OMB Meeting Book - January 8, 2015

61

M C C LURE & L EE : J OURNAL OF AOAC I NTERNATIONAL V OL . 89, N O . 3, 2006 799

conducted ( see Appendix for details). The MC simulation was developed for use with Statistical Analysis System (SAS) software to model a CRM ANOVA assuming L laboratories and n replicates/laboratory to draw a set of simulated data, assuming known laboratory-to-laboratory and within-laboratory standard deviations L r and , respectively, and population mean ( ) or concentration of analyte. The simulated data were then used to obtain an estimate of the sample relative reproducibility standard deviation ( RSD R ). For each set of L , r , and , the cumulative distribution of a total of 10 000 simulated sample relative reproducibility standard deviations was examined to obtain the 95th and 99th percentile values to represent simulated one-tailed 95 and 99% upper limits for future sample relative reproducibility standard deviations. The results of the simulation are presented in Table 1 for values of R ,% 2, 16, and 64; =1/2 and 2/3; number of laboratories = 8 and 20; number of replicates = 2, 5, and 20; and probability levels of 95 and 99%. In general, Table 1 presents one-tailed 95 and 99% upper limits in percent ,% obtained in a collaborative study employing L = 8 and L = 20 laboratories, each performing 2, 5, or 20 replicates. Also presented in Table 1 are the MC simulated one-tailed 95 and 99% upper limit values MC MC 95 99 , , % % and . The probability levels ( p * ) are simulated probability levels that are equivalent to percentiles for the simulated MC values that equal the ,% values. Based on the results in Table 1, it can be seen that there is excellent agreement between the MC p ,% - values and p ,%- values and corresponding p * -values. Hence, the computational formula p provides a satisfactory approximation for obtaining a 100 p % one-tailed upper limit for future sample RSD R ,% values. 0 95. ,% and 0 99. ,% for future sample RSD R 0 95. ,% and 0 99. Consensus Values Assumed for Population Values for R ,% and ,% and ( will not be known. However, in some cases, consensus values, i.e., values obtained on the basis of long-time experience, may be satisfactory approximations. For some analytical methods and materials, consensus values for R ,% and ( may be obtained from the results of research by Horwitz and Albert (7, 8). For example, one might use the “Horwitz equation” to predict a consensus value R C , ,% for the population percent relative reproducibility standard deviation R ,% . The predicted relative reproducibility standard deviation expressed as a percent ( PRSD R ,%) is computed as 0 1505 using for C a known spike or a consensus level of analyte to provide a consensus value for Usually, the population values for R R C , R PRSD C ,% ,% 2 . Determining p

p

R

z

p

1 2/

! " # # $ # #

% & # # ' # #

2

2

2

2

2

2

l

n

R r

r

R

l

n n

2

2

2

2

l

2

2

n L

n L

R

R

p R 2 2

2

r

l

n n

2

nL

R

Letting (

r

(the ratio of the population repeatability

R

and reproducibility standard deviations), we obtained the following:

p

R

z

p

1 2/

2

(

(

2

2

2

(

4

l

l

n n

n n

l

n

p

R

2

2

l

2

2

n L

n L

nL

R be the population relative reproducibility

Letting

R

standard deviation, the following expression was obtained:

p

l

R

z

p

1 2/

(

(

p 2

2

2 2

(

4

l

l

n n

n n

l

n

2

2

nL

l

n L

n L

2

2

Solving this equation for p

we obtained:

1 2 /

2

2

2 2

2

(

(

l

z

n n

n n

l

4

(

n

l

p R

1

2

2

nL

2

2

l

n L

n L

l

z

p

2

2

(

n n

l

R

L

n

p

2 2

2

l (

z

n n

R

p

l

l

nL

R

To reiterate,

p a one-tailed 100 p % upper limit for future

values, (

r

sample RSD R

(the ratio of the population

R

repeatability and reproducibility standard deviations),

R (the population relative reproducibility standard

R

deviation), z p (the abscissa on the standard normal curve that cuts off an area p in the upper tail), and L and n are the number of laboratories and replicates/laboratory, respectively.

,% . To obtain a consensus value for (

R

r

, one might appeal

Accuracy of

p

R

To assess the accuracy of p with respect to the intended probability level, a Monte Carlo ( MC ) simulation study was

to Horwitz’s conclusion based on his observation of several thousand historic collaborative studies (7, 8). That is, Horwitz

30

Recommended to OMB by Committee on Statistics: 07-17-2013 Reviewed and approved by OMB: 07-18-2013