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eduKateSG · Why Mathematics? · Pseudo-Observation Regression for Reliability: Jackknife Values, Survival Probabilities, Cumulative Incidence and RMST
Start with the question, audit the worked example, or jump to the assumptions and practice. Every chapter stays open for uninterrupted reading.
Understand the question
Define the quantity and follow the mechanism.Route 2 · Chapters 6–10
Reproduce the example
Audit the numbers one step at a time.Route 3 · Chapters 11–15
Check assumptions
See where design and calibration matter.Route 4 · Chapters 16–20
Explore applications
Connect the method to a concrete investigation.Route 5 · Chapters 21–25
Study and verify
Practise, inspect failure cases and read further.
Full chapter index · Practice and checking · Mathematics Learning Hub
Why mathematics is important becomes practical when a reliability question is about an interpretable probability or expected operating time rather than an instantaneous hazard. Pseudo-observation regression turns a censored-data summary—such as survival at 1,000 hours, cumulative incidence of one failure mode, restricted mean operating time, or a state-occupation probability—into jackknife-based subject-level values that can be analysed with regression tools.
This guide develops the leave-one-out construction, links, estimating equations, censoring assumptions, repeated horizons, robust uncertainty and validation. Pseudo-observations are mathematical devices, not observed individual outcomes. They can fall outside the natural probability range, and their regression interpretation depends on the estimand, link, censoring process and working correlation.
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 model a direct reliability summary
Engineers may need survival probability, cumulative incidence or expected restricted time at a decision horizon rather than a hazard ratio. A warranty decision may target probability of operating beyond 1000 hours. This is the central mathematical mechanism in why model a direct reliability summary. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. One horizon cannot describe the whole lifetime curve. Choose the estimand before the model. 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 start with a full-sample estimator. Calculate a censoring-aware estimate theta-hat from all n systems. In the illustrative case, Use a Kaplan–Meier survival estimate at 1000 hours or an Aalen–Johansen state probability. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. A biased starting estimator transfers bias into pseudo-values. Document the base estimator.
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 leave-one-out estimates as a model-checking problem rather than a vocabulary exercise. Recalculate the same estimand after deleting each system in turn, producing theta-hat minus-i. With 50 systems there are 50 leave-one-out estimates. Recompute the example from the raw values and label every unit. Deleting event histories rather than complete systems breaks dependence structure. Remove the whole system record.
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 jackknife pseudo-value formula becomes useful only when its mechanism is explicit. Pseudo_i=n theta-hat-(n-1) theta-hat_minus_i measures how subject i influences the full-sample estimand. Consider the worked illustration: If theta-hat=.80 and a leave-one-out estimate is .79 with n=20, the pseudo-value is .99. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. The value is not an observed survival indicator. Keep the algebra and units visible.
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/90.1.15. Read it for the method and its assumptions; do not transfer its performance claims automatically to another model.
Use why pseudo-values may exceed bounds to practise sensitivity analysis. The jackknife transformation is linear, so a pseudo-value for a probability may be below zero or above one even though the population estimate is bounded. An influential early failure can create a negative survival pseudo-value. Before accepting the result, identify the quantity that would change the conclusion most. Clipping values changes estimating equations and can bias results. Treat them as estimating devices.
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 regression mean model. Specify a link g so g(E[pseudo_i|X_i])=beta0+beta^T X_i for the selected estimand. The initial example is: An identity link models absolute survival-probability differences. Then create a nearby case where the usual shortcut becomes unreliable. The conditional mean relationship requires assumptions and support. Match link to interpretation.
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 identity link for absolute differences to a real decision while protecting the boundary between calculation and judgement. With an identity link, a binary coefficient estimates an adjusted difference on the estimand scale. Beta=.06 can mean six percentage points higher fitted 1000-hour survival. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. Linear predictions can leave zero-to-one bounds. Check the full prediction range.
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.
A logit link respects probability bounds for fitted means, while complementary log-log can align with hazard-scale ideas. A logit coefficient exponentiates to an odds ratio for the chosen survival probability. This is the central mathematical mechanism in logit and complementary log-log links. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. An odds ratio is not a risk ratio or hazard ratio. Translate back to probabilities. 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 generalised estimating equations. GEE combines pseudo-values and covariates while accounting for their correlation through a working covariance and robust sandwich variance. In the illustrative case, Repeated horizons for the same machine form correlated responses. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. Robust variance does not repair a wrong mean model. State the working correlation.
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 jackknife calculation as a model-checking problem rather than a vocabulary exercise. For n=5, theta-hat=.70 and theta-hat_minus_i=.65, pseudo_i=5(.70)-4(.65)=.90. A different deletion can produce a value below the full estimate. Recompute the example from the raw values and label every unit. One pseudo-value has no standalone reliability meaning. Verify all deletions reproduce the full average approximately.
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 survival probability at a fixed horizon becomes useful only when its mechanism is explicit. Kaplan–Meier pseudo-values support direct regression of S(tau) at a planned tau. Consider the worked illustration: Compare adjusted 500-hour survival across operating environments. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. Choosing tau after inspecting separation inflates evidence. Pre-specify practical horizons.
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 restricted mean survival time to practise sensitivity analysis. Integrate the survival curve to tau, then jackknife that RMST estimate for regression in time units. A coefficient of 35 hours represents an adjusted restricted-time difference under the model. Before accepting the result, identify the quantity that would change the conclusion most. It does not describe time beyond tau. Report tau beside every effect.
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/10.1177/0962280209105020. Compare the problem definition, estimator and validation rather than quoting one result in isolation.
Build a small counterexample for competing-risk cumulative incidence. Jackknife an Aalen–Johansen cumulative-incidence estimate for a specified cause and horizon. The initial example is: Model corrosion-first probability by 1500 hours. Then create a nearby case where the usual shortcut becomes unreliable. Competing causes alter the target incidence. Report every cause curve.
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 multi-state occupation probabilities to a real decision while protecting the boundary between calculation and judgement. Pseudo-values can target probability of being healthy, degraded or under repair at a horizon. Model expected repair-state occupancy at month 12. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. Separate state regressions may not sum perfectly to one. Check joint probability coherence.
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.
Stack pseudo-values at several planned horizons and include time interactions. Use 500, 1000 and 1500 hours to show changing covariate effects. This is the central mathematical mechanism in repeated time points. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. Many correlated tests invite noisy stories. Use simultaneous inference or a compact model. 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 censoring assumptions. Standard constructions often need independent censoring, sometimes conditional on included variables. In the illustrative case, Retirement caused by hidden deterioration can violate independence. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. A marginal Kaplan–Meier base estimator may be inconsistent under covariate-dependent censoring. Model censoring or stratify when justified.
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 delayed entry and competing structures as a model-checking problem rather than a vocabulary exercise. The base estimator must correctly handle left truncation, competing events or state transitions before jackknifing. A delayed-entry Aalen–Johansen estimate can seed state-probability pseudo-values. Recompute the example from the raw values and label every unit. Pseudo-values do not automatically fix design bias. Build the correct risk sets 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 robust variance and small samples becomes useful only when its mechanism is explicit. Sandwich variance is asymptotic and may need finite-sample correction or bootstrap checks. Consider the worked illustration: Thirty systems with many covariates can yield unstable standard errors. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. Robust does not mean accurate in every small sample. Use parsimonious models and sensitivity checks.
There are three uncertainties to keep apart. Sampling or numerical uncertainty comes from finite computation. Parameter uncertainty comes from estimating inputs. Model-form uncertainty comes from simplified physics, distributions or dependence. Reducing the first does not automatically reduce the other two. A credible report names all three and shows which one drives the decision.
For a practical investigation, create a small reproducible notebook or spreadsheet with a frozen data sheet, formula sheet, diagnostics sheet and decision note. Another student should be able to change one input and obtain the same update without guessing what the model meant.
Use model diagnostics to practise sensitivity analysis. Compare fitted marginal summaries with nonparametric estimates across covariate groups and inspect residual-style pseudo-value diagnostics. Predicted 1000-hour survival should align with held-out Kaplan–Meier estimates. Before accepting the result, identify the quantity that would change the conclusion most. Training agreement can hide misspecification. Use external or temporal validation.
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 interpretation versus causation. Adjusted pseudo-observation regression estimates associations unless design and causal assumptions support intervention effects. The initial example is: High load may be assigned to stronger equipment, creating confounding. Then create a nearby case where the usual shortcut becomes unreliable. A direct probability coefficient is not automatically causal. Separate prediction from intervention.
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 direct adjusted probabilities or time differences with cost, consequence and uncertainty. A five-point survival gain can be compared with redesign cost. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. The regression does not set acceptable risk. State the decision threshold separately.
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.
Compute one full estimate and five leave-one-out estimates, form pseudo-values and fit an identity-link line. Predict how an early failure changes its pseudo-value. 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 are not product evidence. Check averages and bounds. 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. Pseudo-observations connect averages, algebra, probability, functions and the idea of influence. In the illustrative case, Ask what changes when one system is removed. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. The values can look strange outside zero and one. Use a five-card deletion activity.
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. Pseudo-values are not imputed failure times, not observed binary outcomes and not automatically independent. A bounded estimand can generate unbounded pseudo-values. Recompute the example from the raw values and label every unit. A robust standard error cannot fix informative censoring. Name the estimand and base estimator.
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. Andersen, Klein and Rosthøj developed correlated pseudo-observation regression for multi-state probabilities, and Andersen with Pohar Perme reviewed survival applications. Consider the worked illustration: Reproduce the jackknife formula 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. Method papers do not validate a local censoring process. Read assumptions beside the estimand.
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 model a direct reliability summary
Setup: Use this chapter’s illustrative case: A warranty decision may target probability of operating beyond 1000 hours.
Task: Choose the estimand before the model.
Checking target: One horizon cannot describe the whole lifetime curve.
Practice case 2: Start with a full-sample estimator
Setup: Use this chapter’s illustrative case: Use a Kaplan–Meier survival estimate at 1000 hours or an Aalen–Johansen state probability.
Task: Document the base estimator.
Checking target: A biased starting estimator transfers bias into pseudo-values.
Practice case 3: Leave-one-out estimates
Setup: Use this chapter’s illustrative case: With 50 systems there are 50 leave-one-out estimates.
Task: Remove the whole system record.
Checking target: Deleting event histories rather than complete systems breaks dependence structure.
Practice case 4: Jackknife pseudo-value formula
Setup: Use this chapter’s illustrative case: If theta-hat=.80 and a leave-one-out estimate is .79 with n=20, the pseudo-value is .99.
Task: Keep the algebra and units visible.
Checking target: The value is not an observed survival indicator.
Practice case 5: Why pseudo-values may exceed bounds
Setup: Use this chapter’s illustrative case: An influential early failure can create a negative survival pseudo-value.
Task: Treat them as estimating devices.
Checking target: Clipping values changes estimating equations and can bias results.
Practice case 6: Regression mean model
Setup: Use this chapter’s illustrative case: An identity link models absolute survival-probability differences.
Task: Match link to interpretation.
Checking target: The conditional mean relationship requires assumptions and support.
Practice case 7: Identity link for absolute differences
Setup: Use this chapter’s illustrative case: Beta=.06 can mean six percentage points higher fitted 1000-hour survival.
Task: Check the full prediction range.
Checking target: Linear predictions can leave zero-to-one bounds.
Practice case 8: Logit and complementary log-log links
Setup: Use this chapter’s illustrative case: A logit coefficient exponentiates to an odds ratio for the chosen survival probability.
Task: Translate back to probabilities.
Checking target: An odds ratio is not a risk ratio or hazard ratio.
Practice case 9: Generalised estimating equations
Setup: Use this chapter’s illustrative case: Repeated horizons for the same machine form correlated responses.
Task: State the working correlation.
Checking target: Robust variance does not repair a wrong mean model.
Practice case 10: A worked jackknife calculation
Setup: Use this chapter’s illustrative case: A different deletion can produce a value below the full estimate.
Task: Verify all deletions reproduce the full average approximately.
Checking target: One pseudo-value has no standalone reliability meaning.