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eduKateSG · Why Mathematics? · Royston–Sauerbrei D-Index for Reliability Models: Prognostic Separation, Hazard Ratios, Censoring and R²_D
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 is visible when a reliability model produces a continuous prognostic score but a team needs an interpretable statement about how strongly that score separates earlier from later failures. The Royston–Sauerbrei D-index expresses prognostic separation on a log-hazard-ratio scale. Under its framework, D can be read as the log hazard ratio between two equally sized prognostic groups formed around the score distribution, and exp(D) gives the corresponding hazard-ratio interpretation.
This guide develops prognostic indices, score standardisation, the constant kappa, D, hazard-ratio translation, R²_D, censoring, validation and limitations. Larger positive D indicates stronger separation in the assessed data, but it does not prove calibration, causality, proportional hazards or decision value. Distributional assumptions and optimism correction matter.
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
A useful model should separate systems with different event-time experience, not merely fit coefficients with small p-values. High-score turbines fail earlier than low-score turbines in validation data. This is the central mathematical mechanism in why prognostic separation matters. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. Separation is not absolute probability calibration. Define the target event and 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 prognostic index. A prognostic index is the fitted linear predictor or another ordered score intended to summarise relative event risk. In the illustrative case, Eta=.6 for one unit and -.2 for another places the first higher. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. Different software may orient scores in opposite directions. Confirm that larger means higher hazard.
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 from score spread to separation as a model-checking problem rather than a vocabulary exercise. D uses the effective spread of prognostic scores together with survival outcomes to quantify how distinctly the model separates experience. A nearly constant score gives little possible separation. Recompute the example from the raw values and label every unit. Raw standard deviation alone is not the validated D estimate in every implementation. Use the published survival-model procedure.
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 kappa constant becomes useful only when its mechanism is explicit. The framework uses kappa=sqrt(8/pi), approximately 1.596, to connect normal score spread with two equal-sized groups. Consider the worked illustration: sqrt(8/pi)=1.59577. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. The interpretation relies on an approximately normal prognostic-index distribution. Check the score histogram and tails.
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.1621. Read it for the method and its assumptions; do not transfer its performance claims automatically to another model.
Use d on the log-hazard scale to practise sensitivity analysis. D is interpretable as a log hazard ratio between equally sized low- and high-prognostic groups under the method's assumptions. D=.7 means the high group has log hazard ratio .7 against the low group. Before accepting the result, identify the quantity that would change the conclusion most. This is a summary contrast, not a literal fixed cutpoint policy. Report D before exponentiating.
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 translate d into a hazard ratio. Exponentiating converts D to the familiar hazard-ratio scale: HR=exp(D). The initial example is: exp(.7)=2.01375, about 2.01. Then create a nearby case where the usual shortcut becomes unreliable. A hazard ratio is not a ratio of failure probabilities by a date. State the proportional-hazards interpretation.
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 what d=0 means to a real decision while protecting the boundary between calculation and judgement. D=0 corresponds to no prognostic separation on this scale and exp(0)=1. A validation estimate near zero says the ordering has little outcome separation. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. Sampling uncertainty can move an estimate around zero. Report a confidence interval.
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.
In external validation, a negative D can indicate reversal of the intended score direction or serious transport failure. High predicted risk is associated with later rather than earlier failures. This is the central mathematical mechanism in why d can be negative. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. A coding inversion can also create this pattern. Audit direction and data before interpretation. 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 r²_d as a bounded transformation. R²_D transforms D through the method's variance relation to a zero-to-one explained-variation scale. In the illustrative case, With kappa=sqrt(8/pi) and residual variance pi^2/6, D=.7 gives a modest bounded value. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. R²_D is not ordinary linear-regression R-squared. Name the exact formula and software.
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 a worked r²_d calculation as a model-checking problem rather than a vocabulary exercise. Using R²_D=(D²/kappa²)/[(D²/kappa²)+(pi²/6)] gives a transparent transformation. For D=.7, D²/kappa²=.1924 and R²_D=.1924/(.1924+1.6449)=.1047. Recompute the example from the raw values and label every unit. Alternative reported adjustments must not be mixed into this classroom calculation. Keep constants and rounding visible.
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 risk groups are an interpretation becomes useful only when its mechanism is explicit. The equal-sized groups explain D, while estimation should preserve the continuous prognostic score rather than discard information by arbitrary categorisation. Consider the worked illustration: A median split helps explain exp(D) but is not needed to operate the continuous model. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. Actual maintenance bands may use different thresholds. Do not turn the explanation into a policy cutpoint.
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 right censoring to practise sensitivity analysis. Survival-model estimation uses risk sets so censored units contribute while known event-free. A unit censored at 800 hours remains at risk until 800. Before accepting the result, identify the quantity that would change the conclusion most. Informative censoring can bias separation estimates. Describe follow-up and censoring support.
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/14981672/. Compare the problem definition, estimator and validation rather than quoting one result in isolation.
Build a small counterexample for proportional-hazards structure. The hazard-ratio interpretation assumes the prognostic effect is sufficiently stable over time. The initial example is: A score separates burn-in failures but not late wear-out. Then create a nearby case where the usual shortcut becomes unreliable. One D value can average over changing effects. Add time-specific diagnostics.
Counterexamples are positive learning tools. They reveal the conditions that make a method work and help students recognise when a software output deserves a pause. Keep the counterexample small enough to plot or enumerate. Check sign, support, bounds and conservation of probability before adding complexity.
When writing the conclusion, separate a documented fact from an interpretation and from an illustrative scenario. Cite the primary method paper for the factual mechanism, use the local calculation for the illustration, and label any recommendation as conditional on the stated model. This keeps career and engineering advice age-appropriate and honest.
Connect normal-score approximation to a real decision while protecting the boundary between calculation and judgement. The link between D and score spread assumes an approximately normal prognostic-index distribution. A bimodal score from two machine classes violates the simple shape. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. Marked non-normality can affect R²_D. Plot the score and consider stratified reporting.
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.
Extreme scores can alter model coefficients and apparent separation even if the D construction is designed for stable interpretation. One impossible sensor value creates eta=8. This is the central mathematical mechanism in outliers and leverage. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. Deleting an inconvenient unit without a rule biases validation. Check provenance and influence. 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 continuous versus grouped scores. D can evaluate continuous, grouped or rule-based prognostic indices, but information and assumptions differ. In the illustrative case, A five-band risk rule creates discrete score clusters. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. Normality is only approximate for a coarse score. Explain how the score was produced.
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 optimism in development data as a model-checking problem rather than a vocabulary exercise. The same sample used to choose predictors and estimate coefficients typically overstates D. Stepwise selection finds a strong separation by chance. Recompute the example from the raw values and label every unit. Apparent D is not corrected merely by reporting R²_D. Bootstrap the entire modelling process.
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 external validation benchmark becomes useful only when its mechanism is explicit. A frozen score applied to new data tests whether prognostic separation transports. Consider the worked illustration: A laboratory-derived model is evaluated in service conditions. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. Different case mix changes score spread and D. Report the new population rather than comparing blindly.
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 compare d with concordance to practise sensitivity analysis. D measures separation on a log-hazard scale while concordance estimates pairwise ordering probability. Two metrics can rank models differently because their scales and sensitivities differ. Before accepting the result, identify the quantity that would change the conclusion most. Neither replaces calibration. Use a small complementary set of metrics.
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 and baseline risk. D can remain strong when the baseline survival curve is wrong because separation and level are distinct. The initial example is: High and low groups are well separated but both predicted too optimistically. Then create a nearby case where the usual shortcut becomes unreliable. A large D does not validate probabilities. Add calibration-in-the-large and curve 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 interpretation to a real decision while protecting the boundary between calculation and judgement. The hazard-ratio reading helps communicate model strength but does not encode action costs or safety requirements. HR 2.01 does not automatically justify replacement. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. A relative contrast can coexist with low absolute risk. Use absolute risk and consequences for decisions.
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 kappa, exp(.7) and the stated R²_D transformation, then label each result's scale. The values are 1.596, 2.01 and about .105. 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. Rounding kappa to 1.6 is fine only after the exact calculation. Preserve the unrounded worksheet. 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 D-index links exponentials, logarithms, distributions, ratios and transformations to model validation. In the illustrative case, Ask why exp(.7) is about two while .7 itself is not a probability. 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 hazard ratio as double the chance by every horizon. Use a timeline analogy carefully.
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. D is not a universal grade, a causal effect, an absolute-risk measure or a guarantee of stable field performance. Case mix and score distribution affect the estimate. Recompute the example from the raw values and label every unit. A median-group interpretation is not a mandated threshold. Report assumptions, uncertainty and validation design.
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. Royston and Sauerbrei introduced D and R²_D for prognostic separation and discussed optimism correction and validation. Consider the worked illustration: Read the primary paper's simulation and derivation beside the worked .7 example. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. Use validated software for production estimates. Check score direction before everything else.
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 prognostic separation matters
Setup: Use this chapter’s illustrative case: High-score turbines fail earlier than low-score turbines in validation data.
Task: Define the target event and population.
Checking target: Separation is not absolute probability calibration.
Practice case 2: The prognostic index
Setup: Use this chapter’s illustrative case: Eta=.6 for one unit and -.2 for another places the first higher.
Task: Confirm that larger means higher hazard.
Checking target: Different software may orient scores in opposite directions.
Practice case 3: From score spread to separation
Setup: Use this chapter’s illustrative case: A nearly constant score gives little possible separation.
Task: Use the published survival-model procedure.
Checking target: Raw standard deviation alone is not the validated D estimate in every implementation.
Practice case 4: The kappa constant
Setup: Use this chapter’s illustrative case: sqrt(8/pi)=1.59577.
Task: Check the score histogram and tails.
Checking target: The interpretation relies on an approximately normal prognostic-index distribution.
Practice case 5: D on the log-hazard scale
Setup: Use this chapter’s illustrative case: D=.7 means the high group has log hazard ratio .7 against the low group.
Task: Report D before exponentiating.
Checking target: This is a summary contrast, not a literal fixed cutpoint policy.
Practice case 6: Translate D into a hazard ratio
Setup: Use this chapter’s illustrative case: exp(.7)=2.01375, about 2.01.
Task: State the proportional-hazards interpretation.
Checking target: A hazard ratio is not a ratio of failure probabilities by a date.
Practice case 7: What D=0 means
Setup: Use this chapter’s illustrative case: A validation estimate near zero says the ordering has little outcome separation.
Task: Report a confidence interval.
Checking target: Sampling uncertainty can move an estimate around zero.
Practice case 8: Why D can be negative
Setup: Use this chapter’s illustrative case: High predicted risk is associated with later rather than earlier failures.
Task: Audit direction and data before interpretation.
Checking target: A coding inversion can also create this pattern.
Practice case 9: R²_D as a bounded transformation
Setup: Use this chapter’s illustrative case: With kappa=sqrt(8/pi) and residual variance pi^2/6, D=.7 gives a modest bounded value.
Task: Name the exact formula and software.
Checking target: R²_D is not ordinary linear-regression R-squared.
Practice case 10: A worked R²_D calculation
Setup: Use this chapter’s illustrative case: For D=.7, D²/kappa²=.1924 and R²_D=.1924/(.1924+1.6449)=.1047.
Task: Keep constants and rounding visible.
Checking target: Alternative reported adjustments must not be mixed into this classroom calculation.