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eduKateSG · Why Mathematics? · Graf Brier Scores for Censored Reliability Predictions: IPCW Residuals, Survival Probabilities, Time Horizons and 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 practical when a reliability model predicts a probability of surviving beyond a time horizon and the team needs to measure probability error despite right censoring. Graf and colleagues developed prediction-error measures that use inverse probability of censoring weights, allowing observed failures and known survivors to contribute without assigning false outcomes to units whose status is unknown.
This guide develops the fixed-time Brier loss, event and survivor contributions, Kaplan–Meier censoring survival, inverse weights, integrated scores and validation. A complete-data example gives a mean squared probability error of 0.075. The censoring-adjusted score evaluates predictive accuracy under stated assumptions. It is not a causal estimate, a ranking-only measure or proof that a maintenance policy is valuable.
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
At time t, the model predicts S-hat(t given X), the probability a unit remains event-free beyond t. A model assigns survival probabilities .2, .4, .7 and .9 at 500 hours. This is the central mathematical mechanism in the prediction target. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. Predictions must refer to the same origin, event and horizon. Label the probability explicitly. 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 binary outcome at a horizon. The observable target without censoring is I(T greater than t), equal to one for survival beyond t and zero for failure by t. In the illustrative case, Two units failed by 500 and two survived beyond it. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. The status is not known after early censoring. Write the indicator rule.
A student should be able to rebuild the result without hidden settings. Record the sampling rule or optimisation criterion, the exact failure definition, and the evidence used for every distribution or parameter. Compare a transparent baseline with the advanced method. If they disagree, investigate the mechanism before deciding which answer is more credible.
This chapter belongs to a larger decision, so translate the number carefully. State one action the evidence permits and one action it does not. A parent or educator can ask which school ideas are doing the work: algebra organises relationships, probability represents uncertainty, calculus describes local change, and computing repeats the documented rules at scale.
Treat the squared probability residual as a model-checking problem rather than a vocabulary exercise. The Brier loss is the squared difference between the binary outcome and predicted survival probability. A failure with predicted survival .2 has error (0-.2)^2=.04. Recompute the example from the raw values and label every unit. Squared loss combines calibration and discrimination. Keep the prediction on a zero-to-one scale.
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 a complete-data example becomes useful only when its mechanism is explicit. For outcomes 0,0,1,1 and predictions .2,.4,.7,.9, squared errors are .04,.16,.09,.01. Consider the worked illustration: Their mean is .30 divided by 4, or .075. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. Four observations are a teaching example only. Recalculate every square.
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/(SICI)1097-0258(19990915/30)18:17/18%3C2529::AID-SIM274%3E3.0.CO;2-5. Read it for the method and its assumptions; do not transfer its performance claims automatically to another model.
Use why lower is better to practise sensitivity analysis. A smaller Brier score means predicted probabilities lie closer to observed binary outcomes at the chosen horizon. Perfect predictions give zero loss. Before accepting the result, identify the quantity that would change the conclusion most. The attainable and reference values depend on event prevalence. Compare with a baseline model.
A simple baseline is essential. It reveals whether the advanced method is improving accuracy or only adding complexity. Compare model calls, wall time, uncertainty and failure-mode coverage under one declared budget. Repeat stochastic calculations with several seeds; a favourable single run is not evidence of dependable efficiency.
Students can extend the example by adding correlation, a second failure mode or a modest measurement error. Explain which part of the formula changes and which remains fixed. This exercise shows how mathematics transfers from a clean textbook case to engineering and computing without pretending that the classroom model is a certified design tool.
Build a small counterexample for why censoring breaks the raw mean. A unit censored before t has unknown I(T greater than t), so assigning zero or one creates error unrelated to the prediction. The initial example is: Censoring at 350 leaves 500-hour status unknown. Then create a nearby case where the usual shortcut becomes unreliable. Deleting all such units can also bias the sample. Use censoring-aware estimation.
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 the censoring survival function to a real decision while protecting the boundary between calculation and judgement. G(u) is the probability of remaining uncensored beyond time u, estimated by reversing the event indicator in a Kaplan–Meier calculation. G(500)=.8 means about 80 percent remain observable through 500 under the censoring process. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. G is not component survival S. Use separate notation and plots.
The audit should include the original question, input provenance, failure convention, algorithm settings, convergence or stopping rule, and a sensitivity result. For an adaptive method, also preserve the sequence of choices. For an optimisation, preserve every constraint. For Bayesian updating, preserve the prior and observation model. These records turn a result into something another person can challenge and improve.
A final transfer question is useful: where else would the same mathematical idea appear? Density ratios appear in survey correction and machine learning, probabilistic constraints in planning, and conditional probability in medical testing. The context changes, but transparent assumptions and reproducible checks remain valuable.
A unit failing at T_i less than or equal to t contributes its squared error with inverse weight based on G just before T_i. A failure at 300 uses 1/G(300 minus) times the failure residual. This is the central mathematical mechanism in observed failures before t. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. Using G(t) for every failure changes the estimator. Check the time in each weight. The useful habit is to keep the physical question, the probability model and the numerical method in separate columns so that a correct computation is not mistaken for complete evidence.
Did You Know? A result can be numerically stable and still answer the wrong question if the failure event, denominator or transformation changes. Test one nearby case and predict the direction of change before recalculating. Save inputs, intermediate values and units so a second reader can audit the path.
For practice, turn the example into a sensitivity check. Change one defensible input while holding the others fixed, compare the new result with the original and explain why it moved. Then construct a counterexample in which the shortcut fails. End with two sentences: what the mathematics supports, and what remains uncertain.
Start with the operational question behind known survivors beyond t. A unit known event-free beyond t contributes with inverse weight 1/G(t). In the illustrative case, A unit observed through 700 hours can contribute at t=500. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. Its later outcome beyond t is irrelevant to this fixed-time target. Stop the status at the horizon.
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 early-censored units as a model-checking problem rather than a vocabulary exercise. A unit censored before t and before failure has unknown status and receives no direct fixed-time residual under the standard IPCW construction. A censoring time of 350 precedes t=500. Recompute the example from the raw values and label every unit. Zero contribution is balanced by weights on observed comparable outcomes, not a guessed label. Explain the weighting logic.
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 a weighted event contribution becomes useful only when its mechanism is explicit. If G just before a failure is .8, the inverse censoring weight is 1.25. Consider the worked illustration: A raw squared error .16 becomes weighted contribution .20 before normalisation. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. Large weights amplify both signal and noise. Inspect the weight distribution.
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 a weighted survivor contribution to practise sensitivity analysis. If G(t)=.5, a known survivor at t receives weight 2. A residual .09 becomes .18 before averaging. Before accepting the result, identify the quantity that would change the conclusion most. Sparse late follow-up can dominate the estimate. Restrict unsupported horizons.
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/10474158/. Compare the problem definition, estimator and validation rather than quoting one result in isolation.
Build a small counterexample for the independence assumption. The original IPCW construction is consistent under suitable independence between event and censoring processes, with the exact condition tied to the chosen censoring model. The initial example is: Administrative study closure can be plausibly independent of unit risk. Then create a nearby case where the usual shortcut becomes unreliable. Risk-related withdrawal violates simple pooled assumptions. Justify censoring assumptions.
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 positivity and tail support to a real decision while protecting the boundary between calculation and judgement. G must stay meaningfully above zero wherever the score is evaluated. If almost nobody is observed to 2,000 hours, inverse weights explode. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. No weighting method recovers information beyond support. Choose an earlier horizon.
The audit should include the original question, input provenance, failure convention, algorithm settings, convergence or stopping rule, and a sensitivity result. For an adaptive method, also preserve the sequence of choices. For an optimisation, preserve every constraint. For Bayesian updating, preserve the prior and observation model. These records turn a result into something another person can challenge and improve.
A final transfer question is useful: where else would the same mathematical idea appear? Density ratios appear in survey correction and machine learning, probabilistic constraints in planning, and conditional probability in medical testing. The context changes, but transparent assumptions and reproducible checks remain valuable.
A time-integrated score averages fixed-time prediction error over a declared interval using a stated time measure or normalisation. Integrate from 0 to 1,000 hours and divide by the interval length when that is the convention. This is the central mathematical mechanism in integrated brier score. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. Different ranges are not directly comparable. Report limits and normalisation. 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 reference and skill scores. A model’s Brier score is more interpretable beside a prespecified reference such as a marginal survival model. In the illustrative case, Compare .12 with a reference .15 at the same horizon. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. A relative skill score inherits instability from both estimates. Report absolute scores too.
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.
CHAPTER 17 OF 25 · Explore applications
17. Calibration and discrimination together
Treat calibration and discrimination together as a model-checking problem rather than a vocabulary exercise. Squared probability error worsens when probabilities are miscalibrated or fail to separate outcomes. Two models can share ranking but have different probability scales. Recompute the example from the raw values and label every unit. The Brier score alone does not diagnose which component failed. Add calibration and ranking diagnostics.
The strongest check uses evidence the model did not choose for itself. Hold out later observations, an independent simulation set or a known-answer benchmark. Report errors where decisions are sensitive, especially near the failure boundary, instead of relying on one global accuracy score. A method that performs well on easy safe cases can still fail at the rare cases that motivated the analysis.
Now change one assumption and make a directional prediction before running the calculation. If the result moves the opposite way, inspect the sign convention, transformation, density ratio or constraint. This prediction-first routine develops transferable problem-solving skills because it connects symbolic work to a testable expectation.
The mathematics of internal validation becomes useful only when its mechanism is explicit. Flexible models evaluated on their training data have overly favourable prediction error. Consider the worked illustration: Tuning minimises apparent Brier loss. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. IPCW addresses censoring rather than overfitting. Repeat development in each resample.
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 a frozen model to a new fleet and estimate prediction error at prespecified horizons. A supplier model is tested on deployed equipment. Before accepting the result, identify the quantity that would change the conclusion most. The censoring distribution and baseline risk may differ. Describe the target population.
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 uncertainty estimation. Confidence intervals should account for fitted-model uncertainty, censoring estimation and repeated fixed-time observations when curves are integrated. The initial example is: A bootstrap repeats prediction fitting and G estimation. Then create a nearby case where the usual shortcut becomes unreliable. Bootstrapping only final residuals understates uncertainty after model selection. Resample the full pipeline.
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 compare with auc to a real decision while protecting the boundary between calculation and judgement. AUC asks about ranking, while the Brier score evaluates squared probability accuracy at a horizon. A monotone score transform leaves AUC unchanged but can alter Brier error. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. Neither metric replaces the other. Use a diagnostic set.
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.
Compute .04, .16, .09 and .01, average them, then change the last prediction from .9 to .6. The original mean is .075 and the altered last residual becomes .16. 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. The example omits censoring deliberately. Add IPCW only after mastering raw loss. 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 score connects indicators, squared error, probability, weighted averages and incomplete data. In the illustrative case, Ask why censoring at 350 does not prove failure by 500. 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 G with S. Use two colour-coded survival functions.
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 Graf score is not a mean squared error on observed times, not ranking-only and not valid after labelling early censorings as survivors. A low score at one horizon says nothing automatic about another. Recompute the example from the raw values and label every unit. Extreme weights can make tail estimates fragile. Report the horizon and G.
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. Graf and colleagues proposed censoring-adjusted prediction-error measures for comparing prognostic survival models. Consider the worked illustration: Read the primary paper beside the four-unit residual table. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. Production use needs support and censoring diagnostics. Reproduce .075 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: The prediction target
Setup: Use this chapter’s illustrative case: A model assigns survival probabilities .2, .4, .7 and .9 at 500 hours.
Task: Label the probability explicitly.
Checking target: Predictions must refer to the same origin, event and horizon.
Practice case 2: The binary outcome at a horizon
Setup: Use this chapter’s illustrative case: Two units failed by 500 and two survived beyond it.
Task: Write the indicator rule.
Checking target: The status is not known after early censoring.
Practice case 3: The squared probability residual
Setup: Use this chapter’s illustrative case: A failure with predicted survival .2 has error (0-.2)^2=.04.
Task: Keep the prediction on a zero-to-one scale.
Checking target: Squared loss combines calibration and discrimination.
Practice case 4: A complete-data example
Setup: Use this chapter’s illustrative case: Their mean is .30 divided by 4, or .075.
Task: Recalculate every square.
Checking target: Four observations are a teaching example only.
Practice case 5: Why lower is better
Setup: Use this chapter’s illustrative case: Perfect predictions give zero loss.
Task: Compare with a baseline model.
Checking target: The attainable and reference values depend on event prevalence.
Practice case 6: Why censoring breaks the raw mean
Setup: Use this chapter’s illustrative case: Censoring at 350 leaves 500-hour status unknown.
Task: Use censoring-aware estimation.
Checking target: Deleting all such units can also bias the sample.
Practice case 7: The censoring survival function
Setup: Use this chapter’s illustrative case: G(500)=.8 means about 80 percent remain observable through 500 under the censoring process.
Task: Use separate notation and plots.
Checking target: G is not component survival S.
Practice case 8: Observed failures before t
Setup: Use this chapter’s illustrative case: A failure at 300 uses 1/G(300 minus) times the failure residual.
Task: Check the time in each weight.
Checking target: Using G(t) for every failure changes the estimator.
Practice case 9: Known survivors beyond t
Setup: Use this chapter’s illustrative case: A unit observed through 700 hours can contribute at t=500.
Task: Stop the status at the horizon.
Checking target: Its later outcome beyond t is irrelevant to this fixed-time target.
Practice case 10: Early-censored units
Setup: Use this chapter’s illustrative case: A censoring time of 350 precedes t=500.
Task: Explain the weighting logic.
Checking target: Zero contribution is balanced by weights on observed comparable outcomes, not a guessed label.