Probit Spreadsheet

 

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 1-2-3, 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((T29-W29)/X29)

Lotus 1-2-3

 

not supported(1)

   
Lotus 1-2-3 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 2-a. 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 a90/95 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 non-zero 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 a50 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 a90/95 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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