Hazard-Shape Diagnostic¶

Diagnostic — instantiates Survival-Conditioned Persistence Forecasting

Reads whether the exit hazard rises, stays flat, or falls with age — the single fact that decides whether surviving longer is good news or bad news.

The whole archetype turns on one shape question, and this is the mechanism that answers it. Hazard-Shape Diagnostic characterizes how the exit hazard behaves as a function of age: does it rise with age (wear-out — surviving longer brings you closer to the end), stay flat (memoryless — age tells you nothing), or fall (the durable-tail case where surviving longer is genuinely reassuring)? It reads this shape non-parametrically, without yet committing to a distribution family, because the sign of the slope — not the exact curve — is what flips the entire inference from "old means soon-to-go" to "old means likely-to-stay." Its second, easily-missed job is to check whether a comforting declining hazard is real or an illusion manufactured by mixing unlike things together.

Example¶

A storage team has time-to-failure records for a large fleet of disk drives and wants to know what a drive's age implies about its remaining life. The diagnostic estimates the hazard rate across age (from the slope of the cumulative hazard, read off a log-log survival plot) and recovers the classic bathtub: a high early "infant-mortality" hazard from manufacturing defects, a long flat middle where failures arrive at a roughly constant, memoryless rate, then a rising "wear-out" tail as bearings and platters age. That one picture says three different things at three ages — a one-week-old drive is in the risky infancy, a two-year-old sits in the uninformative flat zone, a six-year-old is climbing the wear-out slope — and it warns that survival-conditioned reasoning is only benign in the flat-or-declining regions.

Then the diagnostic does the check that separates it from a naive curve-fit. The pooled hazard for one product line looks like it declines with age — superficially a Lindy-flavored result. Splitting the fleet by manufacturing batch reveals the truth: one batch is flaky and one is robust; the flaky units die early, so the survivors are increasingly the robust ones, and the population hazard falls even though no individual drive is getting safer. The declining shape was a heterogeneity artifact, not durability.

How it works¶

The diagnostic reports a shape, not a fit. What distinguishes it:

Non-parametric shape first — it classifies the hazard as increasing (IFR), constant, decreasing (DFR), or bathtub from the empirical cumulative-hazard slope, a log-log survival plot, or a scaled total-time-on-test transform — before any family is assumed.
The heterogeneity check — it partitions the population into subgroups to test whether an apparently declining pooled hazard is a mixture artifact (frail units dying first) rather than genuine within-unit durability.
A gated verdict — it outputs the shape plus a confidence, which is the pivotal input to the eligibility gate: flat-or-declining keeps survival-conditioning alive, rising kills it.

It does not select a parametric family or turn the shape into a remaining-life number — it reads the sign of the slope and whether that sign survives disaggregation.

Tuning parameters¶

Age resolution — how finely age is binned. Fine bins expose bathtub structure but grow noisy where events are few.
Smoothing bandwidth — how much the hazard estimate is smoothed. More smoothing stabilizes the shape but can erase a real early spike or late upturn.
Subgroup granularity — how finely the population is split for the frailty check. Finer splits catch heterogeneity but thin each group toward noise.
Slope-call threshold — how steep a trend counts as "increasing" versus flat. Conservative thresholds avoid false aging calls but can miss slow wear-out.
Diagnostic lens — whether shape is read from the raw cumulative hazard or through a fitted shape exponent; more structure buys sharper statements at the cost of more assumptions.

When it helps, and when it misleads¶

Its strength is leverage: the shape of the hazard is the one fact that decides the sign of every survival-conditioned conclusion, and this is the cheap check that stops a team from applying "survived-longer-lasts-longer" to a wearing-out system. Reading it before fitting anything is what keeps the rest of the pipeline honest.

Its central trap is that unobserved heterogeneity makes a pooled hazard look decreasing even when every unit is memoryless or aging — frailty faking a Lindy signal — which is why the subgroup check is not optional.^[1] With few exits you often cannot statistically separate "flat" from "gently rising," so a confident shape call on thin data is itself a misuse. And a rising hazard can reflect a regime shift rather than intrinsic aging — the shape is only interpretable within a stable regime. The discipline is to run the heterogeneity split before believing any decline, report the shape with its uncertainty, and hand the "is the regime stable?" question to the stationarity mechanisms rather than reading it off the curve.

How it implements the components¶

This diagnostic fills the shape-characterization side of the modeling machinery:

lifetime_distribution_and_hazard_model — it produces the hazard half of that model: the shape of the hazard over age (rising / flat / falling / bathtub), characterized non-parametrically.
subgroup_lifetime_segment — it partitions the population to test whether a pooled shape is a mixture artifact, surfacing frailty-driven pseudo-decline before it is mistaken for durability.

It does not select a parametric family or convert the tail into a remaining-life estimate (Lifetime Distribution Comparison); it does not turn the shape verdict into an eligibility decision (Non-Aging Eligibility Review); and it consumes, rather than produces, the censoring-corrected record (Censoring and Left-Truncation Audit).

Instantiates: Survival-Conditioned Persistence Forecasting — it supplies the hazard-shape reading that gates whether survival-conditioning even applies.
Consumes: Censoring and Left-Truncation Audit — a survivorship-tilted sample fakes a declining hazard, so the shape must be read on a corrected risk set.
Sibling mechanisms: Lifetime Distribution Comparison · Non-Aging Eligibility Review · Survival or Time-to-Event Analysis · Age-Conditioned Remaining-Life Table · Censoring and Left-Truncation Audit · Reference-Class Forecasting · Stationarity Test · Change-Point Detection · Rolling Window Comparison · Forecast Backtesting · Historical or Holdout Coverage Backtest · Periodic Durability Inspection · Lindy Decision-Horizon Review

References¶

[1] Unobserved heterogeneity (frailty) — when a population mixes units with different constant or rising hazards, the frail ones exit first, so the survivors are progressively selected toward the robust and the population-level hazard falls even though no individual unit is getting safer. It is the standard reason an aggregate "decreasing hazard" can be an artifact rather than evidence of genuine durability. ↩