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eduKateSG · Why Mathematics? · Kent–O’Quigley Explained Randomness for Reliability Cox Models: Information Gain, Entropy, R² and Censoring
Start with the question, audit the worked example, or jump to the assumptions and practice. Every chapter stays open for uninterrupted reading.
Understand the question
Define the quantity and follow the mechanism.Route 2 · Chapters 6–10
Reproduce the example
Audit the numbers one step at a time.Route 3 · Chapters 11–15
Check assumptions
See where design and calibration matter.Route 4 · Chapters 16–20
Explore applications
Connect the method to a concrete investigation.Route 5 · Chapters 21–25
Study and verify
Practise, inspect failure cases and read further.
Full chapter index · Practice and checking · Mathematics Learning Hub
Why mathematics is important becomes visible when a proportional-hazards model needs an R-squared-like summary but ordinary least-squares variance decomposition does not apply to censored event times. Kent and O’Quigley use Kullback–Leibler information gain to compare a fitted survival model with a no-covariate reference, then transform that gain into a bounded coefficient of explained randomness.
This guide develops entropy, information gain, full and null models, the transformation rho-squared = 1 − exp(−Gamma), partial contributions, censoring and validation. The measure is model-based and definition-specific. It is not ordinary linear-regression R-squared, not a causal percentage and not a guarantee of calibration or useful maintenance decisions.
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 ordinary R-squared does not transfer
Censored event times and proportional hazards do not provide the same sum-of-squares decomposition as normal linear regression. A censored lifetime has no observed squared residual after censoring. This is the central mathematical mechanism in why ordinary r-squared does not transfer. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. Borrowing OLS R-squared labels creates false interpretation. Start from the survival likelihood. 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 randomness and information. Entropy-like quantities describe uncertainty in event time, while covariates may reduce that uncertainty under a fitted model. In the illustrative case, Load and temperature explain some variation in failure timing. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. Uncertainty reduction is model-dependent. Name the distributional assumptions.
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 the no-covariate reference as a model-checking problem rather than a vocabulary exercise. A null survival model represents baseline event-time behaviour without the predictors being evaluated. The baseline hazard alone is the comparison for a Cox model. Recompute the example from the raw values and label every unit. A different reference changes the coefficient. Report the null model.
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 the fitted model becomes useful only when its mechanism is explicit. The full model conditions event-time behaviour on the chosen covariates through its risk structure. Consider the worked illustration: A Cox model includes load, temperature and their interaction. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. More predictors can improve apparent fit by chance. Validate selection and tuning.
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/75.3.525. Read it for the method and its assumptions; do not transfer its performance claims automatically to another model.
Use kullback–leibler information gain to practise sensitivity analysis. Gamma=2{I(beta)-I(0)} compares expected log density under the fitted and null models. If Gamma=.4, the fitted model gains .4 units on this scale. Before accepting the result, identify the quantity that would change the conclusion most. The gain is not a percentage. Keep log bases and conventions fixed.
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 transform to rho-squared. The bounded coefficient is rho^2=1-exp(-Gamma). The initial example is: With Gamma=.4, rho^2=1-exp(-.4)=.32968, about .330. Then create a nearby case where the usual shortcut becomes unreliable. The result does not mean 33 percent of every lifetime is predicted. Call it explained randomness.
Counterexamples are positive learning tools. They reveal the conditions that make a method work and help students recognise when a software output deserves a pause. Keep the counterexample small enough to plot or enumerate. Check sign, support, bounds and conservation of probability before adding complexity.
When writing the conclusion, separate a documented fact from an interpretation and from an illustrative scenario. Cite the primary method paper for the factual mechanism, use the local calculation for the illustration, and label any recommendation as conditional on the stated model. This keeps career and engineering advice age-appropriate and honest.
Connect why the transformation is bounded to a real decision while protecting the boundary between calculation and judgement. For nonnegative information gain, exp(-Gamma) lies between zero and one, so rho-squared lies between zero and one. Gamma=0 gives zero and large Gamma approaches one. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. Finite samples can create awkward adjusted estimates. Report estimation details.
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.
For small Gamma, 1-exp(-Gamma) is close to Gamma because the exponential has first-order slope one. Gamma=.05 gives rho^2=.0488. This is the central mathematical mechanism in small-gain approximation. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. The approximation worsens as Gamma grows. Use the exact formula in reports. 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 scale invariance. Information-gain measures are not changed merely by changing predictor measurement units when the model is reparameterised equivalently. In the illustrative case, Celsius and centred Celsius encode the same fitted relationship. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. Changing functional form is not mere reparameterisation. Distinguish units from model structure.
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.
CHAPTER 10 OF 25 · Reproduce the example
10. Simple, multiple and partial coefficients
Treat simple, multiple and partial coefficients as a model-checking problem rather than a vocabulary exercise. The framework can describe one predictor, a set of predictors or the incremental information beyond an existing set. Measure the added contribution of vibration after load and temperature. Recompute the example from the raw values and label every unit. Partial values depend on the base model. State the nesting order.
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 a partial-gain example becomes useful only when its mechanism is explicit. Incremental explained randomness can compare Gamma for a larger model with the baseline information already present. Consider the worked illustration: Gamma rises from .30 to .40 after adding vibration. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. Subtracting transformed rho values is not identical to transforming only the gain difference. Declare the chosen partial 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 censoring in estimation to practise sensitivity analysis. The survival likelihood and risk sets allow censored units to contribute information while observed. A unit censored at 800 hours remains in earlier risk sets. Before accepting the result, identify the quantity that would change the conclusion most. Informative censoring can bias the model and coefficient. Audit withdrawal mechanisms.
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/10.1002/sim.1946. Compare the problem definition, estimator and validation rather than quoting one result in isolation.
Build a small counterexample for proportional-hazards assumption. The Cox interpretation assumes covariate hazard ratios are sufficiently stable over time. The initial example is: Temperature matters more during wear-out. Then create a nearby case where the usual shortcut becomes unreliable. One coefficient can average incompatible regimes. Check time-varying effects.
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-dependent covariates to a real decision while protecting the boundary between calculation and judgement. Later work extended explained-randomness ideas to time-varying covariates within the semiparametric hazard framework. Weekly vibration updates enter start-stop records. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. Future values must not leak into earlier risk sets. Preserve covariate histories.
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.
An estimated coefficient is a noisy sample approximation to a population information contrast. Two validation fleets give .28 and .35. This is the central mathematical mechanism in sample versus population quantity. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. The larger number is not automatically a real improvement. Report uncertainty. 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 internal validation and optimism. Apparent information gain increases when the same data choose and evaluate predictors. In the illustrative case, Stepwise selection among many sensors inflates rho-squared. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. The transformation does not correct overfitting. Bootstrap the full process.
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 external validation as a model-checking problem rather than a vocabulary exercise. Evaluate the frozen model and reference in a new target population. A factory model is applied to offshore units. Recompute the example from the raw values and label every unit. Population shift changes information and baseline risk. Describe transport conditions.
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 compare with schemper–henderson v becomes useful only when its mechanism is explicit. Kent–O’Quigley is based on information gain, while Schemper–Henderson compares integrated prediction inaccuracy. Consider the worked illustration: The same model can have different values on the two scales. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. They answer related but nonidentical questions. Do not rank them by numerical size.
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 compare with r2-d to practise sensitivity analysis. Royston–Sauerbrei R2-D transforms prognostic separation, whereas explained randomness transforms information gain. A model may separate groups well yet leave much event-time uncertainty. Before accepting the result, identify the quantity that would change the conclusion most. Both differ from OLS R-squared. Name the construction.
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 calibration and discrimination. Explained randomness summarises model dependence but does not alone show calibrated probabilities or pair ordering. The initial example is: A model can gain information while miscalibrated in a new fleet. Then create a nearby case where the usual shortcut becomes unreliable. No scalar diagnoses every defect. Use a performance panel.
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. A higher coefficient does not state inspection cost, safety consequence or operational capacity. Rho-squared=.33 does not choose a threshold. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. It has no currency scale. Link models to actions 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.
Calculate rho-squared for Gamma=.4 and .05, then compare exact values with the small-gain approximation. The results are .3297 and .0488. This is the central mathematical mechanism in practice calculation. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. Do not report Gamma itself as R-squared. Show the exponential step. 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. The method connects logarithms, exponentials, probability and information to the question of how much a model reduces uncertainty. In the illustrative case, Ask why doubling Gamma does not double rho-squared. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. Students may treat all R-squared labels as identical. Attach each formula to its definition.
A student should be able to rebuild the result without hidden settings. Record the sampling rule or optimisation criterion, the exact failure definition, and the evidence used for every distribution or parameter. Compare a transparent baseline with the advanced method. If they disagree, investigate the mechanism before deciding which answer is more credible.
This chapter belongs to a larger decision, so translate the number carefully. State one action the evidence permits and one action it does not. A parent or educator can ask which school ideas are doing the work: algebra organises relationships, probability represents uncertainty, calculus describes local change, and computing repeats the documented rules at scale.
Treat common misconceptions as a model-checking problem rather than a vocabulary exercise. Explained randomness is not a causal share, not ordinary variance explained and not comparable across different references without care. A high value can still coexist with poor field calibration. Recompute the example from the raw values and label every unit. Censoring assumptions remain active. Report model, reference and population.
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. Kent and O’Quigley introduced information-gain dependence measures for censored survival data, and O’Quigley, Xu and Stare developed the explained-randomness interpretation. Consider the worked illustration: Read both papers before choosing an implementation. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. The exact coefficient and estimator must be named. Reproduce Gamma=.4 first.
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 ordinary R-squared does not transfer
Setup: Use this chapter’s illustrative case: A censored lifetime has no observed squared residual after censoring.
Task: Start from the survival likelihood.
Checking target: Borrowing OLS R-squared labels creates false interpretation.
Practice case 2: Randomness and information
Setup: Use this chapter’s illustrative case: Load and temperature explain some variation in failure timing.
Task: Name the distributional assumptions.
Checking target: Uncertainty reduction is model-dependent.
Practice case 3: The no-covariate reference
Setup: Use this chapter’s illustrative case: The baseline hazard alone is the comparison for a Cox model.
Task: Report the null model.
Checking target: A different reference changes the coefficient.
Practice case 4: The fitted model
Setup: Use this chapter’s illustrative case: A Cox model includes load, temperature and their interaction.
Task: Validate selection and tuning.
Checking target: More predictors can improve apparent fit by chance.
Practice case 5: Kullback–Leibler information gain
Setup: Use this chapter’s illustrative case: If Gamma=.4, the fitted model gains .4 units on this scale.
Task: Keep log bases and conventions fixed.
Checking target: The gain is not a percentage.
Practice case 6: Transform to rho-squared
Setup: Use this chapter’s illustrative case: With Gamma=.4, rho^2=1-exp(-.4)=.32968, about .330.
Task: Call it explained randomness.
Checking target: The result does not mean 33 percent of every lifetime is predicted.
Practice case 7: Why the transformation is bounded
Setup: Use this chapter’s illustrative case: Gamma=0 gives zero and large Gamma approaches one.
Task: Report estimation details.
Checking target: Finite samples can create awkward adjusted estimates.
Practice case 8: Small-gain approximation
Setup: Use this chapter’s illustrative case: Gamma=.05 gives rho^2=.0488.
Task: Use the exact formula in reports.
Checking target: The approximation worsens as Gamma grows.
Practice case 9: Scale invariance
Setup: Use this chapter’s illustrative case: Celsius and centred Celsius encode the same fitted relationship.
Task: Distinguish units from model structure.
Checking target: Changing functional form is not mere reparameterisation.
Practice case 10: Simple, multiple and partial coefficients
Setup: Use this chapter’s illustrative case: Measure the added contribution of vibration after load and temperature.
Task: State the nesting order.
Checking target: Partial values depend on the base model.