Statistical interpretation of key comparison reference value and degrees of equivalence

Journal of Research of the National Institute of Standards and Technology, Nov-Dec, 2003 by R.N. Kacker, R.U. Datla, A.C. Parr

In order to specify a state-of-knowledge probability distribution for the correction variable C, the laboratory expected values [X.sub.1],..., [X.sub.n] and the value Y of the measurand are regarded as variables with state-of-knowledge distributions and the data [x.sub.1],..., [x.sub.n] and u([x.sub.1]),..., u([x.sub.n]) are regarded as given constants. A state-of-knowledge distribution for [X.sub.i] represents the state of knowledge about the value Y of the measurand in the laboratory labeled i for i = 1, 2,..., n. The expected value E([X.sub.i]) and standard deviation S([X.sub.i]) of the variable [X.sub.i] are assumed to be [x.sub.i] and u([x.sub.i]), respectively, for i = 1, 2,..., n [5], [4]. It follows that [X.sub.UCR] = [[SIGMA].sub.i] [a.sub.i] [X.sub.i] is a variable with a state-of-knowledge probability distribution. The expected value of [X.sub.UCR] is E([X.sub.UCR]) = [[SIGMA].sub.i] [a.sub.i] E([X.sub.i]) = [[SIGMA].sub.i] [a.sub.i] [x.sub.i] = [x.sub.UCR]. In the expression (Y - [X.sub.UCR]) for the negative of bias, treated as a variable, we replace [X.sub.UCR] with its expected value [x.sub.UCR]. Then a probability distribution for C represents belief about the possible values of (Y - [x.sub.UCR]), where [x.sub.UCR] is a constant and Y is the variable. The belief about possible values of Y is based on all available information including results of measurement and scientific judgment. In reference [6], we proposed a triangular distribution for the correction variable C, with peak at [x.sub.UCR] and default limits [[x.sub.(1)] - [x.sub.UCR]] = min {[x.sub.1] - [x.sub.UCR],..., [x.sub.n] - [x.sub.UCR]} and [[x.sub.(n)] - [x.sub.UCR]] = max{[x.sub.1] - [x.sub.UCR],..., [x.sub.n] - [x.sub.UCR]}. A criticism of the proposed triangular distribution with default limits is that it is determined by the extreme results [x.sub.(1)] = min{[x.sub.1],..., [x.sub.n]} and [x.sub.(n)] = max{[x.sub.1],..., [x.sub.n]}, which are sometimes suspected to be in error.

Here, we propose a discrete-equal-probability distribution that is determined by all of the results [x.sub.1],..., [x.sub.n]. The results [x.sub.1],..., [x.sub.n] are plausible values of Y as determined by competent laboratories. (8). So the known constant differences ([x.sub.1] - [x.sub.UCR]),..., ([x.sub.n] - [x.sub.UCR]) are plausible values of (Y - [x.sub.UCR]). These differences are a statistical basis for specifying a probability distribution for C. Let [c.sub.i] = [x.sub.i] - [x.sub.UCR] for i = 1, 2,..., n. Suppose [c.sub.1],..., [c.sub.n] are assigned probabilities [p.sub.1],..., [p.sub.n]. Then the expected value of C is c = E(C) = [[SIGMA].sub.i] [p.sub.i] [c.sub.i] = ([[SIGMA].sub.i] [p.sub.i] [x.sub.i]) - [x.sub.UCR] and the standard deviation of C is u(c) = S(C) =[square root of ([[SIGMA].sub.i][p.sub.i][([c.sub.i] - c)[.sup.2])]. Frequently, the available scientific knowledge is inadequate to assign different probabilities [p.sub.1],..., [p.sub.n] to [c.sub.1],..., [c.sub.n]. Therefore, we propose the discrete-equal-probability distribution for which [p.sub.i] = 1/n for i = 1, 2,..., n. The expected value and standard deviation of C based on discrete-equal-probability distribution are c = [x.sub.A] - [x.sub.UCR] and u(c) = [square root of ([[[SIGMA].sub.i] ([x.sub.i] - [x.sub.A])[.sup.2]/n])], respectively, where [x.sub.A] = [[SIGMA].sub.i] [x.sub.i] / n is the arithmetic mean of the results [x.sub.1],..., [x.sub.n].