OMB Meeting Book - January 8, 2015

59

M C C LURE & L EE : J OURNAL OF AOAC I NTERNATIONAL V OL . 89, N O . 3, 2006 797

STATISTICAL ANALYSIS

Determining a One-Tailed Upper Limit for Future Sample Relative Reproducibility Standard Deviations F OSTER D. M C C LURE and J UNG K. L EE U.S. Food and Drug Administration, Center for Food Safety and Applied Nutrition, Office of Scientific Analysis and Support, Division of Mathematics, Department of Health and Human Services, 5100 Paint Branch Pkwy, College Park, MD 20740-3835

A formula was developed to determine a one-tailed 100 p % upper limit for future sample percent relative reproducibility standard deviations

is the

i , where is an unknown constant and e i

y

e

i

random error associated with y i . After considerable study of the problem, we came to the conclusion that an exact limit for an RSD R was unachievable, primarily because the exact distributions of the sample s R 2 and s R are very complicated, and possibly impossible to obtain. Therefore, we sought to develop a formula to determine an approximate one-tailed 100 p % upper limit p for future sample RSD R values, obtained under a CRM model, by extending Hald’s single sample approximation for p . In doing so, we used a normal approximation and the delta-method ( -method; 1, 3, 4). Here, we will review the CRM used by AOAC to establish background notations. The model represents 2 sources of variation: the first is often referred to as “among-laboratories” and the other as “within-laboratory” variation. For the CRM, an analytical result y ij obtained by laboratory i on test sample j is expressed as y ij i ij , i = 1, 2, …, L and j = 1, 2, …, n , where is the grand mean of all potential analyses for the material, i a constant associated with laboratory i , and ij the random error associated with analysis y ij . It is also assumed that i and ij are independent random variables, such that i is normally distributed (~) with a mean of 0 and variance of L 2 , i.e., i L N ~ ,0 2 . Similarly, ij is Collaborative Study Model

100

s

, where s R is the sample

R

RSD

,%

R

y

reproducibility standard deviation, which is the square root of a linear combination of the sample repeatability variance s r 2 plus the sample laboratory-to-laboratory variance s L 2 , i.e., s R = 2 , and y is the sample mean. The future RSD R ,% is expected to arise from a population of potential RSD R ,% values whose true mean is s s r L 2

100

, where

R and are the population

R

,%

R

reproducibility standard deviation and mean, respectively.

T he sample relative reproducibility standard deviation ( RSD R ), usually expressed as a percent ( RSD R ,%) is obtained using a completely randomized model

100

s

R

(CRM; 1) and is defined as RSD

is the

, where s R

,%

R

y

sample reproducibility standard deviation, which is the square root of a linear combination of the sample repeatability variance s r 2 plus the sample laboratory-to-laboratory

2 , i.e., s

2

2 , and y is the sample mean.

variance s L

s s

R

r

L

The sample RSD R ,% is an important method performance indicator for validation organizations such as AOAC INTERNATIONAL. Therefore, we reasoned that it might be of great value to have a statistical procedure to determine a one-tailed 100 p % upper limit P for future sample RSD R ,% values. A thorough literature search suggested that until now no such procedure, based on a CRM, has existed. However, we did note that Hald (2) had investigated the distribution of the coefficient of variation for the single sample model, i.e.,

normally distributed with a mean of 0 and variance of

2

2

.

, i.e. ,

N ~ , 0

r

ij

r

Given the above model, we note that the expected value of y ij equals the grand mean ( ) E y ij , the variance of y ij equals the reproducibility variance var y ij L r 2 2 , the covariance of y ij and y ik equals the “among-laboratories” component of variation cov , y y ij ik L 2 for j k , and the

L 2

and y ik

is

2 for j k , i.e., within a

correlation between y ij

Received May 17, 2005. Accepted by GL January 31, 2006. Corresponding author's e-mail: foster.mcclure@cfsan.fda.gov

2

r

L

are correlated under the CRM (5, 6).

given laboratory the y ij

28

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