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eduKateSG · Why Mathematics? · Gönen–Heller Concordance for Reliability Cox Models: Pairwise Risk Ordering, Probability, Censoring and Discrimination
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 model must rank which components are likely to fail sooner, even though many units have not yet failed when observation ends. The Gönen–Heller concordance probability uses the Cox model's fitted linear predictors and their distribution to estimate how often the model orders a pair consistently with the proportional-hazards mechanism. It is a model-based discrimination measure, not an absolute failure probability.
This guide develops pairwise ordering, log hazard ratios, the logistic concordance probability, averaging over pairs, censoring, assumptions, validation and responsible engineering interpretation. A high value supports risk ordering under a correctly specified Cox model; it does not prove calibration, causality, proportional hazards or safe maintenance decisions. The worked numbers are illustrative and should be checked against the full model and target fleet.
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
Discrimination asks whether a model gives higher fitted risk to the unit that fails sooner across comparable pairs. A pump scored 1.2 and another .4; the first is modelled as higher hazard. This is the central mathematical mechanism in why pairwise risk ordering matters. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. Ordering quality is different from accurate absolute failure probabilities. Name the decision and the comparison population. 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 cox linear predictor. The Cox score eta=x beta adds covariate contributions on the log-hazard scale, and exp(eta) is relative hazard against the baseline. In the illustrative case, Eta values 1.2 and .4 differ by .8, so their fitted hazard ratio is exp(.8)=2.23. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. The baseline hazard cancels in a pairwise relative-risk comparison. Audit coefficients, units and coding before ranking.
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 concordant and discordant pairs as a model-checking problem rather than a vocabulary exercise. A pair is concordant when the larger fitted score belongs to the member whose event time is earlier under the target event definition. The higher-score bearing fails at 600 hours while the lower-score bearing fails at 900 hours. Recompute the example from the raw values and label every unit. Censoring can make the observed order unknown. Do not force incomparable pairs into an observed-time score.
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 pairwise logistic probability becomes useful only when its mechanism is explicit. Under a proportional-hazards model, a positive score difference d maps to a conditional concordance probability expit(d)=1/(1+exp(-d)). Consider the worked illustration: For d=.8, expit(.8)=.68997, about .690. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. This probability comes from the model and is not an observed pair frequency by itself. Reproduce the exponent, denominator and rounding.
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/10.1093/biomet/92.4.965. Read it for the method and its assumptions; do not transfer its performance claims automatically to another model.
Use why score differences matter to practise sensitivity analysis. Adding the same constant to every linear predictor leaves all differences and therefore concordance unchanged. Scores 1.2 and .4 give the same pair probability as 2.2 and 1.4. Before accepting the result, identify the quantity that would change the conclusion most. Rescaling scores changes differences and changes the measure. Separate harmless shifts from calibration changes.
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 average over ordered pairs. The estimator averages model-based pair concordance probabilities across the empirical covariate distribution. The initial example is: Four units create six unordered pairs whose probabilities can be averaged once each. Then create a nearby case where the usual shortcut becomes unreliable. Duplicating one covariate pattern changes the target population weighting. State whose covariate distribution defines the estimate.
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 four-unit worked example to a real decision while protecting the boundary between calculation and judgement. Compute every absolute score difference, transform it with expit and average the six results. Scores 1.2,.4,.1,-.3 produce pair differences .8,1.1,1.5,.3,.7,.4. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. A small example illustrates arithmetic but has unstable sampling uncertainty. Keep a visible pair table.
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.
Equal fitted scores yield expit(0)=.5, the no-ordering value for that pair. Two identical score values contribute .5 even if their later observed outcomes differ. This is the central mathematical mechanism in the 0.5 reference point. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. A global estimate near .5 can still hide useful ordering in a subgroup. Inspect the score distribution and ties. 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 interpret values below one. Even large score separation maps to probabilities below one because proportional-hazards event ordering remains stochastic. In the illustrative case, A difference of 2 gives expit(2)=.881, not certainty. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. No finite score gap guarantees which unit fails first. Avoid deterministic maintenance language.
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 how this differs from harrell's c as a model-checking problem rather than a vocabulary exercise. Gönen–Heller uses fitted parameters and covariates rather than selecting observed informative event-time pairs. An early-censored unit still contributes through its covariates to the model-based estimate. Recompute the example from the raw values and label every unit. The advantage relies on correct proportional-hazards specification. Report which concordance estimator was used.
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 and model dependence becomes useful only when its mechanism is explicit. Because the final formula does not directly use observed event and censoring times, censoring enters through estimation of the Cox model rather than pair eligibility. Consider the worked illustration: Administrative closure affects coefficient precision but not a later pair-counting step. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. Informative censoring can still bias the fitted coefficients. Audit the censoring process.
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 proportional-hazards assumption to practise sensitivity analysis. The pair probability assumes a score difference has the same hazard-ratio meaning over follow-up. A coating that helps only during burn-in violates a constant effect. Before accepting the result, identify the quantity that would change the conclusion most. A single concordance number can conceal time-varying discrimination. Pair the metric with proportionality diagnostics.
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://itl.nist.gov/div898/handbook/apr/section1/apr167.htm. Compare the problem definition, estimator and validation rather than quoting one result in isolation.
Build a small counterexample for ties in scores and events. Equal risk scores contribute .5, while tied recorded failures require a declared event-time convention during Cox fitting. The initial example is: Weekly inspections can record several failures on the same date. Then create a nearby case where the usual shortcut becomes unreliable. Changing tie handling between fit and validation changes coefficients. Use one documented tie method.
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 continuous and categorical covariates to a real decision while protecting the boundary between calculation and judgement. Both enter the linear predictor, but coding determines distance between scores and therefore pair probabilities. A three-level supplier factor needs explicit reference coding. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. Arbitrary numeric labels for categories create false distances. Preserve the design matrix.
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.
Splines or transformations change eta and can improve ordering when physical risk is not linear in a measurement. Vibration risk may rise sharply only above a threshold. This is the central mathematical mechanism in nonlinear predictor effects. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. Choosing flexibility on the validation set inflates performance. Freeze transformations before external scoring. 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 interactions and operating regimes. An interaction lets the effect of one stress depend on another and can alter pair orderings across regimes. In the illustrative case, Temperature has a stronger effect under high load. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. Sparse combinations yield unstable score differences. Show support for every interaction region.
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 sampling uncertainty as a model-checking problem rather than a vocabulary exercise. The original derivation uses U-statistic theory because many pair contributions share observations and are dependent. Six pairs from four units are not six independent experiments. Recompute the example from the raw values and label every unit. A naive standard error treating pairs as independent is too optimistic. Use validated variance or resampling procedures.
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 and optimism becomes useful only when its mechanism is explicit. Evaluating the same fitted scores on development covariates can overstate transportable discrimination after variable selection or tuning. Consider the worked illustration: A model selected from fifty candidates looks better in its training fleet. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. The metric alone does not correct model-selection optimism. Repeat the entire modelling process in bootstrap validation.
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. Apply the frozen coefficients to a genuinely new fleet and calculate concordance in that target covariate distribution. A factory-A model is assessed on factory-B pumps. Before accepting the result, identify the quantity that would change the conclusion most. Changed score spread can alter concordance even when coefficients transport. Report both distribution shift and performance.
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 separate question. Concordance can remain unchanged if all baseline failure probabilities are badly shifted while the ordering stays intact. The initial example is: A model ranks correctly but predicts 20 percent failure where 8 percent occurs. Then create a nearby case where the usual shortcut becomes unreliable. Good discrimination cannot certify probability accuracy. Add horizon-specific and distribution-level calibration checks.
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 usefulness is separate too to a real decision while protecting the boundary between calculation and judgement. A concordance estimate does not encode inspection cost, missed-failure consequence, capacity or legal rules. A .75 metric does not choose a replacement threshold. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. A higher score can still produce no operational benefit. Evaluate the model-policy pair.
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.
Calculate expit(.8), expit(.3) and expit(0), then explain why each stays between .5 and one for ordered unequal pairs. The values are .690, .574 and .500 after rounding. 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. These are conditional model probabilities, not fleet failure rates. Check with both a calculator and a plot. 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. Pairwise concordance connects algebra, exponentials, inequalities, averages and probability to a real model-checking question. In the illustrative case, Ask a student why adding ten to all scores changes no ordering. 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 relative risk with probability of failure by a date. Keep the event, horizon and scale visible.
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 measure is not censoring-proof under any data process, not a calibration statistic and not guaranteed stable across populations. A correct formula cannot repair biased covariates or informative withdrawal. Recompute the example from the raw values and label every unit. Values from different fleets need not be directly comparable. State assumptions and target population.
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. Gönen and Heller derived the Cox concordance probability and its asymptotic distribution, while NIST explains the proportional-hazards reliability framework. Consider the worked illustration: Read the derivation beside a worked design-matrix calculation. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. The method assumes a correctly specified proportional-hazards model. Reproduce the .8 score-gap example 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 pairwise risk ordering matters
Setup: Use this chapter’s illustrative case: A pump scored 1.2 and another .4; the first is modelled as higher hazard.
Task: Name the decision and the comparison population.
Checking target: Ordering quality is different from accurate absolute failure probabilities.
Practice case 2: The Cox linear predictor
Setup: Use this chapter’s illustrative case: Eta values 1.2 and .4 differ by .8, so their fitted hazard ratio is exp(.8)=2.23.
Task: Audit coefficients, units and coding before ranking.
Checking target: The baseline hazard cancels in a pairwise relative-risk comparison.
Practice case 3: Concordant and discordant pairs
Setup: Use this chapter’s illustrative case: The higher-score bearing fails at 600 hours while the lower-score bearing fails at 900 hours.
Task: Do not force incomparable pairs into an observed-time score.
Checking target: Censoring can make the observed order unknown.
Practice case 4: The pairwise logistic probability
Setup: Use this chapter’s illustrative case: For d=.8, expit(.8)=.68997, about .690.
Task: Reproduce the exponent, denominator and rounding.
Checking target: This probability comes from the model and is not an observed pair frequency by itself.
Practice case 5: Why score differences matter
Setup: Use this chapter’s illustrative case: Scores 1.2 and .4 give the same pair probability as 2.2 and 1.4.
Task: Separate harmless shifts from calibration changes.
Checking target: Rescaling scores changes differences and changes the measure.
Practice case 6: Average over ordered pairs
Setup: Use this chapter’s illustrative case: Four units create six unordered pairs whose probabilities can be averaged once each.
Task: State whose covariate distribution defines the estimate.
Checking target: Duplicating one covariate pattern changes the target population weighting.
Practice case 7: A four-unit worked example
Setup: Use this chapter’s illustrative case: Scores 1.2,.4,.1,-.3 produce pair differences .8,1.1,1.5,.3,.7,.4.
Task: Keep a visible pair table.
Checking target: A small example illustrates arithmetic but has unstable sampling uncertainty.
Practice case 8: The 0.5 reference point
Setup: Use this chapter’s illustrative case: Two identical score values contribute .5 even if their later observed outcomes differ.
Task: Inspect the score distribution and ties.
Checking target: A global estimate near .5 can still hide useful ordering in a subgroup.
Practice case 9: Interpret values below one
Setup: Use this chapter’s illustrative case: A difference of 2 gives expit(2)=.881, not certainty.
Task: Avoid deterministic maintenance language.
Checking target: No finite score gap guarantees which unit fails first.
Practice case 10: How this differs from Harrell's c
Setup: Use this chapter’s illustrative case: An early-censored unit still contributes through its covariates to the model-based estimate.
Task: Report which concordance estimator was used.
Checking target: The advantage relies on correct proportional-hazards specification.