Transform hazard rates to other time intervals

Question

If a population has a hazard rate of $X$ per year, what would be its hazard rate per month or per day?

Answer 1

The hazard rate $h(t)$ at time $t$ is defined in terms of conditional probabilities. For a continuous positive random variable, if $\Delta t$ denotes a small number, then, with $L$ denoting the lifetime of the object under consideration, $$\begin{align} P\left\{t < L \leq t + \Delta t \mid L > t\right\} &= \frac{P\left\{(t < L \leq t + \Delta t) \cap (L > t)\right\}}{P\{L > T\}}\\ & =\frac{P\left\{t < L \leq t + \Delta t\right\}}{P\{L > T\}}\\ &\approx \frac{f_L(t)\Delta t}{1-F_L(t)}\\ &= \frac{f_L(t)}{1-F_L(t)}\Delta t\\ &= h(t)\Delta t \end{align}$$ so that the definition of $h(t)$ is $$h(t) = \frac{f_L(t)}{1-F_L(t)}$$ and the units of $h(t)$ are determined by whatever unit you choose for the measurement of $t$.

As a specific example, if time is measured in years, and the hazard rate has constant value $\lambda$, then the lifetime $L$ of the system is an exponential random variable with parameter $\lambda$ per year and mean $\lambda^{-1}$ years. However, if time is measured in days (assume exactly 365 days per year for simplicity), then the lifetime $\hat{L}$ of the same system, measured in days instead of years, equals $365L$, and thus is an exponential random variable with mean $365\lambda^{-1}$ days and parameter $\hat{\lambda} = \lambda/365$ per day. Thus, the hazard rate $\hat{\lambda}$ of $\hat{L}$, the lifetime of the system measured in days instead of years, is $\lambda/365$ per day

Answer 2

You can make it complicated, but just think of hazard as the expected number of occurrences per unit of time. Half the time, half the expected number of occurrences. Twice the time, twice the number.