Choose your reading route
eduKateSG · Why Mathematics? · Aalen Additive Hazards for Reliability: Absolute Rate Differences, Time-Varying Effects, Cumulative Coefficients 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 visible when an engineer wants an absolute change in failure rate rather than a multiplicative hazard ratio. The Aalen additive hazards model writes intensity as a baseline plus covariate contributions whose coefficients can vary over time. Integrating those coefficients produces cumulative-effect curves that show when an operating condition adds or removes event rate.
This guide develops the additive intensity, risk sets, counting-process estimation, cumulative coefficients, uncertainty and diagnostics. Additivity offers a clear absolute-rate interpretation, but unconstrained linear predictions can become negative and observational covariates do not become causal interventions automatically.
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 additive effects answer a different question
An additive model describes extra failures per unit time associated with a covariate. A hotter regime may add 0.002 failures per operating hour under the fitted model. This is the central mathematical mechanism in why additive effects answer a different question. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. An absolute difference is not a hazard ratio. Choose the scale from the decision. 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 form. A common form is lambda_i(t)=Y_i(t)[beta_0(t)+beta_1(t)x_i1+…]. In the illustrative case, The at-risk indicator removes units not under observation. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. Linear combinations can become negative. Check predicted intensities.
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 time-varying coefficients as a model-checking problem rather than a vocabulary exercise. Each beta_k(t) may change with time, allowing early and late effects to differ. Vibration may matter strongly after wear begins. Recompute the example from the raw values and label every unit. Flexibility raises variance. Plot cumulative effects with bands.
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 cumulative coefficient functions becomes useful only when its mechanism is explicit. B_k(t)=integral_0^t beta_k(u)du is estimated more stably than instantaneous beta_k(t). Consider the worked illustration: A rising B_k indicates accumulated positive rate contribution. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. Slope interpretation depends on local smoothness. Avoid overreading small wiggles.
There are three uncertainties to keep apart. Sampling or numerical uncertainty comes from finite computation. Parameter uncertainty comes from estimating inputs. Model-form uncertainty comes from simplified physics, distributions or dependence. Reducing the first does not automatically reduce the other two. A credible report names all three and shows which one drives the decision.
For a practical investigation, create a small reproducible notebook or spreadsheet with a frozen data sheet, formula sheet, diagnostics sheet and decision note. Another student should be able to change one input and obtain the same update without guessing what the model meant.
Primary source: https://doi.org/10.1002/sim.4780080803. Read it for the method and its assumptions; do not transfer its performance claims automatically to another model.
Use risk sets and design matrices to practise sensitivity analysis. At each event time, covariates of units at risk form the design matrix used for increments. A unit under repair should not remain exposed to operating failure. Before accepting the result, identify the quantity that would change the conclusion most. Rank deficiency makes effects unidentified. Inspect risk-set variation.
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 counting-process increments. Observed dN_i(t) is related to Y_i(t)X_i(t) beta(t)dt plus a martingale residual. The initial example is: Events create increments while exposure supplies denominators. Then create a nearby case where the usual shortcut becomes unreliable. Database event time may differ from physical onset. Audit event definitions.
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 least-squares style estimation to a real decision while protecting the boundary between calculation and judgement. Aalen estimation uses local linear algebra across risk sets to update cumulative coefficients. Generalised inverses may be needed when covariates lack variation. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. A computed solution can be unstable. Track condition numbers.
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 coefficient of 0.001 per hour for a binary exposure means a fitted extra 0.001 instantaneous events per hour, conditional on the model. Across 1000 comparable hours, the cumulative excess scale is about one event if constant. This is the central mathematical mechanism in absolute rate interpretation. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. Intensity is not probability without a time interval. Keep units explicit. 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 comparison with cox models. Cox multiplies a baseline hazard by exp(beta x), while Aalen adds covariate-specific rate components. In the illustrative case, A covariate can have a time-varying additive effect without proportional hazards. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. Neither scale is universally better. Compare scientific interpretability.
A student should be able to rebuild the result without hidden settings. Record the sampling rule or optimisation criterion, the exact failure definition, and the evidence used for every distribution or parameter. Compare a transparent baseline with the advanced method. If they disagree, investigate the mechanism before deciding which answer is more credible.
This chapter belongs to a larger decision, so translate the number carefully. State one action the evidence permits and one action it does not. A parent or educator can ask which school ideas are doing the work: algebra organises relationships, probability represents uncertainty, calculus describes local change, and computing repeats the documented rules at scale.
Treat a worked two-group example as a model-checking problem rather than a vocabulary exercise. If baseline hazard is 0.003/hour and group effect is 0.002/hour, fitted hazards are 0.003 and 0.005/hour. Over a short 10-hour interval, approximate event probabilities are 0.03 and 0.05. Recompute the example from the raw values and label every unit. The approximation deteriorates over long intervals. Use exact survival integration when needed.
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 baseline cumulative hazard becomes useful only when its mechanism is explicit. The intercept cumulative coefficient estimates baseline cumulative hazard under the coding. Consider the worked illustration: Changing reference values changes the intercept interpretation. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. An arbitrary zero covariate may be unrealistic. Centre covariates meaningfully.
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 time-dependent covariates to practise sensitivity analysis. Loads, temperatures and maintenance states can update over time if measured before the risk interval. A load reading at interval start can predict the following event intensity. Before accepting the result, identify the quantity that would change the conclusion most. Post-event measurements can create leakage. Preserve measurement timing.
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/10.1214/aos/1176344247. Compare the problem definition, estimator and validation rather than quoting one result in isolation.
Build a small counterexample for censoring. Independent right censoring is handled through the at-risk process. The initial example is: A survivor contributes exposure until its censoring time. Then create a nearby case where the usual shortcut becomes unreliable. Condition-related retirement can bias effects. Record censoring reasons.
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 robust covariance to a real decision while protecting the boundary between calculation and judgement. Sandwich methods can address some misspecification or within-unit clustering under conditions. Repeated events from one machine share hidden traits. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. Robust standard errors do not fix omitted-variable bias. State the clustering unit.
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.
Differentiate or smooth cumulative coefficient curves to view beta_k(t). Bandwidth controls noise versus temporal detail. This is the central mathematical mechanism in smoothing instantaneous effects. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. Smoothing can manufacture apparent timing. Show cumulative curves first. 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 testing constant effects. A roughly linear cumulative coefficient suggests an approximately constant additive effect. In the illustrative case, Curvature indicates change over time. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. Visual straightness is not a formal test. Combine plots and tests.
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 negative fitted hazards as a model-checking problem rather than a vocabulary exercise. Linear additive predictions can fall below zero for some covariate combinations. Extrapolating far outside observed loads can create impossible values. Recompute the example from the raw values and label every unit. Clipping silently changes the model. Restrict support or choose alternatives.
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 collinearity and identifiability becomes useful only when its mechanism is explicit. Highly correlated operating conditions make separate cumulative effects unstable. Consider the worked illustration: Temperature and load may rise together. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. Large opposing coefficients can cancel. Use design knowledge and diagnostics.
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 causal interpretation limits to practise sensitivity analysis. An additive association is causal only with appropriate design and confounding assumptions. Maintenance may be assigned to already failing units. Before accepting the result, identify the quantity that would change the conclusion most. Adjustment alone may not remove time-dependent confounding. Separate prediction from intervention.
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 prediction and validation. Integrate fitted hazard to produce survival predictions and check them on held-out units. The initial example is: Compare predicted event counts and survival by risk group. Then create a nearby case where the usual shortcut becomes unreliable. Coefficient fit does not ensure calibration. Validate both rate and survival.
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 practice activity to a real decision while protecting the boundary between calculation and judgement. Create two risk sets, calculate a simple additive rate difference and sketch cumulative coefficients. Change the reference coding and explain the intercept shift. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. The sample is illustrative. Check for negative predictions.
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.
Additive hazards connects linear functions, rates, matrices, integrals and graphs. Ask how absolute and relative effects differ. This is the central mathematical mechanism in parent and educator guidance. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. Use two straight rate lines first. Keep units beside coefficients. 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 common misconceptions. Additive hazards is not linear regression on failure time, intensity is not probability and robust variance is not causal proof. In the illustrative case, Negative hazards signal model trouble. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. Cumulative slope needs uncertainty. Translate the rate scale.
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 when the model is unsuitable as a model-checking problem rather than a vocabulary exercise. Strong nonlinearities, sparse risk sets, impossible negative predictions or complex state transitions may need other models. A multistate repair process needs transition-specific intensities. Recompute the example from the raw values and label every unit. One terminal-event hazard can be too narrow. Choose structure from mechanism.
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. Aalen develops a linear regression model for lifetime intensity and counting-process foundations for nonparametric inference. Consider the worked illustration: Read cumulative-coefficient estimation beside risk-set algebra. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. Theory does not validate local covariates. Reproduce a two-group constant-effect benchmark.
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 additive effects answer a different question
Setup: Use this chapter’s illustrative case: A hotter regime may add 0.002 failures per operating hour under the fitted model.
Task: Choose the scale from the decision.
Checking target: An absolute difference is not a hazard ratio.
Practice case 2: Model form
Setup: Use this chapter’s illustrative case: The at-risk indicator removes units not under observation.
Task: Check predicted intensities.
Checking target: Linear combinations can become negative.
Practice case 3: Time-varying coefficients
Setup: Use this chapter’s illustrative case: Vibration may matter strongly after wear begins.
Task: Plot cumulative effects with bands.
Checking target: Flexibility raises variance.
Practice case 4: Cumulative coefficient functions
Setup: Use this chapter’s illustrative case: A rising B_k indicates accumulated positive rate contribution.
Task: Avoid overreading small wiggles.
Checking target: Slope interpretation depends on local smoothness.
Practice case 5: Risk sets and design matrices
Setup: Use this chapter’s illustrative case: A unit under repair should not remain exposed to operating failure.
Task: Inspect risk-set variation.
Checking target: Rank deficiency makes effects unidentified.
Practice case 6: Counting-process increments
Setup: Use this chapter’s illustrative case: Events create increments while exposure supplies denominators.
Task: Audit event definitions.
Checking target: Database event time may differ from physical onset.
Practice case 7: Least-squares style estimation
Setup: Use this chapter’s illustrative case: Generalised inverses may be needed when covariates lack variation.
Task: Track condition numbers.
Checking target: A computed solution can be unstable.
Practice case 8: Absolute rate interpretation
Setup: Use this chapter’s illustrative case: Across 1000 comparable hours, the cumulative excess scale is about one event if constant.
Task: Keep units explicit.
Checking target: Intensity is not probability without a time interval.
Practice case 9: Comparison with Cox models
Setup: Use this chapter’s illustrative case: A covariate can have a time-varying additive effect without proportional hazards.
Task: Compare scientific interpretability.
Checking target: Neither scale is universally better.
Practice case 10: A worked two-group example
Setup: Use this chapter’s illustrative case: Over a short 10-hour interval, approximate event probabilities are 0.03 and 0.05.
Task: Use exact survival integration when needed.
Checking target: The approximation deteriorates over long intervals.