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eduKateSG · Why Mathematics? · Probability of Detection for Nondestructive Evaluation: Hit/Miss Data, Signal Response, Logistic Curves and Confidence Bounds
Start with the question, audit the worked example, or jump to the assumptions and practice. Every chapter stays open for uninterrupted reading.
Understand the question
Define the quantity and follow the mechanism.Route 2 · Chapters 6–10
Reproduce the example
Audit the numbers one step at a time.Route 3 · Chapters 11–15
Check assumptions
See where design and calibration matter.Route 4 · Chapters 16–20
Explore applications
Connect the method to a concrete investigation.Route 5 · Chapters 21–25
Study and verify
Practise, inspect failure cases and read further.
Full chapter index · Practice and checking · Mathematics Learning Hub
Why mathematics is important becomes concrete when an inspection must find a flaw before that flaw becomes dangerous. Nondestructive evaluation does not return a magical yes or no. Its performance depends on flaw size, material, geometry, equipment, procedure, inspector, noise and the rule used to call an indication. Probability of detection, or POD, turns repeated inspection evidence into a curve that can support a cautious inspection decision.
This guide develops hit/miss logistic models, signal-response analysis, false calls, confidence bounds and the meaning of a90/95. It uses worked examples to show why a fitted POD curve is not the same as a guarantee for one future component. The useful question is narrower and more honest: under a documented study design, how did detection probability change with the flaw characteristic, and how uncertain is that estimate?
Full chapter index
Chapters 1–5 · Understand the question
Chapters 6–10 · Reproduce the example
Chapters 11–15 · Check assumptions
Chapters 16–20 · Explore applications
Chapters 21–25 · Study and verify
Repeated inspections of nominally similar flaws can produce different outcomes because detection is affected by noise, access and human judgement. Ten inspectors may not all call the same small crack. This is the central mathematical mechanism in why inspection needs probability. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. A POD curve describes a study population and procedure, not every possible field condition. Define the inspection system and target flaw first. The useful habit is to keep the physical question, the probability model and the numerical method in separate columns so that a correct computation is not mistaken for complete evidence.
Did You Know? A result can be numerically stable and still answer the wrong question if the failure event, denominator or transformation changes. Test one nearby case and predict the direction of change before recalculating. Save inputs, intermediate values and units so a second reader can audit the path.
For practice, turn the example into a sensitivity check. Change one defensible input while holding the others fixed, compare the new result with the original and explain why it moved. Then construct a counterexample in which the shortcut fails. End with two sentences: what the mathematics supports, and what remains uncertain.
Start with the operational question behind the flaw variable a. POD studies relate detection to a declared flaw characteristic, commonly a crack length, depth or area denoted by a. In the illustrative case, A study might use crack length in millimetres as the predictor. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. One size measure may not capture orientation, closure or surface condition. Record geometry and units with every specimen.
A student should be able to rebuild the result without hidden settings. Record the sampling rule or optimisation criterion, the exact failure definition, and the evidence used for every distribution or parameter. Compare a transparent baseline with the advanced method. If they disagree, investigate the mechanism before deciding which answer is more credible.
This chapter belongs to a larger decision, so translate the number carefully. State one action the evidence permits and one action it does not. A parent or educator can ask which school ideas are doing the work: algebra organises relationships, probability represents uncertainty, calculus describes local change, and computing repeats the documented rules at scale.
Treat hit and miss data as a model-checking problem rather than a vocabulary exercise. A hit/miss study records a binary response for each inspection opportunity and models its probability as a function of a. A detected 4 mm flaw is coded 1 and a missed one 0. Recompute the example from the raw values and label every unit. Treating repeated observations as independent can understate uncertainty when inspector or specimen effects are shared. Preserve inspector, specimen and trial identifiers.
The strongest check uses evidence the model did not choose for itself. Hold out later observations, an independent simulation set or a known-answer benchmark. Report errors where decisions are sensitive, especially near the failure boundary, instead of relying on one global accuracy score. A method that performs well on easy safe cases can still fail at the rare cases that motivated the analysis.
Now change one assumption and make a directional prediction before running the calculation. If the result moves the opposite way, inspect the sign convention, transformation, density ratio or constraint. This prediction-first routine develops transferable problem-solving skills because it connects symbolic work to a testable expectation.
The mathematics of the logistic pod curve becomes useful only when its mechanism is explicit. A common generalized linear model is POD(a)=1/(1+exp[-(beta0+beta1 a)]). Consider the worked illustration: With beta0=-5 and beta1=0.1 per mm, POD at 50 mm is 0.5. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. The logistic shape and linear predictor must fit the mechanism and data range. Plot residuals and compare defensible links.
There are three uncertainties to keep apart. Sampling or numerical uncertainty comes from finite computation. Parameter uncertainty comes from estimating inputs. Model-form uncertainty comes from simplified physics, distributions or dependence. Reducing the first does not automatically reduce the other two. A credible report names all three and shows which one drives the decision.
For a practical investigation, create a small reproducible notebook or spreadsheet with a frozen data sheet, formula sheet, diagnostics sheet and decision note. Another student should be able to change one input and obtain the same update without guessing what the model meant.
Primary source: https://ntrs.nasa.gov/citations/20220013822. Read it for the method and its assumptions; do not transfer its performance claims automatically to another model.
Use solving for a target probability to practise sensitivity analysis. For the logistic model, a_p=[log(p/(1-p))-beta0]/beta1 when beta1 is positive. With beta0=-5 and beta1=0.1, a90 is about 71.97 mm because log(9)=2.197. Before accepting the result, identify the quantity that would change the conclusion most. A point estimate of a90 ignores parameter uncertainty. Show the algebra before reporting a size.
A simple baseline is essential. It reveals whether the advanced method is improving accuracy or only adding complexity. Compare model calls, wall time, uncertainty and failure-mode coverage under one declared budget. Repeat stochastic calculations with several seeds; a favourable single run is not evidence of dependable efficiency.
Students can extend the example by adding correlation, a second failure mode or a modest measurement error. Explain which part of the formula changes and which remains fixed. This exercise shows how mathematics transfers from a clean textbook case to engineering and computing without pretending that the classroom model is a certified design tool.
Build a small counterexample for what a90/95 means. The a90/95 size is where a one-sided 95% lower confidence bound on POD reaches 0.90 under the fitted analysis. The initial example is: The lower bound crosses 0.90 at a larger size than the central curve usually does. Then create a nearby case where the usual shortcut becomes unreliable. It is not a claim that 95% of all flaws at that size will certainly be found. State the confidence construction and model.
Counterexamples are positive learning tools. They reveal the conditions that make a method work and help students recognise when a software output deserves a pause. Keep the counterexample small enough to plot or enumerate. Check sign, support, bounds and conservation of probability before adding complexity.
When writing the conclusion, separate a documented fact from an interpretation and from an illustrative scenario. Cite the primary method paper for the factual mechanism, use the local calculation for the illustration, and label any recommendation as conditional on the stated model. This keeps career and engineering advice age-appropriate and honest.
Connect confidence bands to a real decision while protecting the boundary between calculation and judgement. Parameter uncertainty creates a family of plausible POD curves, summarized with a declared confidence band. Sparse data near 90% POD can make the lower bound wide. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. Pointwise and simultaneous bands answer different questions. Identify which band the software computed.
The audit should include the original question, input provenance, failure convention, algorithm settings, convergence or stopping rule, and a sensitivity result. For an adaptive method, also preserve the sequence of choices. For an optimisation, preserve every constraint. For Bayesian updating, preserve the prior and observation model. These records turn a result into something another person can challenge and improve.
A final transfer question is useful: where else would the same mathematical idea appear? Density ratios appear in survey correction and machine learning, probabilistic constraints in planning, and conditional probability in medical testing. The context changes, but transparent assumptions and reproducible checks remain valuable.
When the instrument produces a continuous signal, analysis can model signal versus flaw size before applying a detection threshold. Ultrasonic amplitude may rise with crack size while background responses overlap. This is the central mathematical mechanism in signal-response data. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. Transformations and censoring can change the fitted relationship. Plot raw signal, threshold and residuals. The useful habit is to keep the physical question, the probability model and the numerical method in separate columns so that a correct computation is not mistaken for complete evidence.
Did You Know? A result can be numerically stable and still answer the wrong question if the failure event, denominator or transformation changes. Test one nearby case and predict the direction of change before recalculating. Save inputs, intermediate values and units so a second reader can audit the path.
For practice, turn the example into a sensitivity check. Change one defensible input while holding the others fixed, compare the new result with the original and explain why it moved. Then construct a counterexample in which the shortcut fails. End with two sentences: what the mathematics supports, and what remains uncertain.
Start with the operational question behind detection threshold. A signal-response call rule declares a detection when the measured response exceeds a documented threshold. In the illustrative case, A 6 dB threshold separates calls from non-calls in an illustrative dataset. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. Moving the threshold changes both detection and false-call behaviour. Choose the threshold before evaluating performance.
A student should be able to rebuild the result without hidden settings. Record the sampling rule or optimisation criterion, the exact failure definition, and the evidence used for every distribution or parameter. Compare a transparent baseline with the advanced method. If they disagree, investigate the mechanism before deciding which answer is more credible.
This chapter belongs to a larger decision, so translate the number carefully. State one action the evidence permits and one action it does not. A parent or educator can ask which school ideas are doing the work: algebra organises relationships, probability represents uncertainty, calculus describes local change, and computing repeats the documented rules at scale.
Treat false calls and specificity as a model-checking problem rather than a vocabulary exercise. POD addresses detection of real flaws; unflawed or nonrelevant specimens are needed to study false calls. A very low threshold can increase hits and false alarms together. Recompute the example from the raw values and label every unit. A high POD alone does not describe inspection usefulness. Report false-call evidence beside POD.
The strongest check uses evidence the model did not choose for itself. Hold out later observations, an independent simulation set or a known-answer benchmark. Report errors where decisions are sensitive, especially near the failure boundary, instead of relying on one global accuracy score. A method that performs well on easy safe cases can still fail at the rare cases that motivated the analysis.
Now change one assumption and make a directional prediction before running the calculation. If the result moves the opposite way, inspect the sign convention, transformation, density ratio or constraint. This prediction-first routine develops transferable problem-solving skills because it connects symbolic work to a testable expectation.
The mathematics of a binomial checkpoint becomes useful only when its mechanism is explicit. At one fixed flaw size, 18 detections in 20 independent opportunities give a sample proportion of 0.90. Consider the worked illustration: The count is easy to reproduce and motivates interval estimation. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. Twenty trials are too few for a precise universal claim and may not be independent. Calculate an interval and inspect clustering.
There are three uncertainties to keep apart. Sampling or numerical uncertainty comes from finite computation. Parameter uncertainty comes from estimating inputs. Model-form uncertainty comes from simplified physics, distributions or dependence. Reducing the first does not automatically reduce the other two. A credible report names all three and shows which one drives the decision.
For a practical investigation, create a small reproducible notebook or spreadsheet with a frozen data sheet, formula sheet, diagnostics sheet and decision note. Another student should be able to change one input and obtain the same update without guessing what the model meant.
Use study design and specimen range to practise sensitivity analysis. A useful design places enough flaws across the transition from mostly missed to mostly detected. Only testing very large flaws yields little information about the curve knee. Before accepting the result, identify the quantity that would change the conclusion most. Artificial flaws may not represent service flaws. Justify sizes, fabrication and randomisation.
A simple baseline is essential. It reveals whether the advanced method is improving accuracy or only adding complexity. Compare model calls, wall time, uncertainty and failure-mode coverage under one declared budget. Repeat stochastic calculations with several seeds; a favourable single run is not evidence of dependable efficiency.
Students can extend the example by adding correlation, a second failure mode or a modest measurement error. Explain which part of the formula changes and which remains fixed. This exercise shows how mathematics transfers from a clean textbook case to engineering and computing without pretending that the classroom model is a certified design tool.
A second primary source is https://quicksearch.dla.mil/qsDocDetails.aspx?ident_number=206122. Compare the problem definition, estimator and validation rather than quoting one result in isolation.
Build a small counterexample for inspector variation. Different trained inspectors can have different intercepts, slopes or false-call tendencies. The initial example is: A hierarchical model can represent inspector-to-inspector variation. Then create a nearby case where the usual shortcut becomes unreliable. Pooling without structure can hide a weak subgroup; fitting each inspector separately can be unstable. Report both population and individual diagnostics.
Counterexamples are positive learning tools. They reveal the conditions that make a method work and help students recognise when a software output deserves a pause. Keep the counterexample small enough to plot or enumerate. Check sign, support, bounds and conservation of probability before adding complexity.
When writing the conclusion, separate a documented fact from an interpretation and from an illustrative scenario. Cite the primary method paper for the factual mechanism, use the local calculation for the illustration, and label any recommendation as conditional on the stated model. This keeps career and engineering advice age-appropriate and honest.
Connect repeated measures and dependence to a real decision while protecting the boundary between calculation and judgement. Multiple readings of the same flaw share specimen properties and are not automatically independent. Five passes over one crack are not equivalent to five distinct cracks. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. Naive standard errors can be too small. Use clustered or mixed-effects analysis when justified.
The audit should include the original question, input provenance, failure convention, algorithm settings, convergence or stopping rule, and a sensitivity result. For an adaptive method, also preserve the sequence of choices. For an optimisation, preserve every constraint. For Bayesian updating, preserve the prior and observation model. These records turn a result into something another person can challenge and improve.
A final transfer question is useful: where else would the same mathematical idea appear? Density ratios appear in survey correction and machine learning, probabilistic constraints in planning, and conditional probability in medical testing. The context changes, but transparent assumptions and reproducible checks remain valuable.
Instruments may report only a lower detection limit or an upper saturated value. Signals below a display floor are left-censored rather than observed exactly. This is the central mathematical mechanism in censoring and saturation. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. Replacing every censored value with the limit distorts regression. Encode the observation process explicitly. The useful habit is to keep the physical question, the probability model and the numerical method in separate columns so that a correct computation is not mistaken for complete evidence.
Did You Know? A result can be numerically stable and still answer the wrong question if the failure event, denominator or transformation changes. Test one nearby case and predict the direction of change before recalculating. Save inputs, intermediate values and units so a second reader can audit the path.
For practice, turn the example into a sensitivity check. Change one defensible input while holding the others fixed, compare the new result with the original and explain why it moved. Then construct a counterexample in which the shortcut fails. End with two sentences: what the mathematics supports, and what remains uncertain.
Start with the operational question behind model selection without fishing. Candidate links and transformations should be assessed using diagnostics, mechanism and predeclared criteria. In the illustrative case, Log flaw size may linearise a signal relation better than raw size. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. Choosing the most favourable a90/95 after many fits biases the conclusion. Preserve every candidate and reason for selection.
A student should be able to rebuild the result without hidden settings. Record the sampling rule or optimisation criterion, the exact failure definition, and the evidence used for every distribution or parameter. Compare a transparent baseline with the advanced method. If they disagree, investigate the mechanism before deciding which answer is more credible.
This chapter belongs to a larger decision, so translate the number carefully. State one action the evidence permits and one action it does not. A parent or educator can ask which school ideas are doing the work: algebra organises relationships, probability represents uncertainty, calculus describes local change, and computing repeats the documented rules at scale.
Treat extrapolation beyond the data as a model-checking problem rather than a vocabulary exercise. The fitted curve is best supported inside the tested flaw range and inspection conditions. Predicting POD for a flaw smaller than every tested flaw is extrapolation. Recompute the example from the raw values and label every unit. A smooth equation can conceal weak evidence. Mark the observed range on every plot.
The strongest check uses evidence the model did not choose for itself. Hold out later observations, an independent simulation set or a known-answer benchmark. Report errors where decisions are sensitive, especially near the failure boundary, instead of relying on one global accuracy score. A method that performs well on easy safe cases can still fail at the rare cases that motivated the analysis.
Now change one assumption and make a directional prediction before running the calculation. If the result moves the opposite way, inspect the sign convention, transformation, density ratio or constraint. This prediction-first routine develops transferable problem-solving skills because it connects symbolic work to a testable expectation.
The mathematics of from pod to inspection intervals becomes useful only when its mechanism is explicit. Damage-tolerance decisions combine crack-growth analysis, critical size and inspection capability. Consider the worked illustration: A detectable-size distribution can inform the window before a crack reaches a critical size. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. POD alone does not choose a safe interval or prove structural integrity. Keep growth, consequence and detection models separate.
There are three uncertainties to keep apart. Sampling or numerical uncertainty comes from finite computation. Parameter uncertainty comes from estimating inputs. Model-form uncertainty comes from simplified physics, distributions or dependence. Reducing the first does not automatically reduce the other two. A credible report names all three and shows which one drives the decision.
For a practical investigation, create a small reproducible notebook or spreadsheet with a frozen data sheet, formula sheet, diagnostics sheet and decision note. Another student should be able to change one input and obtain the same update without guessing what the model meant.
Use validation and reproducibility to practise sensitivity analysis. Independent specimens, blind trials and reproducible analysis code test whether the estimated performance transfers. A holdout set can check predicted detection frequencies by size band. Before accepting the result, identify the quantity that would change the conclusion most. A small holdout has low power to reveal rare failure modes. Publish data provenance and code settings.
A simple baseline is essential. It reveals whether the advanced method is improving accuracy or only adding complexity. Compare model calls, wall time, uncertainty and failure-mode coverage under one declared budget. Repeat stochastic calculations with several seeds; a favourable single run is not evidence of dependable efficiency.
Students can extend the example by adding correlation, a second failure mode or a modest measurement error. Explain which part of the formula changes and which remains fixed. This exercise shows how mathematics transfers from a clean textbook case to engineering and computing without pretending that the classroom model is a certified design tool.
Build a small counterexample for practice activity. Create fictional hit/miss data across five flaw sizes and fit or hand-evaluate a simple logistic curve. The initial example is: Check that the predicted probability rises and solve for p=0.5 and p=0.9. Then create a nearby case where the usual shortcut becomes unreliable. A classroom curve is not a certified POD study. Label every observation and assumption.
Counterexamples are positive learning tools. They reveal the conditions that make a method work and help students recognise when a software output deserves a pause. Keep the counterexample small enough to plot or enumerate. Check sign, support, bounds and conservation of probability before adding complexity.
When writing the conclusion, separate a documented fact from an interpretation and from an illustrative scenario. Cite the primary method paper for the factual mechanism, use the local calculation for the illustration, and label any recommendation as conditional on the stated model. This keeps career and engineering advice age-appropriate and honest.
Connect parent and educator guidance to a real decision while protecting the boundary between calculation and judgement. POD joins ratios, graphs, functions, probability, confidence and engineering judgement. Ask why the lower confidence curve matters more than the prettiest fitted line. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. Complex software should not replace a visible count and graph. Begin with repeated coin-style binary trials.
The audit should include the original question, input provenance, failure convention, algorithm settings, convergence or stopping rule, and a sensitivity result. For an adaptive method, also preserve the sequence of choices. For an optimisation, preserve every constraint. For Bayesian updating, preserve the prior and observation model. These records turn a result into something another person can challenge and improve.
A final transfer question is useful: where else would the same mathematical idea appear? Density ratios appear in survey correction and machine learning, probabilistic constraints in planning, and conditional probability in medical testing. The context changes, but transparent assumptions and reproducible checks remain valuable.
POD is not the probability that a called indication is truly a flaw, and a90 is not automatically a90/95. Bayes rule is needed to reverse a detection probability into a post-call probability using prevalence and false calls. This is the central mathematical mechanism in common misconceptions. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. Similar names can hide different conditional probabilities. Write the conditioning event in words. The useful habit is to keep the physical question, the probability model and the numerical method in separate columns so that a correct computation is not mistaken for complete evidence.
Did You Know? A result can be numerically stable and still answer the wrong question if the failure event, denominator or transformation changes. Test one nearby case and predict the direction of change before recalculating. Save inputs, intermediate values and units so a second reader can audit the path.
For practice, turn the example into a sensitivity check. Change one defensible input while holding the others fixed, compare the new result with the original and explain why it moved. Then construct a counterexample in which the shortcut fails. End with two sentences: what the mathematics supports, and what remains uncertain.
Start with the operational question behind when a simple curve is inadequate. Multimodal signals, changing procedures, covariates or strong inspector effects may require richer models and more data. In the illustrative case, Two flaw orientations can create two different response bands. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. Forcing one curve can average away a dangerous subgroup. Stratify or model the mechanism.
A student should be able to rebuild the result without hidden settings. Record the sampling rule or optimisation criterion, the exact failure definition, and the evidence used for every distribution or parameter. Compare a transparent baseline with the advanced method. If they disagree, investigate the mechanism before deciding which answer is more credible.
This chapter belongs to a larger decision, so translate the number carefully. State one action the evidence permits and one action it does not. A parent or educator can ask which school ideas are doing the work: algebra organises relationships, probability represents uncertainty, calculus describes local change, and computing repeats the documented rules at scale.
Treat research and next reading as a model-checking problem rather than a vocabulary exercise. NASA’s 2022 guidebook explains study design and analysis for Standard NDE POD, while MIL-HDBK-1823 provides active DoD guidance. Read the official scope and the statistical chapters together. Recompute the example from the raw values and label every unit. A handbook method still requires application-specific evidence. Reproduce a published benchmark before field use.
The strongest check uses evidence the model did not choose for itself. Hold out later observations, an independent simulation set or a known-answer benchmark. Report errors where decisions are sensitive, especially near the failure boundary, instead of relying on one global accuracy score. A method that performs well on easy safe cases can still fail at the rare cases that motivated the analysis.
Now change one assumption and make a directional prediction before running the calculation. If the result moves the opposite way, inspect the sign convention, transformation, density ratio or constraint. This prediction-first routine develops transferable problem-solving skills because it connects symbolic work to a testable expectation.
Continue with the Mathematics Learning Hub and the previous reliability article. These links own broader probability ideas and the immediately preceding rare-event geometry.
The mathematics of frequently asked questions becomes useful only when its mechanism is explicit. FAQs should cover hits, signals, thresholds, false calls, a90/95, inspector variation and transfer. Consider the worked illustration: Each answer should name the population and procedure. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. Avoid turning a confidence statement into certainty. Publish the full study definition.
There are three uncertainties to keep apart. Sampling or numerical uncertainty comes from finite computation. Parameter uncertainty comes from estimating inputs. Model-form uncertainty comes from simplified physics, distributions or dependence. Reducing the first does not automatically reduce the other two. A credible report names all three and shows which one drives the decision.
For a practical investigation, create a small reproducible notebook or spreadsheet with a frozen data sheet, formula sheet, diagnostics sheet and decision note. Another student should be able to change one input and obtain the same update without guessing what the model meant.
Quick answers
What should I calculate first? Start with the estimand or failure event, then a simple baseline.
Does the advanced method guarantee accuracy? No. It needs input, numerical and model checks.
Which school mathematics matters? Algebra, graphs, probability, functions and careful units all contribute.
How can a student practise safely? Use fictional or open benchmark data and preserve every intermediate value.
What makes a career claim responsible? Describe how the mathematics is used without promising admission, salary or employment.
Extended practice laboratory
Practice case 1: Why inspection needs probability
Setup: Use this chapter’s illustrative case: Ten inspectors may not all call the same small crack.
Task: Define the inspection system and target flaw first.
Checking target: A POD curve describes a study population and procedure, not every possible field condition.
Practice case 2: The flaw variable a
Setup: Use this chapter’s illustrative case: A study might use crack length in millimetres as the predictor.
Task: Record geometry and units with every specimen.
Checking target: One size measure may not capture orientation, closure or surface condition.
Practice case 3: Hit and miss data
Setup: Use this chapter’s illustrative case: A detected 4 mm flaw is coded 1 and a missed one 0.
Task: Preserve inspector, specimen and trial identifiers.
Checking target: Treating repeated observations as independent can understate uncertainty when inspector or specimen effects are shared.
Practice case 4: The logistic POD curve
Setup: Use this chapter’s illustrative case: With beta0=-5 and beta1=0.1 per mm, POD at 50 mm is 0.5.
Task: Plot residuals and compare defensible links.
Checking target: The logistic shape and linear predictor must fit the mechanism and data range.
Practice case 5: Solving for a target probability
Setup: Use this chapter’s illustrative case: With beta0=-5 and beta1=0.1, a90 is about 71.97 mm because log(9)=2.197.
Task: Show the algebra before reporting a size.
Checking target: A point estimate of a90 ignores parameter uncertainty.
Practice case 6: What a90/95 means
Setup: Use this chapter’s illustrative case: The lower bound crosses 0.90 at a larger size than the central curve usually does.
Task: State the confidence construction and model.
Checking target: It is not a claim that 95% of all flaws at that size will certainly be found.
Practice case 7: Confidence bands
Setup: Use this chapter’s illustrative case: Sparse data near 90% POD can make the lower bound wide.
Task: Identify which band the software computed.
Checking target: Pointwise and simultaneous bands answer different questions.
Practice case 8: Signal-response data
Setup: Use this chapter’s illustrative case: Ultrasonic amplitude may rise with crack size while background responses overlap.
Task: Plot raw signal, threshold and residuals.
Checking target: Transformations and censoring can change the fitted relationship.
Practice case 9: Detection threshold
Setup: Use this chapter’s illustrative case: A 6 dB threshold separates calls from non-calls in an illustrative dataset.
Task: Choose the threshold before evaluating performance.
Checking target: Moving the threshold changes both detection and false-call behaviour.
Practice case 10: False calls and specificity
Setup: Use this chapter’s illustrative case: A very low threshold can increase hits and false alarms together.
Task: Report false-call evidence beside POD.
Checking target: A high POD alone does not describe inspection usefulness.