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eduKateSG · Why Mathematics? · Interval-Censored Reliability: Inspection Windows, Turnbull Estimation, Likelihoods and Uncertainty
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 10–18
Reproduce the example
Audit the numbers one step at a time.Route 3 · Chapters 19–27
Check assumptions
See where design and calibration matter.Route 4 · Chapters 28–36
Explore applications
Connect the method to a concrete investigation.Route 5 · Chapters 37–44
Study and verify
Practise, inspect failure cases and read further.
Full chapter index · Practice and checking · Mathematics Learning Hub
Why is mathematics important in interval-censored reliability? Periodic inspections often reveal only that a failure happened between two visits. Treating the midpoint as the event time invents precision. Interval likelihood and Turnbull estimation use the observation window honestly.
This guide develops exact, left, right and interval contributions, inspection design, nonparametric estimation and checked examples. Real decisions require the actual visit process, detection quality, uncertainty and domain review.
Full chapter index
Chapters 1–9 · Understand the question
Chapters 10–18 · Reproduce the example
Chapters 19–27 · Check assumptions
Chapters 28–36 · Explore applications
Chapters 37–44 · Study and verify
A failure is interval censored when the exact time is unknown but lies between two inspections. A device working at day 40 and failed at day 60 has T in (40,60]. The interval is observed evidence; the midpoint is only an assumption.
The monitoring schedule determines interval width. More frequent inspection narrows timing uncertainty but costs time and resources. Reliability analysis must preserve the inspection process rather than pretending failures were continuously observed.
Exact failure records T=t. Left-censored records show T≤R. Right-censored records show T>L. Interval-censored records show L<T≤R. A single dataset can contain all four, and each has a different likelihood contribution.
For lifetime distribution F and survival S, P(L<T≤R)=F(R)-F(L)=S(L)-S(R). This probability, not a density at the midpoint, enters the likelihood.
Let λ=0.02 per day and observe T in (40,60]. S(40)=exp(-0.8)=0.4493 and S(60)=exp(-1.2)=0.3010. The interval probability is 0.1481.
| Observation | Likelihood contribution |
|---|---|
| T in (40,60] | 0.1481 |
| T at or before 20 | 0.3297 |
| T after 100 | 0.1353 |
If failure is already present at the first inspection at day 20, the contribution is F(20)=1-exp(-0.4)=0.3297 under λ=0.02/day. This does not mean failure occurred at day 20.
If a unit still works at day 100, its contribution is S(100)=exp(-2)=0.1353 under the same model. Right censoring provides lower-bound information about lifetime.
An exact continuous failure contributes density f(t), while an increasingly narrow interval contributes approximately f(t) times interval width. They are related but not interchangeable in finite recorded windows.
With one inspection per unit, we know only whether failure occurred before that inspection. This case-1 interval censoring can estimate a distribution when inspection times vary, but each unit supplies coarse information.
With repeated inspections, failure may be bracketed between visits. This is often called case-2 interval censoring. The full visit history determines L and R; after failure detection, later visits do not narrow the original event without more evidence.
Turnbull’s nonparametric maximum-likelihood estimator distributes probability over intervals compatible with observations. It generalises the empirical and Kaplan–Meier ideas to grouped, censored and truncated data.
Turnbull’s algorithm alternates between allocating probability to compatible support intervals and updating their masses. Likelihood should not decrease. Multiple distributions can share the same maximum when data are coarse.
The estimator places mass on Turnbull intervals formed from observation endpoints, not at every arbitrary time. Plotting a smooth curve without labelling smoothing can imply precision the data do not contain.
Interval-censored data may identify fitted probabilities but not a unique allocation inside broad gaps. Report ranges or equivalence classes when relevant. Software output with many decimals does not create exact failure times.
| Type | Known information |
|---|---|
| Exact | T=t |
| Left censored | T<=R |
| Right censored | T>L |
| Interval censored | L<T<=R |
Weibull, lognormal and other lifetime families use interval probabilities in a full likelihood. Parametric shape can improve precision and extrapolation but adds assumptions. Compare diagnostics and sensitivity.
The Weibull reliability article develops hazard and censored likelihood. For interval data, replace exact failure densities with probability between bounds.
Covariates can enter proportional-hazards, proportional-odds or accelerated-failure-time models designed for interval censoring. Do not feed interval midpoints into ordinary Cox software and call the result interval-censored analysis.
Many methods assume inspection timing is independent of failure conditional on modelled information. If warning signs trigger early inspections, intervals are informative. Include the inspection process or perform sensitivity.
Fixed visits can make interval widths predictable and easier to model. Missed visits, early checks and repairs between visits break the schedule. Preserve actual visit dates, not only planned dates.
For repairable systems, more than one event can occur inside one inspection interval. A binary failed/not-failed record cannot recover event count or order. Use grouped-count or recurrent-event models when supported.
If failure cause is learned only at inspection, cause and time are jointly coarsened. Cause-specific interval-censored methods may be needed. Do not treat other causes as ordinary exact censoring.
A unit included only if it survives to entry has truncation in addition to interval censoring. The likelihood conditions on inclusion. Turnbull’s original framework and later corrections address grouped, censored and truncated observations.
Censoring keeps a unit in the sample with incomplete time. Truncation means some units never appear because their times lie outside an inclusion window. The distinction changes denominators and likelihood.
Manufacturing lots can have different reliability. Broad intervals and sparse failures can hide this variation. Use batch covariates or hierarchical models and validate on unseen lots.
A test can miss a failure or falsely declare one. Standard interval censoring assumes status at inspections is correct. Add sensitivity and specificity when detection is imperfect.
Choose visit times to balance cost and information. Equal spacing is simple, but adaptive or quantile-based schedules may target periods of rapid failure accumulation. Planning should be done before seeing outcomes.
Narrower intervals improve timing information only until measurement and operational noise dominate. Inspections may themselves alter failure risk or remove units. Include intervention effects.
Use likelihood intervals, bootstrap methods that respect the observation scheme or Bayesian posterior intervals. Pointwise bands and simultaneous bands answer different questions. Broad inspection windows should widen honest uncertainty.
Compare predicted probabilities for inspection intervals with observed status transitions. Use residual methods designed for interval censoring, posterior predictive checks and parametric sensitivity. Ordinary exact-time residuals may mislead.
Generate true lifetimes, apply the actual inspection schedule, convert times to intervals and fit the model. This checks whether the method recovers reliability under the same coarsening process.
Interval probabilities S(L)-S(R) can suffer cancellation when bounds are close or tails are small. Use stable log-difference calculations and verify against high-precision examples.
Store one row per unit with lower and upper bounds, using clear conventions for zero and infinity. Keep original inspection records separately. Document whether endpoints are open or closed.
Reproduce 0.1481, 0.3297 and 0.1353. Confirm every unit has L<R, exact-time encodings are supported, likelihood increases during iteration and fitted probabilities sum to one.
Hold out later batches or inspection rounds. Compare predicted failures by interval, calibration at mission times and sensitivity to distribution family and visit process.
Calculate S(40)-S(60) for λ=0.02/day. Then compare midpoint 50 with the full interval evidence. Explain why one guessed time discards uncertainty.
Hide an alarm clock behind a screen and look only every ten minutes. The exact ring time is unknown, but a bracket remains. Ask what happens when checks are farther apart.
CHAPTER 37 OF 44 · Study and verify
37. Common misconception: midpoint imputation is neutral
Midpoints can bias shape, hazard and regression, especially when intervals differ or inspections are informative. Use interval likelihood whenever possible.
CHAPTER 38 OF 44 · Study and verify
38. Common misconception: interval censoring is missing data
The bounds are data. They restrict the possible lifetime and contribute probability. Missing the exact time does not mean the record is useless.
A nonparametric maximum-likelihood curve can place mass only where the collection of intervals allows it. The observation schedule helps determine what is statistically visible.
Ordinary Kaplan–Meier handles exact failures with right censoring, not general interval censoring. Turnbull-type estimators are the nonparametric analogue.
CHAPTER 41 OF 44 · Study and verify
41. FAQ: should inspection times be optimised after fitting?
Use the fitted model as one input, but account for parameter uncertainty, detection quality, cost and operational constraints. Revalidate after schedule changes.
Report endpoint convention, inspection process, exact/left/right/interval counts, truncation, detection error, estimator, support, convergence, regression form, uncertainty, diagnostics and validation.
Turnbull’s foundational paper gives the self-consistent nonparametric estimator. Recent reliability work examines doubly truncated and interval-censored data.
Mathematics matters because inspections reveal windows, not invented moments. Interval likelihood, Turnbull estimation and honest uncertainty let coarse observations support useful reliability decisions.
Extended practice laboratory
Practice case 1: Interval probability
Setup: Use lambda=0.02 and bounds 40,60 days.
Task: Calculate S(40)-S(60).
Checking target: Obtain about 0.1481.
Method note: For each case, encode the lower and upper bounds and name exact, left, right or interval censoring. Write the likelihood contribution before calculating. Check endpoint conventions, inspection independence, detection quality and numerical stability. Compare with midpoint imputation only as a sensitivity demonstration and preserve the actual visit history.
Reflection: Explain why the checking target follows, identify the assumption that carries the most consequence, and describe one verification or validation check. Finish with a cautious sentence that distinguishes the illustrative calculation from a real operational approval.
Worked reasoning for Interval probability: Start from this stated setup—Use lambda=0.02 and bounds 40,60 days. Make the requested move explicitly: Calculate S(40)-S(60). The calculation is complete only when it reaches this check—Obtain about 0.1481. Now change one assumption, unit or observation rule and repeat the reasoning. Explain whether the numerical answer, its interpretation, or both would change. This turns a short exercise into a transferable method rather than a number to memorise.
Audit note for Interval probability: Record the formula or algorithm, substitutions, units, rounding rule and result in that order. Add the evidence a real analysis would need, such as failure-time records, censoring indicators, inspection histories, cost definitions or cluster labels. Name one diagnostic that could reject the model and one independent check that could catch a transcription error. Finish by separating the mathematical illustration from an engineering, safety or maintenance decision.
Practice case 2: Left censoring
Setup: Use upper bound 20 days.
Task: Calculate F(20).
Checking target: Obtain about 0.3297.
Method note: For each case, encode the lower and upper bounds and name exact, left, right or interval censoring. Write the likelihood contribution before calculating. Check endpoint conventions, inspection independence, detection quality and numerical stability. Compare with midpoint imputation only as a sensitivity demonstration and preserve the actual visit history.
Reflection: Explain why the checking target follows, identify the assumption that carries the most consequence, and describe one verification or validation check. Finish with a cautious sentence that distinguishes the illustrative calculation from a real operational approval.
Worked reasoning for Left censoring: Start from this stated setup—Use upper bound 20 days. Make the requested move explicitly: Calculate F(20). The calculation is complete only when it reaches this check—Obtain about 0.3297. Now change one assumption, unit or observation rule and repeat the reasoning. Explain whether the numerical answer, its interpretation, or both would change. This turns a short exercise into a transferable method rather than a number to memorise.
Audit note for Left censoring: Record the formula or algorithm, substitutions, units, rounding rule and result in that order. Add the evidence a real analysis would need, such as failure-time records, censoring indicators, inspection histories, cost definitions or cluster labels. Name one diagnostic that could reject the model and one independent check that could catch a transcription error. Finish by separating the mathematical illustration from an engineering, safety or maintenance decision.
Practice case 3: Right censoring
Setup: Use lower bound 100 days.
Task: Calculate S(100).
Checking target: Obtain about 0.1353.
Method note: For each case, encode the lower and upper bounds and name exact, left, right or interval censoring. Write the likelihood contribution before calculating. Check endpoint conventions, inspection independence, detection quality and numerical stability. Compare with midpoint imputation only as a sensitivity demonstration and preserve the actual visit history.
Reflection: Explain why the checking target follows, identify the assumption that carries the most consequence, and describe one verification or validation check. Finish with a cautious sentence that distinguishes the illustrative calculation from a real operational approval.
Worked reasoning for Right censoring: Start from this stated setup—Use lower bound 100 days. Make the requested move explicitly: Calculate S(100). The calculation is complete only when it reaches this check—Obtain about 0.1353. Now change one assumption, unit or observation rule and repeat the reasoning. Explain whether the numerical answer, its interpretation, or both would change. This turns a short exercise into a transferable method rather than a number to memorise.
Audit note for Right censoring: Record the formula or algorithm, substitutions, units, rounding rule and result in that order. Add the evidence a real analysis would need, such as failure-time records, censoring indicators, inspection histories, cost definitions or cluster labels. Name one diagnostic that could reject the model and one independent check that could catch a transcription error. Finish by separating the mathematical illustration from an engineering, safety or maintenance decision.
Practice case 4: Endpoint convention
Setup: Use L<T<=R.
Task: Check every encoded row.
Checking target: Bounds and open sides must be consistent.
Method note: For each case, encode the lower and upper bounds and name exact, left, right or interval censoring. Write the likelihood contribution before calculating. Check endpoint conventions, inspection independence, detection quality and numerical stability. Compare with midpoint imputation only as a sensitivity demonstration and preserve the actual visit history.
Reflection: Explain why the checking target follows, identify the assumption that carries the most consequence, and describe one verification or validation check. Finish with a cautious sentence that distinguishes the illustrative calculation from a real operational approval.
Worked reasoning for Endpoint convention: Start from this stated setup—Use L<T<=R. Make the requested move explicitly: Check every encoded row. The calculation is complete only when it reaches this check—Bounds and open sides must be consistent. Now change one assumption, unit or observation rule and repeat the reasoning. Explain whether the numerical answer, its interpretation, or both would change. This turns a short exercise into a transferable method rather than a number to memorise.
Audit note for Endpoint convention: Record the formula or algorithm, substitutions, units, rounding rule and result in that order. Add the evidence a real analysis would need, such as failure-time records, censoring indicators, inspection histories, cost definitions or cluster labels. Name one diagnostic that could reject the model and one independent check that could catch a transcription error. Finish by separating the mathematical illustration from an engineering, safety or maintenance decision.
Practice case 5: Midpoint sensitivity
Setup: Replace intervals by midpoints only for comparison.
Task: Fit both analyses.
Checking target: Explain the information lost by imputation.
Method note: For each case, encode the lower and upper bounds and name exact, left, right or interval censoring. Write the likelihood contribution before calculating. Check endpoint conventions, inspection independence, detection quality and numerical stability. Compare with midpoint imputation only as a sensitivity demonstration and preserve the actual visit history.
Reflection: Explain why the checking target follows, identify the assumption that carries the most consequence, and describe one verification or validation check. Finish with a cautious sentence that distinguishes the illustrative calculation from a real operational approval.
Worked reasoning for Midpoint sensitivity: Start from this stated setup—Replace intervals by midpoints only for comparison. Make the requested move explicitly: Fit both analyses. The calculation is complete only when it reaches this check—Explain the information lost by imputation. Now change one assumption, unit or observation rule and repeat the reasoning. Explain whether the numerical answer, its interpretation, or both would change. This turns a short exercise into a transferable method rather than a number to memorise.
Audit note for Midpoint sensitivity: Record the formula or algorithm, substitutions, units, rounding rule and result in that order. Add the evidence a real analysis would need, such as failure-time records, censoring indicators, inspection histories, cost definitions or cluster labels. Name one diagnostic that could reject the model and one independent check that could catch a transcription error. Finish by separating the mathematical illustration from an engineering, safety or maintenance decision.
Practice case 6: Turnbull support
Setup: List all interval endpoints.
Task: Construct compatible support intervals.
Checking target: Do not invent exact event locations.
Method note: For each case, encode the lower and upper bounds and name exact, left, right or interval censoring. Write the likelihood contribution before calculating. Check endpoint conventions, inspection independence, detection quality and numerical stability. Compare with midpoint imputation only as a sensitivity demonstration and preserve the actual visit history.
Reflection: Explain why the checking target follows, identify the assumption that carries the most consequence, and describe one verification or validation check. Finish with a cautious sentence that distinguishes the illustrative calculation from a real operational approval.
Worked reasoning for Turnbull support: Start from this stated setup—List all interval endpoints. Make the requested move explicitly: Construct compatible support intervals. The calculation is complete only when it reaches this check—Do not invent exact event locations. Now change one assumption, unit or observation rule and repeat the reasoning. Explain whether the numerical answer, its interpretation, or both would change. This turns a short exercise into a transferable method rather than a number to memorise.
Audit note for Turnbull support: Record the formula or algorithm, substitutions, units, rounding rule and result in that order. Add the evidence a real analysis would need, such as failure-time records, censoring indicators, inspection histories, cost definitions or cluster labels. Name one diagnostic that could reject the model and one independent check that could catch a transcription error. Finish by separating the mathematical illustration from an engineering, safety or maintenance decision.
Practice case 7: Self-consistency
Setup: Run one allocation-update step.
Task: Check likelihood.
Checking target: It should not decrease.
Method note: For each case, encode the lower and upper bounds and name exact, left, right or interval censoring. Write the likelihood contribution before calculating. Check endpoint conventions, inspection independence, detection quality and numerical stability. Compare with midpoint imputation only as a sensitivity demonstration and preserve the actual visit history.
Reflection: Explain why the checking target follows, identify the assumption that carries the most consequence, and describe one verification or validation check. Finish with a cautious sentence that distinguishes the illustrative calculation from a real operational approval.
Worked reasoning for Self-consistency: Start from this stated setup—Run one allocation-update step. Make the requested move explicitly: Check likelihood. The calculation is complete only when it reaches this check—It should not decrease. Now change one assumption, unit or observation rule and repeat the reasoning. Explain whether the numerical answer, its interpretation, or both would change. This turns a short exercise into a transferable method rather than a number to memorise.
Audit note for Self-consistency: Record the formula or algorithm, substitutions, units, rounding rule and result in that order. Add the evidence a real analysis would need, such as failure-time records, censoring indicators, inspection histories, cost definitions or cluster labels. Name one diagnostic that could reject the model and one independent check that could catch a transcription error. Finish by separating the mathematical illustration from an engineering, safety or maintenance decision.
Practice case 8: Informative visits
Setup: Let warning signs trigger earlier inspection.
Task: Draw the dependency.
Checking target: Ordinary independent-inspection analysis may be biased.
Method note: For each case, encode the lower and upper bounds and name exact, left, right or interval censoring. Write the likelihood contribution before calculating. Check endpoint conventions, inspection independence, detection quality and numerical stability. Compare with midpoint imputation only as a sensitivity demonstration and preserve the actual visit history.
Reflection: Explain why the checking target follows, identify the assumption that carries the most consequence, and describe one verification or validation check. Finish with a cautious sentence that distinguishes the illustrative calculation from a real operational approval.
Worked reasoning for Informative visits: Start from this stated setup—Let warning signs trigger earlier inspection. Make the requested move explicitly: Draw the dependency. The calculation is complete only when it reaches this check—Ordinary independent-inspection analysis may be biased. Now change one assumption, unit or observation rule and repeat the reasoning. Explain whether the numerical answer, its interpretation, or both would change. This turns a short exercise into a transferable method rather than a number to memorise.
Audit note for Informative visits: Record the formula or algorithm, substitutions, units, rounding rule and result in that order. Add the evidence a real analysis would need, such as failure-time records, censoring indicators, inspection histories, cost definitions or cluster labels. Name one diagnostic that could reject the model and one independent check that could catch a transcription error. Finish by separating the mathematical illustration from an engineering, safety or maintenance decision.
Practice case 9: Detection error
Setup: Add sensitivity and specificity.
Task: Modify status probabilities.
Checking target: Do not assume every inspection is perfect.
Method note: For each case, encode the lower and upper bounds and name exact, left, right or interval censoring. Write the likelihood contribution before calculating. Check endpoint conventions, inspection independence, detection quality and numerical stability. Compare with midpoint imputation only as a sensitivity demonstration and preserve the actual visit history.
Reflection: Explain why the checking target follows, identify the assumption that carries the most consequence, and describe one verification or validation check. Finish with a cautious sentence that distinguishes the illustrative calculation from a real operational approval.
Worked reasoning for Detection error: Start from this stated setup—Add sensitivity and specificity. Make the requested move explicitly: Modify status probabilities. The calculation is complete only when it reaches this check—Do not assume every inspection is perfect. Now change one assumption, unit or observation rule and repeat the reasoning. Explain whether the numerical answer, its interpretation, or both would change. This turns a short exercise into a transferable method rather than a number to memorise.
Audit note for Detection error: Record the formula or algorithm, substitutions, units, rounding rule and result in that order. Add the evidence a real analysis would need, such as failure-time records, censoring indicators, inspection histories, cost definitions or cluster labels. Name one diagnostic that could reject the model and one independent check that could catch a transcription error. Finish by separating the mathematical illustration from an engineering, safety or maintenance decision.
Practice case 10: Inspection plan
Setup: Compare 10-day and 30-day visits.
Task: Balance width and cost.
Checking target: Report operational burden and uncertainty.
Method note: For each case, encode the lower and upper bounds and name exact, left, right or interval censoring. Write the likelihood contribution before calculating. Check endpoint conventions, inspection independence, detection quality and numerical stability. Compare with midpoint imputation only as a sensitivity demonstration and preserve the actual visit history.
Reflection: Explain why the checking target follows, identify the assumption that carries the most consequence, and describe one verification or validation check. Finish with a cautious sentence that distinguishes the illustrative calculation from a real operational approval.
Worked reasoning for Inspection plan: Start from this stated setup—Compare 10-day and 30-day visits. Make the requested move explicitly: Balance width and cost. The calculation is complete only when it reaches this check—Report operational burden and uncertainty. Now change one assumption, unit or observation rule and repeat the reasoning. Explain whether the numerical answer, its interpretation, or both would change. This turns a short exercise into a transferable method rather than a number to memorise.
Audit note for Inspection plan: Record the formula or algorithm, substitutions, units, rounding rule and result in that order. Add the evidence a real analysis would need, such as failure-time records, censoring indicators, inspection histories, cost definitions or cluster labels. Name one diagnostic that could reject the model and one independent check that could catch a transcription error. Finish by separating the mathematical illustration from an engineering, safety or maintenance decision.
Practice case 11: Validation
Setup: Hold out later inspection rounds.
Task: Compare predicted interval failures.
Checking target: Use the same visit process.
Method note: For each case, encode the lower and upper bounds and name exact, left, right or interval censoring. Write the likelihood contribution before calculating. Check endpoint conventions, inspection independence, detection quality and numerical stability. Compare with midpoint imputation only as a sensitivity demonstration and preserve the actual visit history.
Reflection: Explain why the checking target follows, identify the assumption that carries the most consequence, and describe one verification or validation check. Finish with a cautious sentence that distinguishes the illustrative calculation from a real operational approval.
Worked reasoning for Validation: Start from this stated setup—Hold out later inspection rounds. Make the requested move explicitly: Compare predicted interval failures. The calculation is complete only when it reaches this check—Use the same visit process. Now change one assumption, unit or observation rule and repeat the reasoning. Explain whether the numerical answer, its interpretation, or both would change. This turns a short exercise into a transferable method rather than a number to memorise.
Audit note for Validation: Record the formula or algorithm, substitutions, units, rounding rule and result in that order. Add the evidence a real analysis would need, such as failure-time records, censoring indicators, inspection histories, cost definitions or cluster labels. Name one diagnostic that could reject the model and one independent check that could catch a transcription error. Finish by separating the mathematical illustration from an engineering, safety or maintenance decision.
Practice case 12: Conclusion
Setup: State bounds, likelihood and inspection process.
Task: Report uncertainty and non-uniqueness.
Checking target: Do not convert windows into facts.
Method note: For each case, encode the lower and upper bounds and name exact, left, right or interval censoring. Write the likelihood contribution before calculating. Check endpoint conventions, inspection independence, detection quality and numerical stability. Compare with midpoint imputation only as a sensitivity demonstration and preserve the actual visit history.
Reflection: Explain why the checking target follows, identify the assumption that carries the most consequence, and describe one verification or validation check. Finish with a cautious sentence that distinguishes the illustrative calculation from a real operational approval.
Worked reasoning for Conclusion: Start from this stated setup—State bounds, likelihood and inspection process. Make the requested move explicitly: Report uncertainty and non-uniqueness. The calculation is complete only when it reaches this check—Do not convert windows into facts. Now change one assumption, unit or observation rule and repeat the reasoning. Explain whether the numerical answer, its interpretation, or both would change. This turns a short exercise into a transferable method rather than a number to memorise.
Audit note for Conclusion: Record the formula or algorithm, substitutions, units, rounding rule and result in that order. Add the evidence a real analysis would need, such as failure-time records, censoring indicators, inspection histories, cost definitions or cluster labels. Name one diagnostic that could reject the model and one independent check that could catch a transcription error. Finish by separating the mathematical illustration from an engineering, safety or maintenance decision.