Hellenic Complex Systems Laboratory

Transcription

1 Hellenic Complex Systems Laboratory Technical Report No IV Calculation of the confidence bounds for the fraction nonconforming of normal populations of measurements in clinical laboratory medicine Aristides T. Hatjimihail, M.D., Ph.D. Drama 2015

2 HCSLTR04 Hellenic Complex Systems Laboratory, K. Palamas 21, GR Drama, Greece Copyright Aristides T. Hatjimihail,

3 1. Abstract The fraction nonconforming is a key quality measure used in statistical quality control design in clinical laboratory medicine. The confidence bounds of normal populations of measurements for the fraction nonconforming each of the lower and upper quality specification limits when both the random and the systematic error are unknown can be calculated using the noncentral t-distribution, as it is described in detail and illustrated with examples. 3

4 2. Introduction The quality control (QC) in clinical laboratory medicine is based mainly on the control charts for variables. Although the fraction nonconforming is used in the control charts for attributes, it is also a useful quality measure related to various process capability indices (Pearn & Kotz 2006). It is defined as the ratio of the items nonconforming to the total number of items in a subgroup (Juran et al. 1999). The quality control design process (Hatjimihail 1992) is based on the quality specifications of the maximum total allowable analytical error, the upper acceptable bound for the fraction nonconforming, the lower acceptable bounds for the probabilities for the critical random and systematic error detection, and the upper acceptable bound for the false rejection (CLSI 2012; Linnet 1989). It includes the following steps: 1. Estimation of the error of the measurement procedure based on a sample of n control measurements. 2. Calculation of the critical errors of the measurement procedure based on an upper acceptable bound for the expected fraction nonconforming the maximum total allowable analytical error quality specification. This bound is usually set equal to Definition of the decision bounds for the quality control rules based on the lower acceptable bounds for the probabilities for the critical errors detection and the upper acceptable bound for false rejection. These steps are performed at typically one to three levels of the measurand. Therefore, the expected fraction nonconforming is a key measure of the quality control design process. While this measure is routinely calculated in the clinical laboratories, very little attention has been given to its confidence bounds. 4

5 In general, for the quality control estimations in laboratory medicine it is assumed that the error of a measurement procedure is normally distributed with known mean and standard deviation. Usually in practice we only know the mean x and the standard deviation s of a sample of n control measurements at each of various levels of the measurand. As both the systematic and the random error of the measurement procedure are unknown lower and upper bounds for the tail probabilities of the lower (x LSL ), and the upper (x USL ) quality specifications of the measurement x can be calculated with confidence level γ using the noncentral t-distribution (1987), as described in detail in the Materials and Methods section. Therefore, the purpose of this technical report is to introduce the noncentral t-distribution into clinical laboratory medicine and to describe its application for the calculation of the confidence bounds of the fraction nonconforming each of the lower and upper quality specification limits of populations of normal measurements in a clinical laboratory setting. 3. Materials and Methods If Z a standard normal N (0, 1) deviate and V a chi-square deviate with v degrees of freedom, the distribution of the ratio is the noncentral t-distribution with ν degrees of freedom and noncentrality parameter δ. T, Z (3.1) V The cumulative density function of a noncentral t-distribution with ν degrees of freedom and noncentrality parameter δ is defined as (Owen 1965): 5

6 2 tx v1 F, ( t) x ( x) dx 1/2 ( 2)/2 ( / 2) 2 0 v (3.2) where ( x) the gamma function and: 2 x 2 1 () x e (3.3) 2 and x ( x) ( t) dt (3.4) Scholz (Scholz 1994) presents an elegant derivation of the lower and upper bounds for the lower tail probability p(x LSL ) of a normal population with unknown mean μ and unknown standard deviation σ. Briefly, xlsl pxlsl pxlsl,, P, X xlsl Φ σ (3.5) If we denote with pˆul an upper bound for p(x LSL ) with confidence level γ then ˆ, UL LSL P p p x (3.6) Let / Z n X (3.7) a standard normal deviate, and 6

7 2 2 1 / V n S (3.8) a chi-square deviate with n-1 degrees of freedom. If n x LSL 1 nφ p xlsl (3.9) then, n x LSL s n X n x LSL x 2 n X 2 n n 1 n x LSL n1 S s Z T V 1 n1, (3.10) where the Tn 1, has by definition a noncentral t-distribution with n-1 degrees of freedom and noncentrality parameter δ. According to the probability integral transformation (Qeuesenberry 2006) the random variable U is uniformly distributed over (0,1). Therefore, n x X U F F T S LSL n1, n1, n1, (3.11) PU ( 1 ) (3.12) As x Fn 1, is a decreasing function of δ we have 7

8 F n1, n xlsl X 1 ˆ S (3.13) where ˆ is estimated by the following equation: F n x X LSL 1, ˆ 1 n S (3.14) Consequently, the upper bound with confidence level γ for the fraction non conforming the lower quality specification limit is calculated as ˆ p UL ˆ Φ n (3.15) As pˆ ˆ UL LSL pul LSL P p x P p x (3.16),, 1 the lower bound with confidence level γ for the fraction non conforming the lower quality specification limit is calculated as p pˆ ˆ 1 (3.17) LL UL In addition, if we denote with pˆuu fraction non conforming the upper quality specification limit then the upper bound with confidence level γ for the P, pˆ UU p xusl p x P ˆ, 1 puu 1 USL 1 (3.18) Therefore, the lower and upper bounds with confidence level γ for the fraction non conforming the upper quality specification limit are calculated respectively as 8

9 pˆ 1 pˆ (3.19) ' LU UU and pˆ ' UU 1 pˆ 1 (3.20) UU If we denote with c the true value of the measurand of the control material, and e max the maximum total clinically allowable analytical error (Linnet 1989), expressed as a ratio to the true value, then x c(1 e ) (3.21) LSL max and x c(1 e ) (3.22) USL max To illustrate the application of the noncentral t-distribution for the calculation of the confidence bounds of the fraction nonconforming of normal populations of measurements in a clinical laboratory setting I calculated upper and lower confidence bounds for the fraction nonconforming the upper and lower quality specification limits of populations of normal measurements, assuming unknown systematic and random errors of the measurement procedure. The calculations were performed by MATLAB, ver. 2012b, using the method described by Owen (Owen 1965). It was assumed that a reference control material with a measurand with an assigned value of units has been measured repeatedly by the measuring procedure and that the sample mean and standard deviation of the measurand were estimated from samples of 20 and 40 control measurements. It was also assumed that both the systematic and the random error of the measurement procedure were unknown and that the uncertainty of the assigned value of 9

10 the measurand of the control material was negligible. The lower and upper confidence bounds for the fraction nonconforming each of the lower and upper quality specifications of the population of the measurements were calculated by the noncentral t-distribution as described previously. The parameters of these calculations are presented in the Table 1. figures tables assigned value of the measurand μ of the control material sample mean x increments of the sample mean sample s.d. s increments of the sample s.d total allowable analytical error e max confidence level γ sample size n Table 1: The parameters of the calculations of the confidence bounds for the fraction nonconforming of the populations of the normal control measurements. 4. Results The results of the study are described in the following subsections 4.1 and 4.2. They are presented in the tables 2-5 and in the figures 1-8. They are also presented in the respective MATLAB figures included in the supporting information file MATLAB_Figures.zip so that the data can be explored interactively in the MATLAB environment. 10

11 parameters of the sample confidence bounds for the fraction nonconforming at a 95% confidence level size of the control measurements sample n = 20 total allowable analytical error taae = 0.10 fraction nonconforming the lower quality specification limit fraction nonconforming the upper quality specification limit x s expected lower bound upper bound expected lower bound upper bound ,62E-24 2,14E-33 2,85E-15 7,62E-24 2,14E-33 2,85E ,87E-07 8,69E-10 5,32E-05 2,87E-07 8,69E-10 5,32E ,29E-04 2,63E-05 5,36E-03 4,29E-04 2,63E-05 5,36E ,21E-03 1,09E-03 2,99E-02 6,21E-03 1,09E-03 2,99E ,28E-02 6,56E-03 7,01E-02 2,28E-02 6,56E-03 7,01E ,91E-28 5,78E-40 4,15E-18 1,13E-19 1,92E-27 1,07E ,90E-08 1,85E-11 9,75E-06 3,40E-06 2,88E-08 2,51E ,23E-04 4,53E-06 2,42E-03 1,35E-03 1,31E-04 1,12E ,98E-03 3,94E-04 1,86E-02 1,22E-02 2,79E-03 4,66E ,39E-02 3,33E-03 5,07E-02 3,59E-02 1,22E-02 9,50E ,78E-33 3,76E-47 3,30E-21 6,22E-16 4,14E-22 2,19E ,87E-10 2,78E-13 1,54E-06 3,17E-05 6,72E-07 1,02E ,17E-05 6,72E-07 1,02E-03 3,83E-03 5,59E-04 2,19E ,35E-03 1,31E-04 1,12E-02 2,28E-02 6,56E-03 7,01E ,20E-03 1,61E-03 3,58E-02 5,48E-02 2,17E-02 1,26E ,12E-39 5,81E-55 1,43E-24 1,28E-12 2,17E-17 2,48E ,02E-11 2,93E-15 2,10E-07 2,33E-04 1,11E-05 3,63E ,34E-06 8,54E-08 4,07E-04 9,82E-03 2,06E-03 4,03E ,77E-04 3,98E-05 6,47E-03 4,01E-02 1,42E-02 1,02E ,66E-03 7,34E-04 2,48E-02 8,08E-02 3,67E-02 1,64E ,79E-45 2,18E-63 3,37E-28 9,87E-10 2,78E-13 1,54E ,28E-12 2,17E-17 2,48E-08 1,35E-03 1,31E-04 1,12E ,53E-06 9,31E-09 1,52E-04 2,28E-02 6,56E-03 7,01E ,33E-04 1,11E-05 3,63E-03 6,68E-02 2,84E-02 1,44E ,56E-03 3,18E-04 1,68E-02 1,15E-01 5,88E-02 2,10E ,67E-51 1,96E-72 4,33E-32 2,87E-07 8,69E-10 5,32E ,19E-14 1,13E-19 2,51E-09 6,21E-03 1,09E-03 2,99E ,87E-07 8,69E-10 5,32E-05 4,78E-02 1,81E-02 1,15E ,84E-05 2,85E-06 1,96E-03 1,06E-01 5,25E-02 1,98E ,35E-03 1,31E-04 1,12E-02 1,59E-01 8,99E-02 2,63E-01 Table 2: Confidence bounds for the fraction nonconforming the lower and upper quality specification limits assuming a true value of the measurand of the control material equal to measurement units, a sample of 20 control measurements, and a total allowable analytical error equal to

12 parameters of the sample confidence bounds for the fraction nonconforming at a 95% confidence level size of the control measurements sample n = 40 total allowable analytical error taae = 0.10 fraction nonconforming the lower quality specification limit fraction nonconforming the upper quality specification limit x s expected lower bound upper bound expected lower bound upper bound ,62E-24 1,85E-30 1,05E-17 7,62E-24 1,85E-30 1,05E ,87E-07 5,14E-09 1,20E-05 2,87E-07 5,14E-09 1,20E ,29E-04 6,17E-05 2,61E-03 4,29E-04 6,17E-05 2,61E ,21E-03 1,86E-03 1,91E-02 6,21E-03 1,86E-03 1,91E ,28E-02 9,58E-03 5,09E-02 2,28E-02 9,58E-03 5,09E ,91E-28 2,01E-36 4,82E-21 1,13E-19 4,68E-25 1,12E ,90E-08 1,55E-10 1,65E-06 3,40E-06 1,24E-07 7,38E ,23E-04 1,25E-05 1,04E-03 1,35E-03 2,67E-04 6,12E ,98E-03 7,31E-04 1,10E-02 1,22E-02 4,38E-03 3,18E ,39E-02 5,15E-03 3,51E-02 3,59E-02 1,70E-02 7,21E ,78E-33 5,97E-43 1,08E-24 6,22E-16 3,27E-20 5,83E ,87E-10 3,41E-12 1,91E-07 3,17E-05 2,19E-06 3,81E ,17E-05 2,19E-06 3,81E-04 3,83E-03 1,01E-03 1,33E ,35E-03 2,67E-04 6,12E-03 2,28E-02 9,58E-03 5,09E ,20E-03 2,64E-03 2,36E-02 5,48E-02 2,88E-02 9,97E ,12E-39 4,90E-50 1,19E-28 1,28E-12 6,34E-16 1,50E ,02E-11 5,46E-14 1,85E-08 2,33E-04 2,82E-05 1,66E ,34E-06 3,35E-07 1,30E-04 9,82E-03 3,32E-03 2,70E ,77E-04 9,02E-05 3,25E-03 4,01E-02 1,95E-02 7,84E ,66E-03 1,29E-03 1,54E-02 8,08E-02 4,66E-02 1,34E ,79E-45 1,10E-57 6,39E-33 9,87E-10 3,41E-12 1,91E ,28E-12 6,34E-16 1,50E-09 1,35E-03 2,67E-04 6,12E ,53E-06 4,45E-08 4,11E-05 2,28E-02 9,58E-03 5,09E ,33E-04 2,82E-05 1,66E-03 6,68E-02 3,69E-02 1,16E ,56E-03 6,01E-04 9,84E-03 1,15E-01 7,22E-02 1,77E ,67E-51 6,80E-66 1,67E-37 2,87E-07 5,14E-09 1,20E ,19E-14 5,35E-18 1,02E-10 6,21E-03 1,86E-03 1,91E ,87E-07 5,14E-09 1,20E-05 4,78E-02 2,43E-02 8,97E ,84E-05 8,17E-06 8,12E-04 1,06E-01 6,50E-02 1,66E ,35E-03 2,67E-04 6,12E-03 1,59E-01 1,07E-01 2,28E-01 Table 3: Confidence bounds for the fraction nonconforming the lower and upper quality specification limits assuming a true value of the measurand of the control material equal to measurement units, a sample of 40 control measurements, and a total allowable analytical error equal to

13 parameters of the sample confidence bounds for the fraction nonconforming at a 95% confidence level size of the control measurements sample n = 20 total allowable analytical error taae = 0.20 fraction nonconforming the lower quality specification limit fraction nonconforming the upper quality specification limit x s expected lower bound upper bound expected lower bound upper bound ,75E-89 5,76E-127 1,62E-55 2,75E-89 5,76E-127 1,62E ,62E-24 2,14E-33 2,85E-15 7,62E-24 2,14E-33 2,85E ,31E-11 5,94E-16 1,05E-07 1,31E-11 5,94E-16 1,05E ,87E-07 8,69E-10 5,32E-05 2,87E-07 8,69E-10 5,32E ,17E-05 6,72E-07 1,02E-03 3,17E-05 6,72E-07 1,02E ,28E-98 9,84E-140 5,34E-61 8,53E-81 8,09E-115 2,66E ,32E-26 1,33E-36 1,17E-16 1,05E-21 2,42E-30 5,95E ,28E-12 2,17E-17 2,48E-08 1,20E-10 1,39E-14 4,16E ,60E-08 1,32E-10 2,32E-05 1,02E-06 5,22E-09 1,18E ,33E-05 1,99E-07 5,94E-04 7,23E-05 2,15E-06 1,73E ,44E-107 4,03E-153 9,58E-67 9,74E-73 2,72E-103 2,37E ,91E-28 5,78E-40 4,15E-18 1,13E-19 1,92E-27 1,07E ,12E-13 6,79E-19 5,48E-09 9,87E-10 2,78E-13 1,54E ,90E-08 1,85E-11 9,75E-06 3,40E-06 2,88E-08 2,51E ,41E-06 5,55E-08 3,36E-04 1,59E-04 6,52E-06 2,85E ,33E-117 3,95E-167 9,31E-73 4,11E-65 2,19E-92 1,15E ,60E-31 1,76E-43 1,26E-19 9,48E-18 1,06E-24 1,65E ,83E-15 1,82E-20 1,13E-09 7,28E-09 4,75E-12 5,36E ,46E-09 2,37E-12 3,95E-06 1,07E-05 1,45E-07 5,16E ,11E-06 1,47E-08 1,86E-04 3,37E-04 1,87E-05 4,59E ,39E-127 9,26E-182 4,91E-79 6,39E-58 4,23E-82 3,03E ,78E-33 3,76E-47 3,30E-21 6,22E-16 4,14E-22 2,19E ,22E-16 4,14E-22 2,19E-10 4,82E-08 6,94E-11 1,74E ,87E-10 2,78E-13 1,54E-06 3,17E-05 6,72E-07 1,02E ,93E-07 3,67E-09 1,01E-04 6,87E-04 5,08E-05 7,24E ,06E-138 5,20E-197 1,41E-85 3,67E-51 1,96E-72 4,33E ,73E-36 5,56E-51 7,40E-23 3,19E-14 1,13E-19 2,51E ,93E-17 8,08E-24 3,98E-11 2,87E-07 8,69E-10 5,32E ,05E-10 2,98E-14 5,80E-07 8,84E-05 2,85E-06 1,96E ,87E-07 8,69E-10 5,32E-05 1,35E-03 1,31E-04 1,12E-02 Table 4: Confidence bounds for the fraction nonconforming the lower and upper quality specification limits assuming a true value of the measurand of the control material equal to measurement units, a sample of 20 control measurements, and a total allowable analytical error equal to

14 parameters of the sample confidence bounds for the fraction nonconforming at a 95% confidence level size of the sample of the measurements n = 40 total allowable analytical error taae = 0.20 fraction nonconforming the lower quality specification limit fraction nonconforming the upper quality specification limit x s expected lower bound upper bound expected lower bound upper bound ,75E-89 2,23E-115 4,21E-65 2,75E-89 2,23E-115 4,21E ,62E-24 1,85E-30 1,05E-17 7,62E-24 1,85E-30 1,05E ,31E-11 1,28E-14 8,16E-09 1,31E-11 1,28E-14 8,16E ,87E-07 5,14E-09 1,20E-05 2,87E-07 5,14E-09 1,20E ,17E-05 2,19E-06 3,81E-04 3,17E-05 2,19E-06 3,81E ,28E-98 5,78E-127 1,47E-71 8,53E-81 2,34E-104 5,88E ,32E-26 2,26E-33 2,46E-19 1,05E-21 1,09E-27 3,74E ,28E-12 6,34E-16 1,50E-09 1,20E-10 2,24E-13 4,10E ,60E-08 9,30E-10 4,56E-06 1,02E-06 2,63E-08 3,05E ,33E-05 7,21E-07 2,02E-04 7,23E-05 6,32E-06 7,01E ,44E-107 4,11E-139 2,48E-78 9,74E-73 6,75E-94 4,00E ,91E-28 2,01E-36 4,82E-21 1,13E-19 4,68E-25 1,12E ,12E-13 2,72E-17 2,55E-10 9,87E-10 3,41E-12 1,91E ,90E-08 1,55E-10 1,65E-06 3,40E-06 1,24E-07 7,38E ,41E-06 2,26E-07 1,04E-04 1,59E-04 1,73E-05 1,25E ,33E-117 7,98E-152 2,04E-85 4,11E-65 5,33E-84 1,32E ,60E-31 1,28E-39 7,90E-23 9,48E-18 1,45E-22 2,78E ,83E-15 1,01E-18 4,01E-11 7,28E-09 4,51E-11 8,21E ,46E-09 2,40E-11 5,74E-07 1,07E-05 5,42E-07 1,71E ,11E-06 6,74E-08 5,21E-05 3,37E-04 4,53E-05 2,18E ,39E-127 4,24E-165 8,18E-93 6,39E-58 1,15E-74 2,13E ,78E-33 5,97E-43 1,08E-24 6,22E-16 3,27E-20 5,83E ,22E-16 3,27E-20 5,83E-12 4,82E-08 5,17E-10 3,27E ,87E-10 3,41E-12 1,91E-07 3,17E-05 2,19E-06 3,81E ,93E-07 1,91E-08 2,54E-05 6,87E-04 1,13E-04 3,70E ,06E-138 6,16E-179 1,59E-100 3,67E-51 6,80E-66 1,67E ,73E-36 2,01E-46 1,24E-26 3,19E-14 5,35E-18 1,02E ,93E-17 9,16E-22 7,82E-13 2,87E-07 5,14E-09 1,20E ,05E-10 4,49E-13 6,07E-08 8,84E-05 8,17E-06 8,12E ,87E-07 5,14E-09 1,20E-05 1,35E-03 2,67E-04 6,12E-03 Table 5: Confidence bounds for the fraction nonconforming the lower and upper quality specification limits assuming a true value of the measurand of the control material equal to measurement units, a sample of 40 control measurements, and a total allowable analytical error equal to Fraction nonconforming the lower quality specification limit Figures 1, 3, 5, and 7 present the expected fraction nonconforming the lower quality specification limit (surface II) and the lower (surface III) and upper (surface I) confidence bounds for the fraction nonconforming at a 95% confidence level versus the sample mean and the sample standard deviation. The confidence bounds were calculated using the noncentral t- distribution. It was assumed a known true value of the measurand of the control material equal to measurement units. For the figures 1 and 3 it was assumed a total allowable analytical error equal to 0.10 while for the figures 5 and 7 it was assumed a total allowable 14

15 analytical error equal to The respective samples were samples of 20 control measurements for the figures 1 and 5, and samples of 40 control measurements for the figures 3 and Fraction nonconforming the upper quality specification limit Figures 2, 4, 6, and 8 present the expected fraction nonconforming the upper quality specification limit (surface II) and the lower (surface III) and upper (surface I) confidence bounds for the fraction nonconforming at a 95% confidence level versus the sample mean and the sample standard deviation. The confidence bounds were calculated using the noncentral t- distribution. It was assumed a known true value of the measurand of the control material equal to measurement units. For the figures 2 and 4 it was assumed a total allowable analytical error equal to 0.10 while for the figures 6 and 8 it was assumed a total allowable analytical error equal to The respective samples were samples of 20 control measurements for the figures 2 and 6, and samples of 40 control measurements for the figures 4 and 8. The results are summarized in the tables 2 and 3, assuming a total allowable analytical error equal to 0.10, and in the tables 4 and 5, assuming a total allowable analytical error equal to The respective samples were samples of 20 control measurements for the tables 2 and 4, and samples of 40 control measurements for the tables 3 and 5. The confidence bounds increase with the estimated systematic and random error of the measurements and decrease with the sample size and the total allowable analytical error. Although there have been assumed samples with a positive systematic error, it can be easily shown that the bounds for the fraction nonconforming the upper quality specification limit for a systematic error se are equal to the respective bounds for the fraction nonconforming the lower quality specification limit for a systematic error se. 15

16 Figure 1: Confidence bounds for the fraction nonconforming the lower quality specification limit. The expected fraction nonconforming the lower quality specification limit (surface II) and the lower (surface III) and upper (surface I) confidence bounds for the fraction nonconforming at a 95% confidence level versus the sample mean and the sample standard deviation. It is assumed a known true value of the measurand of the control material equal to measurement units, a sample of 20 control measurements, and a total allowable analytical error equal to

17 Figure 2: Confidence bounds for the fraction nonconforming the upper quality specification limit. The expected fraction nonconforming the upper quality specification limit (surface II) and the lower (surface III) and upper (surface I) confidence bounds for the fraction nonconforming at a 95% confidence level versus the sample mean and the sample standard deviation. It is assumed a known true value of the measurand of the control material equal to measurement units, a sample of 20 control measurements, and a total allowable analytical error equal to

18 Figure 3: Confidence bounds for the fraction nonconforming the lower quality specification limit. The expected fraction nonconforming the lower quality specification limit (surface II) and the lower (surface III) and upper (surface I) confidence bounds for the fraction nonconforming at a 95% confidence level versus the sample mean and the sample standard deviation. It is assumed a known true value of the measurand of the control material equal to measurement units, a sample of 40 control measurements, and a total allowable analytical error equal to

19 Figure 4: Confidence bounds for the fraction nonconforming the upper quality specification limit. The expected fraction nonconforming the upper quality specification limit (surface II) and the lower (surface III) and upper (surface I) confidence bounds for the fraction nonconforming at a 95% confidence level versus the sample mean and the sample standard deviation. It is assumed a known true value of the measurand of the control material equal to measurement units, a sample of 40 control measurements, and a total allowable analytical error equal to

20 Figure 5: Confidence bounds for the fraction nonconforming the lower quality specification limit. The expected fraction nonconforming the lower quality specification limit (surface II) and the lower (surface III) and upper (surface I) confidence bounds for the fraction nonconforming at a 95% confidence level versus the sample mean and the sample standard deviation. It is assumed a known true value of the measurand of the control material equal to measurement units, a sample of 20 control measurements, and a total allowable analytical error equal to

21 Figure 6: Confidence bounds for the fraction nonconforming the upper quality specification limit. The expected fraction nonconforming the upper quality specification limit (surface II) and the lower (surface III) and upper (surface I) confidence bounds for the fraction nonconforming at a 95% confidence level versus the sample mean and the sample standard deviation. It is assumed a known true value of the measurand of the control material equal to measurement units, a sample of 20 control measurements, and a total allowable analytical error equal to

22 Figure 7: Confidence bounds for the fraction nonconforming the lower quality specification limit. The expected fraction nonconforming the lower quality specification limit (surface II) and the lower (surface III) and upper (surface I) confidence bounds for the fraction nonconforming at a 95% confidence level versus the sample mean and the sample standard deviation. It is assumed a known true value of the measurand of the control material equal to measurement units, a sample of 40 control measurements, and a total allowable analytical error equal to

23 Figure 8: Confidence bounds for the fraction nonconforming the upper quality specification limit. The expected fraction nonconforming the upper quality specification limit (surface II) and the lower (surface III) and upper (surface I) confidence bounds for the fraction nonconforming at a 95% confidence level versus the sample mean and the sample standard deviation. It is assumed a known true value of the measurand of the control material equal to measurement units, a sample of 40 control measurements, and a total allowable analytical error equal to

24 5. Discussion In clinical laboratory medicine the fraction nonconforming besides being a quality measure related to various process capability indices, it is used for the definition of the critical errors during the QC design process. To estimate the fraction nonconforming the upper and lower quality specification bounds for the measurement error are needed. These are defined by the maximum total clinically allowable analytical error (Linnet 1989). As both the systematic and the random error of any measurement procedure are actually unknown, we can calculate the respective confidence bounds for the fraction nonconforming each of the upper and lower quality specification limits, using the noncentral t-distribution. The calculation is mathematically complicated and computationally intensive. It is not possible to calculate the confidence bounds for the total fraction nonconforming either the upper or the lower quality specification limits, without additional assumptions (Owen 1965). The results of the study show that the difference of the confidence bounds for the fraction nonconforming from the expected fraction nonconforming is considerable for greater random and systematic errors and for smaller sample sizes, even when we assume an insignificant uncertainty of the assigned value of the measurand of the control material. Consequently it is considerable when the expected fraction nonconforming equals its acceptable upper bound that is usually set equal to Therefore, the uncertainty of the calculation of the critical errors is considerable as well, as Gernand and Wojciechit have pointed out previously (Gernand 2006). Accordingly, the uncertainty of the whole quality control design process is significant, even assuming a negligible uncertainty of the assigned value of the measurand of the control material. Although it was assumed a reference control material with a measurand with an assigned value of , the tables and figures of this study can be used for the estimation of the 24

25 confidence bounds for the fraction nonconforming for any measurand with any assigned value. 6. Conclusion The confidence bounds for the fraction nonconforming are useful quality measures, as the expected fraction nonconforming is a key measure of the quality design process. The confidence bounds for the fraction nonconforming each of the upper and lower quality specification limits can be calculated using the noncentral t-distribution. 7. Acknowledgements I am deeply grateful to Dr. Fritz Scholz for providing me with the technical report on the applicatios of the noncentral t- distribution (Scholz 1994), that was the basis of this technical report, as well as for his suggestions and corrections. 25

26 8. References Sampling distributions. In: Ito K, ed. The encyclopedic dictionary of mathematics. 2nd ed. Boston, Massachusetts: The MIT Press, CLSI Expression of Uncertainty in Measurement in Laboratory Medicine. Wayne, Pennsylvania: Clinical and Laboratory Standards Institute. p Gernand W How Reliable Are Critical Error Calculations? Clinical Chemistry 52: Hatjimihail AT A tool for the design and evaluation of alternative quality-control procedures [published erratum appears in Clin Chem 1992 Nov;38(11):2360] [see comments]. Clin Chem 38: Juran JM, Godfrey AB, Hoogstoel RE, and Schilling EG Juran's quality handbook: McGraw Hill. Linnet K Choosing quality-control systems to detect maximum clinically allowable analytical errors. Clin Chem 35: Owen DB A special case of a bivariate non-central t-distribution. Biometrika 52: Pearn W, and Kotz S Encyclopedia and handbook of process capability indices: World Scientific Publishing. Qeuesenberry CP Probability integral transformations. In: Kotz S, Balakrishnan N, Read C, and Vidakovic B, eds. Encyclopedia of statistical sciences. 2nd ed. Hoboken, New Jersey: John Wiley and Sons, Scholz FW Applications of the Noncentral t-distribution. Mathematics and Engineering Analysis Library Boeing Computer Services. p