Survival Analysis
Censoring
Censoring
- Surivial without Censoring
- Surivial with Censoring
Kaplan Meier Curve
More individual in each group, better sepration of the group, better p-value
- Takes censoring into account
- Estimates probabilitu of “survival” on a given day
- Conditional probability of surviving on a given day:
$$ \frac {N_{ \text{“alive” day before}} - N_{ \text{“dying” nextday}}} { \text{“alive” day before}} $$
Kaplan-Meier survival curve
- Survival times $t_1 \leq t_2 \leq \cdots \leq t_n$
- The proportion of subjects, $S(t)$, surviving beyoind any follow up time $t$ is estimated by (conditional probability):
$$ S(t) = \frac {r_1 - d_1}{r_1} \times \frac {r_2 - d_2}{r_2} \times \cdots \times \frac{r_p - d_p}{r_p} $$
where
- $t_p$ is the largest survival time less han or equal to $t$
- $r_i$ is the number of subjects alive just before time $t_i$
- $d_i$ = numebr who died at time $t_i$
- for censored obeservations $d_i = 0$
Statistic
Log Rank Test
- Compares survival times of two independent groups.
- Assumes that the relative risk of event (e.g. death) between the two groups is constant (proportional hazards)
- Ranks the survial times combined and compared observed and expected rates
Null hypothesis: the rates of events (death) in the two groups are equal
under $H_0$,
$$ X^2 = \frac { (O_A - E_A)^2}{E_A} + \frac { (O_B - E_B)^2}{E_B} \sim \chi^2 $$
- $O_A$: observed events in group A
- $E_A$: expected events in gorup A under null hypohesis
expect = (proportion in risk set) * (# of failures over both groups)
$$ e_{1j} = ( \frac{ n_{1j}}{ n_{1j} + n_{2j}}) \times ( m_{1j} + m_{2j}) $$
$$ e_{2j} = ( \frac{ n_{2j}}{ n_{1j} + n_{2j}}) \times ( m_{1j} + m_{2j}) $$
Cox Regression
- Extends comparison of survial times to allow different predictors (estimate k variable together)
- Models the
hazard: probability of dying at a point in time, given survival to that point in time
$$ H(t) = H_0(t) \times \exp(b_1X_1 + b_2X_2 + \cdots + b_kX_k ) $$
- Model links to a baseline hazard, $H_0(t)$
- Can accomodate many variables, both discrete and continuous measures of event times
- Proportional hazards assumption: the hazard for any individual is a fixed proportion of the hazard for any other individual
Hazard ratio
- Exp(B) give the hazard ratio (or relative hazard/risk)