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eduKateSG · Why Mathematics? · Cramér–von Mises Tests for Reliability Curves: Integrated Squared Differences, Censoring and Omnibus Comparisons
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 clear when reliability curves differ through many moderate discrepancies rather than one sharp excursion. Cramér–von Mises-type procedures accumulate squared differences or squared score-process departures over a declared follow-up range. Because positive and negative gaps are squared, they do not cancel, and because the whole path contributes, the test can respond to broad departures missed by a terminal rank score.
This guide develops integrated squared discrepancy, censoring-aware weighting, information-time integration, null calibration, crossing curves and interpretation. The statistic is an omnibus measure of evidence, not a direct effect size. Its sensitivity depends on the integration measure, follow-up range and censoring assumptions, so a significant result still needs graphs and decision-scale summaries.
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
Many small curve gaps can matter even if no single time produces a dramatic maximum. A material can be modestly less reliable across most of 0–1000 hours. This is the central mathematical mechanism in why integrate squared differences. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. Adding signed gaps can cancel. Square or otherwise norm the path. 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 cramér–von mises principle. A CvM statistic integrates the squared discrepancy between empirical objects over their domain. In the illustrative case, For two curves, accumulate [S1(t)-S2(t)]^2 with a chosen measure. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. The integration measure changes emphasis. State the exact statistic.
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-aware survival comparison as a model-checking problem rather than a vocabulary exercise. Right-censored data require Kaplan–Meier or score-process constructions with appropriate risk-set and censoring treatment. A withdrawn system informs the curve only to its censoring time. Recompute the example from the raw values and label every unit. An uncensored two-sample formula is not automatically valid. Use a censored-data method.
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 squared score-process form becomes useful only when its mechanism is explicit. Some procedures integrate the squared standardised observed-minus-expected score path. Consider the worked illustration: Both positive early and negative late departures add positively after squaring. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. Standardisation and covariance matter. Show the process 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.
Primary source: https://doi.org/10.1002/bimj.4710200608. Read it for the method and its assumptions; do not transfer its performance claims automatically to another model.
Use a discrete approximation to practise sensitivity analysis. On a grid, approximate an integral by summing squared gaps times interval widths or information increments. Gaps .1 and -.1 each contribute .01 before weighting. Before accepting the result, identify the quantity that would change the conclusion most. Unequal grid spacing needs weights. Keep width and gap columns.
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 a worked curve-gap calculation. For gaps .10,.05,-.05 over equal unit intervals, the sum of squares is .01+.0025+.0025=.015. The initial example is: A signed sum would be .10 and tells a different story. Then create a nearby case where the usual shortcut becomes unreliable. The toy number is not a calibrated test statistic. Separate discrepancy from p-value.
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 why signs do not cancel to a real decision while protecting the boundary between calculation and judgement. Squaring converts deviations in either direction into positive contributions. Crossing curves can produce a large integrated discrepancy despite a small average signed gap. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. Direction is lost in the total. Plot signed differences too.
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.
CvM tends to reward persistent moderate gaps, while supremum tests focus on the largest local excursion. A broad .05 gap can accumulate substantial evidence. This is the central mathematical mechanism in broad versus sharp departures. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. No test dominates every alternative. Match method to plausible patterns. 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 choice of integration measure. Integrate over clock time, event times, pooled distribution mass or variance time depending on the formal procedure. In the illustrative case, Event-time weighting concentrates on observed information. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. Clock-time integration can overemphasise long empty intervals. Follow the cited method exactly.
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 pooled risk-set information as a model-checking problem rather than a vocabulary exercise. Censoring-aware tests often scale discrepancies using the systems at risk in each group. A gap supported by 100 systems carries more reliable information than one supported by four. Recompute the example from the raw values and label every unit. Raw curve distance ignores precision. Report risk sets.
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 right censoring assumptions becomes useful only when its mechanism is explicit. Independent censoring within groups is central to ordinary null calibration. Consider the worked illustration: Condition-driven retirement can create apparent curve differences. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. Equal censoring percentages do not prove independence. Record reasons and histories.
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 differential censoring to practise sensitivity analysis. Groups may have different administrative cutoffs yet remain valid under conditional independence, but late support differs. One site may stop observing at 800 hours. Before accepting the result, identify the quantity that would change the conclusion most. A common integration range can become narrow. Use shared supported follow-up.
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://doi.org/10.2307/1403046. Compare the problem definition, estimator and validation rather than quoting one result in isolation.
Build a small counterexample for tied event times. Tied failures affect Kaplan–Meier steps and score covariance. The initial example is: Rounded inspection data can create multiple events at one age. Then create a nearby case where the usual shortcut becomes unreliable. Arbitrary ordering is incorrect. Use joint tie handling.
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 null distribution to a real decision while protecting the boundary between calculation and judgement. Because the statistic combines correlated discrepancies, its null law is not a simple one-degree chi-square in general. Critical values may come from asymptotic eigen-expansions or resampling. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. Guessing a chi-square reference gives wrong p-values. Record calibration.
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 group labels are exchangeable under the design, permutation can approximate the null distribution. Randomised design labels can be shuffled as whole-system units. This is the central mathematical mechanism in permutation calibration. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. Observational groups may not be exchangeable. Justify permutation conditions. 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 bootstrap calibration. Resample complete systems and recompute the full curve statistic to preserve within-record structure. In the illustrative case, The empirical upper tail supplies a critical value. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. Naive row resampling breaks repeated histories. Resample the independent unit.
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 crossing reliability curves as a model-checking problem rather than a vocabulary exercise. Squared discrepancy accumulates both sides of a crossing instead of cancellation. Early disadvantage and late advantage both add evidence. Recompute the example from the raw values and label every unit. The total does not say which interval matters most. Plot a time-resolved contribution curve.
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 goodness-of-fit use becomes useful only when its mechanism is explicit. One-sample CvM variants compare an observed censored survival pattern with a specified model. Consider the worked illustration: Test a Weibull survival curve against censored fleet data. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. A non-rejection does not prove the model true. Inspect residual and tail diagnostics.
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 two-sample and k-sample extensions to practise sensitivity analysis. Censored-data variants compare two or several group distributions using quadratic discrepancies. Three suppliers can receive one global omnibus test. Before accepting the result, identify the quantity that would change the conclusion most. A global result does not identify each pair. Plan adjusted follow-ups.
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 effect size remains separate. The CvM statistic has integrated squared units or a standardised scale, not a direct survival-probability interpretation. The initial example is: Report maximum absolute gap, RMST and milestones alongside it. Then create a nearby case where the usual shortcut becomes unreliable. A large test value can arise from small widespread differences. Translate to decision units.
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. Use an omnibus difference as a signal to examine where and how reliability changed. A broad modest gap may matter for high-volume systems. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. The test does not declare one design uniformly superior. Combine with signed effect plots.
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.
Create two step curves on five intervals, compute signed gaps, squared gaps and a width-weighted sum. Compare with the maximum gap and terminal score. 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. Toy sums are not calibrated p-values. Label every integration weight. 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. CvM tests connect area, squares, graphs, accumulation and the idea that direction can cancel. In the illustrative case, Ask why squaring preserves distance from zero. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. Use shaded gap rectangles. Then discuss censoring.
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. The statistic is not the area between curves unless defined that way, its p-value is not a probability one design is better, and a global rejection gives no automatic direction. Resampling assumptions matter. Recompute the example from the raw values and label every unit. Late unsupported gaps can mislead. Report range and effect summaries.
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. Koziol developed a two-sample CvM test for randomly censored data, and Schumacher studied CvM and Kolmogorov–Smirnov two-sample procedures under censoring. Consider the worked illustration: Reproduce a discrete squared-gap sum before software. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. Omnibus evidence does not explain physical cause. Read calibration beside power comparisons.
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 integrate squared differences
Setup: Use this chapter’s illustrative case: A material can be modestly less reliable across most of 0–1000 hours.
Task: Square or otherwise norm the path.
Checking target: Adding signed gaps can cancel.
Practice case 2: Cramér–von Mises principle
Setup: Use this chapter’s illustrative case: For two curves, accumulate [S1(t)-S2(t)]^2 with a chosen measure.
Task: State the exact statistic.
Checking target: The integration measure changes emphasis.
Practice case 3: Censoring-aware survival comparison
Setup: Use this chapter’s illustrative case: A withdrawn system informs the curve only to its censoring time.
Task: Use a censored-data method.
Checking target: An uncensored two-sample formula is not automatically valid.
Practice case 4: Squared score-process form
Setup: Use this chapter’s illustrative case: Both positive early and negative late departures add positively after squaring.
Task: Show the process definition.
Checking target: Standardisation and covariance matter.
Practice case 5: A discrete approximation
Setup: Use this chapter’s illustrative case: Gaps .1 and -.1 each contribute .01 before weighting.
Task: Keep width and gap columns.
Checking target: Unequal grid spacing needs weights.
Practice case 6: A worked curve-gap calculation
Setup: Use this chapter’s illustrative case: A signed sum would be .10 and tells a different story.
Task: Separate discrepancy from p-value.
Checking target: The toy number is not a calibrated test statistic.
Practice case 7: Why signs do not cancel
Setup: Use this chapter’s illustrative case: Crossing curves can produce a large integrated discrepancy despite a small average signed gap.
Task: Plot signed differences too.
Checking target: Direction is lost in the total.
Practice case 8: Broad versus sharp departures
Setup: Use this chapter’s illustrative case: A broad .05 gap can accumulate substantial evidence.
Task: Match method to plausible patterns.
Checking target: No test dominates every alternative.
Practice case 9: Choice of integration measure
Setup: Use this chapter’s illustrative case: Event-time weighting concentrates on observed information.
Task: Follow the cited method exactly.
Checking target: Clock-time integration can overemphasise long empty intervals.
Practice case 10: Pooled risk-set information
Setup: Use this chapter’s illustrative case: A gap supported by 100 systems carries more reliable information than one supported by four.
Task: Report risk sets.
Checking target: Raw curve distance ignores precision.