OMB Meeting Book - January 8, 2015

63

M C C LURE & L EE : J OURNAL OF AOAC I NTERNATIONAL V OL . 89, N O . 3, 2006 801

Substituting R,C 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: 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. = 0.88398 for R 0 99 . 2 2 1 2326 005566 007644 . . . 1 059175 . R R R Example 2

observed from his research that the estimate of (

r

, i.e.,

R

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: 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 PRSD C ,% ,% 2 . 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 R C , R C , ,% . . 0088398. 100 88398 100

1 2 /

2

2

2

z p R 2 2

4

n

n

1

(

(

l

n n

(

l

n

l

2

2

l

nL

n L

n L

2

2

l

z p

2

2

( l

R n n

nL

p

2 2

2

l (

z p n n

l

R

l

nL

R

we obtained an easier-to-use formula for computing 0.95 , given the above special case values as follows:

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).

2

1 1645 005566 009293 . . .

R

R

0 95 .

2

1 029597 .

R

32

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