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eduKateSG · Why Mathematics? · Heagerty–Lumley–Pepe Time-Dependent ROC Curves for Reliability Models: Cumulative Cases, Dynamic Controls, Censoring and AUC
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 a reliability team asks whether a score separates units that fail by a chosen time from units that remain event-free beyond it. Heagerty, Lumley and Pepe defined cumulative/dynamic time-dependent ROC curves for censored event-time data, so sensitivity, specificity and AUC can be interpreted at an explicit horizon instead of pretending every outcome is already known.
This guide develops cumulative cases, dynamic controls, marker thresholds, censoring-aware estimation, monotonicity, AUC and validation. A worked complete-data example makes the classification geometry visible before censoring weights enter. The curve measures time-specific discrimination. It does not establish calibration, causality, maintenance value or safe performance outside the observed follow-up support.
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
Event status changes with the horizon, so the same unit can be a control early and a case later. A bearing working at 300 hours but failed by 700 hours changes status across horizons. This is the central mathematical mechanism in why a fixed roc curve is insufficient. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. A single binary label erases time. Declare the prediction horizon. 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 cumulative case definition. At time t, a cumulative case has experienced the event by t: T less than or equal to t. In the illustrative case, At 500 hours, failures at 120 and 430 hours are cases. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. The definition combines early and later failures within the window. Write the event inequality.
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 the dynamic control definition as a model-checking problem rather than a vocabulary exercise. A dynamic control is known to remain event-free beyond t: T greater than t. A unit observed healthy through 800 hours is a control at 500. Recompute the example from the raw values and label every unit. A unit censored at 350 has unknown status at 500. Separate controls from unknowns.
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 marker and its direction becomes useful only when its mechanism is explicit. The marker M must have a declared direction, commonly larger values indicating greater near-term failure risk. Consider the worked illustration: Scores .9 and .7 are treated as riskier than .2. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. Reversing direction reverses the ROC interpretation. Test one known high-risk case.
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://doi.org/10.1111/j.0006-341X.2000.00337.x. Read it for the method and its assumptions; do not transfer its performance claims automatically to another model.
Use time-specific sensitivity to practise sensitivity analysis. Cumulative sensitivity at threshold c is P(M greater than c given T less than or equal to t). Three of four cases exceed .55, so complete-data sensitivity is .75. Before accepting the result, identify the quantity that would change the conclusion most. Censoring can hide case membership. Show numerator and denominator.
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 time-specific specificity. Dynamic specificity is P(M less than or equal to c given T greater than t). The initial example is: Three of four controls are at or below .55, so specificity is .75. Then create a nearby case where the usual shortcut becomes unreliable. Threshold equality conventions must be fixed. State the comparison signs.
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 a complete-data threshold example to a real decision while protecting the boundary between calculation and judgement. With case scores .9, .7, .6, .3 and control scores .8, .5, .2, .1, threshold .55 gives three true positives and three true negatives. Sensitivity and specificity both equal 3 divided by 4, or .75. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. This teaching set has no censoring and is not a production estimate. Recount the eight units.
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.
Varying c from above every score to below every score creates pairs of false-positive rate and true-positive rate. Moving below .8 admits the high-scoring control and raises false positives. This is the central mathematical mechanism in tracing the roc curve. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. Coarse or tied scores produce steps. Plot every distinct threshold. 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 auc as pairwise discrimination. AUC(t) is the probability that a randomly selected cumulative case has a larger marker than a randomly selected dynamic control at t, under the declared tie rule. In the illustrative case, The .9 case outranks all four controls. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. AUC is not the probability that a prediction is correct for one unit. Explain the sampled pair.
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 censoring creates unknown labels as a model-checking problem rather than a vocabulary exercise. Right censoring before t prevents direct classification as a case or dynamic control at t. A unit last observed at 350 hours cannot be labelled at 500. Recompute the example from the raw values and label every unit. Calling it a control biases the curve. Retain the censoring time.
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 nearest-neighbour estimator becomes useful only when its mechanism is explicit. The original paper proposed a nearest-neighbour estimator of the bivariate marker and event-time distribution that preserves valid ROC properties. Consider the worked illustration: Nearby marker values inform the conditional survival estimate. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. Bandwidth choices affect smoothing. Report the estimator and tuning.
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 the kaplan–meier subset estimator to practise sensitivity analysis. A simpler approach estimates survival within marker-defined subsets using Kaplan–Meier calculations. Units above a threshold form one subset for a conditional event estimate. Before accepting the result, identify the quantity that would change the conclusion most. Naive versions may yield nonmonotone ROC curves when assumptions fail. Inspect monotonicity and assumptions.
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://pubmed.ncbi.nlm.nih.gov/10877287/. Compare the problem definition, estimator and validation rather than quoting one result in isolation.
Build a small counterexample for why monotonicity matters. Sensitivity and specificity should move consistently as the threshold changes, and the ROC curve should not fold backward. The initial example is: Lowering a high-risk threshold should not reduce the set called positive. Then create a nearby case where the usual shortcut becomes unreliable. An estimator can violate the expected geometry. Plot the full curve.
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 marker-dependent censoring to a real decision while protecting the boundary between calculation and judgement. If censoring depends on the marker, pooled censoring assumptions can distort time-specific operating characteristics. High-risk machines are withdrawn for overhaul earlier. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. Measured-marker adjustment cannot solve unmeasured informative censoring. Model the observation process.
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.
Each relevant marker region needs enough chance of observation through t to identify case and control probabilities. No high-score unit remains observed beyond 450 hours. This is the central mathematical mechanism in positivity before the horizon. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. Weights or smoothing cannot create absent support. Shorten the horizon or collect follow-up. 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 choosing the horizon. The horizon should reflect an operational decision and data support rather than whichever time gives the best AUC. In the illustrative case, A 500-hour inspection decision motivates t=500. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. Post hoc horizon selection exaggerates performance. Pre-specify primary times.
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 confidence intervals as a model-checking problem rather than a vocabulary exercise. Estimated ROC coordinates and AUC need uncertainty calculations that respect censoring and repeated use of units across thresholds. A bootstrap resamples units and repeats the full estimation procedure. Recompute the example from the raw values and label every unit. Treating threshold points as independent is invalid. Use validated unit-level inference.
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 internal validation becomes useful only when its mechanism is explicit. A score selected and assessed on the same fleet has optimistic discrimination. Consider the worked illustration: Feature selection uses noise that later enters the reported AUC. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. Censoring correction does not correct overfitting. Resample model development.
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 external validation to practise sensitivity analysis. A frozen score should be evaluated at the same stated horizon in a new target fleet. A laboratory score is tested on field equipment at 500 hours. Before accepting the result, identify the quantity that would change the conclusion most. Different case mix can change AUC. Describe the validation population.
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 calibration is a different question. ROC analysis ranks cases and controls but does not check whether predicted survival probabilities are numerically accurate. The initial example is: AUC=.80 can coexist with risks that are all twice too high. Then create a nearby case where the usual shortcut becomes unreliable. Threshold performance does not validate probability scale. Add calibration plots and scores.
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 decision value is a different question to a real decision while protecting the boundary between calculation and judgement. A time-specific AUC does not encode inspection cost, missed-failure harm, capacity or intervention effect. Two thresholds with similar ROC positions can have different operational costs. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. High discrimination alone does not select a policy. Add consequence-aware analysis.
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.
Reproduce the .55 threshold table, then lower the threshold to .45 and recount cases and controls. At .45 all but the .3 case are positive while controls .8 and .5 are positive. This is the central mathematical mechanism in practice calculation. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. One threshold cannot summarise the full curve. Build a confusion table at each threshold. 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 parent and educator guidance. The method connects inequalities, conditional probability, graphs and changing definitions to incomplete observation. In the illustrative case, Ask why one unit changes from control to case when t moves. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. Students may confuse censoring with survival. Use timelines beside the score axis.
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 common misconceptions as a model-checking problem rather than a vocabulary exercise. Cumulative/dynamic AUC is not ordinary binary AUC with censored units deleted, not calibration and not proof of benefit. A censored unit before t has unknown status rather than a negative label. Recompute the example from the raw values and label every unit. Different definitions of time-dependent AUC are not interchangeable. Name the case-control convention.
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, the eduKate Mathematics Learning System, the approach to learning mathematics, the Singapore Mathematics curriculum overview, the cross-subject learning approach and the previous reliability article. Each destination has its own role: subject navigation, learning method, school-to-advanced progression, or continuity with the latest reliability mechanism.
The mathematics of research and next reading becomes useful only when its mechanism is explicit. Heagerty, Lumley and Pepe formalised time-dependent ROC curves for censored survival data and compared two estimators. Consider the worked illustration: Read the primary paper beside the eight-unit worksheet. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. Implementation details determine valid curve behaviour. Reproduce the complete-data geometry first.
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 a fixed ROC curve is insufficient
Setup: Use this chapter’s illustrative case: A bearing working at 300 hours but failed by 700 hours changes status across horizons.
Task: Declare the prediction horizon.
Checking target: A single binary label erases time.
Practice case 2: The cumulative case definition
Setup: Use this chapter’s illustrative case: At 500 hours, failures at 120 and 430 hours are cases.
Task: Write the event inequality.
Checking target: The definition combines early and later failures within the window.
Practice case 3: The dynamic control definition
Setup: Use this chapter’s illustrative case: A unit observed healthy through 800 hours is a control at 500.
Task: Separate controls from unknowns.
Checking target: A unit censored at 350 has unknown status at 500.
Practice case 4: The marker and its direction
Setup: Use this chapter’s illustrative case: Scores .9 and .7 are treated as riskier than .2.
Task: Test one known high-risk case.
Checking target: Reversing direction reverses the ROC interpretation.
Practice case 5: Time-specific sensitivity
Setup: Use this chapter’s illustrative case: Three of four cases exceed .55, so complete-data sensitivity is .75.
Task: Show numerator and denominator.
Checking target: Censoring can hide case membership.
Practice case 6: Time-specific specificity
Setup: Use this chapter’s illustrative case: Three of four controls are at or below .55, so specificity is .75.
Task: State the comparison signs.
Checking target: Threshold equality conventions must be fixed.
Practice case 7: A complete-data threshold example
Setup: Use this chapter’s illustrative case: Sensitivity and specificity both equal 3 divided by 4, or .75.
Task: Recount the eight units.
Checking target: This teaching set has no censoring and is not a production estimate.
Practice case 8: Tracing the ROC curve
Setup: Use this chapter’s illustrative case: Moving below .8 admits the high-scoring control and raises false positives.
Task: Plot every distinct threshold.
Checking target: Coarse or tied scores produce steps.
Practice case 9: AUC as pairwise discrimination
Setup: Use this chapter’s illustrative case: The .9 case outranks all four controls.
Task: Explain the sampled pair.
Checking target: AUC is not the probability that a prediction is correct for one unit.
Practice case 10: Censoring creates unknown labels
Setup: Use this chapter’s illustrative case: A unit last observed at 350 hours cannot be labelled at 500.
Task: Retain the censoring time.
Checking target: Calling it a control biases the curve.