Using a90/95 as a single-number summary of
an entire POD(a) relationship is not without its detractors, I among them.
Still, with all its shortcomings it is the currency for comparing NDE systems, and quality
practitioners need to be able to estimate it. It also provides a specific example of a
general procedure for determining an entire confidence contour. The confidence contour is
the locus of (L, S) pairs expected to encompass the true (L, S)
in 95% of similar experiments. Unfortunately, the corresponding POD(a) lower
bound does NOT have a simple functional form like logit or probit, but depends on the
shape of the (L, S) confidence contour, which itself depends on the
experimental design (selection of cracksizes) and the inspection results. The idea is to
select plausible values (values within a 95% confidence region) for which the cracksize at
90% POD is as large as possible, such that any further increase would require
parameter estimates no longer supported by the data, according to some criterion.
Hosmer and Lemeshow (1989) observe that some
pharmacological applications may be concerned only with the model's "slope"
(interpreted as incremental improvement per incremental change in the independent
variable) and are thus interested in confidence bounds for a single parameter only, and
Normal theory methods are usually used to establish confidence bounds on individual
parameter estimates. Simultaneous confidence intervals for multiple parameters also often
rely on Normal theory methods due to the computational simplification afforded by the
asymptotic behavior of the likelihood function (e.g.: McCallagh and Nelder, 1989).
However, for different applications, Ostrouchov and Meeker (1988) and Vander Wiel and
Meeker (1990) compare normal theory and likelihood ratio methods for constructing
confidence intervals for parameter estimates and suggest the increased computational
demands of likelihood ratio methods are warranted by their superior performance. The
likelihood ratio statistic is L = -2ln(l/l0),
where l0 is the maximum likelihood, and l is the likelihood
evaluated at other parameter values. This statistic has an asymptotic c2 distribution with
degrees of freedom equal to parameter rank, (c.f.: Kendall and Stuart, 1979) which for a
two parameter model is 2. We will use this criterion to establish our confidence region.
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