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I am confused about the equation that serves as the definition of the hazard rate. I get the idea of what the hazard rate is, but I just don't see how the equation expresses that intuition.

If $x$ is a random variable which represents the point of time of death of someone on a time interval $[0,T]$. Then the hazard rate is:

$$h(x)=\frac{f(x)}{1-F(x)}$$

Where $F(x)$ represents the probability of death until time point $x\in[0,T]$,
$1-F(x)$ represents the probability of having survived up until time point $x\in[0,T]$,
and $f(x)$ is the probability of death at point $x$.

How does dividing $f(x)$ by the survival rate explain the intuition of the probability of instantaneous death in the next $\Delta t$? Shouldn't it just be $f(x)$, making the calculation of the hazard rate trivial?

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Let $X$ denote the time of death (or time of failure if you prefer a less morbid description). Suppose that $X$ is a continuous random variable whose density function $f(t)$ is nonzero only on $(0,\infty)$. Now, notice that it must be the case that $f(t)$ decays away to $0$ as $t \to \infty$ because if $f(t)$ does not decay away as stated, then $\displaystyle \int_{-\infty}^\infty f(t)\,\mathrm dt = 1$ cannot hold. Thus, your notion that $f(T)$ is the probability of death at time $T$ (actually, it is $f(T)\Delta t$ that is (approximately) the probability of death in the short interval $(T, T+\Delta t]$ of length $\Delta t$) leads to implausible and unbelievable conclusions such as

You are more likely to die within the next month when you are thirty years old than when you are ninety-eight years old.

whenever $f(t)$ is such that $f(30) > f(98)$.

The reason why $f(T)$ (or $f(T)\Delta t$) is the "wrong" probability to look at is that the value of $f(T)$ is of interest only to those who are alive at age $T$ (and still mentally alert enough to read stats.SE on a regular basis!) What ought to be looked at is the probability of a $T$-year old dying within the next month, that is,

\begin{align}P\{(X \in (T, T+\Delta t] \mid X \geq T\} &= \frac{P\{\left(X \in (T, T+\Delta t]\right) \cap \left(X\geq T\right)\}}{P\{X\geq T\}} & \\ \scriptstyle{ \text{ definition of conditional probability}}\\ &= \frac{P\{X \in (T, T+\Delta t]\}}{P\{X\geq T\}}\\ &= \frac{f(T)\Delta t}{1-F(T)} & \\ \scriptstyle{ \text{because }X\text{ is a continuous rv}} \end{align}

Choosing $\Delta t$ to be a fortnight, a week, a day, an hour, a minute, etc. we come to the conclusion that the (instantaneous) hazard rate for a $T$-year old is

$$h(T) = \frac{f(T)}{1-F(T)}$$

in the sense that the approximate probability of death in the next femtosecond $(\Delta t)$ of a $T$-year old is $\displaystyle \frac{f(T)\Delta t}{1-F(T)}.$

Note that in contrast to the density $f(t)$ integrating to $1$, the integral $\displaystyle \int_0^\infty h(t)\, \mathrm dt$ must diverge. This is because the CDF $F(t)$ is related to the hazard rate through

$$F(t) = 1 - \exp\left(-\int_0^t h(\tau)\, \mathrm d\tau\right)$$ and since $\lim_{t\to \infty}F(t) = 1$, it must be that $$\lim_{t\to \infty} \int_0^t h(\tau)\, \mathrm d\tau = \infty,$$ or stated more formally, the integral of the hazard rate must diverge: there is no potential divergence as a previous edit claimed.

Typical hazard rates are increasing functions of time, but constant hazard rates (exponential lifetimes) are possible. Both of these kinds of hazard rates obviously have divergent integrals. A less common scenario (for those who believe that things improve with age, like fine wine does) is a hazard rate that decreases with time but slowly enough that the integral diverges.

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Imagine that you are interested in the incidence of (first) marriage for men. To look at the incidence of marriage at age 20, say, you would select a sample of people who are not married at that age and see if they get married within the next year (before they turn 21).

The you could get a rough estimate for $$ P(\mathrm{marry\,\, before\,\, 21}| \mathrm{not\,\, married \,\,at\,\, 20}) $$ as the proportion of individuals who got married from your sample of single 20 year olds, i.e. $$ \frac{N(\mathrm{married \,\,before \,\,21\,\, and \,\, not\,\,married\,\, at \,\, 20})}{N(\mathrm{not\,\, married\,\, at\,\, 20})} $$

So basically this is just using the definition of conditional probability, $$ P(X|Y) = \frac{P(X,Y)}{P(Y)}. $$ Now imagine we make the age unit smaller and smaller, up to days for example. I.e. what is the incidence of marriage at age of 7300 days? Then you would do the same, but survey all individuals of 7300 days and look who gets married before the end of the day. If $T$ is a random variable age at marriage, then we could write $$ P(T \leq 7301)| T \geq 7300) = \frac{P(T \in [7300, 7301))}{P(T \geq 7300)} $$ by the same logic as before.

The hazard would then be the instantaneous probability of marriage at age $t$, for a non-married individual. We can write this as $$ h(t) dt= P(T \in [t, t+dt) | T\geq t) = \frac{P(T \in[t, t+dt))}{P(T \geq t)} $$

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$f(x)$ is not the probability of death, but the probability density; the expected number of times you die within the next unit of time if the probability density remained constant during that unit of time.

Notice there is a problem: your probability of dying when you already died before is rather problematic. So it makes more sense to compute the probability of dying conditional on having survived thus far. $1-F(t)$ it the probability of having survived until $t$, so dividing the probabilty density by that probability, will get us the expected number of times we will die within the next unit of time conditional on not having died before. That is the hazard rate.

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