,
n
= 2,
(
= 0.5, and
z
0.99
= 2.326 (the standard normal deviate
for
p
= 0.99) into
p
above, and obtained the following:
0 99
2
2
1 2326 005566 007644
1 059175
.
.
.
.
.
R
R
R
Example 2
Those familiar with the results from the “Horwitz
equation” or predicted relative reproducibility standard
deviation,
PRSD
R
, may recognize that the
R
,%= 2, 16, and 64
in Table 1 coincide with
PRSD
R
,% = 2, 16, and 64 when the
concentrations
C
= 10
0
, 10
–6
, and 10
–10
, respectively, are used
in
PRSD
R
,% = 2
C
–0.1505
. This implies that
p
may also be used
to obtain one-tailed 100
p
% upper limits for future sample
RSD
R
obtained from a population with known
RSD
R
=
PRSD
R
using the “Horwitz equation.”
Figure 1 presents plots of
PRSD
R
,% and one-tailed 95 and
99% upper limits, assuming
L
= 8,
n
= 2, and
(
= 0.5, for
future sample
RSD
R
,% on predefined concentrations
transformed to Log
10
(C). In Figure 1, the lower curve
represents a plot of the
PRSD
R
,% values on Log
10
(C) of
analyte. This curve is called the “Horwitz curve." The 2 upper
curves reflect, respectively, one-tailed 95 and 99% upper
limits for future sample
RSD
R
,% values.
M
C
C
LURE
& L
EE
: J
OURNAL OF
AOAC I
NTERNATIONAL
V
OL
. 89, N
O
. 3, 2006
801
Figure 1. Predicted relative reproducibility standard
deviation (PRSD_R%), 95% upper limits (95% U_Lim)
and 99% upper limits (99% U_Lim) for future sample
relative reproducibility standard deviations (RSD_R%)
on log
10
(concentration).
134