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A regime exit can be timed precisely while its physical cause remains uncertain. Honest mechanism inference carries that uncertainty through the event ledger instead of turning the reviewer’s best guess into truth.
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. Why an observed label is not automatically a mechanism
A tracked polymer domain may leave a regime through disentanglement, chain scission or crystallisation. Imaging contrast, spectroscopy and trajectory features may not identify the route perfectly. The observed label is produced by a measurement and classification process; it is not the latent physical cause itself.
Hard labels hide borderline cases and can bias mechanism shares, especially when one cause is easier to see. A useful analysis separates event time, latent cause, observed evidence and review decision. It then asks what cause probabilities are supported rather than forcing every exit into a certain box.
CHAPTER 2 OF 12 · FRAME THE PROBLEM
2. A misclassification matrix links causes to labels
Let M[j,k] be the probability of observing label j when the true cause is k. Each column sums to one under this convention. Diagonal entries are sensitivities for correct recognition; off-diagonal entries describe confusions. The matrix can come from a blinded validation subset with stronger microscopy, spectroscopy or destructive follow-up.
The direction matters. A row-normalised table answers a different question: the probability of the true cause given the observed label, which also depends on cause prevalence. Record the convention beside every matrix and keep an “uncertain” label if reviewers genuinely use one.
CHAPTER 3 OF 12 · WORK THE LEDGERS
3. A 100-exit observed-label ledger
Suppose 100 regime exits receive observed labels: 45 disentanglement, 35 scission and 20 crystallisation. Review alone suggests those percentages, but validation finds systematic confusion between disentanglement and scission. The observed ledger is evidence to be corrected, not a final mechanism census.
| Observed label | Count | Main plausible true causes | Evidence needed |
|---|---|---|---|
| Disentanglement | 45 | Disentanglement or mild scission | Chain continuity and recovery |
| Scission | 35 | Scission or abrupt tracking break | Molecular-weight or bond evidence |
| Crystallisation | 20 | Crystallisation or contrast change | Diffraction or thermal evidence |
| Total | 100 | Mutually exclusive first-exit labels | Common time window |
If missing or unresolved causes are discarded, the denominator becomes selected by observability. Preserve all exits in the risk set and represent unresolved attribution explicitly.
CHAPTER 4 OF 12 · WORK THE LEDGERS
4. A worked probabilistic attribution
Assume a fitted model assigns each observed disentanglement label probabilities 0.80, 0.15 and 0.05 across the three true causes. For observed scission, use 0.20, 0.75 and 0.05; for observed crystallisation, use 0.05, 0.05 and 0.90. Summing posterior cause probabilities across all 100 exits yields expected counts of 44, 34 and 22.
| Observed group | Expected disentanglement | Expected scission | Expected crystallisation |
|---|---|---|---|
| 45 labelled disentanglement | 36.0 | 6.75 | 2.25 |
| 35 labelled scission | 7.0 | 26.25 | 1.75 |
| 20 labelled crystallisation | 1.0 | 1.0 | 18.0 |
| Expected totals | 44.0 | 34.0 | 22.0 |
The corrected totals are close to the hard labels here, yet the uncertainty is now visible. In another protocol with different mechanism prevalence, the posterior probabilities would change even if the measurement matrix stayed fixed.
CHAPTER 5 OF 12 · WORK THE LEDGERS
5. Validation data identify the matrix
A misclassification matrix cannot usually be learned from noisy labels alone without strong assumptions. Select a validation subset before seeing the desired result, apply a more credible cause-ascertainment method and blind reviewers to protocol when possible. Include difficult and low-signal exits, not only clean textbook examples.
Estimate matrix uncertainty with intervals or a hierarchical model. Small validation cells produce wide uncertainty that should propagate to cumulative incidence and protocol contrasts. Pooling all protocols may be inappropriate if temperature, imaging quality or strain amplitude changes recognisability.
CHAPTER 6 OF 12 · TRANSFER THE METHOD
6. Carry probabilities into competing-risk estimates
The preceding first-exit analysis used cause-specific hazards and cumulative incidence. With uncertain causes, one approach weights each exit by its posterior cause probabilities; another uses multiple imputation, drawing a plausible cause for every unresolved event and combining results across completed data sets. Both approaches require an explicit model for why causes are missing or misclassified.
Do not merely split uncertain exits equally. Equal allocation is a substantive assumption that can erase protocol-specific evidence. Compare complete-case, hard-label, probabilistic and sensitivity analyses so readers can see how attribution changes the mechanism conclusion.
CHAPTER 7 OF 12 · TRANSFER THE METHOD
7. Changed-context transfer: oxidative ageing
Transfer the method to polymer films aged under heat and oxygen. An exit from optical stability may reflect chain scission, additive loss or crystallinity change. The event time comes from a monitoring threshold, while confirmatory chemical assays exist for only a subset.
Build the validation subset across exposure conditions, because oxidation can change both true prevalence and spectral appearance. The ledger structure transfers; the cause set, measurement matrix and missingness assumptions must be rebuilt for the ageing experiment.
CHAPTER 8 OF 12 · TRANSFER THE METHOD
8. Connecting to the regime-transition route
Use the competing exit-risks guide to define the shared risk set and the first-exit estimands. The hidden semi-Markov guide connects uncertain states with dwell-time dynamics.
Use duration covariates and protocol-dependent persistence when exposure changes the timing distribution. Cause uncertainty should be handled before interpreting a protocol coefficient as a physical mechanism effect.
CHAPTER 9 OF 12 · CHECK LIMITS AND EVIDENCE
9. Limits and misconceptions
Probabilistic attribution does not discover mechanisms from labels alone. Its conclusions inherit the validation design, conditional-independence assumptions and cause list. An omitted physical route can be redistributed across the listed causes and create confident but wrong probabilities.
A high reviewer agreement rate is not enough when all reviewers share the same blind spot. Destructive validation may itself alter the specimen. Posterior cause probabilities are model outputs, not measured fractions for each event. Use sensitivity analyses for plausible matrices and missing-cause mechanisms.
CHAPTER 10 OF 12 · CHECK LIMITS AND EVIDENCE
10. Practical evidence checklist
Archive the event definition, risk set, raw observations, reviewer labels, confidence scores, validation sampling rule, gold-standard procedure and matrix convention. Preserve disagreement and “unresolved” outcomes. Confirm that all matrix columns or rows sum as declared.
Report observed and corrected counts, validation cell sizes, matrix uncertainty, complete-case bias checks and protocol-specific sensitivity. Refit cumulative incidence under alternative plausible confusion rates. Seek orthogonal evidence such as molecular-weight, diffraction or recovery data before naming a mechanism.
CHAPTER 11 OF 12 · ASK AND CONTINUE
11. Frequently asked questions
Why not use the majority reviewer label?
Majority voting reduces individual noise but does not correct shared systematic confusion or selected missing causes.
Are fractional expected counts physically real?
They summarise uncertainty across events. Each event has one physical first cause in the model, while the analysis does not know it with certainty.
Can external validation supply the matrix?
It can inform a sensitivity analysis, but transport requires comparable materials, instruments, protocols and cause prevalence. Internal validation is usually stronger.
CHAPTER 12 OF 12 · ASK AND CONTINUE
12. Sources and next reading
Lu and Tsiatis develop multiple imputation for competing risks with missing causes. Lu studies inverse-probability and doubly robust estimation under missing causes. Fine and Gray define a regression framework for subdistribution hazards in competing risks. The attribution ledgers above are original teaching examples.
Continue through the Polymer Science root, competing exit risks and Bayesian change points and transition uncertainty. The Gold Standard of Education and How X Works provide the wider route.
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