Figure 1 appears to suggest that if one were to use the
95%_U_Lim or 99%_U_Lim values to define method
acceptability, when the variability is higher, usually for low
concentrations, the limits are wider, as they should be,
allowing a greater degree of leniency for a method to be
classified as acceptable than when the variability is lower for
the higher concentrations.
Summary
A formula was developed for use in computing an upper
limit for future sample relative reproducibility standard
deviations obtained using a given method to analyze a given
material in a collaborative study. This formula, and to a degree
the results in Table 1, will prove useful to Study Directors in
the design of collaborative studies because they can use the
formula calculations or the results in Table 1 as a barometer
for the worst that can be expected, given a specified level of
confidence, with respect to reproducibility precision prior to
conducting a study. The one drawback in using the formula is
that it assumes that the relative reproducibility standard
deviation and the ratio of the repeatability standard deviation
to the reproducibility standard deviation are known
population parameters. However, in practice this assumption
may be relaxed by accepting and using the research results by
Horwitz and Albert (7, 8) with respect to reproducibility
precision. The results of that research, particularly that
relating to the "Horwitz equation," appear useful for obtaining
reproducibility precision consensus values for the above
mentioned parameters that are generally accepted as
standards.
Acknowledgements
The authors are grateful to Robert Blodgett (FDA/CFSAN,
College Park, MD) for assistance in developing the SAS
simulation procedure. In addition, we thank the referee for
comments, which have assisted in improving the paper.
References
(1) McClure, F.D., & Lee, J.K. (2005)
J. AOAC Int.
88
, 1503–1510
(2) Hald, A. (1952)
Statistical Theory with Engineering
Applications
, John Wiley & Sons Inc., New York, NY
(3) Bishop, Y., Fienberg, S., & Holland, P. (1975)
Discrete
Multivariate Analysis: Theory and Practice
, MIT Press,
Cambridge, MA
(4) ISO 5725 (2000)
Statistical Methods for Quality Control, Vol.
2 Measurement Methods and Results Interpretation of
Statistical Data Process Control
, 5th Ed., International
Organization for Standardization, Geneva, Switzerland
(5) Stuart, A., Ord, K., & Arnold, S. (1999)
Kendall’s Advanced
Theory of Statistics,
6th Ed., Vol. 2A, Oxford University
Press Inc., New York, NY
(6) Scheffe, H. (1959)
The Analysis of Variance,
John Wiley &
Sons, Inc., New York, NY
(7) Horwitz, W.H., & Albert, R.A. (1996)
J. AOAC Int.
79
,
589–621
(8) Horwitz, W. (1999) Personal Communication on the
Magnitude of Historic Ratios of the Repeatability Standard
Deviation to the Reproducibility Standard Deviation
s
s
r
R
(9)
Official Methods of Analysis
(2005) Appendix D: Guidelines
for Collaborative Study Procedures to Validate
Characteristics of a Method of Analysis; Part 6: Guidelines
for Collaborative Study, AOAC Official Methods Program
Manual (OMA Program Manual): A Policies and Procedures
Guide for the Official Methods Program (OMA), AOAC
INTERNATIONAL (2003)
(10) Mood, A., Graybill, F., & Boes, D. (1974)
Introduction to the
Theory of Statistics
, McGraw-Hill, Inc., New York, NY
Appendix
The following Statistical Analysis System (SAS) program
was written and executed to obtain a simulated distribution of
sample RSD
R
values. It is an unabridged version of the
program used to generate the simulation results presented
earlier.
SAS Program to Determine a One-Tailed 100p%
Upper Limit for Future Sample Relative
Reproducibility Standard Deviations
OPTIONS NODATE NONUMBER;
%LET TEST = 10000;
/*INPUT NUMBER OF SAMPLE RSD
R
SIMULATIONS*/
%LET N_LABS = 8;
/*INPUT NUMBER OF LABORATORIES*/
%LET REPS= 2;
/*INPUT NUMBER OF REPLICATES*/
%LET C = 1;
/*INPUT VALUE FOR CONCENTRATION LEVEL*/
%LET XI_R = .02;
/*INPUT CONSENSUS VALUE FOR POP. */
%LET THETA = 0.5;
/*INPUT
(
r
R
*/
DATA FSIM (KEEP=X LAB I RHO N_LABS REPS ); /* NEEDED FOR GLM**/
ARRAY XG{&N_LABS.} XG1 - XG&N_LABS.;
ARRAY SLGP{&N_LABS.} SLGP1 - SLGP&N_LABS.;
SIG_L = SQRT((&C.*&XI_R.)**2 - (&THETA.*&XI_R.*&C.)**2); /*LAB STD*/
RHO = 1 - &THETA**2; /**ICC CALC.***/
SIG_R = &THETA*&XI_R.*&C; /*REPEATABILITY STANDARD DEVIATION*/
N_LABS = &N_LABS.;
REPS = &REPS.;
DO I = 1 TO &TEST.;
DO J = 1 TO &N_LABS.;
SLGP{J} = SIG_L*RANNOR(0); /*LABORATORY SELECTION*/
END;
DO J = 1 TO &REPS.;
DO LAB = 1 TO &N_LABS.;
X = &C + SLGP{LAB} + SIG_R*RANNOR(0); /*REPLICATE SELECTION*/
OUTPUT FSIM;
END;
END;END;
RUN;
PROC GLM DATA=FSIM NOPRINT OUTSTAT=STATS;
BY I;
CLASSES LAB;
MODEL X= LAB;
RUN; QUIT;
802
M
C
C
LURE
& L
EE
: J
OURNAL OF
AOAC I
NTERNATIONAL
V
OL
. 89, N
O
. 3, 2006
Recommended to OMB by Committee on Statistics: 07-17-2013
Reviewed and approved by OMB: 07-18-2013
33
64
