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eduKateSG · Why Mathematics? · Schoenfeld Residuals for Reliability Hazard Models: Proportional Hazards, Time Trends, Scaling and Diagnostics
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 fitted Cox reliability model reports one hazard ratio but the true effect changes through burn-in, useful life or wear-out. A Schoenfeld residual is formed at an event time by subtracting the risk-set weighted covariate average from the covariate value of the failed system. If the proportional-hazards assumption is reasonable, scaled residuals should not show a systematic trend with time.
This guide develops event-time risk sets, weighted averages, scaling, time transforms, smooth curves, formal tests, ties and interpretation. A residual trend is evidence that one constant coefficient may be inadequate; it is not a physical diagnosis by itself. The useful response is to inspect data quality, follow-up support and plausible time-varying mechanisms before changing the model.
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
CHAPTER 1 OF 25 · Understand the question
1. Why proportional hazards needs a check
A Cox coefficient assumes a covariate multiplies the hazard by a constant ratio over the modelled time range. A protective coating may help early but lose effect after thermal cycling. This is the central mathematical mechanism in why proportional hazards needs a check. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. One average hazard ratio can conceal timing changes. Plot diagnostics before relying on one coefficient. 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 residuals are defined at failures. Schoenfeld residuals arise at observed event times because the partial likelihood compares the failed system with its current risk set. In the illustrative case, A data set with 80 systems but 25 failures supplies about 25 event-time residual contributions per covariate, adjusted for ties. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. Censored times do not create their own Schoenfeld residuals. Count events rather than rows.
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 risk-set weighted covariate average as a model-checking problem rather than a vocabulary exercise. At event time t, the Cox model weights each at-risk covariate by exp(beta x) and normalises those weights. If risk-set values are 0,1,2 and their risk scores are 1,2,1, the weighted mean is one. Recompute the example from the raw values and label every unit. This is not the ordinary sample mean unless risk scores are equal. Reproduce one weighted mean by hand.
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 observed minus expected covariate becomes useful only when its mechanism is explicit. For one failure, the residual is x_event minus the model-weighted risk-set mean xbar(beta,t). Consider the worked illustration: If the failed unit has x=2 and xbar=1.2, the raw residual is 0.8. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. Its sign depends on covariate coding. State units and reference level.
There are three uncertainties to keep apart. Sampling or numerical uncertainty comes from finite computation. Parameter uncertainty comes from estimating inputs. Model-form uncertainty comes from simplified physics, distributions or dependence. Reducing the first does not automatically reduce the other two. A credible report names all three and shows which one drives the decision.
For a practical investigation, create a small reproducible notebook or spreadsheet with a frozen data sheet, formula sheet, diagnostics sheet and decision note. Another student should be able to change one input and obtain the same update without guessing what the model meant.
Primary source: https://doi.org/10.1093/biomet/69.1.239. Read it for the method and its assumptions; do not transfer its performance claims automatically to another model.
Use why the fitted score balances to practise sensitivity analysis. At the partial-likelihood estimate, event-time score contributions sum to zero or nearly zero under the chosen tie algorithm. Positive early residuals can be offset by negative late residuals. Before accepting the result, identify the quantity that would change the conclusion most. A zero total does not rule out a time trend. Plot residuals in event-time order.
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 scaled schoenfeld residuals. Scaling uses information-matrix structure to place residual patterns on a coefficient-related scale and account for covariate covariance. The initial example is: Two correlated temperature measures require joint scaling rather than separate arbitrary rescaling. Then create a nearby case where the usual shortcut becomes unreliable. Software definitions can differ by constant factors. Record the implementation.
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.
CHAPTER 7 OF 25 · Reproduce the example
7. Residuals and a time-varying coefficient
Connect residuals and a time-varying coefficient to a real decision while protecting the boundary between calculation and judgement. Grambsch–Therneau reasoning links a smooth of scaled residuals plus the fitted coefficient to beta(t), a diagnostic view of changing effect. A downward smooth suggests a protective coefficient weakening with time under the coding used. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. The smooth is not a guaranteed causal trajectory. Add uncertainty and risk-set support.
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.
Tests may regress residuals on event time, rank time, logarithmic time or a survival-based transform. Rank time spreads event locations more evenly when calendar times are highly skewed. This is the central mathematical mechanism in choosing a time transform. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. Different transforms target different alternatives. Choose the transform before inspection. 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 a zero-slope idea. Under proportional hazards, the expected association between properly scaled residuals and the chosen time transform is zero. In the illustrative case, A fitted slope near zero supports but does not prove proportionality. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. Low power can hide a real change. Report slope, test and plot together.
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 formal covariate-specific test as a model-checking problem rather than a vocabulary exercise. A weighted regression or score test converts the residual-time association into a statistic for one coefficient. A small p-value flags time variation for a pressure indicator. Recompute the example from the raw values and label every unit. It does not identify the best replacement function. Use the test to guide a planned diagnostic review.
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 global proportional-hazards test becomes useful only when its mechanism is explicit. Covariate-specific components can be combined with their covariance into one global test of the model's proportional-hazards structure. Consider the worked illustration: Five predictors can receive one joint diagnostic. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. A global rejection does not identify every failing term. Inspect component plots after the global result.
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 tied failure times to practise sensitivity analysis. Several failures at one recorded time require residual calculations consistent with the Breslow, Efron or exact partial-likelihood convention. Four units fail during the same daily inspection interval. Before accepting the result, identify the quantity that would change the conclusion most. Mixing tie methods can change residuals and covariance. Use one documented tie algorithm.
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://doi.org/10.1093/biomet/81.3.515. Compare the problem definition, estimator and validation rather than quoting one result in isolation.
Build a small counterexample for time-dependent covariates. The event-time covariate and risk-set average must use values current just before that event. The initial example is: Load changes after a repair and therefore differs across later risk sets. Then create a nearby case where the usual shortcut becomes unreliable. Using final values leaks future information. Store covariate histories with start-stop intervals.
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 stratified cox models to a real decision while protecting the boundary between calculation and judgement. Strata have separate baseline hazards, so each event is compared within its stratum while coefficients may remain shared. Different test chambers form strata with their own baseline rates. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. Sparse strata weaken diagnostics. Check event counts within strata.
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.
Schoenfeld trends primarily assess time variation in coefficients, while an incorrect nonlinear covariate shape can require martingale or spline diagnostics. A curved temperature effect may exist even with proportional hazards. This is the central mathematical mechanism in nonlinearity is a different question. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. One residual plot cannot test every assumption. Match the diagnostic to the model component. 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 outlying event covariates. Large raw residuals occur when a failed system's covariate is unusual relative to its risk set. In the illustrative case, A low-risk unit fails while high-risk units remain. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. An unusual point can be valid rather than erroneous. Verify records before excluding anything.
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 sparse late risk sets as a model-checking problem rather than a vocabulary exercise. Late event residuals can be unstable because few systems remain and covariate variation may collapse. Only three units remain after 2,000 hours. Recompute the example from the raw values and label every unit. A dramatic tail smooth can be driven by little information. Show numbers at risk beneath the plot.
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 multiple comparisons becomes useful only when its mechanism is explicit. Inspecting many covariates, transforms and smooth settings increases the chance of an apparently interesting pattern. Consider the worked illustration: Twenty unplanned tests can produce a small p-value by chance. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. The global test does not license unlimited post-hoc claims. Separate planned checks from exploration.
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 repairing non-proportionality to practise sensitivity analysis. Possible responses include a time interaction, stratification, piecewise coefficient or a different effect summary such as RMST. Use x times log(t) when a scientifically plausible log-time change is supported. Before accepting the result, identify the quantity that would change the conclusion most. A more flexible model can overfit. Validate the repaired 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 effect size after diagnostics. Report time-specific hazard ratios or survival differences with uncertainty instead of replacing one constant estimate with a residual p-value. The initial example is: Hazard ratios at 100 and 1,000 hours can explain the change. Then create a nearby case where the usual shortcut becomes unreliable. Post-hoc time points can exaggerate contrast. Pre-plan decision horizons where possible.
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. Translate a time-varying effect into commissioning, inspection, warranty or replacement decisions. A coating that helps only during burn-in may still be valuable for launch reliability. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. Residuals do not choose the policy. Combine timing with consequences.
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.
Build a three-unit risk set, calculate risk scores, xbar and one raw residual, then repeat after changing beta. Predict the residual direction before calculating. This is the central mathematical mechanism in practice activity. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. A toy risk set cannot validate a full model. Keep event and censoring roles distinct. 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. Schoenfeld residuals connect weighted averages, subtraction, graphs, slopes and the idea that a constant rule must be checked over time. In the illustrative case, Ask why a zero overall sum can still hide a pattern. 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 balance with randomness. Use an early-positive late-negative sequence.
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 flat plot does not prove the model, a small p-value does not explain the physical cause, censored rows do not each create residuals and a smooth is not a causal coefficient path. Tail patterns can be unstable. Recompute the example from the raw values and label every unit. Test choice and time transform matter. Report plot, test and support.
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. Schoenfeld defined partial residuals for proportional-hazards regression, while Grambsch and Therneau linked weighted residual plots and tests to time-varying coefficients. Consider the worked illustration: Read the residual definition beside the scaled diagnostic theory. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. Large-sample tests still need data-quality checks. Reproduce one event-time residual 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 proportional hazards needs a check
Setup: Use this chapter’s illustrative case: A protective coating may help early but lose effect after thermal cycling.
Task: Plot diagnostics before relying on one coefficient.
Checking target: One average hazard ratio can conceal timing changes.
Practice case 2: Residuals are defined at failures
Setup: Use this chapter’s illustrative case: A data set with 80 systems but 25 failures supplies about 25 event-time residual contributions per covariate, adjusted for ties.
Task: Count events rather than rows.
Checking target: Censored times do not create their own Schoenfeld residuals.
Practice case 3: Risk-set weighted covariate average
Setup: Use this chapter’s illustrative case: If risk-set values are 0,1,2 and their risk scores are 1,2,1, the weighted mean is one.
Task: Reproduce one weighted mean by hand.
Checking target: This is not the ordinary sample mean unless risk scores are equal.
Practice case 4: Observed minus expected covariate
Setup: Use this chapter’s illustrative case: If the failed unit has x=2 and xbar=1.2, the raw residual is 0.8.
Task: State units and reference level.
Checking target: Its sign depends on covariate coding.
Practice case 5: Why the fitted score balances
Setup: Use this chapter’s illustrative case: Positive early residuals can be offset by negative late residuals.
Task: Plot residuals in event-time order.
Checking target: A zero total does not rule out a time trend.
Practice case 6: Scaled Schoenfeld residuals
Setup: Use this chapter’s illustrative case: Two correlated temperature measures require joint scaling rather than separate arbitrary rescaling.
Task: Record the implementation.
Checking target: Software definitions can differ by constant factors.
Practice case 7: Residuals and a time-varying coefficient
Setup: Use this chapter’s illustrative case: A downward smooth suggests a protective coefficient weakening with time under the coding used.
Task: Add uncertainty and risk-set support.
Checking target: The smooth is not a guaranteed causal trajectory.
Practice case 8: Choosing a time transform
Setup: Use this chapter’s illustrative case: Rank time spreads event locations more evenly when calendar times are highly skewed.
Task: Choose the transform before inspection.
Checking target: Different transforms target different alternatives.
Practice case 9: A zero-slope idea
Setup: Use this chapter’s illustrative case: A fitted slope near zero supports but does not prove proportionality.
Task: Report slope, test and plot together.
Checking target: Low power can hide a real change.
Practice case 10: Formal covariate-specific test
Setup: Use this chapter’s illustrative case: A small p-value flags time variation for a pressure indicator.
Task: Use the test to guide a planned diagnostic review.
Checking target: It does not identify the best replacement function.