Hazard and density function in survival analysis with discrete time

Question

I am running a survival analysis with descrete time. For that purpose I use the R package survival with this function

surv.km <- survfit(formula = Surv(analyse$Time, analyse$Event) ~ 1, conf.type = "log",
  conf.int = 0.95, type = "kaplan-meier", error = "greenwood", data = analyse)

In the following the terms, notation and symbols from Wikipedia are used.

I can plot the survival function S(t), the event function resp. cumulative density function F(t) (fun="event"), the cumulativ hazard function H(t) (fun="cumhaz") and some other functions.

However, is there a way to calculate the density function f(t) or the hazard function h(t)? Both are actually defined for continuous time. At the moment I use the following formulas:

$f(t) = F(t+1) - F(t) = S(t) - S(t+1)$ where $t$ is discrete

$h(t) = \frac{f(t)}{S(t)} = 1 - \frac{S(t+1)}{S(t)}$ where $t$ is discrete

Does this make sense and is mathematical well founded? References to books or papers are welcome!

Answer 1

According to Survival Analysis (Techniques for Censored and Truncated Data ), Klein and Moeschberger pg. 30-31, 1997, the discrete time density function would be

$P(X=t)= P(t-1<X\leq t) = P(X\leq t) - P(X\leq t-1) = F(t) - F(t-1) = (1-S(t))- (1-S(t-1)) = S(t-1)-S(t)$ S(0)=1

Likewise, the discrete hazard is defined to be

$P(X=t \mid X \geq t) = P( X=t \mid X> t-1) = \dfrac{P(X=t , X>t-1)}{P(X>t-1)} = \dfrac{P(X=t )}{S(t-1)}= \dfrac{S(t-1)-S(t)}{S(t-1)}=1-\dfrac{S(t)}{S(t-1)}$