184 / 274
Data Analysis
To obtain the sample estimate of the repeatability and
reproducibility variances
s
s
r
R
2
2
and , respectively, the data
from the CRM are analyzed to obtain the mean squares
reflecting the "among-laboratories" and “within-laboratory”
variations. Using an analysis of variance (ANOVA) technique
for analyzing the data, the sample mean for the
i
th laboratory
y
i
y
n
ij
n
1
and the sample grand mean
y
y
nL
L n
ij
l l
are
used in computing the “among-laboratories” mean square
MS
n
L
y y s ns
L
L
i
r
L
l
1
2
2
2
and the “within-laboratory” mean square
MS
L n
y y
s
r
L n
ij
i
r
l
l
1
1
2
2
The sample reproducibility variance
s
n
MS MS MS s s
R
L
r
r
r
L
2
2
2
l
is an estimate of the population reproducibility variance
R
r
L
2
2
2
. The sample reproducibility standard
deviation (
s
R
) is the square root of
s s
s
R R
R
2
2
and is an
estimate of the population reproducibility standard deviation
(
R
). The sample
RSD
s
y
R
R
is an estimate of the population
relative reproducibility standard deviation
R
R
, where
is the population mean.
Statistical Distribution and Independence of s
R
and y
In developing a formula for
p
, it is important to establish
that the distribution and independence of
s
R
and
y
exist. In an
earlier paper, McClure and Lee (1) detailed the derivation of
the asymptotic distribution of
s
R
, assuming that the
reproducibility variance
s
R
2
was approximately normally
distributed (~) with mean
R
2
and variance
V s
R
2
, i.e.,
s N V s
R
R
R
2
2
2
~
,
, by finding
V s
R
2
and applying the
-method (3, 4). Thus, the distribution of
s
R
is asymptotically
normal with mean (
R
) and variance
V s
R
, i.e.,
s N V s
R
R
R
~
,
, where
V s
n
n L
n
n L
R
R
r
r
L
l
2
l
l
2
2
4
2
2
2
2
. Also, based on the
CRM, the sample mean
y
is normally distributed with a
mean ( ) and variance
V y
n
nL
r
L
2
2
,
i.e.,
y N V y
~ ,
.
In establishing the independence of
s
R
and
y
, we direct
attention to the work of Stuart et al. (5), who have shown the
mean, “among-groups” and “within-groups” sums of squares,
which are analogous to our mean
y
, “among-laboratories”
sum of squares (
SS
L
) and “within-laboratory” sum of squares
(
SS
r
), are statistically independent under the CRM, and,
hence, the mean
y
and reproducibility standard deviation
s
s
SS
Ln
SS
n L
R
R
r
L
2
l
are independent.
100p% One-Tailed Upper Limits for Future Sample
RSD
R
Values
In approximating the distribution of the sample
RSD
R
, we
want the probability that the sample
RSD
R
is less than the
p
th
percentile value
p
to equal
p
,
i.e.,
Pr
or Pr
RSD
p
s y
p
R p
R
p
0 . Here we note
that the variable
z s
y
R p
in the probability statement
Pr
0
s
y
p
R p
is approximately normally distributed
with mean
E z
R p
and variance
V z V s
V y
R
p
2
.
We chose the variable
z s
y
R p
because it is known that
a linear function of a normal and an approximately normal
variable will usually deviate less from the normal distribution
than the distribution of the ratio of the 2 variables (2).
Substituting the variances
V s
V y
V z
R
and
into , we
obtained the following:
V z
n
n L
n
n L
nL
R
r
r
L
p
r
l
2
l
l
2
4
2
2
2 2
2
2
2
n
L
2
Hence, we obtained
Pr
Pr
s
y
s
y
V z
V z
R
p
R
p
R
p
R
p
0
1 2
1 2
/
/
p
R
Var z
p
1 2/
where
represents the cumulative standard normal
distribution. Therefore,
p
R
p
V z
z
1 2/
, where
z
p
is the
abscissa on the standard normal curve that cuts off an area
p
in
the upper tail. Substituting the expression for
V(z)
in the above
formula, we have
z
V z
n
n L
n
n L
p
p
R
p
R
R
r
r
L
1 2
2
4
2
2
2 2
2
1
2
1
/
l
!
"
#
$#
%
&
#
'#
p
r
L
nL
n
2
2
2
1 2/
Performing some algebra on the right-most expression above,
we obtained the following:
798
M
C
C
LURE
& L
EE
: J
OURNAL OF
AOAC I
NTERNATIONAL
V
OL
. 89, N
O
. 3, 2006
131