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eduKateSG · Why Mathematics? · Tarone–Ware Tests for Reliability Curves: Square-Root Risk-Set Weights, Early–Late Balance, Censoring and Power
Start with the question, audit the worked example, or jump to the assumptions and practice. Every chapter stays open for uninterrupted reading.
Understand the question
Define the quantity and follow the mechanism.Route 2 · Chapters 6–10
Reproduce the example
Audit the numbers one step at a time.Route 3 · Chapters 11–15
Check assumptions
See where design and calibration matter.Route 4 · Chapters 16–20
Explore applications
Connect the method to a concrete investigation.Route 5 · Chapters 21–25
Study and verify
Practise, inspect failure cases and read further.
Full chapter index · Practice and checking · Mathematics Learning Hub
Why mathematics is important becomes visible when an analyst wants more early sensitivity than ordinary log-rank but less concentration on the earliest risk sets than Gehan–Breslow. The Tarone–Ware test uses the square root of the pooled number at risk as an event-time weight. This intermediate growth rate creates a principled compromise between equal weights and full risk-set weights.
This guide develops square-root weighting, observed-minus-expected failures, variance, censoring, power and interpretation. “Compromise” does not mean universally safest or most powerful. The procedure still answers a global equality question, depends on risk-set quality and needs planned effect summaries to explain when and by how much reliability differs.
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
Log-rank can underweight an early departure for some alternatives, while full risk-set weighting can concentrate strongly on the earliest events. A component defect creates a moderate early gap that persists into mid-life. This is the central mathematical mechanism in why an intermediate weight exists. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. No compromise weight is optimal for every pattern. Describe the plausible timing before choosing. 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 tarone–ware weight formula. At each pooled failure time use a weight proportional to the square root of the pooled risk-set size, w(t)=sqrt(Y(t)). In the illustrative case, Y values 100, 64 and 25 give weights 10, 8 and 5. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. Some software rescales all weights without changing the standardised result. Record the formula, not only the name.
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 observed-minus-expected foundation as a model-checking problem rather than a vocabulary exercise. The unweighted event-time increment compares observed group failures with the null expectation from current risk-set proportions. With 30 of 50 units in A and one failure, A expects 0.6 under equality. Recompute the example from the raw values and label every unit. This allocation is conditional on the risk set. Build the risk table first.
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 weighted score construction becomes useful only when its mechanism is explicit. Multiply each O-E increment by sqrt(Y) and add across event times. Consider the worked illustration: An increment 0.4 at Y=100 contributes 4; an increment -0.5 at Y=16 contributes -2. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. Positive and negative phases can cancel. Keep a cumulative score column.
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/64.1.156. Read it for the method and its assumptions; do not transfer its performance claims automatically to another model.
Use a worked three-event example to practise sensitivity analysis. For Y=100,64,25 and deviations .2,-.3,.4, weighted contributions are 2,-2.4,2, giving a total of 1.6. The same deviations sum to .3 under equal raw weights. Before accepting the result, identify the quantity that would change the conclusion most. Neither raw total is a p-value. Compute the standard error next.
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 between log-rank and gehan–breslow. Equal weights are 1, Tarone–Ware weights grow as sqrt(Y), and Gehan–Breslow weights grow as Y. The initial example is: At Y=100 versus 25, relative weights are 1:1, 2:1 and 4:1 respectively. Then create a nearby case where the usual shortcut becomes unreliable. Relative weighting depends on the actual risk-set path. Compare weights on one table.
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 why square roots matter to a real decision while protecting the boundary between calculation and judgement. The square root grows with information support but more slowly than the risk-set count, tempering extreme early dominance. Quadrupling Y doubles sqrt(Y) rather than quadrupling the weight. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. The square root is a test design choice, not a law of physical reliability. Explain its statistical role.
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.
Each event allocation variance is multiplied by w(t)^2=Y(t), with covariance and tie terms handled consistently. A sqrt(Y)=8 weight contributes a factor 64 to the corresponding variance term. This is the central mathematical mechanism in variance of the weighted sum. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. Using sqrt(Y) again in the variance is wrong. Check weight versus squared weight. 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 standardised z statistic. Divide the weighted score by its estimated standard deviation and use its planned one- or two-sided null approximation. In the illustrative case, A total 1.6 with SE 0.8 gives z=2. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. Few failures can make normal calibration rough. Report event counts and sensitivity checks.
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 tied event blocks as a model-checking problem rather than a vocabulary exercise. When several failures occur at the same recorded time, compute observed totals, expectations and variance jointly for that block. Four weekly inspection failures share time 12 weeks. Recompute the example from the raw values and label every unit. Ordering tied failures one by one changes Y artificially. Use one pre-event risk set for the block.
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 right censoring becomes useful only when its mechanism is explicit. Censored units remain in Y until their last observed working time and leave without a failure count. Consider the worked illustration: A unit lost at 300 hours supports earlier risk sets only. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. Condition-driven removal can bias all rank tests. Compare reasons and censoring paths.
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 differential follow-up to practise sensitivity analysis. If one group has shorter observation, later risk sets can become unbalanced and the usable common horizon shrinks. Supplier B data stop six months earlier. Before accepting the result, identify the quantity that would change the conclusion most. Square-root weighting cannot create missing late evidence. Set and justify the analysis horizon.
A simple baseline is essential. It reveals whether the advanced method is improving accuracy or only adding complexity. Compare model calls, wall time, uncertainty and failure-mode coverage under one declared budget. Repeat stochastic calculations with several seeds; a favourable single run is not evidence of dependable efficiency.
Students can extend the example by adding correlation, a second failure mode or a modest measurement error. Explain which part of the formula changes and which remains fixed. This exercise shows how mathematics transfers from a clean textbook case to engineering and computing without pretending that the classroom model is a certified design tool.
A second primary source is https://doi.org/10.1093/biomet/65.1.167. Compare the problem definition, estimator and validation rather than quoting one result in isolation.
Build a small counterexample for early-versus-late sensitivity. Because Y generally declines, sqrt(Y) favours earlier events, but less sharply than Y weighting. The initial example is: The method can retain more mid-life sensitivity than Gehan–Breslow. Then create a nearby case where the usual shortcut becomes unreliable. A late-only separation may still be weakly represented. Plot the actual weight trajectory.
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 non-proportional hazards to a real decision while protecting the boundary between calculation and judgement. Tarone–Ware can gain power when hazards differ more early or during a broad early-to-middle interval. A manufacturing improvement removes infant mortality but not wear-out. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. A significant test does not estimate a time-varying hazard ratio. Add interval-specific effects.
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.
Signed weighted contributions before and after a crossing may cancel, depending on where sqrt(Y) places mass. Design A is worse during burn-in and better after 500 hours. This is the central mathematical mechanism in crossing curves. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. One global statistic cannot declare uniform superiority. Use curve and cumulative-score plots. 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 two-sided and directional questions. A two-sided statistic detects either direction; a one-sided version requires a pre-specified better-reliability direction. In the illustrative case, The same absolute z can arise after swapping labels. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. Choosing the tail after inspection is invalid. Declare group coding and alternative.
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 k-sample extension as a model-checking problem rather than a vocabulary exercise. Weighted group-score vectors and their covariance matrices support a global comparison of several reliability distributions. Three manufacturing lines can be assessed together. Recompute the example from the raw values and label every unit. A global result does not rank every line. Plan multiplicity-controlled contrasts.
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 power comparisons by simulation becomes useful only when its mechanism is explicit. Simulate proportional, early, delayed and crossing alternatives with the intended censoring pattern to compare candidate tests. Consider the worked illustration: Tarone–Ware may sit between log-rank and Gehan–Breslow across early scenarios. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. One favourable simulation does not establish dominance. Use a broad scenario grid.
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 effect estimates after testing to practise sensitivity analysis. Report milestone reliability, restricted mean lifetime, quantiles or hazard differences with confidence intervals. A 200-hour absolute reliability difference explains practical magnitude. Before accepting the result, identify the quantity that would change the conclusion most. Selecting only the largest post-hoc effect exaggerates evidence. Pre-plan key horizons.
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 multiplicity and test shopping. Running log-rank, Gehan–Breslow and Tarone–Ware and reporting only the smallest p-value is multiple testing. The initial example is: Three unadjusted choices raise false-positive risk. Then create a nearby case where the usual shortcut becomes unreliable. A story about the winning test written later is not pre-specification. Use one planned test or calibrated combination.
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. The test is useful when early and middle operating periods matter and the square-root compromise matches the decision. Commissioning plus the first service interval may define the critical window. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. The statistic cannot assign replacement cost or safety consequence. Link time weights to mission needs.
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.
Calculate weights for Y=144,81,36 and 9 under log-rank, Tarone–Ware and Gehan–Breslow, then multiply the same O-E increments. Predict which event dominates each total. 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. The toy table is not a calibrated analysis. Keep weights and effects distinct. The useful habit is to keep the physical question, the probability model and the numerical method in separate columns so that a correct computation is not mistaken for complete evidence.
Did You Know? A result can be numerically stable and still answer the wrong question if the failure event, denominator or transformation changes. Test one nearby case and predict the direction of change before recalculating. Save inputs, intermediate values and units so a second reader can audit the path.
For practice, turn the example into a sensitivity check. Change one defensible input while holding the others fixed, compare the new result with the original and explain why it moved. Then construct a counterexample in which the shortcut fails. End with two sentences: what the mathematics supports, and what remains uncertain.
Start with the operational question behind parent and educator guidance. Tarone–Ware offers a clear lesson in square roots, scale, proportional change, expected values and how a middle choice can still have assumptions. In the illustrative case, Ask why four times the risk set produces only twice the square-root weight. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. Students may call compromise automatically fair. Test the idea with counterexamples.
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. The method is not an average of other p-values, sqrt(Y) is not an effect size, early emphasis is not early causation and a non-significant result is not proof of equality. Software rescaling can change raw scores. Recompute the example from the raw values and label every unit. Unplanned test selection needs multiplicity control. Report the planned formula and curve effects.
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. Tarone and Ware showed that log-rank and modified Wilcoxon statistics differ through risk-set weights and proposed an intermediate statistic; Prentice developed the wider linear-rank framework. Consider the worked illustration: Reproduce the 1:1, 2:1 and 4:1 relative-weight comparison. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. Power claims depend on alternatives and censoring. Read the primary articles beside simulations.
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 an intermediate weight exists
Setup: Use this chapter’s illustrative case: A component defect creates a moderate early gap that persists into mid-life.
Task: Describe the plausible timing before choosing.
Checking target: No compromise weight is optimal for every pattern.
Practice case 2: Tarone–Ware weight formula
Setup: Use this chapter’s illustrative case: Y values 100, 64 and 25 give weights 10, 8 and 5.
Task: Record the formula, not only the name.
Checking target: Some software rescales all weights without changing the standardised result.
Practice case 3: Observed-minus-expected foundation
Setup: Use this chapter’s illustrative case: With 30 of 50 units in A and one failure, A expects 0.6 under equality.
Task: Build the risk table first.
Checking target: This allocation is conditional on the risk set.
Practice case 4: Weighted score construction
Setup: Use this chapter’s illustrative case: An increment 0.4 at Y=100 contributes 4; an increment -0.5 at Y=16 contributes -2.
Task: Keep a cumulative score column.
Checking target: Positive and negative phases can cancel.
Practice case 5: A worked three-event example
Setup: Use this chapter’s illustrative case: The same deviations sum to .3 under equal raw weights.
Task: Compute the standard error next.
Checking target: Neither raw total is a p-value.
Practice case 6: Between log-rank and Gehan–Breslow
Setup: Use this chapter’s illustrative case: At Y=100 versus 25, relative weights are 1:1, 2:1 and 4:1 respectively.
Task: Compare weights on one table.
Checking target: Relative weighting depends on the actual risk-set path.
Practice case 7: Why square roots matter
Setup: Use this chapter’s illustrative case: Quadrupling Y doubles sqrt(Y) rather than quadrupling the weight.
Task: Explain its statistical role.
Checking target: The square root is a test design choice, not a law of physical reliability.
Practice case 8: Variance of the weighted sum
Setup: Use this chapter’s illustrative case: A sqrt(Y)=8 weight contributes a factor 64 to the corresponding variance term.
Task: Check weight versus squared weight.
Checking target: Using sqrt(Y) again in the variance is wrong.
Practice case 9: Standardised z statistic
Setup: Use this chapter’s illustrative case: A total 1.6 with SE 0.8 gives z=2.
Task: Report event counts and sensitivity checks.
Checking target: Few failures can make normal calibration rough.
Practice case 10: Tied event blocks
Setup: Use this chapter’s illustrative case: Four weekly inspection failures share time 12 weeks.
Task: Use one pre-event risk set for the block.
Checking target: Ordering tied failures one by one changes Y artificially.