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

(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 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 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; Appendix 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 */

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. 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. Summary

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

CLASSES LAB; MODEL X= LAB; RUN; QUIT;

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