OMB Meeting Book - January 8, 2015
60
798 M C C LURE & L EE : J OURNAL OF AOAC I NTERNATIONAL V OL . 89, N O . 3, 2006
2
L 2
Data Analysis
n
r
mean ( ) and variance
,
i.e.,
V y
nL
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 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 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 . 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: We chose the variable z s y R p s s SS Ln SS n L R R r L 2 l are independent.
n
L n
y
y
ij
ij
1
l l
and the sample grand mean
are
y
y
i
nL
n
used in computing the “among-laboratories” mean square
n
L
2
2
2
MS
y y s ns
L
i
r
L
l 1
L
and the “within-laboratory” mean square
l
L n
2
2
MS
y y
s
r
ij
i
r
L n
1
l
1
The sample reproducibility variance
l
2
2
2
s
MS MS MS s s
R
L
r
r
r
L
n
is an estimate of the population reproducibility variance
2
2
2 . The sample reproducibility standard
R
r
L
2
2
) is the square root of s s
and is an
deviation ( s R
s
R R
R
estimate of the population reproducibility standard deviation
2
2 2
p 2
4
n
l
n
l
r
L
s
r
2
L 2
V z
n
R is an estimate of the population
). The sample RSD
( R
r
2
2
2
nL
2
l
n L
n L
R
y
R
R , where
relative reproducibility standard deviation R
Hence, we obtained
is the population mean.
s
y V z
R
p
R
p
R
p
Pr
s
y
0
Pr
R
p
1 2 /
1 2 /
V z
Statistical Distribution and Independence of s R and y
p
R
p
In developing a formula for p
, it is important to establish
1 2/
Var z
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
where
represents the cumulative standard normal
p
R
distribution. Therefore,
, where z p
is the
z
p
1 2/
V z
R 2
and variance V s R 2
, i.e.,
distributed (~) with mean
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
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
) and variance
, i.e.,
normal with mean ( R
V s
R
p
R
p
R
z
p
% & # '# 1 2/
1 2 /
! " # $#
V z
, where
s N V s R R R ~ ,
2
2
2 2
4
n
n
1
1
p
r
L
2
L 2
r
n
2
r
2
L 2
2
2
2
n L
n L
2
l
nL
n
l
l
n
R
r
4
. Also, based on the
V s
R
r
2
2
2
2
l
n L
n L
R
Performing some algebra on the right-most expression above, we obtained the following:
y is normally distributed with a
CRM, the sample mean
29
Recommended to OMB by Committee on Statistics: 07-17-2013 Reviewed and approved by OMB: 07-18-2013
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