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eduKateSG · Why Mathematics? · Grønnesby–Borgan Goodness-of-Fit Tests for Reliability Cox Models: Risk-Score Groups, Martingale Residuals, Chi-Square Checks and Calibration
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 Cox reliability model assigns risk scores to components but the grouped predictions do not match the failures that actually accumulate. The Grønnesby–Borgan method orders systems by fitted risk score, forms groups and studies grouped martingale-residual processes. It turns many difficult individual residuals into a graphical and formal overall goodness-of-fit check.
This guide develops risk-score grouping, observed and expected event accumulation, covariance adjustment, chi-square reasoning, group sensitivity and interpretation. A rejection says the fitted model is inconsistent with the grouped event experience; it does not identify one unique physical cause or guarantee that a particular repair is correct. The useful response is to inspect group support, covariate form, proportionality, dependence and validation data.
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
A coefficient table can look plausible while predicted risk is systematically too high for some risk ranges and too low for others. A compressor model ranks high-temperature units correctly but underpredicts failures in the middle-risk group. This is the central mathematical mechanism in why an overall fit test is useful. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. Good coefficient signs do not establish calibration. Check fitted risk against grouped event experience. 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 the cox risk score. The linear predictor eta=x beta and relative-risk score exp(eta) order units by modelled hazard multiplier. In the illustrative case, Scores .5, 1, 2 and 4 place four units from lower to higher fitted relative risk. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. A risk score is not an absolute failure probability. Keep ranking and calibration questions separate.
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 sort systems before grouping as a model-checking problem rather than a vocabulary exercise. The method orders observations by fitted risk score so each group represents a neighbouring range of modelled risk. One hundred systems can be split into five groups of about twenty after sorting. Recompute the example from the raw values and label every unit. Ties and unequal group sizes can affect boundaries. Record the sorting and cut rules.
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 group formation becomes useful only when its mechanism is explicit. Pre-specified quantile groups make the comparison interpretable and stop the data analyst from choosing cut points that maximise disagreement. Consider the worked illustration: Quintiles create five groups with similar counts but not necessarily similar event totals. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. Too many groups produce sparse cells; too few can hide local misfit. Choose groups before viewing residual departures.
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/10.1007/BF00127305. Read it for the method and its assumptions; do not transfer its performance claims automatically to another model.
Use martingale residual foundation to practise sensitivity analysis. For each unit the martingale residual is observed event indicator minus fitted cumulative hazard at its observed time. A failure with fitted cumulative hazard .4 contributes .6; a censored unit at .3 contributes -.3. Before accepting the result, identify the quantity that would change the conclusion most. Residuals depend on the fitted model and censoring history. Reproduce delta minus cumulative hazard.
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 grouped residual totals. Summing martingale residuals within a risk-score group compares observed event accumulation with model-expected accumulation for that group. The initial example is: Group totals 1.2, -.8, .3 and -.7 sum to zero in a toy balanced example. Then create a nearby case where the usual shortcut becomes unreliable. Zero overall sum can conceal offsetting group errors. Inspect each group and the joint pattern.
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 observed and expected events to a real decision while protecting the boundary between calculation and judgement. The grouped residual can be read as observed failures minus expected cumulative events under the model, with risk-set structure retained in the formal process. Twelve observed events against expected 9.5 give a positive discrepancy of 2.5. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. Expected accumulation is not simply group size times one probability when follow-up differs. Use individual cumulative hazards.
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.
Grouped residual totals are correlated because all coefficients and the baseline hazard were estimated from the same risk sets. A positive discrepancy in one group can be paired with a compensating negative discrepancy elsewhere. This is the central mathematical mechanism in why covariance matters. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. Adding squared residuals without covariance gives the wrong reference distribution. Use the derived covariance matrix. 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 quadratic-form statistic. A goodness-of-fit statistic takes the grouped discrepancy vector and scales it by an estimated covariance inverse, producing a chi-square-type measure under regularity conditions. In the illustrative case, With identity covariance only as a classroom simplification, residuals 1.2,-.8,.3,-.7 give 2.66. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. Real covariance is not the identity and may be singular after constraints. Let validated software handle the covariance.
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 degrees of freedom as a model-checking problem rather than a vocabulary exercise. The reference degrees of freedom depend on the number of effective groups and parameter-estimation constraints in the chosen implementation. Five groups do not automatically mean five independent discrepancies. Recompute the example from the raw values and label every unit. Quoting the wrong degrees of freedom changes the p-value. Report implementation and reference distribution.
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 graphical grouped process becomes useful only when its mechanism is explicit. Plotting cumulative grouped residual processes can reveal where low-, middle- or high-risk groups diverge over follow-up. Consider the worked illustration: A high-risk group that drifts upward after 800 hours suggests underpredicted late failures there. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. Visual selection of the most dramatic interval is exploratory. Show simultaneous or simulated reference bands where available.
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 right censoring to practise sensitivity analysis. The Cox fit and martingale residuals rely on risk sets that correctly retain censored systems until their censoring times. Administrative closure at 1,000 hours contributes exposure without an observed failure. Before accepting the result, identify the quantity that would change the conclusion most. Informative withdrawal can create apparent misfit or mask it. Audit censoring mechanisms separately.
A simple baseline is essential. It reveals whether the advanced method is improving accuracy or only adding complexity. Compare model calls, wall time, uncertainty and failure-mode coverage under one declared budget. Repeat stochastic calculations with several seeds; a favourable single run is not evidence of dependable efficiency.
Students can extend the example by adding correlation, a second failure mode or a modest measurement error. Explain which part of the formula changes and which remains fixed. This exercise shows how mathematics transfers from a clean textbook case to engineering and computing without pretending that the classroom model is a certified design tool.
A second primary source is https://doi.org/10.1093/biomet/77.1.147. Compare the problem definition, estimator and validation rather than quoting one result in isolation.
Build a small counterexample for tied failures. Daily or inspection-based recording creates tied failures that require a consistent Breslow, Efron or exact partial-likelihood convention. The initial example is: Six units discovered failed at one weekly inspection share a recorded time. Then create a nearby case where the usual shortcut becomes unreliable. Changing tie handling between fitting and diagnostics corrupts the comparison. Use one documented tie method.
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. For changing loads or temperatures, risk scores and expected event accumulation must use values current over each start-stop interval. A vibration level changes after maintenance and alters later risk sets. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. Using a final value leaks future information. Store complete 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.
Grouped risk-score misfit can arise when a continuous predictor needs a nonlinear term or two stresses interact. A linear temperature term can underfit both cool and very hot systems. This is the central mathematical mechanism in nonlinearity and omitted interactions. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. The omnibus test does not select the spline or interaction. Use scientific reasoning for targeted repairs. 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 proportional-hazards violations. A constant coefficient that changes with time can create grouped departures even if the baseline ranking looks reasonable. In the illustrative case, A coating helps during burn-in but not during wear-out. Draw the relevant graph, density, boundary or timeline before calculating; the picture should use the same variables and sign convention as the equations. One grouped test cannot replace Schoenfeld-residual timing checks. Combine overall and component 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 group-number sensitivity as a model-checking problem rather than a vocabulary exercise. Reasonable choices such as four, five or ten groups may have different power against different patterns. A narrow middle-risk defect can disappear when merged into four broad groups. Recompute the example from the raw values and label every unit. Trying many groupings and reporting only the smallest p-value inflates evidence. Pre-specify one primary grouping and label sensitivity 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 sparse groups and tail risk becomes useful only when its mechanism is explicit. Extreme risk-score groups may contain few failures or little covariate variation, weakening asymptotic calibration. Consider the worked illustration: The highest decile has three systems and one failure. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. A large residual there can be unstable. Report counts, events and follow-up by group.
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 external validation to practise sensitivity analysis. Apply the fitted model and planned grouping rule to a new fleet so calibration evidence is not optimistically reused from model development. A model built in chamber A is checked in chamber B. Before accepting the result, identify the quantity that would change the conclusion most. New fleets may have shifted predictor distributions. Recalibrate only after documenting transport differences.
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 from rejection to repair. A disciplined sequence checks data, time origin, censoring, covariate transformations, interactions, proportionality and dependence before refitting. The initial example is: A unit-conversion error can create a dramatic high-risk departure. Then create a nearby case where the usual shortcut becomes unreliable. A flexible repair can overfit the same residual pattern. Preserve a validation set.
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. Grouped misfit matters when a maintenance or warranty decision relies on predicted hazard across the same risk ranges. Underprediction in the highest risk group can delay inspection for the units needing it most. First verify the arithmetic, then translate it into an allowed action and an action the evidence does not support. The test does not set the inspection threshold. Combine calibration with consequences and capacity.
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 four risk-score groups, sum toy martingale residuals, calculate a classroom quadratic form under identity covariance and explain why production analysis needs the real covariance. Use residuals 1.2,-.8,.3,-.7 and verify 2.66. This is the central mathematical mechanism in practice activity. Write the quantities and assumptions before using software, then reproduce the calculation or geometric argument in a small example. The toy calculation is not a valid chi-square test. Separate arithmetic from inferential calibration. 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 sorting, grouping, expected values, residuals, covariance and the question of whether a rule works equally well across a range. In the illustrative case, Ask why opposite group errors can cancel overall. 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 think one average proves every subgroup fits. Use grouped classroom predictions as an analogy.
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. A non-significant test does not prove the model, a significant test does not identify causation, risk-score groups are not natural engineering classes and group counts are not free tuning parameters. Censoring and sparse tails still matter. Recompute the example from the raw values and label every unit. The formal statistic is joint, not a list of independent cells. Report grouping, support and follow-up diagnostics.
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. Grønnesby and Borgan developed grouped martingale-residual processes for checking Cox and Aalen regression models, building on martingale-based survival diagnostics. Consider the worked illustration: Read the formal covariance and simulation study beside the original martingale residual paper. Preserve the original inputs, calculate step by step and compare with a second method whenever an analytic or high-budget reference exists. Asymptotic validity needs adequate events and correctly constructed risk sets. Reproduce grouped observed-minus-expected totals 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 an overall fit test is useful
Setup: Use this chapter’s illustrative case: A compressor model ranks high-temperature units correctly but underpredicts failures in the middle-risk group.
Task: Check fitted risk against grouped event experience.
Checking target: Good coefficient signs do not establish calibration.
Practice case 2: The Cox risk score
Setup: Use this chapter’s illustrative case: Scores .5, 1, 2 and 4 place four units from lower to higher fitted relative risk.
Task: Keep ranking and calibration questions separate.
Checking target: A risk score is not an absolute failure probability.
Practice case 3: Sort systems before grouping
Setup: Use this chapter’s illustrative case: One hundred systems can be split into five groups of about twenty after sorting.
Task: Record the sorting and cut rules.
Checking target: Ties and unequal group sizes can affect boundaries.
Practice case 4: Group formation
Setup: Use this chapter’s illustrative case: Quintiles create five groups with similar counts but not necessarily similar event totals.
Task: Choose groups before viewing residual departures.
Checking target: Too many groups produce sparse cells; too few can hide local misfit.
Practice case 5: Martingale residual foundation
Setup: Use this chapter’s illustrative case: A failure with fitted cumulative hazard .4 contributes .6; a censored unit at .3 contributes -.3.
Task: Reproduce delta minus cumulative hazard.
Checking target: Residuals depend on the fitted model and censoring history.
Practice case 6: Grouped residual totals
Setup: Use this chapter’s illustrative case: Group totals 1.2, -.8, .3 and -.7 sum to zero in a toy balanced example.
Task: Inspect each group and the joint pattern.
Checking target: Zero overall sum can conceal offsetting group errors.
Practice case 7: Observed and expected events
Setup: Use this chapter’s illustrative case: Twelve observed events against expected 9.5 give a positive discrepancy of 2.5.
Task: Use individual cumulative hazards.
Checking target: Expected accumulation is not simply group size times one probability when follow-up differs.
Practice case 8: Why covariance matters
Setup: Use this chapter’s illustrative case: A positive discrepancy in one group can be paired with a compensating negative discrepancy elsewhere.
Task: Use the derived covariance matrix.
Checking target: Adding squared residuals without covariance gives the wrong reference distribution.
Practice case 9: Quadratic-form statistic
Setup: Use this chapter’s illustrative case: With identity covariance only as a classroom simplification, residuals 1.2,-.8,.3,-.7 give 2.66.
Task: Let validated software handle the covariance.
Checking target: Real covariance is not the identity and may be singular after constraints.
Practice case 10: Degrees of freedom
Setup: Use this chapter’s illustrative case: Five groups do not automatically mean five independent discrepancies.
Task: Report implementation and reference distribution.
Checking target: Quoting the wrong degrees of freedom changes the p-value.