| Model
Interpretation Those engaged in pharmacological studies
prefer the logit because S can be interpreted in terms of the odds of a treatment's
efficacy (cf.: Hosmer and Lemeshow, 1989, or Agresti, 1990). In NDE the probit
arises naturally in the analysis of some inspection methods, such as eddy current and to a
lesser extent ultrasound, which provide an indication of a defect's size in addition to
the hit or miss result. The analysis of such data is outside the scope of this discussion
except to note that when the signal strength and defect size are proportional
logarithmically and the error structure is lognormal, the resulting POD(a)
curve has a probit form. It is convenient to use the same form for hit/miss
inspections to facilitate comparisons among NDE methods. Further discussion can be found
in Annis, et.al., (1989). The probit Location and Scale parameters also have a
physical interpretation. L is log(cracksize) at 50% POD. Cracks smaller than
L (actually, 10L) are more often missed than found; those larger
than L are more often found than missed. S is related to the
"slope" of the POD(a) relationship at its midpoint, the smaller S,
the steeper the slope, and the more discriminating the NDE. For these reasons an earlier
version of this paper used the probit model as an example and spreadsheets were easily
constructed using EXCEL and Quattro Pro.
Unfortunately, Lotus 1-2-3
does not support the @NORMAL(•) function when the solver is used, making the probit
less attractive here as an example(3). The logit link is therefore used. The (x-L)/S
parameterization, where L0 + S0 x = (x-L)/S
= x/S - L/S ?L0 = -L/S, S0 =
1/S, is retained however because of the physical interpretation of the parameters.
Building spreadsheets using the both logit and probit links will be described later.
Example Data:
The data in this example result from an industrial experiment to
determine the effectiveness of fluorescent penetrant inspection, FPI. FPI is a method for
detecting cracks whereby the surface is wetted with a penetrating fluid which fluoresces
under ultraviolet light. The capillary action of the crack attracts the fluid which
remains in the crack after the excess has been removed from the surface. Then, as the
fluid seeps from the crack, it appears to glow under UV light. The brightness is not
well correlated with cracksize, nor is the apparent length of the indication. The
inspection provides little information other than its binary (pass/fail) outcome, making
GLM attractive for describing the relationship between size and detectability.
The data were produced by an ensemble of 33
flat-plate specimens containing 58 fatigue cracks varying in length from 0.003 inches to
0.44 inches. The specimens began as 1?" x
5" x ?" plates and small starter slots
(0.005" x 0.007") were electro-discharge-machined (edm)
into the surfaces to facilitate fatigue cracking. The specimens were then fatigued in
three-point bending until a crack of desired dimensions was produced. The surface was
ground to remove the starter edm slot, and 1?" x 3?" specimens were then cut from the plates so the cracks
were no longer positioned at the center. For smaller cracks the final cracksize was
determined using cellulose acetate replication.
Each specimen has two surfaces, and each surface is
mapped into a grid of three rows by nine columns using a transparent overlay template. The
inspector must correctly identify the row-column coordinates of any "find" for
it to be considered legitimate. Some surfaces had more than one crack and there were 19
sides with no cracks. (More recent procedures include specimens with no cracks on either
side.) The inspector was not told which side was cracked. These data are for one
inspection by a single inspector.
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