Choose your reading route
eduKateSG · Why Mathematics? · Piecewise Exponential Reliability Models: Time Bands, Constant Hazards, Exposure Totals and Poisson Likelihoods
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 constant failure rate is too rigid but a fully smooth hazard is unnecessary. A piecewise exponential model divides time into intervals and assigns one constant hazard to each. The survival curve is continuous, while its slope on the cumulative-hazard scale changes at declared breakpoints.
This guide develops time splitting, exposure totals, Poisson likelihood equivalence, censoring, covariates, breakpoint choice and validation. The model is transparent and useful, but arbitrary intervals can overfit or hide real transitions. Each band must have enough exposure and a defensible relationship to the failure mechanism or reporting resolution.
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
Different life stages can have different approximately constant hazards. Use 0–100,100–500 and 500–1000 hours for early, middle and late operation. This is the central mathematical mechanism in why divide time into bands. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. Bands chosen after seeing failures can exaggerate patterns. Pre-specify or validate breakpoints. 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 constant hazard within a band. For interval j, hazard lambda_j is constant over that interval. In the illustrative case, Survival decays exponentially while the unit remains in the band. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. Physical hazard need not jump at boundaries. Treat steps as approximation.
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 cumulative hazard across bands as a model-checking problem rather than a vocabulary exercise. Add lambda_j times exposure accumulated in each completed or partial band. A unit surviving two bands carries both hazard contributions. Recompute the example from the raw values and label every unit. Resetting cumulative hazard at each band is wrong. Keep a running total.
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 continuous survival function becomes useful only when its mechanism is explicit. S(t)=exp[-H(t)] remains continuous even when hazard changes at breakpoints. Consider the worked illustration: Only the slope of H(t) changes. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. A discontinuous survival curve would imply instantaneous failure mass. Check continuity numerically.
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/10.1080/01621459.1981.10477634. Read it for the method and its assumptions; do not transfer its performance claims automatically to another model.
Use a worked calculation to practise sensitivity analysis. With hazards 0.001/hour to 100 hours and 0.003/hour after, H(200)=0.1+0.3=0.4 and S(200)=exp(-0.4)≈0.6703. Both intervals contribute exposure. Before accepting the result, identify the quantity that would change the conclusion most. The numbers are illustrative. Verify units cancel in H.
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 person-time splitting. Split each unit record at every breakpoint it crosses. The initial example is: A unit observed to 250 hours contributes 100 hours to band one and 150 to band two. Then create a nearby case where the usual shortcut becomes unreliable. Duplicated split rows are not independent units. Retain unit identifiers.
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 events and exposure totals to a real decision while protecting the boundary between calculation and judgement. For a no-covariate band, MLE hazard is event count divided by total exposure in that band. Three failures over 2000 unit-hours give 0.0015 per hour. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. Zero events do not prove zero hazard. Report intervals or bounds.
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.
Piecewise exponential survival likelihood can be written as Poisson counts with log exposure offsets. Event counts by band use expected count exposure times rate. This is the central mathematical mechanism in poisson likelihood equivalence. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. Ordinary Poisson independence needs careful unit clustering with split records. Use offsets and robust methods as needed. 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 right censoring. A censored unit contributes exposure through censoring and no event count. In the illustrative case, A survivor to 600 hours supplies all earlier-band exposure. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. Dropping survivors inflates hazard. Include every observed interval.
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 covariate effects as a model-checking problem rather than a vocabulary exercise. Log-linear terms can multiply band-specific hazards using Poisson regression. Temperature group can have a rate ratio within the model. Recompute the example from the raw values and label every unit. Proportional covariate effects may still fail over time. Add interactions when justified.
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 time-varying covariates becomes useful only when its mechanism is explicit. Split records when a covariate changes as well as at hazard breakpoints. Consider the worked illustration: A load shift at hour 180 creates a new row. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. Using future values creates leakage. Use start–stop timing.
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 choosing breakpoints from engineering to practise sensitivity analysis. Maintenance schedules, known phases or inspection times can motivate boundaries. A burn-in period may define an early band. Before accepting the result, identify the quantity that would change the conclusion most. Convenient calendar cuts may not match mechanism. Document rationale.
A simple baseline is essential. It reveals whether the advanced method is improving accuracy or only adding complexity. Compare model calls, wall time, uncertainty and failure-mode coverage under one declared budget. Repeat stochastic calculations with several seeds; a favourable single run is not evidence of dependable efficiency.
Students can extend the example by adding correlation, a second failure mode or a modest measurement error. Explain which part of the formula changes and which remains fixed. This exercise shows how mathematics transfers from a clean textbook case to engineering and computing without pretending that the classroom model is a certified design tool.
A second primary source is https://pubmed.ncbi.nlm.nih.gov/7407317/. Compare the problem definition, estimator and validation rather than quoting one result in isolation.
Build a small counterexample for choosing breakpoints statistically. Information criteria, cross-validation or penalisation can compare candidate partitions. The initial example is: More bands improve in-sample fit. Then create a nearby case where the usual shortcut becomes unreliable. Searching many partitions creates selection uncertainty. Validate on held-out data.
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 sparse intervals to a real decision while protecting the boundary between calculation and judgement. Late bands often contain little exposure and few failures. One event over 20 hours gives a noisy large rate. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. Fine partitions can become unstable. Merge or regularise weak bands.
The audit should include the original question, input provenance, failure convention, algorithm settings, convergence or stopping rule, and a sensitivity result. For an adaptive method, also preserve the sequence of choices. For an optimisation, preserve every constraint. For Bayesian updating, preserve the prior and observation model. These records turn a result into something another person can challenge and improve.
A final transfer question is useful: where else would the same mathematical idea appear? Density ratios appear in survey correction and machine learning, probabilistic constraints in planning, and conditional probability in medical testing. The context changes, but transparent assumptions and reproducible checks remain valuable.
Poisson or likelihood methods quantify uncertainty in band hazards and rate ratios. A zero-event band can receive a one-sided upper bound. This is the central mathematical mechanism in confidence intervals. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. Wald intervals can behave poorly with small counts. Prefer likelihood or exact checks. 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 relationship to nelson–aalen. Nelson–Aalen estimates hazard jumps nonparametrically; piecewise exponential models smooth by pooling exposure inside bands. In the illustrative case, Wider bands trade temporal detail for precision. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. The two answer related but not identical estimation questions. Compare cumulative hazards.
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 relationship to weibull as a model-checking problem rather than a vocabulary exercise. Weibull imposes a smooth parametric hazard shape, while piecewise exponential uses local constants. A step model can approximate a rising Weibull hazard. Recompute the example from the raw values and label every unit. Many bands can mimic data without extrapolating well. Compare supported horizons.
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 change-point interpretation becomes useful only when its mechanism is explicit. A fitted breakpoint may mark a statistical change, not a physical threshold. Consider the worked illustration: A maintenance policy change can coincide with a rate shift. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. Confounding calendar changes can mimic ageing. Check operational history.
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 model diagnostics to practise sensitivity analysis. Compare observed and expected counts by band, residuals, cumulative hazard and held-out survival. Systematic within-band trend suggests bands are too wide. Before accepting the result, identify the quantity that would change the conclusion most. Good count fit can hide dependence. Inspect unit clusters.
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 extrapolation. Within observed bands prediction is transparent; beyond the last band requires an explicit final-rate assumption. The initial example is: Holding the last hazard constant is one choice. Then create a nearby case where the usual shortcut becomes unreliable. It may be wrong during wear-out. Label extrapolation clearly.
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. Split three unit histories, total exposure and calculate band hazards and survival. Move one breakpoint and predict which exposures change. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. Small counts are unstable. Reconcile with direct likelihood.
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.
The model connects intervals, rates, exponentials, tables and data organisation. Ask why one unit can occupy several rows but fail once. 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 coloured timelines. Check event/exposure units. 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. A piecewise constant hazard does not make survival discontinuous, split rows are not new units and zero events do not mean zero risk. In the illustrative case, Breakpoints are modelling choices. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. Late bands need uncertainty. Publish exposure totals.
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. Continuous physics, strong dependence, recurrent repairs or many competing transitions may need smoother or multistate models. A degradation signal can provide richer state information than age bands. Recompute the example from the raw values and label every unit. Overpartitioning wastes information. Use the simplest adequate structure.
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. Laird and Olivier connect censored survival data with log-linear techniques, while Holford develops log-linear rate and survivorship analysis. Consider the worked illustration: Read the exposure-offset derivation carefully. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. Software convenience does not justify arbitrary bands. Reproduce one band before several.
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 divide time into bands
Setup: Use this chapter’s illustrative case: Use 0–100,100–500 and 500–1000 hours for early, middle and late operation.
Task: Pre-specify or validate breakpoints.
Checking target: Bands chosen after seeing failures can exaggerate patterns.
Practice case 2: Constant hazard within a band
Setup: Use this chapter’s illustrative case: Survival decays exponentially while the unit remains in the band.
Task: Treat steps as approximation.
Checking target: Physical hazard need not jump at boundaries.
Practice case 3: Cumulative hazard across bands
Setup: Use this chapter’s illustrative case: A unit surviving two bands carries both hazard contributions.
Task: Keep a running total.
Checking target: Resetting cumulative hazard at each band is wrong.
Practice case 4: Continuous survival function
Setup: Use this chapter’s illustrative case: Only the slope of H(t) changes.
Task: Check continuity numerically.
Checking target: A discontinuous survival curve would imply instantaneous failure mass.
Practice case 5: A worked calculation
Setup: Use this chapter’s illustrative case: Both intervals contribute exposure.
Task: Verify units cancel in H.
Checking target: The numbers are illustrative.
Practice case 6: Person-time splitting
Setup: Use this chapter’s illustrative case: A unit observed to 250 hours contributes 100 hours to band one and 150 to band two.
Task: Retain unit identifiers.
Checking target: Duplicated split rows are not independent units.
Practice case 7: Events and exposure totals
Setup: Use this chapter’s illustrative case: Three failures over 2000 unit-hours give 0.0015 per hour.
Task: Report intervals or bounds.
Checking target: Zero events do not prove zero hazard.
Practice case 8: Poisson likelihood equivalence
Setup: Use this chapter’s illustrative case: Event counts by band use expected count exposure times rate.
Task: Use offsets and robust methods as needed.
Checking target: Ordinary Poisson independence needs careful unit clustering with split records.
Practice case 9: Right censoring
Setup: Use this chapter’s illustrative case: A survivor to 600 hours supplies all earlier-band exposure.
Task: Include every observed interval.
Checking target: Dropping survivors inflates hazard.
Practice case 10: Covariate effects
Setup: Use this chapter’s illustrative case: Temperature group can have a rate ratio within the model.
Task: Add interactions when justified.
Checking target: Proportional covariate effects may still fail over time.