The traditional ‘decision matrix‘ of statistical hypothesis testing is augmented (above figure) with an economic (loss function or more generally impact) approach. Instead of arbitrary % risks, the decision-maker can assess real costs – which can even be a profit or a loss!
Customer risk cost (lower specification limit, LSL):
The consequence costs, C, of incorrect decisions for test result, ym, (distribution gtest) can be balanced against measurement & test costs, D:
A consideration is how to model the higher cost of increased efforts made to reduce measurement uncertainty. There are of course a number of conceivable models of how test costs could vary with measurement uncertainty.
In the present work, the test cost is assumed to depend inversely on the squared (standard) measurement uncertainty, that is,
, where D is the test cost at nominal test (standard) uncertainty umeasure. Such a model was suggested [Fearn et al. 2002], based mainly on the argument that N repeated measurements would reduce the standard deviation in the measurement result by √N while costing (at most) N times as much as each individual measurement. In the present work, this model of measurement costs is not only used where the statistical distribution associated with measurement uncertainties are known (such a type A evaluations), but is also extended to cover more generally even other components of uncertainty (including the expanded uncertainty in the overall final measurement result) where the underlying statistical distribution is often not known (type B evaluation).
T. Fearn, S. Fisher, M. Thompson and S. Ellison 2002, “A decision-theory approach to fitness for purpose in analytical measurement,” Analyst, vol. 127, pp. 818 – 824
L R Pendrill 2007, “Optimised Measurement Uncertainty and Decision-Making in Conformity Assessment“, NCSLi Measure, Vol. 2 No. 2 • June 2007, pp. 76 – 86
- Conformity assessment:
- Specifications of Process and Measurement Capabilities
- Risks for incorrect decisions
- Customer and suppler risks
- Introducing impact and cost into conformity assessment risks