Survival Function, Hazard and Hazard Ratio

Let us first discuss the survival function. The formula of the function is defined as

S(t) = P(T > t)

and represents the probability that the object survives past time t. The survival function is a non-increasing function with t ranges from 0 to ∞. When t = 0, S(t) = 1 and when t = ∞ , S(t) = S(∞) = 0. It is a smooth function theoretically but practically, events occur on a discrete time scale (days, weeks, years).

Figure 13.4 – Survival function illustrated

In this example, we go back to the cancer study that spanned 5 years. At time zero, when the study started, the survival probability was 1 or 100% but at year 5, the probability of survival was close to 0.2 or 20%.

Now we consider the Stanford heart transplant dataset. The dataset contains the information of 103 patients who participated in an experimental heart transplant program (see Figure 13.5). The patients were...