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MIL-HDBK-1823A
"Nondestructive Evaluation System
Reliability Assessment"

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POD and PFP are inextricably joined

Making the detection threshold size smaller will improve the probability of detection (POD), since more of the signals will be above the lowered threshold (Figure 1).  Unfortunately, lowering the detection threshold size will also increase the probability of false positive (PFP), a fact too often ignored. 

Thus the "obvious" fix for an inadequate inspection - lower the decision threshold - won't work in practice.  If the threshold is lowered by fiat, the inspection throughput will plummet as false positives unnecessarily remove parts. 

Sometimes this results in a serious communications breakdown:  The engineering staff thinks the inspection threshold is set low, where they mandated it, while the inspections are using a higher threshold to make throughput quotas.  ("Not in MY shop," I can hear you say.  Are you sure?)

 

How POD and PFP depend on decision:

Look closely at Figure 1.  Imagine what happens when the horizontal line representing the decision threshold (‚ decision = 200) is lowered.  What happens? 

  1. The POD curve in the inset plot moves to the left as the probability of detection increases for any given size.
  2. The Probability of a False Positive, the tail of the noise probability distribution plotted on the left y-axis, also increases (gets worse).

example 2 ‚ vs.a summary plot

Figure 1: How POD is related to size through the signal vs. size (‚ vs. a) relationship.

 

Probability of Detection and Probability of False Positive are unavoidably related to detection threshold.   Both will increase or decrease together as the decision threshold moves up or down, but while an increase in POD is good, an increase in PFP is bad:  Decreasing the detection threshold will improve Probability of Detection, and make the Probability of a False Positive worse.  That is unavoidable.  (There's no "free lunch.")

Figure 2 graphically illustrates this by plotting the size at 90% POD (a90) on the left vertical axis and PFP on the right vertical axis, as functions of the decision threshold size.

Figure G-28

Figure 2:   ‚ decision†influences PFP, and the size at 90% POD, in a trade-off.