Hazard Rate Ratio⚠ unverified

Understanding the Parameters

• The two times $t_1, t_2$ — the ages being compared; only their ratio $t_2/t_1$ matters. Doubling both times leaves the hazard ratio unchanged (for $t_2/t_1$ fixed) — a scale-free comparison.

• Shape parameter $\beta$ — the whole story of the ratio's behaviour:

• $\beta = 1$: ratio $= 1$ for any times — constant hazard, no aging (the exponential/memoryless case).

• $\beta > 1$: ratio $> 1$ for $t_2 > t_1$ — the item is aging; failure is more likely later. A wear-out item with $\beta = 3$ is $2^{2} = 4\times$ more hazardous at twice the age.

• $\beta < 1$: ratio $< 1$ for $t_2 > t_1$ — the item is improving (infant mortality burning off); failure is less likely later.

• Scale $\eta$ cancels — a key feature. Because both hazards carry the same $\beta/\eta$ and $\eta$-scaling, the ratio depends only on $\beta$ and $t_2/t_1$. So you can characterise aging without knowing the absolute timescale — useful when $\eta$ is uncertain but the shape is known. (The registry keeps $\eta$ as an input though it does not affect the result.)

• The output ratio — a dimensionless multiplier on risk. It quantifies aging directly, complementing the absolute hazard rate.

Derivation (Approaching a Proof)

Take the Weibull hazard rate at each time, $h(t) = \frac{\beta}{\eta}(t/\eta)^{\beta-1}$ (Weibull Failure Rate), and form the ratio. Every factor except the time-power is common to numerator and denominator:

$$\frac{h(t_2)}{h(t_1)} = \frac{\frac{\beta}{\eta}(t_2/\eta)^{\beta-1}}{\frac{\beta}{\eta}(t_1/\eta)^{\beta-1}} = \frac{(t_2/\eta)^{\beta-1}}{(t_1/\eta)^{\beta-1}} = \left(\frac{t_2}{t_1}\right)^{\beta-1}. \qquad\blacksquare$$

The prefactor $\beta/\eta$ and the $\eta$ inside each power cancel, leaving a pure function of $t_2/t_1$ and $\beta - 1$. This is the mathematical statement that the Weibull hazard is a power law: its ratio over any two times is set by the exponent alone, the signature of scale-invariance in the failure process.

Dimensional check. $t_2/t_1$ is dimensionless, so its power $(\cdot)^{\beta-1}$ is dimensionless — a pure ratio. $\checkmark$

History and Development

• Aging in one number. The hazard rate ratio distils the aging behaviour of a Weibull item into a single scale-free quantity, making it easy to communicate "twice as old, four times as likely to fail" to non-specialists.

• Proportional hazards. The idea that hazards can be compared as ratios is the foundation of Cox proportional-hazards models in survival analysis and reliability, where covariates (stress, temperature, design) multiply a baseline hazard — the ratio being the effect size.

• Maintenance timing. Comparing hazards across ages informs age-based replacement: when the hazard ratio relative to a new part climbs past a threshold, preventive replacement becomes worthwhile — a direct application of the $\beta > 1$ increasing-hazard regime.

Related Concepts: Weibull Failure Rate, Weibull Reliability, Weibull Distribution PDF, Bathtub Curve, Exponential Reliability, B10 Life

Notes: Registry calculator hazard-rate-ratio (unverified). $h(t_2)/h(t_1) = (t_2/t_1)^{\beta-1}$ — correct as shipped; independent of $\eta$ (which the registry keeps as an unused input). $\beta=1$ ⇒ ratio $1$ (no aging); $\beta>1$ ⇒ aging; $\beta<1$ ⇒ improving. Scale-free measure of aging. Returns $+\infty$ if $h(t_1)=0$.