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eduKateSG · Why Mathematics? · Inverse Probability of Censoring Weighting for Reliability: Censoring Models, Weights, Positivity and Diagnostics
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 systems that remain under observation are not a fair sample of the original fleet. Inverse probability of censoring weighting, or IPCW, estimates the chance of still being observed and gives more weight to comparable histories that remain visible. The weighted sample can recover a target quantity when censoring is conditionally independent given the measured history and every relevant history has a positive chance of continued observation.
This guide develops censoring survival functions, stabilised weights, time-varying histories, weighted estimating equations, positivity, truncation and diagnostics. IPCW cannot recover information from groups with zero follow-up, and a sophisticated weight model cannot remove bias from unmeasured causes of both censoring and failure.
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 censoring can bias reliability evidence
If withdrawal depends on deterioration, observed survivors differ systematically from withdrawn systems. Machines with rising vibration may be retired before failure is recorded. This is the central mathematical mechanism in why censoring can bias reliability evidence. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. Ordinary independent-censoring methods may then misrepresent risk. Record why and when observation ends. 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 define the censoring time. Let C be the last time a system could provide the outcome under the observation plan. In the illustrative case, A contract ending at 800 hours creates administrative censoring. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. Failure, repair and censoring are different events. Write an explicit censoring rule.
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 censoring survival function as a model-checking problem rather than a vocabulary exercise. G(t|H)=P(C>=t|H) is the conditional probability of remaining observable through t given measured history H. A fitted G=.60 means 60% of comparable histories remain observed. Recompute the example from the raw values and label every unit. It is not equipment survival. Label observation survival separately.
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 basic inverse weight becomes useful only when its mechanism is explicit. A retained observation receives weight 1/G so it represents itself and comparable histories lost earlier. Consider the worked illustration: If G=.50, a visible record receives weight 2. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. Large weights increase variance and sensitivity. Inspect the entire weight 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/10.1111/j.0006-341X.2000.00779.x. Read it for the method and its assumptions; do not transfer its performance claims automatically to another model.
Use a worked weight calculation to practise sensitivity analysis. For G=.8, .5 and .25, inverse weights are 1.25, 2 and 4. The last record has four times unit influence before stabilisation. Before accepting the result, identify the quantity that would change the conclusion most. Arithmetic correctness does not establish a correct censoring model. Link each probability to measured history.
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 stabilised weights. Multiply by a simpler numerator probability to reduce variance while preserving the intended contrast under conditions. The initial example is: A marginal numerator divided by conditional G can moderate extremes. Then create a nearby case where the usual shortcut becomes unreliable. Stabilisation changes scale and interpretation of totals. Report numerator and denominator models.
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 time-varying censoring models to a real decision while protecting the boundary between calculation and judgement. Update G using information known before each interval, such as sensor readings and maintenance status. Vibration measured at hour 400 can predict observation from 400 to 500. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. Using future measurements creates leakage. Use start–stop data and lagged history.
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.
Remaining observed across many intervals has probability equal to a product of interval-specific retention probabilities. Probabilities .9, .8 and .7 yield G=.504. This is the central mathematical mechanism in product of conditional probabilities. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. Multiplication can create very small G over long follow-up. Compute on log scale when needed. 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 weighted risk sets. At each event time, IPC weights modify contributions to the risk set or estimating equation. In the illustrative case, A system with weight 1.8 contributes more than one nominal unit. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. Weights do not create actual extra systems. Use effective sample size diagnostics.
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 ipcw log-rank logic as a model-checking problem rather than a vocabulary exercise. A weighted log-rank comparison balances observed and expected events after correcting for modeled censoring. Robins and Finkelstein used IPCW in a censored time-to-event comparison. Recompute the example from the raw values and label every unit. The result depends on the observation and treatment models. Report unweighted and weighted analyses.
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 ipcw prediction scores becomes useful only when its mechanism is explicit. Brier scores under censoring weight observable event and survival contributions by inverse censoring probabilities. Consider the worked illustration: A system known failed before t and a system known event-free beyond t receive different denominators. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. Wrong G biases model evaluation. Use the same horizon and event definition.
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 conditional exchangeability for censoring to practise sensitivity analysis. Given measured history, future failure must be independent of censoring for the weighted identification argument. Conditioning on current degradation may make retirement comparable within strata. Before accepting the result, identify the quantity that would change the conclusion most. Unmeasured damage can violate the assumption. List plausible common causes.
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://www.jmlr.org/papers/volume24/19-1030/19-1030.pdf. Compare the problem definition, estimator and validation rather than quoting one result in isolation.
Build a small counterexample for positivity. Every history relevant to the target must have a nonzero probability of remaining observed. The initial example is: If all severely degraded units are immediately removed, their later failure risk is unidentified. Then create a nearby case where the usual shortcut becomes unreliable. No weight can recover a zero-probability path. Restrict the target or redesign follow-up.
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 correct censoring-model specification to a real decision while protecting the boundary between calculation and judgement. The model for G needs appropriate covariates, functional forms and time structure. Nonlinear temperature effects may need splines. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. Adding variables blindly can worsen instability. Use subject knowledge and diagnostics.
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.
Capping extreme weights trades some bias for lower variance and numerical stability. Limit weights above the 99th percentile in a sensitivity analysis. This is the central mathematical mechanism in weight truncation. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. Truncation is not a hidden cleaning step. Report thresholds and alternatives. 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 effective sample size. A common diagnostic is (sum w)^2/sum w^2, showing how unequal weights reduce information. In the illustrative case, One huge weight can make 100 records behave like far fewer. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. This summary does not diagnose model correctness. Pair it with covariate balance.
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 weight and covariate balance as a model-checking problem rather than a vocabulary exercise. After weighting, measured histories among retained records should represent the intended risk set. Compare weighted sensor distributions across censoring patterns. Recompute the example from the raw values and label every unit. Balance on measured variables cannot prove exchangeability. Plot standardised differences over time.
The strongest check uses evidence the model did not choose for itself. Hold out later observations, an independent simulation set or a known-answer benchmark. Report errors where decisions are sensitive, especially near the failure boundary, instead of relying on one global accuracy score. A method that performs well on easy safe cases can still fail at the rare cases that motivated the analysis.
Now change one assumption and make a directional prediction before running the calculation. If the result moves the opposite way, inspect the sign convention, transformation, density ratio or constraint. This prediction-first routine develops transferable problem-solving skills because it connects symbolic work to a testable expectation.
The mathematics of variance estimation becomes useful only when its mechanism is explicit. Robust sandwich, bootstrap or influence-function methods should account for estimated weights and repeated contributions. Consider the worked illustration: Treating weights as fixed can understate uncertainty. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. Naive bootstrapping may break clustered histories. Resample whole systems.
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.
CHAPTER 19 OF 25 · Explore applications
19. Sensitivity to unmeasured censoring causes
Use sensitivity to unmeasured censoring causes to practise sensitivity analysis. Vary assumptions about an omitted degradation factor or compare bounds when possible. Ask how strong hidden damage would need to be to reverse the conclusion. Before accepting the result, identify the quantity that would change the conclusion most. Sensitivity analysis does not identify the truth. Report the range of conclusions.
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 validation and transport. A censoring model fitted in one service contract may not describe another. The initial example is: Different sites may retire equipment under different thresholds. Then create a nearby case where the usual shortcut becomes unreliable. Weights can fail under policy shift. Re-estimate and validate G locally.
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 weighted estimates only within the supported target population and follow-up range. Weighted 1000-hour failure risk can inform monitoring policy if positivity holds. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. IPCW does not set a maintenance rule. Combine with consequences and cost.
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.
Create interval retention probabilities, multiply them into G, compute weights and examine truncation. Predict which history receives the largest weight. 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 weights cannot validate real censoring. Check positivity first. 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. IPCW connects fractions, conditional probability, products and fair representation. In the illustrative case, Ask why a rare visible history represents more lost histories. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. Avoid presenting a weight as a duplicated machine. Use coloured observation cards.
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. IPCW is not imputation, large weights are not more trustworthy records and weighting cannot fix unmeasured confounding or zero follow-up. A correct outcome model does not guarantee a correct censoring model. Recompute the example from the raw values and label every unit. Stabilisation does not remove positivity problems. Audit G before the result.
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. Robins and Finkelstein developed IPC-weighted survival comparison under dependent censoring, and later prediction work formalised IPCW scores. Consider the worked illustration: Reproduce a three-interval product before using software. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. The method requires measured-history support. Read assumptions beside every weight.
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 censoring can bias reliability evidence
Setup: Use this chapter’s illustrative case: Machines with rising vibration may be retired before failure is recorded.
Task: Record why and when observation ends.
Checking target: Ordinary independent-censoring methods may then misrepresent risk.
Practice case 2: Define the censoring time
Setup: Use this chapter’s illustrative case: A contract ending at 800 hours creates administrative censoring.
Task: Write an explicit censoring rule.
Checking target: Failure, repair and censoring are different events.
Practice case 3: Censoring survival function
Setup: Use this chapter’s illustrative case: A fitted G=.60 means 60% of comparable histories remain observed.
Task: Label observation survival separately.
Checking target: It is not equipment survival.
Practice case 4: Basic inverse weight
Setup: Use this chapter’s illustrative case: If G=.50, a visible record receives weight 2.
Task: Inspect the entire weight distribution.
Checking target: Large weights increase variance and sensitivity.
Practice case 5: A worked weight calculation
Setup: Use this chapter’s illustrative case: The last record has four times unit influence before stabilisation.
Task: Link each probability to measured history.
Checking target: Arithmetic correctness does not establish a correct censoring model.
Practice case 6: Stabilised weights
Setup: Use this chapter’s illustrative case: A marginal numerator divided by conditional G can moderate extremes.
Task: Report numerator and denominator models.
Checking target: Stabilisation changes scale and interpretation of totals.
Practice case 7: Time-varying censoring models
Setup: Use this chapter’s illustrative case: Vibration measured at hour 400 can predict observation from 400 to 500.
Task: Use start–stop data and lagged history.
Checking target: Using future measurements creates leakage.
Practice case 8: Product of conditional probabilities
Setup: Use this chapter’s illustrative case: Probabilities .9, .8 and .7 yield G=.504.
Task: Compute on log scale when needed.
Checking target: Multiplication can create very small G over long follow-up.
Practice case 9: Weighted risk sets
Setup: Use this chapter’s illustrative case: A system with weight 1.8 contributes more than one nominal unit.
Task: Use effective sample size diagnostics.
Checking target: Weights do not create actual extra systems.
Practice case 10: IPCW log-rank logic
Setup: Use this chapter’s illustrative case: Robins and Finkelstein used IPCW in a censored time-to-event comparison.
Task: Report unweighted and weighted analyses.
Checking target: The result depends on the observation and treatment models.