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eduKateSG · Why Mathematics? · Schemper–Henderson Explained Variation for Reliability Models: Prediction Error, Censoring, Time Weights and R²
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
The importance of mathematics appears when a survival model must be judged by how closely its predicted survival curves follow individual event histories, not merely by whether one coefficient is statistically significant. Schemper and Henderson compare an integrated prediction-error quantity from a covariate model with the corresponding error from a no-covariate reference. Their explained-variation measure asks what fraction of that reference error is reduced.
This guide develops individual survival processes, fitted curves, time integration, inverse censoring weights, null and full-model error, explained variation, worked arithmetic and interpretation. Smaller prediction error is better, while a larger explained fraction indicates more improvement over the declared reference. Neither quantity guarantees causal validity, discrimination, safe extrapolation or useful maintenance action.
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 significant hazard ratio can coexist with weak individual prediction, so performance needs a direct comparison between observed survival processes and fitted curves. A temperature coefficient is nonzero but predicted curves remain close to the fleet average. This is the central mathematical mechanism in why prediction error matters. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. Statistical significance is not predictive accuracy. Evaluate predictions on validation data. 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 individual survival process. For each unit, the observed survival process is one while it remains event-free and becomes zero after an observed failure. In the illustrative case, A bearing that fails at 700 hours has process one before 700 and zero after. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. After right censoring its later state is unknown. Draw the process before calculating distance.
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 predicted survival curve as a model-checking problem rather than a vocabulary exercise. The fitted model supplies S_hat(t|Z) for each covariate vector Z across time. A pump may have predicted survival .9 at 200 hours and .55 at 800 hours. Recompute the example from the raw values and label every unit. A Cox risk score alone is incomplete without baseline survival. Validate monotonicity and range.
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 distance between process and curve becomes useful only when its mechanism is explicit. Prediction inaccuracy accumulates the absolute difference between the individual process and the fitted survival probability over relevant times. Consider the worked illustration: Before failure, a prediction of .8 differs from observed state one by .2. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. Distance depends on the time-weighting rule. Keep the integrand and limits explicit.
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/10.1111/j.0006-341X.2000.00249.x. Read it for the method and its assumptions; do not transfer its performance claims automatically to another model.
Use why integrate across follow-up to practise sensitivity analysis. A single horizon ignores errors earlier and later, whereas integration summarises the curve over the supported time region. Two models tie at 500 hours but differ greatly before 300 hours. Before accepting the result, identify the quantity that would change the conclusion most. Long unsupported tails can dominate if weights are careless. Declare the evaluation interval.
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 right-censoring weights. Inverse estimates of the censoring survival distribution give more weight where fewer comparable units remain observable. The initial example is: If G_hat(t)=.8, a contribution at t is weighted by 1/.8=1.25. Then create a nearby case where the usual shortcut becomes unreliable. Near-zero G_hat produces unstable weights. Truncate evaluation before support collapses.
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 event-time weighting to a real decision while protecting the boundary between calculation and judgement. The estimator combines time-specific distances using observed event counts and censoring weights, then normalises by their total. Three event times with different d_j/G_hat(t_j) do not contribute equally. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. Replacing these weights with equal spacing changes the estimand. Use the published definition.
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.
D_z summarises distance from individual processes to covariate-specific fitted survival curves. A validated model has integrated inaccuracy .18. This is the central mathematical mechanism in full-model inaccuracy d_z. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. Smaller is better but the scale depends on follow-up and weights. Report D_z with its definition. 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 inaccuracy d. D uses a fitted survival curve without covariate information and provides the denominator for explained variation. In the illustrative case, The null-model integrated inaccuracy is .24. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. A weak or mismatched reference can distort interpretation. Name the reference 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 explained-variation formula as a model-checking problem rather than a vocabulary exercise. The proportion explained is V=1-D_z/D, often reported as 100V percent. With D_z=.18 and D=.24, V=1-.18/.24=.25, or 25 percent. Recompute the example from the raw values and label every unit. This is not ordinary least-squares R-squared. Show both errors and the ratio.
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 interpret zero and negative values becomes useful only when its mechanism is explicit. V=0 means no reduction against the reference, while negative V means the fitted model predicts worse under the chosen measure. Consider the worked illustration: D_z=.27 and D=.24 give V=-.125. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. Negative values are possible and should not be silently clipped. Investigate transport failure or overfitting.
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 absolute versus relative accuracy to practise sensitivity analysis. D_z and D describe absolute inaccuracy on the chosen scale, while V describes relative reduction. Two fleets can share V=.25 but have different D_z values. Before accepting the result, identify the quantity that would change the conclusion most. Relative improvement can hide a large remaining error. Report all three quantities.
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://support.sas.com/documentation/cdl/en/statug/68162/HTML/default/statug_phreg_details48.htm. Compare the problem definition, estimator and validation rather than quoting one result in isolation.
Build a small counterexample for censoring assumptions. The weighting approach assumes the censoring distribution is estimated appropriately for the target data process. The initial example is: Administrative closure may be plausibly independent after measured covariates. Then create a nearby case where the usual shortcut becomes unreliable. Condition-dependent withdrawal can bias the distance estimate. Model or stratify censoring when justified.
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 left truncation and delayed entry to a real decision while protecting the boundary between calculation and judgement. Units entering observation after time zero should contribute only while they are actually at risk. A machine first inspected at 300 hours cannot inform earlier prediction error. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. Ignoring delayed entry creates immortal-time distortion. Use risk sets with entry times.
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.
Comparisons need one clinically or operationally meaningful interval with adequate follow-up. Evaluate to 1,000 hours because at-risk support remains acceptable there. This is the central mathematical mechanism in choice of time window. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. Changing the cutoff after viewing results invites selective reporting. Pre-specify the horizon. 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 model misspecification. Incorrect baseline shape, nonlinearity or time-varying effects can increase curve distance even when ranking looks fair. In the illustrative case, A constant temperature effect misses late wear-out acceleration. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. The scalar measure does not identify the defect. Inspect residuals and calibration plots.
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 discrimination remains distinct as a model-checking problem rather than a vocabulary exercise. Two models can have similar pairwise ranking but different survival-curve accuracy and therefore different D_z. Both order pumps correctly, but one predicts levels closer to observed experience. Recompute the example from the raw values and label every unit. Explained variation is not a concordance index. Report complementary metrics.
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. Computing D_z on the development data favours the same choices used to fit the curves. Consider the worked illustration: Spline knots selected on the full fleet reduce apparent error. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. A single split can be noisy. Use bootstrap or repeated validation of the full process.
There are three uncertainties to keep apart. Sampling or numerical uncertainty comes from finite computation. Parameter uncertainty comes from estimating inputs. Model-form uncertainty comes from simplified physics, distributions or dependence. Reducing the first does not automatically reduce the other two. A credible report names all three and shows which one drives the decision.
For a practical investigation, create a small reproducible notebook or spreadsheet with a frozen data sheet, formula sheet, diagnostics sheet and decision note. Another student should be able to change one input and obtain the same update without guessing what the model meant.
Use external validation to practise sensitivity analysis. Frozen predictions should be scored on a new cohort using that cohort's supported follow-up and censoring structure. A model from laboratory tests is applied to field units. Before accepting the result, identify the quantity that would change the conclusion most. Different censoring can change precision and weighting. Describe both populations.
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 subgroup checks. Overall error can average over opposite miscalibration for suppliers, environments or component classes. The initial example is: Supplier A predictions are too optimistic and supplier B too pessimistic. Then create a nearby case where the usual shortcut becomes unreliable. Small strata yield unstable weighted errors. Pre-specify important groups.
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. Reduced prediction error supports probability quality but does not choose an inspection threshold or encode consequences. A 25 percent reduction may or may not change a maintenance plan. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. V has no direct currency or safety scale. Add decision and capacity analysis.
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 V for D_z=.18 and D=.24, then recompute after D_z rises to .21. The values are .25 and .125, showing less error reduction. 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 the component errors too early can alter V. Keep full precision. 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 method connects fractions, weighted averages, graphs, areas and the idea of comparing a model with a baseline. In the illustrative case, Ask why .18 is better than .24 but does not mean 18 percent error in every case. 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 assume every R-squared has the same definition. Attach the formula to words.
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. A large V does not prove causation, perfect calibration, good ranking or transportability, and a small V does not make every predictor useless. Performance depends on horizon, reference and censoring support. Recompute the example from the raw values and label every unit. Published values are not universal benchmarks. Report context and uncertainty.
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. Schemper and Henderson defined D_z, D and V for censored Cox prediction, and official SAS documentation gives the event and censoring-weighted estimators. Consider the worked illustration: Compare the paper's definitions with a software output table. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. Validated implementation is essential for censoring weights. Reproduce the .18/.24 example first.
There are three uncertainties to keep apart. Sampling or numerical uncertainty comes from finite computation. Parameter uncertainty comes from estimating inputs. Model-form uncertainty comes from simplified physics, distributions or dependence. Reducing the first does not automatically reduce the other two. A credible report names all three and shows which one drives the decision.
For a practical investigation, create a small reproducible notebook or spreadsheet with a frozen data sheet, formula sheet, diagnostics sheet and decision note. Another student should be able to change one input and obtain the same update without guessing what the model meant.
Quick answers
What should I calculate first? Start with the estimand or failure event, then a simple baseline.
Does the advanced method guarantee accuracy? No. It needs input, numerical and model checks.
Which school mathematics matters? Algebra, graphs, probability, functions and careful units all contribute.
How can a student practise safely? Use fictional or open benchmark data and preserve every intermediate value.
What makes a career claim responsible? Describe how the mathematics is used without promising admission, salary or employment.
Extended practice laboratory
Practice case 1: Why prediction error matters
Setup: Use this chapter’s illustrative case: A temperature coefficient is nonzero but predicted curves remain close to the fleet average.
Task: Evaluate predictions on validation data.
Checking target: Statistical significance is not predictive accuracy.
Practice case 2: Individual survival process
Setup: Use this chapter’s illustrative case: A bearing that fails at 700 hours has process one before 700 and zero after.
Task: Draw the process before calculating distance.
Checking target: After right censoring its later state is unknown.
Practice case 3: Predicted survival curve
Setup: Use this chapter’s illustrative case: A pump may have predicted survival .9 at 200 hours and .55 at 800 hours.
Task: Validate monotonicity and range.
Checking target: A Cox risk score alone is incomplete without baseline survival.
Practice case 4: Distance between process and curve
Setup: Use this chapter’s illustrative case: Before failure, a prediction of .8 differs from observed state one by .2.
Task: Keep the integrand and limits explicit.
Checking target: Distance depends on the time-weighting rule.
Practice case 5: Why integrate across follow-up
Setup: Use this chapter’s illustrative case: Two models tie at 500 hours but differ greatly before 300 hours.
Task: Declare the evaluation interval.
Checking target: Long unsupported tails can dominate if weights are careless.
Practice case 6: Right-censoring weights
Setup: Use this chapter’s illustrative case: If G_hat(t)=.8, a contribution at t is weighted by 1/.8=1.25.
Task: Truncate evaluation before support collapses.
Checking target: Near-zero G_hat produces unstable weights.
Practice case 7: Event-time weighting
Setup: Use this chapter’s illustrative case: Three event times with different d_j/G_hat(t_j) do not contribute equally.
Task: Use the published definition.
Checking target: Replacing these weights with equal spacing changes the estimand.
Practice case 8: Full-model inaccuracy D_z
Setup: Use this chapter’s illustrative case: A validated model has integrated inaccuracy .18.
Task: Report D_z with its definition.
Checking target: Smaller is better but the scale depends on follow-up and weights.
Practice case 9: Reference inaccuracy D
Setup: Use this chapter’s illustrative case: The null-model integrated inaccuracy is .24.
Task: Name the reference model.
Checking target: A weak or mismatched reference can distort interpretation.
Practice case 10: Explained-variation formula
Setup: Use this chapter’s illustrative case: With D_z=.18 and D=.24, V=1-.18/.24=.25, or 25 percent.
Task: Show both errors and the ratio.
Checking target: This is not ordinary least-squares R-squared.