Survival Analysis P2: Survival and Hazard Functions

Self-Study Notes for Survival Analysis

Survival Function

We often work with the survival function, the probability that an individual has not experienced an event by time \(t\). It is defined \[S(t) = 1 - F(t) = P(T > t)\] where \(F(t)\) is the CDF for a random event time \(T\).

At \(t = 250\) days, we can calculate the survival function \(S(250) = 0.153\), indicating the probability of survival past \(250\) days is only \(15.3 \%\). Equivalently, \(84.7 \%\) of patients died prior to day \(t = 250\). Out of the initial \(n = 119\) patients, only \(18\) are still alive and participating the study.

summary(fit, times = 250)

Call: survfit(formula = Surv(time, status) ~ 1, data = veteran, conf.type = "plain")

 time n.risk n.event survival std.err lower 95% CI upper 95% CI
  250     18     111    0.153  0.0326       0.0892        0.217

Note the staircase pattern to the survival function. At each occurrence of event, the survival function drops by \(1/119\). The crosses indicates the censor times. The survival function does not drop at censored times immediately, but it does influence the size of the drop at the next observed event time as they are no longer in the denominator (the risk set). As time moves forward, the survival probability can either drop or stay the same.

Hazard function

We are often interested in how the instantaneous risk of an event, aka the hazard of an event, changes over time, how it differs between groups, and how it depends on covariates. The individual-level hazard function at time \(t\), conditioned on covariates \(x_i\), and parameterized by a vector \(\theta\), is defined as \[h_i(t; \theta) \equiv \lim_{h \to 0} \frac{P(t \leq \tilde{T}_i < t + h \mid \tilde{T}_i \geq t, x_i; \theta)}{h} \tag{1}\]

We may also refer to its more generalized version for some derivation \[h(t) = \lim_{\Delta t \to 0} \frac{P(t \leq T < t + \Delta t \mid T \geq t)}{\Delta t} \tag{2}\]

This definition captures the instantaneous risk that an event occurs at time \(t\) for individual \(i\),assuming the individual has survived up to that point. It is: - Conditional on the individual’s covariates \(x_i\), such as age, sex, treatment status - Parameterized by a vector \(\theta\), and - Defined as a limit, where the length of the time interval approaches zero, yielding an instantaneous event rate at time \(t\)

Using the same dataset, note the hazard function is flat when the survival function is also flat (\(t = 620\) onward) and is greater whenever the survival function is steeper.

Cumulative Hazard Function

The cumulative hazard function \(H(t)\) is the accumulated risk of an event occuring up to time \(t\). It is defined by \[H(t) = \int_0^t h(x) dx\]

Relationships between Survival and Hazard Functions

Note

Exercise: Hazard and survival function.

Show that the hazard function \(h(t) = \dfrac{f(t)}{S(t)}\).

Note

Exercise: Hazard and survival function.

Show that the survival function \(S(t) = \exp{\{- \int_0^t h(x)dx\}}\).