Introduction

Survival analysis is one of the most important branches of biostatistics and actuarial science, as it focuses on the study of data related to the time until the occurrence of a specific event. Traditionally, this event was death, and the analysis was mainly used in medical research to estimate the time from a patient’s treatment until their death. For this reason, the method was named “survival analysis.” In modern science, survival analysis has gained much broader applications. Apart from medicine, where it is used to estimate prognosis in chronic and fatal diseases, it is also applied in engineering to evaluate the time until a machine fails, in agriculture to analyze the time until a tree produces fruit, and in many other scientific fields. Its usefulness lies in its ability to provide tools for prediction and comparison of different interventions or conditions.

Survival Analysis Functions

In survival analysis, two main functions describe the behavior of lifetime duration: the survival function and the hazard function.

Survival Function

The survival function ST(x)S_T(x)ST​(x) is defined as the probability that an individual survives beyond age xxx. Mathematically, it is expressed as ST(x)=P(T>x)S_T(x) = P(T > x)ST​(x)=P(T>x) for every 0≤x≤ω0 \leq x \leq \omega0≤x≤ω, where ω\omegaω represents the maximum possible lifespan. This function is decreasing, since the probability of survival reduces as age increases. It is also continuous, with two known boundary values: ST(0)=1S_T(0) = 1ST​(0)=1, since at age 0 all newborns are alive, and ST(ω)=0S_T(\omega) = 0ST​(ω)=0, since it is considered certain that beyond the limiting age death occurs. The graphical representation of this function is known as the survival curve.

Hazard Function

The hazard function expresses the probability that the event, such as death, will occur immediately after a given time point, provided that the individual has survived up to that time. It represents the intensity with which the event occurs and is crucial in the mathematical modeling of risk.

Notation

In actuarial science and biostatistics, specific notations are established to ensure clarity. The variable xxx represents the age of an individual, while the random variable TTT represents lifetime, that is, age at death. The set of values of TTT is the interval [0,ω][0, \omega][0,ω]. The age ω\omegaω is called the limiting age and is usually considered to be around 110 or 120 years. These notations allow the development of mathematical models and the construction of survival tables that are widely used in practice.

Survival Tables

Survival tables are the main tool of survival analysis, as they provide a detailed picture of the probability of survival of a population at each point in time. They are widely used in clinical trials and can be constructed with different methods. The actuarial method relies on data uniformly distributed over time and requires a large number of participants. However, it faces difficulties when the time intervals are large or when there are many censored observations, which may reduce accuracy. On the other hand, the Kaplan-Meier method, the most widely used technique for survival estimation, does not rely on predefined time intervals. It calculates the survival probability each time an event occurs, providing more accurate results even in small samples. Although it does not take into account censored data in the same way, its simplicity and reliability make it a fundamental tool in biostatistics.

Methods of Analysis and Comparison

For comparing different therapeutic interventions or patient groups, survival curves and statistical tests are employed. The most commonly used is the log-rank test, which allows statistical comparison between two survival curves and the identification of significant differences in prognosis between groups. In addition, the Cox proportional hazards model provides a more generalized method, allowing for the inclusion of multiple covariates such as treatment type, age, or gender in survival estimation. This model provides hazard ratio estimates and is considered a cornerstone tool in epidemiological research and clinical data analysis.

Conclusions

Survival analysis is a fundamental tool for scientific research, as it enables the study and prediction of the time until important events occur. It is applied not only in medicine and epidemiology but also across many other scientific and technological domains. By employing functions such as the survival and hazard functions, as well as methods like Kaplan-Meier and the Cox model, researchers can better understand survival dynamics and evaluate therapeutic interventions or varying conditions. Thus, survival analysis is not merely a statistical technique but also a decision-making tool of great importance for the advancement of science.