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## Survival function

The survival function (often denoted as S(t)) defines the probability of death that hasn't occurred. In simpler terms, a subject will not die before a specific period of time. The survival function ranges from 0 to 1, and it's a decreasing function.

## Hazard function

This function is the exact opposite of the survival function. It's the probability that the subject under study will die before a period. The hazard function can also be referred the instantaneous death rate or instantaneous failure rate. Medics often use the hazard function to model a patient's probability of death as a function of their age. This function is not only limited to medicine, it can be applied in the modeling of any event of interest that is time-dependent.

## Kaplan-Meier plots

The Kaplan-Meier plot is used to visualize the survival chances. It provides a graphical description of the survival function. The curve takes a step shape, given the discrete nature of survival data. On the y-axis are the survival probabilities and on the x-axis the total number of subjects. The curve decreases from the left side to the right side depicting the non-decreasing nature of survival probabilities.

## Log-rank test

Let's assume that we were testing the effects of chemotherapy treatment on cancer patients against the effect of applying no treatment. To analyze the two categories (chemotherapy and non-treatment), we would need to use the log-rank test. It's a non-parametric test ideal for comparing two survival curves. The advantage of using the log-rank test is that it's easy to calculate and has few assumptions.