# Questions

### 1. Explaining the derivation of the hazard function

A bit of context

The hazard function is often found stated in brevity as:

$$h(t)=\frac{f(t)}{S(t)}$$

where $$f(\cdot)$$ is the probability density function, and $$S(\cdot)$$ is the survival function. Throughout this question I will be referring the descriptions given by Rodríguez and Tian.

Traditionally the survival and hazard functions come into play when the random variable $$T$$ is non-negative and continuous. In this sense, at least the concept of the survival function is remarkably straight forward being the probability that $$T$$ is greater than $$t$$.

$$S(t)= 1-F(t) = P(T>t)$$

This is especially intuitive when put in context, e.g. the time following diagnosis of a disease until death.

From the definition of the hazard function above, it is clear that it is not a probability distribution as it allows for values greater than one.

My confusion comes in at Rodríguez's definition:

$$h(t) = \lim\limits_{dt\rightarrow0}\frac{P(t\leq T

This to me, really only reads in a manner that makes sense in context, e.g. the numerator being the probability that the diagnosed person dies in some increment of time ($$dt$$) following some passage of time $$t$$, given that they have lived at least so long as the passage of time $$t$$ (or simpler, if it has been $$t$$ time since diagnosis, the probability that you'll die within the next $$dt$$ time).

Confusion starts here:

Prior to the definition of equation (7.3) he states:

"The conditional probability in the numerator may be written as the ratio of the joint probability that $$T$$ is in the interval $$[t,t+dt)$$ and $$T\geq t$$ (which is, of course, the same as the probability that $$t$$ is in the interval), to the probability of the condition $$T\geq t$$. The former may be written as $$f(t)dt$$ for small $$dt$$, while the latter is $$S(t)$$ by definition"

My confusion comes from the following:

1. in my exposure, joint distortions come from two random variables, not one as is the case here, $$T$$.

If I just accept that can be the case, I can then use a rule from conditional probability $$P(A\cap B)=P(A|B)P(B)$$ to restructure the numerator:

$$P(t \leq T < t+dt | T \geq t) = \frac{P(t \leq T < t+dt \cap T\geq t)}{P(T\geq t)}$$

then substitute back in to get: $$h(t) = \lim\limits_{dt\rightarrow0} = \frac{P(t \leq T < t+dt \cap T\geq t)}{P(T\geq t)dt}$$

1. it is stated matter of fact that $$P(t \leq T < t+dt \cap T\geq t)$$ may be written as $$f(t)dt$$ for small $$dt$$. How?

2. What does passing to the limit mean?

3. The claim is made that $$h(t) = -\frac{d}{dt}\log{S(t)}$$, while possibly trivial I would appreciate to see this calculation.

4. What are the units of the hazard function (other than a vaguely defined likelihood)?

### 2. Time-independent random variables for hazard functions?

Since the hazard function is often used in a time-dependent manner, can one use it for a time-indenepent continuous random variable?

2. By definition $$f_T(t) = \frac {d} {dt} F_T(t) = \lim_{\Delta t \to 0} \frac {F_T(t+\Delta t) - F_T(t)} {\Delta t} = \lim_{\Delta t \to 0} \frac {\Pr\{t < T \leq t + \Delta t\}} {\Delta t}$$ Therefore you claim that $\Pr\{t < T \leq t + \Delta t\} \approx f_T(t)\Delta t$ as $\Delta t$ is small. Also note $$\Pr\{t < T \leq t + \Delta t \cap T > t\} = \Pr\{t < T \leq t + \Delta t\}$$ as $t < T \leq t + \Delta t$ is a subset of $T > t$
4. It depends on your fundamental definition of $h(t)$: $$h(t) = \frac {f(t)} {S(t)} = \frac {1} {S(t)} \frac {d} {dt} F(t) = \frac {1} {S(t)} \frac {d} {dt} [1 - S(t)] = -\frac {1} {S(t)} \frac {d} {dt} S(t) = - \frac {d} {dt} \ln S(t)$$