Elements of Survival Analysis

TODO:

[x] General Context

[x] Survival Function

[ ] Hazard Function

[ ] Examples

General Context

Fix some condition. We have a current best treatment method (e.g. placebo) along with a new treatment. We’d like to determine a context in which the new treatment is better than the current best method. The introduction of a new treatment is costly on multiple levels. Therefore, we want to strong, significant evidence that the the new treatment is an improvement. We should expect that we will typically say that we cannot conclude that the new treatment is an improvement.

This is of course an appropriate context for hypothesis testing.

The notion of “better” is ambiguous. For simplicity, we adopt the definition that better means an increase in survival time. Here, survival time means is the duration between the onset of symptoms and death under “real-life circumstances”.

We should keep in mind that there’s also an observation period. This observation period might not contain the time of onset and the time death. For example, the patient might have been withdrawn from the study due to negative side effects or the patient might have stopped the taking the treatment.

Note that the survival time and duration of the observation period may be dependent.

The Data

We pause to note that We should have data for each patient. This consists of a time of onset and a time of death. In other words the data of:

\[\begin{split}I &\longrightarrow \mathbb{R} \times \mathbb{R} \\ i &\longmapsto (\tau_i^0, \tau_i^1)\end{split}\]

Note that we’d like everything we save should be time-invariant. Therefore, we can use the affine structure to think of this data as being a single real number:

\[\begin{split}I &\longrightarrow \mathbb{R} \times \mathbb{R}^{>0} \\ i &\longmapsto t_i = \tau_i^0 - \tau_i^1\end{split}\]

This generates a probability distribution \(\rho\) on \(\mathbb{R}\), by pushing forward the uniform distribution on \(I\).

Survival Function

The survival function is the probability of surviving beyond a certain duration. We can express this as:

\[\begin{split}\mathbb{R}^{>0} &\overset{S}\longrightarrow \mathbb{R} \\ t &\longmapsto S(t) = 1 - \int_0^t \rho\end{split}\]

This function satisfies a few obvious properties. For example, death cannot occur instantaneously, and the probability of survival cannot increase as time progresses.

\[\begin{split}S(0) = 1 \\ \frac{\partial S}{\partial t} \leq 0\end{split}\]

Note that it’s not necessarily the case that \(S(\infty) = 0\) as the condition might not be uniformly fatal.

Moreover, the fundamental theorem of calculus asserts:

\[\rho = - \frac{\partial S}{\partial t}\]

Hazard Function

The hazard function is another approach to survival analysis. As we’ll see, in many interesting cases the hazard function admits a much simpler description.

The cumulative hazard function is the information in surviving beyond a certain duration:

\[H(t) = - \log S(t)\]

The hazard function is defined as:

\[\begin{split}h(t) &= \frac{\partial H}{\partial t} \\ &= \frac{ \rho }{S}\end{split}\]

Examples

Weibull distribution:

Hazard function

Cumulative hazard function

Survival Function