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eduKateSG · Why Mathematics? · Calibration-in-the-Large and Calibration Slope for Reliability Survival Models: Baseline Risk, Spread, Recalibration and External Validation
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 is transferred to a new fleet and two different questions must be separated. Calibration-in-the-large asks whether predictions are systematically too high or too low overall. The calibration slope asks whether the spread of the prognostic index is too wide or too narrow. Crowson, Atkinson and Therneau described a model-based framework, including Poisson regression, for calibration of prognostic risk scores with survival outcomes.
This guide develops observed-to-expected ratios, log calibration intercepts, prognostic-index slopes, baseline risk, shrinkage, censoring and external validation. If 24 events are observed when 30 are expected, O/E = 0.8 and log(O/E) ≈ −0.223. A slope of 0.7 maps prognostic scores −1, 0 and 1 to −0.7, 0 and 0.7 before baseline recalibration. Neither statistic alone proves full calibration; curves remain essential.
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
Overall level and prediction spread can fail separately when a model moves populations. A fleet has fewer failures overall but similar risk ordering. This is the central mathematical mechanism in two calibration dimensions. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. One statistic cannot describe both. Report level and slope. 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 expected events. Expected counts come from the original model’s predicted cumulative hazards or risks over each unit’s follow-up. In the illustrative case, The model expects 30 failures by the study cutoff. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. Expected values require the baseline hazard, not coefficients alone. Preserve the full prediction model.
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 observed events as a model-checking problem rather than a vocabulary exercise. Observed failures are counted over the compatible follow-up design, with censoring handled by the survival framework. Twenty-four failures are observed under the validation protocol. Recompute the example from the raw values and label every unit. Raw counts can be incomparable with varying exposure. Use the declared framework.
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 observed-to-expected ratio becomes useful only when its mechanism is explicit. O/E divides observed events by model-expected events. Consider the worked illustration: 24/30=.8. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. The exact construction can be time-specific or exposure-based. Define the denominator.
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/10.1177/0962280213497434. Read it for the method and its assumptions; do not transfer its performance claims automatically to another model.
Use interpreting o/e to practise sensitivity analysis. O/E below one means fewer observed events than expected under the model, while above one means more. O/E .8 indicates twenty percent fewer observed than expected in this aggregate sense. Before accepting the result, identify the quantity that would change the conclusion most. It does not locate subgroup error. Add a calibration curve.
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 the log calibration intercept. In a Poisson offset formulation, the fitted intercept exponentiates to O/E. The initial example is: log(.8)=-.2231. Then create a nearby case where the usual shortcut becomes unreliable. Sign interpretation depends on modelling event versus survival scale. Write the model equation.
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 numerical level example to a real decision while protecting the boundary between calculation and judgement. Compute 24 divided by 30, then take the natural logarithm. The results are .8 and approximately -.223. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. Small counts need uncertainty. 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.
For a Cox model, the prognostic index is the original linear predictor beta-transpose-X without the baseline hazard. Scores -1,0,1 encode relative hazard spread. This is the central mathematical mechanism in the prognostic index. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. A score alone cannot produce absolute risk. Retain baseline information. 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 calibration slope. Fit the original prognostic index as the only covariate in a validation survival model; its coefficient is the calibration slope. In the illustrative case, A fitted coefficient .7 compresses the score spread. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. The slope is not complete calibration. Interpret with level and curve.
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 slope example as a model-checking problem rather than a vocabulary exercise. Multiplying scores -1,0,1 by .7 gives -.7,0,.7. Hazard contrasts become less extreme. Recompute the example from the raw values and label every unit. This arithmetic omits baseline recalibration. Keep level and spread separate.
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 slope below one becomes useful only when its mechanism is explicit. A slope below one often indicates predictions are too extreme in the validation setting, consistent with development overfitting or case-mix change. Consider the worked illustration: Slope .7 shrinks high and low scores toward zero. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. Other mechanisms can also cause it. Avoid diagnosing cause from slope alone.
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 slope above one to practise sensitivity analysis. A slope above one means the validation association is steeper than the original score implies. Slope 1.3 expands score contrasts. Before accepting the result, identify the quantity that would change the conclusion most. Sampling variation can produce values above one. Report uncertainty.
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://pmc.ncbi.nlm.nih.gov/articles/PMC3933449/. Compare the problem definition, estimator and validation rather than quoting one result in isolation.
Build a small counterexample for why slope one is insufficient. A slope of one can coexist with a wrong baseline risk and therefore systematic over- or underprediction. The initial example is: Correct spread sits on an offset calibration curve. Then create a nearby case where the usual shortcut becomes unreliable. Calling slope alone calibration is misleading. Always assess overall level.
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 o/e one is insufficient to a real decision while protecting the boundary between calculation and judgement. Balanced aggregate predictions can hide opposing errors across risk groups. Low-risk units are overpredicted and high-risk units underpredicted. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. Errors cancel in total expected counts. Plot flexible 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.
Poisson regression with suitable offsets can estimate overall calibration and extensions for survival risk scores. Observed event counts are related to expected hazards. This is the central mathematical mechanism in poisson calibration framework. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. Exposure and time construction must match the model. Follow the published setup. 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 censoring and follow-up. Validation must account for incomplete observation and varying follow-up rather than treating censored units as event-free forever. In the illustrative case, A unit leaves observation at 400 hours. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. Naive event proportions distort level. Use survival-aware expected counts.
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 baseline hazard availability as a model-checking problem rather than a vocabulary exercise. Absolute calibration cannot be reconstructed from relative-risk coefficients alone. A published Cox score lacks its baseline cumulative hazard. Recompute the example from the raw values and label every unit. Approximating it from another population changes the model. Obtain complete model specifications.
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 recalibration options becomes useful only when its mechanism is explicit. Updating baseline level can correct broad event-rate shift; applying a slope can shrink or expand the prognostic index. Consider the worked illustration: Use new baseline hazard and beta times original PI. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. Updating on the test set changes the estimand. Show performance before and after.
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 internal validation to practise sensitivity analysis. Apparent slopes estimated on development data are biased toward favourable values. Variable selection inflates score spread. Before accepting the result, identify the quantity that would change the conclusion most. A fitted slope on the same data is not external evidence. Use bootstrap correction.
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 external validation. Freeze the original score and baseline, then estimate level, slope and curves in the target fleet. The initial example is: A model from factory A is tested in factory B. Then create a nearby case where the usual shortcut becomes unreliable. Different workload can shift both dimensions. Describe case mix and follow-up.
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 uncertainty and sample size to a real decision while protecting the boundary between calculation and judgement. O/E and slope need enough events and follow-up for useful precision. Twenty-four events give wider intervals than 240. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. Point estimates can look decisive when noisy. Plan validation size.
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 24/30, log(.8), and .7 times scores -1,0,1. The answers are .8, -.2231 and -.7,0,.7. 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 examples teach separate dimensions. Do not merge them into one grade. 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 topic connects ratios, logarithms, linear transformations and graphs to checking transferred predictions. In the illustrative case, Ask how a correct slope can still have a wrong intercept. 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 ranking with probability accuracy. Draw two shifted lines.
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. Calibration slope is not full calibration, O/E one is not local calibration, and recalibration does not prove causal transport. A good pair of summaries can still hide nonlinear error. Recompute the example from the raw values and label every unit. Censoring and baseline data remain essential. Keep a flexible 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.
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. Crowson, Atkinson and Therneau presented a model-based calibration framework for prognostic scores, including survival extensions with Poisson regression. Consider the worked illustration: Read the open paper beside the O/E and slope worksheets. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. Complete model information is necessary. Reproduce .8 and -.2231.
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: Two calibration dimensions
Setup: Use this chapter’s illustrative case: A fleet has fewer failures overall but similar risk ordering.
Task: Report level and slope.
Checking target: One statistic cannot describe both.
Practice case 2: Expected events
Setup: Use this chapter’s illustrative case: The model expects 30 failures by the study cutoff.
Task: Preserve the full prediction model.
Checking target: Expected values require the baseline hazard, not coefficients alone.
Practice case 3: Observed events
Setup: Use this chapter’s illustrative case: Twenty-four failures are observed under the validation protocol.
Task: Use the declared framework.
Checking target: Raw counts can be incomparable with varying exposure.
Practice case 4: The observed-to-expected ratio
Setup: Use this chapter’s illustrative case: 24/30=.8.
Task: Define the denominator.
Checking target: The exact construction can be time-specific or exposure-based.
Practice case 5: Interpreting O/E
Setup: Use this chapter’s illustrative case: O/E .8 indicates twenty percent fewer observed than expected in this aggregate sense.
Task: Add a calibration curve.
Checking target: It does not locate subgroup error.
Practice case 6: The log calibration intercept
Setup: Use this chapter’s illustrative case: log(.8)=-.2231.
Task: Write the model equation.
Checking target: Sign interpretation depends on modelling event versus survival scale.
Practice case 7: A numerical level example
Setup: Use this chapter’s illustrative case: The results are .8 and approximately -.223.
Task: Report a confidence interval.
Checking target: Small counts need uncertainty.
Practice case 8: The prognostic index
Setup: Use this chapter’s illustrative case: Scores -1,0,1 encode relative hazard spread.
Task: Retain baseline information.
Checking target: A score alone cannot produce absolute risk.
Practice case 9: The calibration slope
Setup: Use this chapter’s illustrative case: A fitted coefficient .7 compresses the score spread.
Task: Interpret with level and curve.
Checking target: The slope is not complete calibration.
Practice case 10: A slope example
Setup: Use this chapter’s illustrative case: Hazard contrasts become less extreme.
Task: Keep level and spread separate.
Checking target: This arithmetic omits baseline recalibration.