This summarizes some very basic concepts in survival analysis from this amazing tutorial, which helps me a lot.

Chapter 1: Basic Knowledge and Survival Data

Basic Definitions

Failure/Event time random variables $T > 0$. T can be both discrete or continuos.

Censored random variables $U$ is a non-negative variable indicating the censored time.

Censored failure/event time random variable $X$ is defined as

$$X = \min(T, U)$$

Issues with Survival Data

• staggered entry:
  • people do not attend the study at the same time
• censoring
  • people may experience no event until the end of study
  • people may drop out in the middle of the study

Censoring is the main problem we care about.

Tyeps of Censoring

There are two ways of categorizing censoring.

The first way is

• Right censoring
• Left censoring
• Interval censoring

The second way is

• Independent censoring
• Informative censoring

Right Censoring

This is the most common type of censorng.

Only $X_i = \min(T_i, U_i)$ is observed.

In addition the $X_i$, we also know the failure indicator $\delta_i$

$$\delta_i = \begin{cases} 1,& \text{if } T_i\leq U_i \text{ failure }\\ 0, & \text{if } T_i > U_i \text{ censored } \end{cases}$$

Left Censoring

Only $X_i = \max(T_i, U_i)$ is observed and the failure indicater is defined as

$$\delta_i = \begin{cases} 1,& \text{if } U_i \leq T_i \text{ failure }\\ 0, & \text{if } U_i > T_i \text{ censored } \end{cases}$$

That is, we know the event happens before a certain time point, but have no idea of the exact event time.

Interval Censoring

$(L_i, R_i)$ is observed, where $T_i \in (L_i, R_i)$.

Independent Censoring

Censoring is independent if $U_i$ is independent o $T_i$.

Informative Censoring

Distrobution of $U_i$ caontains information about the parameters characterizing the distribution of $T_i$.

Chapter 2 Survival Distribution

Definitions

These four terms below are the keys in survival analysis.

• density function: $f(t)$
• survival function: $S(t)$
• hazard function: $\lambda(t)$
• cumulative hazard function: $\Lambda(t)$

Density Function $f(t)$

For Discrete R.V.

Suppose $T \in \{a_1, a_2,...,a_n\}$

$$f(t) = Pr(T=t) = \begin{cases} f_j,& \text{if } t = a_j, j = 1,2,..,n \\ 0,& \text{if } t \neq a_j, j = 1,2,..,n \\ \end{cases}$$

For Continuos R.V.

$$f(t) = \lim_{t\rightarrow0} \frac{Pr(t \leq T \leq t + \Delta t)}{\Delta t}$$

Survival Function $S(t)$

$S(t) = Pr(T \geq t) = 1 - F(t)$.

For Discrete R.V.

$$S(t) = \int_{t}^{\infty} f(u) du$$

For Continuos R.V.

We have

$$S(t) = \sum_{\{j\vert a_j \geq t\}} f_j$$

Hazard Function $\delta(t)$

Hazard function is sometimes called instantaneous failure rate. It represents the probability of some event’s happening right now, given the fact that the event has not happened yet. The hazard function is restricted to be non-negative $\delta(t) \geq 0$.

It is define as

For Continuos R.V.

$$\begin{split} \lambda(t) & = \lim_{\Delta t \rightarrow 0} \frac{1}{\Delta t} Pr(t \leq T < t + Delta t \vert T \geq t) \\ & = \lim_{\Delta t \rightarrow 0} \frac{1}{\Delta t} \frac{Pr(t \leq T < t + Delta t \cap T \geq t)}{Pr(T \geq t)} \\ & = \frac{f(t)}{S(t)} \end{split}$$

For Discrete R.V.

$$\begin{split} \lambda(a_j) & = Pr(T = a_j|T \geq a_j) \\ & = \frac{Pr(T = a_j)}{T \geq a_j} \\ & = \frac{f(a_j)}{S(a_j)} \end{split}$$

Some example hazard shapes

• Increasing: age after 65
• Decreasing: after effective treatment
• Bathtub: age-specific mortality
• Constant: patients with advanced chronic disease

Cumulative Hazard Function $\Delta(t)$

As time $t \rightarrow$, $\Delta(t) \rightarrow \infty$.

For Continuos R.V.

$$\Lambda(t) = \int_{0}^t \lambda(u) du$$

For Discrete R.V.

$$\Lambda(t) = \sum_{\{j \vert a_j \leq t\}} \lambda_j$$

Relationship: $(S(t), \lambda(t))$ and $(S(t), \Lambda(t))$

$S(t)$ and $\lambda(t)$

For continuos r.v., We know that

• $\lambda(t) = \frac{f(t)}{S(t)}$.
• for left-continuos survival function $S(t)$, $f(t) = - S'(t)$

Then it is easy to show that

$$- \frac{d (log S(t)) }{d t} = \frac{f(t)}{S(t)} = \lambda{t}$$

That is, $\lambda{t} = - \frac{d (log S(t)) }{d t}$

For discrete r.v.,

$$\lambda_j = \frac{f(a_j)}{S(a_j)} = \frac{S(a_j) - S(a_{j+1})}{S(a_j)}$$

$S(t)$ and $\Lambda(t)$

Continuous case

$$\begin{split} \Lambda(t) & = \int_{0}^t \lambda(u) du\\ & = \int_{0}^t \frac{f(u)}{S(u)} du\\ & = \int_{0}^t - \frac{d(log S(u))}{du} du \\ & = -log S(t) + log S(0) \\ & = -log S(t) \end{split}$$

That is

$S(t) = e^{-\Lambda(t)}$.

Remember that we require that $\Lambda(t \rightarrow \infty) = \infty$, then we can guanrantee that $S(t \rightarrow \infty) = e^{-\Lambda(t \rightarrow \infty)} \rightarrow 0$. This makes lots of sense, as people cannot live forever.

In summary, More hazard cumulative, less chance of survival.

Discrete case

Suppose that $a_j < t \leq a_{j+1}$

$$\begin{split} S(t) & = Pr(T \geq a_{j+1}) \\ & = Pr(T \geq a_{1}, T \geq a_{2}... T \geq a_{j+1}) \\ & = Pr(T \geq a_{1}) Pr(T \geq a_{2}\vert \geq a_{1})... Pr(T \geq a_{j+1} \vert \geq a_{j}) \\ & = 1 \times \prod_{k=1...j} (1 - \lambda_{k})\\ & = \prod_{\{k\vert a_k < t \}} (1 - \lambda_{k}) \end{split}$$

The equations for discrete case and continuos case here are different. Therefore, instead of using the definition that $\Lambda(t) = \sum_{\{j \vert a_j \leq t\}} \lambda_j$, Cox defines that

$$\Lambda(t) = \sum_{\{j\vert a_j < t\}} log(1-\lambda_k)$$

so that $S(t) = e^{-\Lambda(t)}$ holds for discrete case too.

Reference

[1] http://www.amstat.org/chapters/northeasternillinois/pastevents/presentations/summer05_Ibrahim_J.pdf [2] http://en.wikipedia.org/wiki/Failure_rate