mathematics//statistics//survival analysis

Survival analysis is the branch of statistics that studies the time until an event happens (a machine fails, a patient relapses, a customer leaves), and it is used wherever the data are durations, many of which are still running when the data are collected. Engineers call the same methods reliability or life data analysis; medicine calls them survival; the mathematics is one.

Survival analysis is the branch of statistics that studies the time until an event happens (a machine fails, a patient relapses, a customer leaves), and it is used wherever the data are durations, many of which are still running when the data are collected. Engineers call the same methods reliability or life data analysis; medicine calls them survival; the mathematics is one.

Its central object is the survival function, the fraction of the population still free of the event at time ttt:

S(t)=P(T>t),S(t)=P(T>t),S(t)=P(T>t),

with TTT the random time to the event. It starts at 1 and falls toward 0, and its shape says when the population thins out. Its companion, the hazard function, gives the instantaneous risk for those still alive; the two carry the same information, one as a stock and the other as a rate, and in reliability SSS is written RRR (reliability).

What makes the field a field of its own is that most durations are incomplete. A bearing replaced at 8,000 hours while still healthy, a sensor still running when the study ends, a pump sold with the plant: each is known to have lasted at least that long, and nothing more. Treating such censored data as failures, or throwing them away, biases every estimate, and the methods of the field exist to use them correctly.

The Kaplan-Meier estimator draws the survival function from data without assuming a shape: at each failure time it multiplies the running survival by the share of units at risk that survived that instant, counting censored units in the risk set until they leave. It is the honest first plot of any life dataset.

Parametric models assume a shape and fit its parameters by maximum likelihood, with censored units contributing their survival probability instead of a density: the exponential distribution for a constant hazard, the Weibull distribution for one that rises or falls.

When covariates matter (load, temperature, supplier), the Cox proportional-hazards model scales a baseline hazard by a factor per covariate, which says how much a hot environment shortens life without assuming the baseline's shape. Python's lifelines library implements all three.