Find your next learning step
When a polymer regime exit occurs between observations and its mechanism is uncertain, two separate shortcuts can compound. A joint model keeps the time bracket and the cause evidence together.
Open the chapter contents · Polymer Science root · Gold Standard of Education · How X Works
Chapter contents
Choose the route closest to the question, or read from Chapter 1 for the complete argument. The chapters remain open on the page; these grouped routes are an index.
Route 1 · Frame the problem
Route 2 · Work the ledgers
Route 3 · Transfer the method
Route 4 · Check limits and evidence
Route 5 · Ask and continue
CHAPTER 1 OF 12 · FRAME THE PROBLEM
1. Two uncertainties belong in one event record
A tracked domain may be in regime A at 20 minutes and outside it at 30 minutes, so its exit time lies in (20,30]. Imaging and spectroscopy may suggest either scission or disentanglement. Replacing the interval with 25 and the cause with a hard label invents both time and mechanism precision.
A joint record contains the last confirmed in-regime time, first confirmed exit time, observed evidence, possible latent causes and protocol. The likelihood integrates over the time interval and sums over causes compatible with the evidence.
CHAPTER 2 OF 12 · FRAME THE PROBLEM
2. Interval likelihoods with competing causes
Let Fj(t) be the cumulative incidence of first exit by cause j. A specimen known to exit by cause j in (L,R] contributes Fj(R)−Fj(L). If the cause is unknown, its contribution sums the interval probabilities across possible causes, modified by the cause-observation model when labels are imperfect.
A right-censored specimen contributes the probability of no first exit by its last visit. These terms preserve the risk-set logic: once one mechanism produces the first exit, the specimen no longer remains available for another first-exit mechanism.
CHAPTER 3 OF 12 · WORK THE LEDGERS
3. A six-domain joint observation ledger
Consider six domains observed under two protocols. Some have precise cause evidence; others have ambiguous labels or no observed exit by the final visit.
| Domain | Protocol | Time information | Cause evidence |
|---|---|---|---|
| D1 | A | (10,20] | Scission probability 0.85 |
| D2 | A | (20,30] | Disentanglement probability 0.70 |
| D3 | A | (20,40] | Cause unresolved |
| D4 | B | (0,10] | Scission probability 0.55 |
| D5 | B | (30,50] | Crystallisation probability 0.80 |
| D6 | B | Beyond 60 | Right-censored |
D3 should not be deleted, and D4 should not be forced to scission merely because 0.55 is the largest value. Both can contribute probabilistically to protocol-specific cumulative incidence.
CHAPTER 4 OF 12 · WORK THE LEDGERS
4. A worked interval-and-cause contribution
Suppose model A assigns interval probabilities over (10,20] of 0.08 for scission, 0.03 for disentanglement and 0.01 for crystallisation. If D1’s observed assay has probabilities 0.85, 0.10 and 0.05 conditional on those causes, the joint contribution is proportional to 0.08×0.85 + 0.03×0.10 + 0.01×0.05 = 0.0715.
| Latent cause | Interval probability | Evidence factor | Joint term |
|---|---|---|---|
| Scission | 0.080 | 0.85 | 0.0680 |
| Disentanglement | 0.030 | 0.10 | 0.0030 |
| Crystallisation | 0.010 | 0.05 | 0.0005 |
| Summed contribution | — | — | 0.0715 |
The calculation preserves alternatives until the data distinguish them. Posterior cause probabilities can then be derived from all terms, rather than inserted as fixed truth before fitting.
CHAPTER 5 OF 12 · WORK THE LEDGERS
5. Protocol-specific exit maps answer a physical question
A protocol-specific exit map shows, for each time, the estimated probability of first exit through each mechanism under that protocol. It is not merely a coloured state timeline. The curves share a common no-exit probability and therefore must be interpreted jointly.
Compare protocols using cumulative incidence differences at prespecified times or integrated contrasts. A cause-specific hazard contrast answers a different question from a cumulative-incidence contrast when competing mechanisms change.
CHAPTER 6 OF 12 · TRANSFER THE METHOD
6. The observation process can depend on the protocol
Protocols may use different inspection intervals, imaging contrast or destructive assays. Those choices affect interval width and cause recognisability. A joint model can include protocol-specific observation matrices and visit schedules, but it cannot identify them without validation or defensible assumptions.
Report whether the schedule was fixed, adaptive or interrupted. If unstable specimens were inspected more often, the visit process may be informative and require sensitivity analysis.
CHAPTER 7 OF 12 · TRANSFER THE METHOD
7. Changed-context transfer: coating degradation
Transfer the method to a coating that can first fail by cracking, delamination or chemical embrittlement. Chamber openings bracket failure time, while microscopy and spectroscopy supply uncertain cause evidence.
The transfer check asks whether opening the chamber changes degradation, whether destructive assays are selected by visible damage and whether temperature changes label accuracy. A joint statistical model supports, but does not replace, material-mechanism validation.
CHAPTER 8 OF 12 · TRANSFER THE METHOD
8. Connecting time brackets and cause uncertainty
The interval-censoring guide keeps the event time inside its observed bracket. The uncertain-cause guide carries label ambiguity through a response matrix. This article joins those two evidence layers.
Use competing exit risks and cause-specific hazards to keep the estimand clear when one first-exit mechanism removes another from the risk set.
CHAPTER 9 OF 12 · CHECK LIMITS AND EVIDENCE
9. Limits and misconceptions
A joint model does not observe the latent time or cause. It distributes evidence under assumptions about event dynamics, censoring and label generation. Flexible models can remain weakly identified when intervals are wide and cause validation is sparse.
Protocol-specific curves are not molecular rate constants. Misclassification may change with time, signal intensity or protocol. Treating an external validation matrix as fixed can understate uncertainty when the new experiment has different imaging conditions.
CHAPTER 10 OF 12 · CHECK LIMITS AND EVIDENCE
10. Practical evidence checklist
Archive planned and actual visits, time brackets, censoring rules, assay features, reviewer labels, validation subsets and protocol conditions. State whether causes are mutually exclusive first exits and how repeated or reversible transitions are handled.
Report interval-width distributions, cause-evidence calibration, protocol-specific cumulative incidence, model and bootstrap uncertainty, missed-visit sensitivity and alternative label matrices. Preserve the original joint ledger so later assays can update attribution without rewriting event times.
CHAPTER 11 OF 12 · ASK AND CONTINUE
11. Frequently asked questions
Can separate time and cause models be combined afterward?
Only if their assumptions produce a coherent joint distribution. Two convenient marginal models do not automatically define a valid joint likelihood.
Are fractional cause counts physically meaningful?
They are expected latent counts used for inference. Each realised event still has one physical first-exit cause under the model.
What if mechanisms can occur together?
Then a mutually exclusive first-cause model is incomplete. Define multistate, recurrent or composite events that match the physics.
CHAPTER 12 OF 12 · ASK AND CONTINUE
12. Sources and next reading
Mao, Lin and Zeng develop semiparametric regression for interval-censored competing risks. Li studies the Fine–Gray model under interval censoring. Park and colleagues address interval-censored competing risks with missing event types. The polymer ledgers above are original teaching examples.
Continue through the Polymer Science root, interval-censored exit times and probabilistic mechanism attribution. The Gold Standard of Education and How X Works provide the wider route.
Top · Contents · Previous · How X Works