Technical publications concerning ILC
Beta risk in proficiency testing in relation with the number of participants
The Monte Carlo method was applied to PT schemes to investigate their efficiency. Probabilities that the computed z values are over 3 while the true value is less than 2 and that the computed z values are less than 2 while the true values are over 3 are computed for a series of situations: number of participants from 5 to 30, various ratios of repeatability over reproducibility and number of test results per participant, introduction or not of outliers with z from 3,5 to 10. For each situation, the probabilities of not detecting true outliers and to trigger false alerts are discussed. Guidance and keys are proposed to check and improve the efficiency of real PT programs.
Abstract of conclusions:
This study demonstrates that:
- The ratio
is of main importance to control the efficiency of a PT scheme, even more than the number of participants. The PT providers should then care Nr, number of test results per participant that they request; λ = σ r /( σ L × N r )
- Even in adverse conditions, the α-risk is always very low (less than 0,7%);
- Robust algorithms improve the efficiency of the PT program (i.e. β-risk) at a slight expense on α-risk (which always remain very low). This comes from a significantly better estimation of the standard deviation of reference when an outlier is present among the participants when these algorithms are used;
- A number of 6 participants is large enough to detect a strongly outlying participant provided that good PT conditions (i.e. low value of λ) are present;
- PT with a low number of participants is (almost) always better than no PT at all.
ISO 5725-1 and ISO 13528 recommend not to organise an ILC with less than 12 participants. This makes sense for ISO 5725-1, which goal is to determine the performance of a test method. It makes less sense for ISO 13528, which goal is to check the performance of a lab. Obviously, when no PT is organised, β-risk is 100%: any lab having a problem can never at all realise it! Consequently, for test methods that are performed by a little number of labs, it is obviously better to organise PT with 6 participants than nothing. In those cases, the PT provider should specially care the Nr it requests, to ensure a proper λ value and consequently assure an efficiency as good as possible.
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Appropriate rankits to use for normal probability plots and Standard deviation probability plots
Normal probability plots are usually used to check whether a distribution can be regarded as Gaussian, to visualise whether some figures are likely to be outliers and, using a linear regression, to estimate its mean value and its standard deviation. In the same way, “SD probability plots”, based on the distribution of standard deviation estimates, could be quite useful to reach similar goals: check whether a hypothesis of homoscedasticity can be accepted or not, visualise estimates that are likely to be outliers, and estimate the true underlying standard deviation. In practice, a change of variable is necessary to change the rank of each value into a corresponding cumulated probability and inverse Gaussian transformation to get a “rankit” to be used as ordinates for these plots. Equations in the form of (i-a)/(N+1-2a) with 0 ≤ a ≤ 1 are usually used to determine the adequate cumulated probabilities. As a matter of fact, at least for small values of N, the choice of the “a” value has an important impact on the conclusions that are drawn afterwards. This document:
- Discusses the grounds of these equations;
- Evaluates their adequacy for a series of situations and types of distribution laws;
- Proposes equations to determine “a” values as function of N, that provide better rankits than usually used and enable to estimate mean values and/or standard deviations without any bias for a series of situations;
- Proposes an accurate way to determine envelope curves of confidence for normal probability plots and probability plots of any distribution which cumulative function is known.
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