| title | Introduction to Hazard Function Technology |
|---|---|
| description | Conceptual overview of parametric hazard analysis and the motivation for multi-phase models |
Some of the most relevant outcomes of medical procedures — or the life-history of machines — are time-related events. The raw data for such events is the time interval between a defined time zero (t = 0) and the occurrence of the event. The distribution of these time intervals can be visualized as:
- A cumulative distribution (or its complement, the survivorship function)
- A probability density function (histogram form)
- A rate of occurrence — the most natural domain for studying biological or mechanical phenomena across time
This rate of occurrence is the hazard function. The term was introduced by John Graunt in the 17th century, borrowing from the language of dice games. It is also known as the force of mortality in demography, and as the inverse of Mills ratio in finance.
The methodology is applicable to any positively distributed variable — not only survival times.
The nature of living things and real machines is such that lifetimes and other time-related events often follow low-order distributions. This supports a parametric approach: fitting a compact, interpretable functional form rather than a purely non-parametric one.
The parametric approach taken in the HAZARD procedures — developed in the early 1980s at the University of Alabama at Birmingham — uses a decompositional strategy. The distribution of intervals is viewed as consisting of one or more overlapping phases (early, constant, and late), additive in hazard (competing risks). A generic functional form is used for each phase that encompasses a large number of hierarchically nested models.
The hazard function is decomposed as:
h(t | X) = μ₁(X) · SG₁(t) [early phase]
+ μ₂(X) [constant phase]
+ μ₃(X) · SG₃(t) [late phase]
Each phase is scaled by a log-linear function of concomitant information (covariates):
μⱼ(X) = exp(intercept_j + β_j' X)
This allows the model to be non-proportional in hazards — an assumption often made with Cox models but frequently unrealistic in clinical data.
The HAZARD model handles three common complexities beyond simple right-censored data:
- Censoring types: Right censoring, left censoring, and interval censoring are all supported.
- Repeating events: The observation window can be longitudinally segmented, which also accommodates a wide class of time-varying covariates — specifically those that change at discrete intervals.
- Weighted events: Events can carry a positive weight (such as cost or severity), enabling analysis of, e.g., time-related repeated cost data.
At its most general, HAZARD models time-related repeating weighted cost data with time-varying covariates and a non-proportional hazard structure.
A key theoretical property of the model is the Conservation of Events theorem (Turner et al.): one of the phase intercepts can be solved in closed form from a constraint equation, reducing optimization dimensionality by one. This substantially improves numerical stability.
| Feature | Cox (semi-parametric) | HAZARD (parametric) |
|---|---|---|
| Proportional hazards | Required | Not required |
| Baseline hazard | Unspecified | Fully parametric |
| Multiple hazard phases | Not supported | Supported |
| Patient-specific prediction | Approximate | Exact (plug in covariates) |
| Event conservation check | Not available | Built-in |
| Interval censoring | Not standard | Supported |
The fully parametric nature of HAZARD means that once parameter estimates are available, the resulting equation can be solved for any set of risk factors — enabling patient-specific survival prediction and goodness-of-fit checks via conservation of events.
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Contact: hazard@bio.ri.ccf.org