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eduKateSG · Why Mathematics? · MaxCombo Tests for Non-Proportional Reliability Hazards: Weighted Components, Correlation, Multiplicity 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 clear when engineers expect that reliability curves could separate early, late or throughout follow-up but cannot justify one timing pattern in advance. A MaxCombo procedure calculates several pre-specified correlated weighted log-rank statistics and uses their maximum as one global statistic. Its p-value comes from their joint null distribution, so it accounts for searching across the planned component tests.
This guide develops component weights, correlation, multivariate-normal calibration, one- and two-sided choices, power trade-offs and interpretation. MaxCombo is not permission to add weights after seeing the data. It tests equality of curves and still needs effect estimates, follow-up support, censoring checks and a decision-relevant explanation of when the difference occurs.
Full chapter index
Chapters 1–5 · Understand the question
Chapters 6–10 · Reproduce the example
Chapters 11–15 · Check assumptions
Chapters 16–20 · Explore applications
Chapters 21–25 · Study and verify
CHAPTER 1 OF 25 · Understand the question
1. Why combine several timing alternatives
A single weight is powerful for one expected pattern but can be weak if separation occurs elsewhere. A redesign might help immediately, after a delay or during wear-out. This is the central mathematical mechanism in why combine several timing alternatives. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. Including every imaginable weight dilutes planning and complicates interpretation. Choose a compact component set. 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 pre-specified component tests. Define the weighted log-rank statistics before observing group outcomes. In the illustrative case, A common set includes ordinary, early and late weights. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. Adding a favourable component after inspection invalidates the planned calibration. Freeze weights and directions.
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 standardise every component as a model-checking problem rather than a vocabulary exercise. Each weighted score is divided by its own estimated standard deviation to create comparable z statistics. Raw scores on different weight scales cannot be maximised fairly. Recompute the example from the raw values and label every unit. Small component variance can cause instability. Inspect every SE.
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 maximum combination statistic becomes useful only when its mechanism is explicit. For a one-sided design, Z_max is the maximum of the planned z values; two-sided versions require an explicit absolute-value or tail convention. Consider the worked illustration: If z values are 1.1,2.0,.7, the one-sided maximum is 2.0. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. The maximum alone is not a valid ordinary-normal test. Use the joint null distribution.
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/10.1177/1536867X1601600308. Read it for the method and its assumptions; do not transfer its performance claims automatically to another model.
CHAPTER 5 OF 25 · Understand the question
5. Why component statistics are correlated
Use why component statistics are correlated to practise sensitivity analysis. All components use the same failures and risk sets with different weights, so their scores share information. Ordinary and late-weighted statistics may correlate strongly. Before accepting the result, identify the quantity that would change the conclusion most. Bonferroni ignores useful correlation and can be conservative. Estimate the covariance matrix.
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 multivariate-normal calibration. Under regularity conditions the vector of standardised scores is approximated by a correlated multivariate normal distribution. The initial example is: The p-value is the probability that any planned component exceeds the observed maximum. Then create a nearby case where the usual shortcut becomes unreliable. Numerical integration accuracy matters. Record algorithm and tolerances.
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 a worked two-component probability to a real decision while protecting the boundary between calculation and judgement. If two independent one-sided z tests each use threshold 1.96, chance either exceeds it is 1-(.975)^2≈.0494. Positive correlation lowers the search penalty relative to independence. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. Real component tests are not independent. Use the estimated joint distribution.
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.
Components may use (rho,gamma)=(0,0),(1,0) and (0,1) for broad, early and late sensitivity. The set spans three plausible timing patterns. This is the central mathematical mechanism in typical fleming–harrington set. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. It is not automatically optimal for crossing or middle-only alternatives. Justify each component. 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 including a middle-weighted component. Adding (1,1) can target mid-curve differences. In the illustrative case, The weight S(1-S) peaks near pooled survival .5. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. More components raise the multiplicity penalty and interpretation burden. Add only with design rationale.
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 one-sided versus two-sided tests as a model-checking problem rather than a vocabulary exercise. One-sided MaxCombo targets a declared favourable direction; two-sided procedures protect both directions. Reliability superiority may justify a directional alternative only when harm is handled separately. Recompute the example from the raw values and label every unit. Selecting the tail after results doubles opportunism. Declare the direction before data.
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 non-proportional hazards becomes useful only when its mechanism is explicit. MaxCombo retains sensitivity across several time-varying alternatives without assuming one constant hazard ratio. Consider the worked illustration: Early damage followed by later improvement can defeat ordinary log-rank. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. A significant combination does not summarise the pattern. Plot curves and time-specific effects.
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 crossing curves to practise sensitivity analysis. Some component scores can cancel while a maximum or supremum-type component detects a sustained excursion. One curve can be worse early and better late. Before accepting the result, identify the quantity that would change the conclusion most. A one-sided component set may miss reversals. Match the set to plausible crossings.
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/10.1080/19466315.2021.2008485. Compare the problem definition, estimator and validation rather than quoting one result in isolation.
Build a small counterexample for censoring and risk sets. Every component relies on correct risk sets and independent-censoring assumptions. The initial example is: Late components become fragile if most systems are censored before wear-out. Then create a nearby case where the usual shortcut becomes unreliable. Combination cannot create late information. Set minimum follow-up support.
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 ties and event-time weights to a real decision while protecting the boundary between calculation and judgement. Tied failures require consistent expected-count and covariance formulas across components. Hourly recording can create a shared tied block. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. Inconsistent tie handling corrupts correlation estimates. Use one declared convention.
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 joint p-value controls the planned family as one test under its null approximation. The search is internal to the specified maximum statistic. This is the central mathematical mechanism in type-i error control. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. Reporting the best component p-value alone loses that control. Report the MaxCombo p-value. 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 power trade-offs. MaxCombo usually sacrifices some power versus a perfectly chosen single weight but gains robustness across uncertain patterns. In the illustrative case, A true late-only effect may be detected nearly as well as by its late component. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. No method dominates all alternatives. Compare scenarios by simulation.
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 sample-size planning as a model-checking problem rather than a vocabulary exercise. Design requires event timing, censoring, component correlation and anticipated non-proportional alternatives. Total event count alone may not describe late information. Recompute the example from the raw values and label every unit. Planning under only proportional hazards can misstate power. Simulate accrual and follow-up.
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 component attribution becomes useful only when its mechanism is explicit. The largest component helps describe which timing emphasis drove the statistic but is post-selection information. Consider the worked illustration: A late component may be largest. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. Its unadjusted p-value is not a separate confirmed finding. Treat attribution as descriptive.
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 estimation after the test to practise sensitivity analysis. Report survival differences, RMST, milestone reliability or flexible hazard estimates with intervals. A combined test can be followed by a 1000-hour absolute difference. Before accepting the result, identify the quantity that would change the conclusion most. Post-test effect selection can bias emphasis. Pre-plan effect summaries too.
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 regulatory and engineering transfer. FDA-authored evaluations concern clinical trials; engineering use must redefine events, censoring and mission consequences. The initial example is: A machine failure curve shares mathematics but not treatment context. Then create a nearby case where the usual shortcut becomes unreliable. Regulatory conclusions do not certify an engineering product. Transfer methods, not claims.
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 MaxCombo when timing uncertainty is real and the component set maps to decisions. Detecting any curve difference may trigger diagnostic investigation. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. The test does not identify an optimal maintenance action. Combine with consequence analysis.
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 three correlated-looking weighted z values, take their maximum and compare naive versus joint calibration conceptually. Increase correlation and predict the multiplicity penalty. 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 normal calculations are illustrative. Keep the component set fixed. 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. MaxCombo connects maxima, correlation, normal distributions and multiple-testing logic. In the illustrative case, Ask why choosing the largest of three needs a different cutoff. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. Use three linked thermometers as a model. Separate test choice from effect size.
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. MaxCombo is not the smallest component p-value, not free post-hoc searching and not proof of benefit at every time. Correlation matters to calibration. Recompute the example from the raw values and label every unit. A maximum statistic does not show when curves differ. Report components and effect plots.
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. Karrison developed versatile maximum combinations of weighted log-rank statistics, and an FDA team evaluated MaxCombo under non-proportional hazards. Consider the worked illustration: Reproduce the two-test search probability before software. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. Evaluation scenarios do not replace local design simulation. Read power and type-I error together.
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 combine several timing alternatives
Setup: Use this chapter’s illustrative case: A redesign might help immediately, after a delay or during wear-out.
Task: Choose a compact component set.
Checking target: Including every imaginable weight dilutes planning and complicates interpretation.
Practice case 2: Pre-specified component tests
Setup: Use this chapter’s illustrative case: A common set includes ordinary, early and late weights.
Task: Freeze weights and directions.
Checking target: Adding a favourable component after inspection invalidates the planned calibration.
Practice case 3: Standardise every component
Setup: Use this chapter’s illustrative case: Raw scores on different weight scales cannot be maximised fairly.
Task: Inspect every SE.
Checking target: Small component variance can cause instability.
Practice case 4: Maximum combination statistic
Setup: Use this chapter’s illustrative case: If z values are 1.1,2.0,.7, the one-sided maximum is 2.0.
Task: Use the joint null distribution.
Checking target: The maximum alone is not a valid ordinary-normal test.
Practice case 5: Why component statistics are correlated
Setup: Use this chapter’s illustrative case: Ordinary and late-weighted statistics may correlate strongly.
Task: Estimate the covariance matrix.
Checking target: Bonferroni ignores useful correlation and can be conservative.
Practice case 6: Multivariate-normal calibration
Setup: Use this chapter’s illustrative case: The p-value is the probability that any planned component exceeds the observed maximum.
Task: Record algorithm and tolerances.
Checking target: Numerical integration accuracy matters.
Practice case 7: A worked two-component probability
Setup: Use this chapter’s illustrative case: Positive correlation lowers the search penalty relative to independence.
Task: Use the estimated joint distribution.
Checking target: Real component tests are not independent.
Practice case 8: Typical Fleming–Harrington set
Setup: Use this chapter’s illustrative case: The set spans three plausible timing patterns.
Task: Justify each component.
Checking target: It is not automatically optimal for crossing or middle-only alternatives.
Practice case 9: Including a middle-weighted component
Setup: Use this chapter’s illustrative case: The weight S(1-S) peaks near pooled survival .5.
Task: Add only with design rationale.
Checking target: More components raise the multiplicity penalty and interpretation burden.
Practice case 10: One-sided versus two-sided tests
Setup: Use this chapter’s illustrative case: Reliability superiority may justify a directional alternative only when harm is handled separately.
Task: Declare the direction before data.
Checking target: Selecting the tail after results doubles opportunism.