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eduKateSG · Why Mathematics? · Fleming–Harrington Weighted Log-Rank Tests for Reliability Curves: Early, Middle and Late Failure Differences
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 two reliability curves may differ mainly during burn-in, useful life or wear-out. The ordinary log-rank test weights event times equally on its standard information scale. Fleming–Harrington tests use the pooled survival estimate to place planned emphasis on early, middle or late failures through weights of the form S(t-)^rho[1-S(t-)]^gamma.
This guide develops event-time scores, pooled-survival weights, covariance, parameter choice, censoring and interpretation. A weighted test is most credible when the expected timing pattern and weight family are chosen before results are seen. A small p-value does not measure the size of a reliability difference, and a late-weighted test can be fragile when few systems remain.
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 sustained hazard difference, an early burn-in gap and a late wear-out gap create different score patterns. Two coatings may perform similarly for 500 hours and separate after 1000. This is the central mathematical mechanism in why event timing changes the question. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. One test cannot be uniformly most powerful for every pattern. Describe the expected timing before analysis. 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 ordinary log-rank baseline. At each failure time, compare observed group failures with the allocation expected from the current risk set, then sum. In the illustrative case, With 60% of units in group A, one pooled failure gives A an expected count of .6. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. Equal standard weighting can lose power under changing effects. Compute the unweighted result as a baseline.
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 weighted score construction as a model-checking problem rather than a vocabulary exercise. Multiply each observed-minus-expected contribution by a pre-specified weight w(t_j) before summing. A late failure can count more under a late-emphasis design. Recompute the example from the raw values and label every unit. Weights change sensitivity, not the underlying event data. Show weights beside event times.
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 fleming–harrington weight family becomes useful only when its mechanism is explicit. A common family uses w(t)=S_pool(t-)^rho[1-S_pool(t-)]^gamma. Consider the worked illustration: The pooled Kaplan–Meier curve supplies a label-blind time-varying weight. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. Using group-specific survival would entangle weighting with group labels. Use the pooled pre-event estimate.
There are three uncertainties to keep apart. Sampling or numerical uncertainty comes from finite computation. Parameter uncertainty comes from estimating inputs. Model-form uncertainty comes from simplified physics, distributions or dependence. Reducing the first does not automatically reduce the other two. A credible report names all three and shows which one drives the decision.
For a practical investigation, create a small reproducible notebook or spreadsheet with a frozen data sheet, formula sheet, diagnostics sheet and decision note. Another student should be able to change one input and obtain the same update without guessing what the model meant.
Primary source: https://doi.org/10.1093/biomet/69.3.553. Read it for the method and its assumptions; do not transfer its performance claims automatically to another model.
Use why the left limit matters to practise sensitivity analysis. S(t-) is evaluated just before failures at t so the weight is determined before allocating tied events. At the first event, the pooled left-limit survival is usually one. Before accepting the result, identify the quantity that would change the conclusion most. Using the post-event value changes the score. Document the convention.
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 rho and gamma equal zero. With rho=0 and gamma=0, every event weight is one and the test reduces to ordinary log-rank. The initial example is: This provides a reproducible reference within the same family. Then create a nearby case where the usual shortcut becomes unreliable. The equality does not make log-rank optimal under every alternative. Report the parameter pair.
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 early-emphasis weights to a real decision while protecting the boundary between calculation and judgement. Positive rho with gamma zero gives larger weights while pooled survival is high, emphasising early failures. Rho=1,gamma=0 weights a time with S=.9 by .9 and a time with S=.3 by .3. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. Early emphasis can miss a delayed difference. Use only with a justified early alternative.
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.
Rho zero with positive gamma increases weight as 1-S grows, emphasising later failures. For gamma=1, S=.9 gives weight .1 while S=.3 gives .7. This is the central mathematical mechanism in late-emphasis weights. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. Late risk sets may be small and unstable. Set a supported follow-up range. 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 middle-emphasis weights. Positive rho and gamma can peak when survival is neither near one nor zero. In the illustrative case, With rho=gamma=1, w=S(1-S) peaks at S=.5. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. The location is on survival scale, not fixed clock time. Translate the peak to the observed curve.
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 weight table as a model-checking problem rather than a vocabulary exercise. For S values .9,.6,.3, late weights with gamma=1 are .1,.4,.7. Multiply each by its O-E contribution before summing. Recompute the example from the raw values and label every unit. A large weight does not create more events. Keep score and weight columns separate.
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 variance of the weighted score becomes useful only when its mechanism is explicit. Event-time allocation variances are multiplied by squared weights and summed, with tie corrections when needed. Consider the worked illustration: A weight of .5 multiplies a variance term by .25. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. Ignoring squared weights gives a wrong standardisation. Audit the covariance formula.
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 standardised statistic to practise sensitivity analysis. Divide the weighted score by its estimated standard deviation to form a z-type statistic under the null. A score 2.4 with SE 1.2 gives z=2. Before accepting the result, identify the quantity that would change the conclusion most. Asymptotic calibration can be weak with few events. Report event counts and approximation.
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/10.1080/03610928108828073. Compare the problem definition, estimator and validation rather than quoting one result in isolation.
Build a small counterexample for tied failures. When several failures share a time, expected counts and hypergeometric variance use the tied event total. The initial example is: Three failures among 20 at risk are allocated jointly. Then create a nearby case where the usual shortcut becomes unreliable. Coarse time recording can create many ties. Preserve timestamp resolution.
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 right censoring to a real decision while protecting the boundary between calculation and judgement. Censored systems contribute to risk sets until censoring under independent-censoring assumptions. A system working at cutoff remains at risk through its last observed time. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. Condition-driven withdrawal can bias every weighted test. Compare censoring patterns.
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.
Choosing rho and gamma after seeing which produces the smallest p-value is multiple testing. Trying six weights and reporting one favourable result inflates false positives. This is the central mathematical mechanism in pre-specification protects error rates. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. A plausible story invented after inspection is not pre-specification. Freeze the weight in the analysis plan. 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 direction and sign. The sign of the score depends on group coding, while a two-sided squared or absolute statistic does not. In the illustrative case, Swapping A and B reverses z. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. A positive sign has no universal better-or-worse meaning. State coding and event definition.
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 effect size remains necessary as a model-checking problem rather than a vocabulary exercise. A test quantifies evidence against equal curves but not hours gained or probability difference. Pair a late-weighted result with survival differences at supported horizons. Recompute the example from the raw values and label every unit. Significance can coexist with a tiny practical gap. Report curves and intervals.
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.
CHAPTER 18 OF 25 · Explore applications
18. Non-proportional hazards interpretation
The mathematics of non-proportional hazards interpretation becomes useful only when its mechanism is explicit. Weighted tests can gain power for planned time-varying differences without producing one constant hazard ratio. Consider the worked illustration: A delayed benefit violates the simple proportional-hazards story. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. The test alone does not estimate the time-varying effect. Show interval-specific effects or RMST.
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 more than two groups to practise sensitivity analysis. Rank-test vectors and covariance matrices can extend weighted comparisons to several groups. Three materials can receive a global late-weighted test. Before accepting the result, identify the quantity that would change the conclusion most. A significant global result does not identify every pair. Plan multiplicity-aware contrasts.
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 simulation for design. Simulate plausible burn-in, delayed and wear-out patterns with censoring to compare power and type-I error before choosing weights. The initial example is: A planned late effect may favour gamma=1. Then create a nearby case where the usual shortcut becomes unreliable. Simulation is only as credible as its scenarios. Include adverse alternatives.
Counterexamples are positive learning tools. They reveal the conditions that make a method work and help students recognise when a software output deserves a pause. Keep the counterexample small enough to plot or enumerate. Check sign, support, bounds and conservation of probability before adding complexity.
When writing the conclusion, separate a documented fact from an interpretation and from an illustrative scenario. Cite the primary method paper for the factual mechanism, use the local calculation for the illustration, and label any recommendation as conditional on the stated model. This keeps career and engineering advice age-appropriate and honest.
Connect decision interpretation to a real decision while protecting the boundary between calculation and judgement. Use the test with absolute reliability, consequence and mission timing. A late difference matters only if systems commonly operate into that range. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. A p-value cannot set replacement policy. Translate timing into the real decision.
The audit should include the original question, input provenance, failure convention, algorithm settings, convergence or stopping rule, and a sensitivity result. For an adaptive method, also preserve the sequence of choices. For an optimisation, preserve every constraint. For Bayesian updating, preserve the prior and observation model. These records turn a result into something another person can challenge and improve.
A final transfer question is useful: where else would the same mathematical idea appear? Density ratios appear in survey correction and machine learning, probabilistic constraints in planning, and conditional probability in medical testing. The context changes, but transparent assumptions and reproducible checks remain valuable.
Build a five-event risk table and compute weights for (0,0),(1,0),(0,1) and (1,1). Predict which event dominates each score. This is the central mathematical mechanism in practice activity. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. Toy data cannot validate a product. Check left-limit survival. The useful habit is to keep the physical question, the probability model and the numerical method in separate columns so that a correct computation is not mistaken for complete evidence.
Did You Know? A result can be numerically stable and still answer the wrong question if the failure event, denominator or transformation changes. Test one nearby case and predict the direction of change before recalculating. Save inputs, intermediate values and units so a second reader can audit the path.
For practice, turn the example into a sensitivity check. Change one defensible input while holding the others fixed, compare the new result with the original and explain why it moved. Then construct a counterexample in which the shortcut fails. End with two sentences: what the mathematics supports, and what remains uncertain.
Start with the operational question behind parent and educator guidance. Weighted log-rank tests connect powers, products, expected values, graphs and fair comparisons. In the illustrative case, Ask how changing rho and gamma moves attention along a curve. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. Start with three survival values. Keep timing and magnitude separate.
A student should be able to rebuild the result without hidden settings. Record the sampling rule or optimisation criterion, the exact failure definition, and the evidence used for every distribution or parameter. Compare a transparent baseline with the advanced method. If they disagree, investigate the mechanism before deciding which answer is more credible.
This chapter belongs to a larger decision, so translate the number carefully. State one action the evidence permits and one action it does not. A parent or educator can ask which school ideas are doing the work: algebra organises relationships, probability represents uncertainty, calculus describes local change, and computing repeats the documented rules at scale.
Treat common misconceptions as a model-checking problem rather than a vocabulary exercise. A weighted test is not a weighted average lifetime, late weighting does not prove late benefit and post-hoc weight choice is not one planned test. Weights affect sensitivity, not effect size. Recompute the example from the raw values and label every unit. Censoring assumptions still matter. Report the full analysis plan.
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. Harrington and Fleming developed a broad class of censored-data rank tests whose survival-based weights include early and late emphasis. Consider the worked illustration: Reproduce three weights before using software. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. The method does not choose the scientifically relevant alternative. Read power arguments beside effect plots.
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 event timing changes the question
Setup: Use this chapter’s illustrative case: Two coatings may perform similarly for 500 hours and separate after 1000.
Task: Describe the expected timing before analysis.
Checking target: One test cannot be uniformly most powerful for every pattern.
Practice case 2: Ordinary log-rank baseline
Setup: Use this chapter’s illustrative case: With 60% of units in group A, one pooled failure gives A an expected count of .6.
Task: Compute the unweighted result as a baseline.
Checking target: Equal standard weighting can lose power under changing effects.
Practice case 3: Weighted score construction
Setup: Use this chapter’s illustrative case: A late failure can count more under a late-emphasis design.
Task: Show weights beside event times.
Checking target: Weights change sensitivity, not the underlying event data.
Practice case 4: Fleming–Harrington weight family
Setup: Use this chapter’s illustrative case: The pooled Kaplan–Meier curve supplies a label-blind time-varying weight.
Task: Use the pooled pre-event estimate.
Checking target: Using group-specific survival would entangle weighting with group labels.
Practice case 5: Why the left limit matters
Setup: Use this chapter’s illustrative case: At the first event, the pooled left-limit survival is usually one.
Task: Document the convention.
Checking target: Using the post-event value changes the score.
Practice case 6: Rho and gamma equal zero
Setup: Use this chapter’s illustrative case: This provides a reproducible reference within the same family.
Task: Report the parameter pair.
Checking target: The equality does not make log-rank optimal under every alternative.
Practice case 7: Early-emphasis weights
Setup: Use this chapter’s illustrative case: Rho=1,gamma=0 weights a time with S=.9 by .9 and a time with S=.3 by .3.
Task: Use only with a justified early alternative.
Checking target: Early emphasis can miss a delayed difference.
Practice case 8: Late-emphasis weights
Setup: Use this chapter’s illustrative case: For gamma=1, S=.9 gives weight .1 while S=.3 gives .7.
Task: Set a supported follow-up range.
Checking target: Late risk sets may be small and unstable.
Practice case 9: Middle-emphasis weights
Setup: Use this chapter’s illustrative case: With rho=gamma=1, w=S(1-S) peaks at S=.5.
Task: Translate the peak to the observed curve.
Checking target: The location is on survival scale, not fixed clock time.
Practice case 10: A worked weight table
Setup: Use this chapter’s illustrative case: Multiply each by its O-E contribution before summing.
Task: Keep score and weight columns separate.
Checking target: A large weight does not create more events.