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eduKateSG · Why Mathematics? · Restricted Mean Survival Time for Reliability: Survival-Curve Area, Time Horizons, Censoring and Fair Comparisons
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 engineers need one understandable summary of reliable operating time over a declared horizon. Restricted mean survival time, or RMST, is the area under a survival curve from time zero to a chosen time tau. It equals the expected operating time accumulated before tau when lifetime is capped at that horizon.
This guide develops the integral, step-function calculation, group differences, ratios, censoring, uncertainty and horizon choice. RMST does not require proportional hazards, but it does require enough follow-up and a defensible comparison window. A larger restricted mean is not a guarantee for an individual component, and changing tau changes the question.
Full chapter index
Chapters 1–5 · Understand the question
Chapters 6–10 · Reproduce the example
Chapters 11–15 · Check assumptions
Chapters 16–20 · Explore applications
Chapters 21–25 · Study and verify
A finite horizon converts a survival curve into expected time alive or operating within a window. A 1000-hour mission can be summarised by expected operating time before 1000 hours. This is the central mathematical mechanism in why use a restricted mean. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. The result does not describe time beyond the restriction. State tau in every result. 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 definition as a capped lifetime. RMST(tau)=E[min(T,tau)] for nonnegative lifetime T. In the illustrative case, A component lasting 1300 hours contributes 1000 hours when tau=1000. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. Capping is a design choice, not data loss. Match tau to the decision.
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 area under the survival curve as a model-checking problem rather than a vocabulary exercise. RMST(tau)=integral from 0 to tau of S(t)dt. A survival curve staying near one creates an area near tau. Recompute the example from the raw values and label every unit. Area is expressed in time units, not probability units. Check dimensions.
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 step-function calculation becomes useful only when its mechanism is explicit. With a Kaplan–Meier curve, sum rectangle widths times survival levels between event times. Consider the worked illustration: Survival 1.0 for 100 hours and 0.9 for the next 200 contributes 280 hours. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. Using endpoint values inconsistently changes area. Declare left- or right-continuous convention.
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.4274. Read it for the method and its assumptions; do not transfer its performance claims automatically to another model.
Use a complete worked example to practise sensitivity analysis. For intervals 0–2,2–5,5–8 with survival 1,0.8,0.5, the RMST to 8 is 2+2.4+1.5=5.9. The units follow the time axis. Before accepting the result, identify the quantity that would change the conclusion most. The toy curve is illustrative. Rebuild it in a table.
A simple baseline is essential. It reveals whether the advanced method is improving accuracy or only adding complexity. Compare model calls, wall time, uncertainty and failure-mode coverage under one declared budget. Repeat stochastic calculations with several seeds; a favourable single run is not evidence of dependable efficiency.
Students can extend the example by adding correlation, a second failure mode or a modest measurement error. Explain which part of the formula changes and which remains fixed. This exercise shows how mathematics transfers from a clean textbook case to engineering and computing without pretending that the classroom model is a certified design tool.
Build a small counterexample for right censoring. A censoring-aware survival estimator supplies the area under independent-censoring assumptions. The initial example is: A unit working at study end still contributes survival evidence to its last observed time. Then create a nearby case where the usual shortcut becomes unreliable. Naively averaging observed failures biases results. Show numbers at risk.
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 choosing tau to a real decision while protecting the boundary between calculation and judgement. A fair tau should be supported by follow-up in all groups and relevant to the mission or planning window. Use 500 hours if both designs have credible observation to 500 hours. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. Choosing tau after seeing the most favourable gap inflates evidence. Pre-specify or report sensitivity.
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 difference in group RMSTs has direct time units. A difference of 35 hours to tau=1000 means 35 additional expected operating hours within that window under the comparison design. This is the central mathematical mechanism in rmst difference. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. It is not a median difference. Give confidence intervals. 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 rmst ratio. A ratio compares expected restricted time multiplicatively. In the illustrative case, 980 divided by 940 is about 1.043. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. Ratios can look stable while absolute gains are small. Report difference and ratio 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 comparison with hazard ratios as a model-checking problem rather than a vocabulary exercise. RMST remains interpretable when hazards cross because it integrates survival over time. Early harm and later benefit can produce nonproportional hazards. Recompute the example from the raw values and label every unit. A single area can still hide when differences occur. Plot the curves too.
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 comparison with mean lifetime becomes useful only when its mechanism is explicit. Full mean lifetime integrates survival to infinity, while RMST stops at tau. Consider the worked illustration: A limited test can estimate a supported restricted mean without inventing the far tail. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. RMST is not the unrestricted mean. Use the full name.
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 comparison with mean residual life to practise sensitivity analysis. RMST starts at time zero and caps lifetime; mean residual life conditions on surviving to age t. The two integrals answer different questions. Before accepting the result, identify the quantity that would change the conclusion most. Interchanging them confuses populations and horizons. Write conditioning explicitly.
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/10.1186/1471-2288-13-152. Compare the problem definition, estimator and validation rather than quoting one result in isolation.
Build a small counterexample for variance and confidence intervals. Greenwood-based, influence-function, bootstrap or model-based methods can quantify sampling uncertainty. The initial example is: Uncertainty grows when risk sets shrink before tau. Then create a nearby case where the usual shortcut becomes unreliable. One interval does not include model-form uncertainty automatically. Document the method.
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 restricted mean time lost to a real decision while protecting the boundary between calculation and judgement. RMTL(tau)=tau-RMST(tau), the area under the cumulative failure curve to tau. If RMST is 940 hours at tau=1000, expected restricted time lost is 60 hours. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. Time lost is not downtime after repair unless the state process is defined that way. Name the lifetime event.
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.
For cause-specific questions, use cumulative incidence and restricted mean time in states rather than ordinary all-cause survival alone. One failure mode can prevent another from occurring first. This is the central mathematical mechanism in competing risks caution. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. Treating competing events as ordinary censoring changes the estimand. Match method to event structure. 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 covariate adjustment. Regression or standardisation can estimate covariate-adjusted RMST under declared assumptions. In the illustrative case, Design imbalance in load or environment may require adjustment. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. Adjustment does not create randomisation. List covariates and overlap.
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 milestone survival versus area as a model-checking problem rather than a vocabulary exercise. Survival at one milestone ignores earlier curve differences that RMST captures. Two designs can both have S(1000)=0.9 but different early failures. Recompute the example from the raw values and label every unit. Area also compresses timing into one summary. Show milestone and area.
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 horizon sensitivity becomes useful only when its mechanism is explicit. Plot RMST differences over several defensible tau values to reveal when benefit accumulates. Consider the worked illustration: A gap may grow early and flatten later. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. Repeatedly searching horizons needs multiplicity care. Distinguish planned from exploratory.
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 validation with later cohorts to practise sensitivity analysis. Check restricted-time predictions on units observed after model development. Compare predicted and observed area to the same tau. Before accepting the result, identify the quantity that would change the conclusion most. Training agreement is optimistic. Use a temporal holdout.
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 decision interpretation. Combine expected operating-time gain with cost, consequence and uncertainty. The initial example is: Thirty extra hours may matter for a short mission but not justify a costly redesign. Then create a nearby case where the usual shortcut becomes unreliable. RMST alone does not optimise policy. Translate time into the real decision.
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. Compute RMST from a four-step survival curve and compare two designs at two horizons. Predict how extending tau changes each area. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. Toy numbers do not support product claims. Audit every rectangle.
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.
RMST connects graphs, area, expected value, units and fair comparison. Ask why the same curve yields different answers at different tau. 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 rectangles under a curve. Start without calculus. 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. RMST is not median lifetime, not full mean life and not a guarantee for one item. In the illustrative case, Changing tau changes the estimand. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. A larger area can hide early disadvantages. Always show curves.
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 rmst is unsuitable as a model-checking problem rather than a vocabulary exercise. Weak follow-up before tau, informative censoring, dynamic repairs or multi-state service may need other summaries. A repairable system can accumulate several up periods. Recompute the example from the raw values and label every unit. A single terminal lifetime can miss restoration. Use availability or multistate reward models.
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. Royston and Parmar develop RMST comparisons when proportional hazards are doubtful and discuss design and interpretation. Consider the worked illustration: Transfer the mathematics carefully from survival studies to engineering lifetimes. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. Context changes the event and horizon. Reproduce a step-curve area 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 use a restricted mean
Setup: Use this chapter’s illustrative case: A 1000-hour mission can be summarised by expected operating time before 1000 hours.
Task: State tau in every result.
Checking target: The result does not describe time beyond the restriction.
Practice case 2: Definition as a capped lifetime
Setup: Use this chapter’s illustrative case: A component lasting 1300 hours contributes 1000 hours when tau=1000.
Task: Match tau to the decision.
Checking target: Capping is a design choice, not data loss.
Practice case 3: Area under the survival curve
Setup: Use this chapter’s illustrative case: A survival curve staying near one creates an area near tau.
Task: Check dimensions.
Checking target: Area is expressed in time units, not probability units.
Practice case 4: Step-function calculation
Setup: Use this chapter’s illustrative case: Survival 1.0 for 100 hours and 0.9 for the next 200 contributes 280 hours.
Task: Declare left- or right-continuous convention.
Checking target: Using endpoint values inconsistently changes area.
Practice case 5: A complete worked example
Setup: Use this chapter’s illustrative case: The units follow the time axis.
Task: Rebuild it in a table.
Checking target: The toy curve is illustrative.
Practice case 6: Right censoring
Setup: Use this chapter’s illustrative case: A unit working at study end still contributes survival evidence to its last observed time.
Task: Show numbers at risk.
Checking target: Naively averaging observed failures biases results.
Practice case 7: Choosing tau
Setup: Use this chapter’s illustrative case: Use 500 hours if both designs have credible observation to 500 hours.
Task: Pre-specify or report sensitivity.
Checking target: Choosing tau after seeing the most favourable gap inflates evidence.
Practice case 8: RMST difference
Setup: Use this chapter’s illustrative case: A difference of 35 hours to tau=1000 means 35 additional expected operating hours within that window under the comparison design.
Task: Give confidence intervals.
Checking target: It is not a median difference.
Practice case 9: RMST ratio
Setup: Use this chapter’s illustrative case: 980 divided by 940 is about 1.043.
Task: Report difference and ratio together.
Checking target: Ratios can look stable while absolute gains are small.
Practice case 10: Comparison with hazard ratios
Setup: Use this chapter’s illustrative case: Early harm and later benefit can produce nonproportional hazards.
Task: Plot the curves too.
Checking target: A single area can still hide when differences occur.