Building a Probit Spreadsheet: Building a Probit spreadsheet from the Logit is straightforward since they
differ only in the link and related cells. First make a copy of the Logit sheet from which
to work. Change columns F, G, and T, and U (also columns N and O), to reflect the Probit
link, as summarized in Table 3. Build Cell G6, the probit link, then copy into the
appropriate rows of the data sheet on page one of the spreadsheet. Build Cell T29 and copy
into the appropriate rows in the Model Evaluation, Right and Left Confidence Contour
tables on the second page of the spreadsheet; similarly for Cell U29. Again notice that
log(cracksize) at a given POD is used in EXCEL, while POD at a given
cracksize is used in Quattro Pro and Lotus 123, because of the plotting conventions in
the different spreadsheets.
Table 3 (probit link)

column F 
column G 
Cell T29
(also column N) 
Cell U29
(also column O) 

not used 
Link 
log(a) 
POD 
EXCEL 

=NORMSDIST($E6) 
=NORMSINV(U29)*X29+W29 
0.9 
Quattro Pro 

@NORMSDIST($E6) 
0.5 (say) 
@NORMSDIST((T29W29)/X29) 
Lotus 123 

not supported(^{1}) 


Lotus 123 does not support some @ functions, including
@NORMAL(•), when using the Solver. 
Interpreting the Results:
It is interesting to note the degree to which the
conventional quadratic form (an ellipse, Kendall and Stuart (1979)) approximates the
confidence region shown in Figure 2a. The shape seems about right  it looks like
an ellipse  but whereas the quadratic form is centered at the mle's, here the geometric
center of the confidence region and the mle's are far from coincident. Thus confidence
bounds based on the quadratic form would be conservative in some regions and
anticonservative in others. (In this example the resulting a_{90/95} for
the quadratic approximation  performed elsewhere  is 0.168 inches, or about half the
0.318 inches determined here using the likelihood ratio criterion.) The tilt of the
"ellipse" is due to the nonzero covariance between the location and scale
parameters, which itself results from an experiment with twice as many hits as misses. (Of
course designing an experiment with equal numbers of hits and misses would require knowing
the location parameter in advance.)
It is also interesting to see that unlike conventional regression
where observations at the extremes of the experimental space have greater influence than
those near the center, the opposite is true of binary data. The contributions to the total
likelihood of tests with very high probability of success  or failure  are minimal
unless there is an unexpected outcome, as can be seen by examining the spreadsheet, column
H. Not surprisingly the more influential tests are those for which the outcome is not a
foredrawn conclusion. For this reason experience suggests that NDE experiments should be
designed with about a quarter of the cracksizes small and likely to be missed, a quarter
large and likely to be found, and half in a cracksize range where the outcome is uncertain
beforehand. A discussion of the sensitivity of a maximum likelihood fit of a logistic
regression model and other similar models to extreme points in the design space can be
found in Pregibon (1981).
Finally, it is worthwhile to compare the logit and probit
descriptions of this FPI experiment. Not surprisingly, they are indistinguishable near the
center of the data, and the a_{50} is 0.033" for both the logit and
probit. At 90% POD they differ only slightly, 0.090" for the logit,
0.087" for the probit. (That difference, 0.003 inches, 3 mils, is about the diameter
of a human hair.) But the confidence statements by the different models begin to
illustrate the heavier tails exhibited by the logistic function. For the logit a_{90/95}
is 0.318" and is 0.266" for the probit. This provides a reminder of the
meaninglessness of confidence statements "correctly" estimated for an
inappropriate model.
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