Degrees of Equivalence in a Key Comparison

DEGREES OF EQUIVALENCE IN A KEY COMPARISON 1 Thang H. L., Nguyen D. D. Vietnam Metrology Institute, Address: 8 Hoang Quoc Viet, Hanoi, Vietnam Abstract: In an interlaboratory key co mparison, a data analysis pro cedure for this comparison was proposed and recommended by CIPM [1, 2, 3], therein the degrees of equivalence of measurement standards of the la boratories participated in the comparison and the ones between each two laboratories were introduced but a corresponding clear an d plausible measurement model was not given. Authors in [4] offere d possible measurement m odels for a given comparison and a suitable m odel was selected out after rigorous analyzi ng steps for expectation values of these degrees of equivalence. The systematic laborato ry-effects model was then selected as a right one in this report. Those models were all base d on the one true value existence assumption. However in the year 2008, a new ve rsion of the Vocabulary for International Metrology (VIM) [7] was issued where the true value of a given measurement standard should be now perceived as multiple true values wh ich following a given statistics distribution. Applying this perception of true values of a measurement standard with combination of the steps in [4], measurement models hav e been deve loped and degrees of e quivalence have been analyzed. The results show that although with new definition, the systematic laboratory-effects model is still the reasonable one in a given key comparison. I. Introduction In reference [2], concept of degrees of equi valence between laboratories was stated as one of important criteria in Mutual Recognition Arrangem ent (MRA) between National Metrology Institutes (NMIs). Degrees of equivale nce are defined in [1] as following: Degree of equivalence of a measurement standard : the degr ee to which the value of a measurement standa rd is consistent with the key com parison reference value. This is expressed quantitatively by the deviation from the key comp arison reference value and the uncertainty of this deviation. The degree of e quivalence between two measuremen t standards is expressed as the difference between their respective deviatio ns from the key comparison reference value and the uncertainty of this difference. Mathematically, the degree is expressed as d i = x i - x K and u 2 (d i ) = u 2 (x i ) - u 2 (x K ) . The degree of equivalence betw een two measurem ent standards is expressed as d ij = x i - x j and u 2 (d ij ) = u 2 (x i ) + u 2 (x j ) [3]. To illuminate th e statistics natures of those quantities, m easurement models for a key comparison have been offered and an alyzed in [4]. In those models, a given measurement standard is assumed having only one tr ue value. Actually, as discussed in [7], a    1  Email:  thanglh@vmi.gov.vn  more general view should be of understanding that for a given m easurement standard, there exist a set of true values which we then assume following a given stat istics distribution. II. Mathematical modeling Let consider a given key comparison where a m easurement quantity having a set of true values Y i , i = 1 to N (N is the number of participants) which is f ollowing a unique stable distribution during the co mparison time. The expectation and varian ce of Y i will be E(Y i ) = Y and V(Y i ) = s 2 (Y i ). Call X 1 , X 2 … X N and x 1 , x 2 … x N are expectation values and measured values of the measurem ent quantity measured and provided by the i th laboratory. Each measured value will have a re liable m easurement uncertainty u(x i ). Call b 1 = (X 1 – Y 1 ), b 2 = (X 2 – Y 2 ),…, b N = (X N – Y N ). The set of b 1 , b 2 ,…, b N are not always zero due to some unrecognizable errors during the measurement but all of the measurement values of a certain laboratory should still have the same expectation value. Next some m easurement models with different assumptions will be developed a nd their analysis will be carried out. 1. None laboratory effect In this case the measurement equation will be of the form: x i = Y i + e i (1) The equation for expectation values will be: E(x i ) = X i = Y. Here b i = 0 implies the participating laboratory makes no errors on the measurem ent or all the errors were r ecognizable and corrected. The corresponding variance equation will be: V(x i ) = V(Y i ) + V(e i ) or V(x i ) = s 2 (Y i ) + u 2 (e i ) (2) 2. Random laboratory effect The measurement equation will b e: x i = Y i + b i + e i (3) The expectation equation: E(x i ) = E(Y i ) + E(b i ) + E(e i ) or E(x i ) = Y (4) where b i is assumed to follow a statistics di stribution with zero expectation. The variance equation: V(x i ) = V(Y i ) + V(b i ) + V(e i ) or V(x i ) = s 2 (Y i ) + s 2 (b i ) + u 2 (e i ) (5) 3. Systematic laboratory effect The measurement equation will b e: x i = Y i + b i + e i (6) where b i becomes a constant now. The expectation and variance equation: E(x i ) = E(Y i ) + E(b i ) + E(e i ) (7) V(x i ) = V(Y i ) + V(b i ) + V(e i ) (8) or E(x i ) = Y + b i (9) V(x i ) = V(Y i ) + u 2 (e i ) (10) III. Key reference values 1. None laboratory effect The key reference value: x K = ( Σ i x i / (s 2 (Y i ) + u 2 (e i )))/( Σ i 1/ (s 2 (Y i ) + u 2 (e i ))), u(x K ) = 1/ √ ( Σ i 1/ (s 2 (Y i ) + u 2 (e i ))) (11) 2. Random laboratory effect The key reference value: x K = ( Σ i x i / (s 2 (Y i ) + s 2 (b i ) + u 2 (e i )))/( Σ i 1/ (s 2 (Y i ) + s 2 (b i ) + u 2 (e i ))), u(x K ) = 1/ √ ( Σ i 1/ (s 2 (Y i ) + s 2 (b i ) + u 2 (e i ))) (12) 3. Systematic laboratory effect The key reference value: x K = ( Σ i x i / (s 2 (Y i ) + u 2 (e i )))/( Σ i 1/ (s 2 (Y i ) + u 2 (e i ))), u(x K ) = 1/ √ ( Σ i 1/ (s 2 (Y i ) + u 2 (e i ))) (13) IV. Degrees of equivalence 1. None laboratory effect Measurement models of any two participating laboratories: x i = Y i + e i and x j = Y j + e j (14) Deviation of measured values of two laboratories: d ij = x i - x j = Y i - Y j + e i - e j (15) Deviation of a measured value and the key reference value: d i = x i - x K = Y i + e i - ( Σ j x j / (s 2 (Y j ) + u 2 (e j )))/( Σ j 1/ (s 2 (Y j ) + u 2 (e j ))) (16) The expectation values: E(d ij ) = E(Y i ) - E(Y j ) + E(e i ) - E(e j ) = 0, E(d i ) = E(x i ) - E(x K ) = E(Y i ) + E(e i ) - ( Σ j E(x j )/ (s 2 (Y j ) + u 2 (e j )))/( Σ j 1/ (s 2 (Y j ) + u 2 (e j ))) = E(Y i ) + E(e i ) - ( Σ j E(Y j + e j )/ (s 2 (Y j ) + u 2 (e j )))/( Σ j 1/ (s 2 (Y j ) + u 2 (e j ))) = E(Y i ) + E(e i ) - ( Σ j E(Y j )/ (s 2 (Y j ) + u 2 (e j )))/( Σ j 1/ (s 2 (Y j ) + u 2 (e j ))) = E(Y i ) - ( Σ j E(Y j )/ (s 2 (Y j ) + u 2 (e j )))/( Σ j 1/ (s 2 (Y j ) + u 2 (e j ))) = E(Y i ) - E(Y j ) = 0 (17) 2. Random laboratory effect Measurement models of any two participating laboratories: x i = Y i + b i + e i and x j = Y j + b j + e j (18) Deviation of measured values of two laboratories: d ij = x i - x j = Y i - Y j + b i - b j + e i - e j (19) Deviation of a measured value and the key reference value: d i = x i - x K = Y i + b i + e i - ( Σ j x j / (s 2 (Y j ) + s 2 (b j ) + u 2 (e j )))/( Σ j 1/ (s 2 (Y j ) + s 2 (b j ) + u 2 (e j ))) (20) The expectation values: E(d i ) = E(Y i ) + E(b i ) + E(e i ) - ( Σ j E(x j )/ (s 2 (Y j ) + s 2 (b j ) + u 2 (e j )))/( Σ j 1/ (s 2 (Y j ) + s 2 (b j ) + u 2 (e j ))) = E(Y i ) + E(b i ) + E(e i ) - E(Y j ) = 0 and d ij = 0 (21) 3. Systematic laboratory effect Measurement models of any two participating laboratories: x i = Y i + b i + e i và x j = Y j + b j + e j (22) Deviation of measured values of two laboratories: d ij = x i - x j = Y i - Y j + b i - b j + e i - e j (23) Deviation of a measured value and the key reference value: d i = x i - x K = Y i + b i + e i - ( Σ j x j / u 2 (e j ))/( Σ j 1/ (u 2 (Y j ) + u 2 (e j ))) (24) The expectation values: E(d i ) = E(Y i ) + E(b i ) + E(e i ) - ( Σ j E(Y j + b j + e j )/ u 2 (e j ))/( Σ j 1/ (u 2 (Y j ) + u 2 (e j ))) = E(Y i ) + E(b i ) + E(e i ) - ( Σ j E(Y j ) + E(b j ) + E(e j ))/ u 2 (e j ))/( Σ j 1/ (u 2 (Y j ) + u 2 (e j ))) = E(Y i ) + E(b i ) + E(e i ) - E(Y j )- Σ j b j / u 2 (e j ))/( Σ j 1/ (u 2 (Y j ) + u 2 (e j ))) = E(b i ) - Σ j b j / u 2 (e j ))/( Σ j 1/ (u 2 (Y j ) + u 2 (e j ))) = b i - ( Σ j b j / u 2 (e j ))/( Σ j 1/ (u 2 (Y j ) + u 2 (e j ))) and E(d ij ) = E(Y i ) - E(Y j ) + E(b i ) - E(b j ) + E(e i ) - E(e j ) = b i - b j (25) V. Discussion The approach in this report accepted the assump tion of existence of a set of true values instead of the existence of only one unique true value for a given measurem ent standard of the artifact in a key comparison. Those true values are distributed in a common probabilistic density function. The corresponding degrees of equivalence, or in other wo rds, the deviations and their measurement uncertainties are then analyzed. It is then s een that if a given participating laboratory did not contribute any error to the measurem ent or th e error contributed of this laboratory to the measurement is random in nature as seen in equations (17) and (21), then the expectations are always zero. These imply th at the laboratories under question are always equivalent which is not a reasonable acceptance. Th is fact im plies that they should not be good models for a key comparison. In contrast, if a participating labo ratory contributed to the measurem ent a systematic error then the expectations of deviati ons are not possibly zero in all cases as seen in equation (25). The systematic errors committed by each one b i and b j and their uncertainties will definitely decide if they are equivalent or not. And then this mode l could be assigned to be a good model to describe the measurement process. It is worthy to notice th at this conclusion is coincident to the one in [4]. VI. Conclusion In this report, the degree of equivalence is considered in three different models. The explicit deviations of each labor atory pairs and that of one la boratory with the key reference value are derived. The expectations of the devia tions and then the degrees of equivalence are analyzed for each model with the assumption of multiple true values. Th e result support that the laboratory’s systematic e rror model is the accepte d one. The result is similar to the one in [4]. Acknowledgement : Dr. Nguyen Duc Dung and Dr. Tran Bao have contributed to the discussion and minutes of the report. References [1] International Committee for Weights and Measures (CIPM): Mutual recognition of national measurement standards and of calibration and me asurement certificates issued by national metrology institutes , Technical Report, 1999 ( w ww.bipm.org/pdf/mra.pdf ). [2] International Committee of W eights and Measures (CIPM): Guidelines for CIPM key comparisons ,1 March 1999, (http://www1.bipm.org /utils/en/pdf/guidelines.pdf). [3] M. G. Cox: The evaluation of key comparison data , Metrologia 39, pp 589-595, 2002. [4] R. N. Kacker, R. U. Datla , and A. C. Parr, National Ins titute of Standards and Technology, Gaithersburg, MD 20899-0001 USA: Statistical Interpretation of Key Comparison Reference Value and Degrees of Equivalence , J. Res. Natl. Inst. Stand. Technol. 108, 439-446 (2003). [5] BIPM key comparison data base ( http://kcdb.bipm.org ). [6] Evaluation of measurement data — Guide to the expression of uncertainty in measurem ent (http://www.bipm.org/utils/comm on/documents/jcgm/JCGM_100_2008_E.pdf). [7] International vocabulary of metrology — Basi c and general concepts and associated terms (VIM) (http://www .bipm.org/utils/common/docum ents/jcgm /JCGM_200_2008.pdf).