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eduKateSG · Why Mathematics? · Antolini’s Time-Dependent Concordance for Reliability Survival Curves: Event-Time Comparisons, Crossing Predictions and Censoring
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
The importance of mathematics is clear when a reliability model predicts an entire survival curve for each unit and those curves may cross. Antolini’s time-dependent concordance index evaluates whether the unit that fails earlier had a lower predicted probability of surviving beyond that earlier failure time than a unit observed longer. The comparison uses the curve at the time where the ordering becomes known.
This guide develops individual survival functions, comparable pairs, event-time evaluation, curve crossing, censoring, jackknife uncertainty and external validation. The index measures discrimination of survival distributions. It does not establish calibration, causality, safe extrapolation or decision benefit, and it should not be mixed casually with score-based concordance measures.
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
CHAPTER 1 OF 25 · Understand the question
1. Why full curves need a concordance rule
A single static risk score cannot express changing rankings when individual survival curves cross over time. Unit A is riskier early while unit B becomes riskier late. This is the central mathematical mechanism in why full curves need a concordance rule. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. One summary still compresses time variation. Plot representative curves. 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 individual survival predictions. Each unit needs a valid monotone function S_i(t) between zero and one. In the illustrative case, A model gives S_A(200)=.70 and S_B(200)=.88. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. A bare Cox linear predictor is not a full curve. Check range and monotonicity.
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 earlier event time as a model-checking problem rather than a vocabulary exercise. For a pair with T_i<T_j and an observed event for i, evaluate both predicted curves at T_i. A fails at 200 hours while B is observed to 500 hours. Recompute the example from the raw values and label every unit. Later information must not be used to define the earlier comparison. Anchor the worksheet at T_i.
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 antolini concordance condition becomes useful only when its mechanism is explicit. The pair is concordant when S_i(T_i)<S_j(T_i): the earlier-failing unit had lower predicted survival at that time. Consider the worked illustration: At 200 hours .70<.88, so the pair is concordant. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. Equality requires a tie convention. Write the inequality explicitly.
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/10.1002/sim.2427. Read it for the method and its assumptions; do not transfer its performance claims automatically to another model.
Use a four-pair example to practise sensitivity analysis. Count concordant event-time comparisons among eligible pairs. Three concordant pairs out of four give C_td=.75. Before accepting the result, identify the quantity that would change the conclusion most. Four pairs are far too few for inference. List every pair and curve value.
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 why curves are compared at one time. Using the earlier event time asks whether the model distinguished the pair at the moment the event ordering became observable. The initial example is: The same two curves may reverse at 600 hours. Then create a nearby case where the usual shortcut becomes unreliable. A fixed later time can misrepresent the earlier decision. Use the event-time rule consistently.
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 crossing survival curves to a real decision while protecting the boundary between calculation and judgement. C_td remains usable when rankings change because each pair is evaluated at its relevant time. A time-varying temperature effect makes curves cross. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. Crossing can make the overall index hard to localise. Add time-window summaries.
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.
A censored unit may be the longer-observed member when another unit fails before its censoring time. A fails at 200 and B is censored at 400, so their order is known to 200. This is the central mathematical mechanism in censored units can contribute. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. B cannot anchor an event comparison at 400. Track which member supplies the event. 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 pairs that cannot be ordered. If one unit is censored before the other's event, the true order may be unknown and the pair is not comparable under the rule. In the illustrative case, A is censored at 100 and B fails at 200. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. Treating A as a survivor beyond 200 is false. Exclude only by the declared rule.
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 tied predictions as a model-checking problem rather than a vocabulary exercise. Equal survival probabilities at the earlier event time commonly receive half credit or another documented tie treatment. Both curves equal .82 at 300 hours. Recompute the example from the raw values and label every unit. Rounding can create artificial ties. Calculate with unrounded predictions.
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 tied observed times becomes useful only when its mechanism is explicit. Simultaneous recorded failures need a convention because there is no strict earlier event. Consider the worked illustration: Two units fail between the same monthly inspections. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. Arbitrary order inflates information. Report interval and tie handling.
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 jackknife confidence intervals to practise sensitivity analysis. The original method derives uncertainty using a jackknife for correlated one-sample U-statistics. Delete one unit at a time and recompute the index. Before accepting the result, identify the quantity that would change the conclusion most. Deleting pairs instead of units ignores dependence. Use unit-level resampling.
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/16320281/. Compare the problem definition, estimator and validation rather than quoting one result in isolation.
Build a small counterexample for time-dependent covariates. Predicted curves must use covariate histories available up to each prediction origin without leaking future measurements. The initial example is: Vibration is updated weekly before later risk predictions. Then create a nearby case where the usual shortcut becomes unreliable. A final sensor value leaks information backward. Freeze landmark and update rules.
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 dynamic prediction origins to a real decision while protecting the boundary between calculation and judgement. A model updated at several landmarks needs concordance definitions aligned to each prediction time and future horizon. At 300 hours, predict survival from 300 to 800. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. Mixing baseline and landmark curves changes the question. State the prediction origin.
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.
Overall survival curves do not validate cause-specific cumulative-incidence predictions. Corrosion and electronics failure compete. This is the central mathematical mechanism in competing risks. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. Using one-minus overall survival for one cause is wrong. Match the curve to the event estimand. 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 curve calibration. Well-ranked curves can all be too optimistic or pessimistic at a time point. In the illustrative case, The earlier-failing unit has lower S, but both values exceed observed survival. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. C_td is not calibration. Add fixed-time and curve-level checks.
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 integrated versus local discrimination as a model-checking problem rather than a vocabulary exercise. A single C_td averages pairwise ordering over observed event times, while a time-specific measure can show where performance changes. Discrimination is .8 early and .55 late. Recompute the example from the raw values and label every unit. An average can hide a weak late period. Plot performance over 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 internal validation becomes useful only when its mechanism is explicit. Flexible survival models can overfit curve shapes and apparent concordance. Consider the worked illustration: A neural model tuned on the full fleet separates its training curves strongly. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. Jackknife precision does not remove selection optimism. Validate the full modelling pipeline.
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. Use frozen model parameters and curve-generation rules on a new fleet. A laboratory model is assessed on field equipment. Before accepting the result, identify the quantity that would change the conclusion most. Different follow-up affects which pairs are observed. Describe comparability and censoring.
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 case mix and transport. Wider predictor variation can raise discrimination even if model coefficients are unchanged. The initial example is: A mixed fleet has more separable risk profiles than one model family. Then create a nearby case where the usual shortcut becomes unreliable. Raw C_td values are not universal grades. Report predictor distributions.
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 interpretation to a real decision while protecting the boundary between calculation and judgement. C_td says whether survival distributions order pairs, not whether a thresholded maintenance action improves outcomes. C_td=.75 does not choose inspection timing. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. Costs and capacity are absent. Add utility or decision 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.
Draw two monotone curves, mark the earlier failure and compare both survival values at that time. With .70 and .88 the pair is concordant; three of four gives .75. This is the central mathematical mechanism in practice activity. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. The drawing must use the same time axis. Check curve monotonicity 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 parent and educator guidance. The method connects functions, graphs, inequalities and conditional comparisons to a changing real-world process. In the illustrative case, Ask why the comparison occurs at the earlier failure time. 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 read a curve as an actual path rather than a probability function. Label axes and units.
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. Antolini C is not a fixed-time AUC, not calibration and not interchangeable with every C-index. Curve crossing is allowed but complicates interpretation. Recompute the example from the raw values and label every unit. Censoring still limits comparable evidence. Name the exact index.
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. Antolini, Boracchi and Biganzoli proposed C_td using whole predicted survival functions and jackknife inference. Consider the worked illustration: Read the primary paper beside a plotted pair example. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. Production use requires valid individual survival curves. Reproduce the .70 versus .88 comparison.
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 full curves need a concordance rule
Setup: Use this chapter’s illustrative case: Unit A is riskier early while unit B becomes riskier late.
Task: Plot representative curves.
Checking target: One summary still compresses time variation.
Practice case 2: Individual survival predictions
Setup: Use this chapter’s illustrative case: A model gives S_A(200)=.70 and S_B(200)=.88.
Task: Check range and monotonicity.
Checking target: A bare Cox linear predictor is not a full curve.
Practice case 3: The earlier event time
Setup: Use this chapter’s illustrative case: A fails at 200 hours while B is observed to 500 hours.
Task: Anchor the worksheet at T_i.
Checking target: Later information must not be used to define the earlier comparison.
Practice case 4: Antolini concordance condition
Setup: Use this chapter’s illustrative case: At 200 hours .70<.88, so the pair is concordant.
Task: Write the inequality explicitly.
Checking target: Equality requires a tie convention.
Practice case 5: A four-pair example
Setup: Use this chapter’s illustrative case: Three concordant pairs out of four give C_td=.75.
Task: List every pair and curve value.
Checking target: Four pairs are far too few for inference.
Practice case 6: Why curves are compared at one time
Setup: Use this chapter’s illustrative case: The same two curves may reverse at 600 hours.
Task: Use the event-time rule consistently.
Checking target: A fixed later time can misrepresent the earlier decision.
Practice case 7: Crossing survival curves
Setup: Use this chapter’s illustrative case: A time-varying temperature effect makes curves cross.
Task: Add time-window summaries.
Checking target: Crossing can make the overall index hard to localise.
Practice case 8: Censored units can contribute
Setup: Use this chapter’s illustrative case: A fails at 200 and B is censored at 400, so their order is known to 200.
Task: Track which member supplies the event.
Checking target: B cannot anchor an event comparison at 400.
Practice case 9: Pairs that cannot be ordered
Setup: Use this chapter’s illustrative case: A is censored at 100 and B fails at 200.
Task: Exclude only by the declared rule.
Checking target: Treating A as a survivor beyond 200 is false.
Practice case 10: Tied predictions
Setup: Use this chapter’s illustrative case: Both curves equal .82 at 300 hours.
Task: Calculate with unrounded predictions.
Checking target: Rounding can create artificial ties.