Big picture on survival analysis and life data analysis

I have heard of survival analysis and life data analysis, but don't quite get the big picture.

I was wondering what topics they are covering?

Is it pure statistics, or just application of statistics on some specific area?

Is life date analysis part of survival analysis?

Thanks and regards!

• Life data is sufficient but not necessary. Case and death counts are statistically sufficient for nonparametric estimates of survival functions. sites.google.com/site/fieldreliability Commented Apr 24, 2021 at 21:42

The concept of censoring is the key to survival analysis and life data analysis. This issue can also enter via industrial statistics. When monitoring the length of time it takes for a sample of units to fail, you can have

• Complete data: the exact time a unit fails is known
• Censored to the right: the time to fail for a unit is beyond the present run time
• Censored to the left: the known time is after the time a unit failed

Other issues that enters the data mix are

• Singly censored: all unfailed units have a common run time
• Multiply censored: the unfailed units have different run times
• Interval censored: the time to fail is known to be between a particular set of times.
• Time censored: the censoring time is fixed
• Failure censored: a test is stopped when a fixed number of units fail
• Competing failure modes: the sample units fail for different reasons

Common distributions capable of handling these situations are: lognormal, Weibull, and extreme value. The issues become interesting because there are graphical procedures to handle analysis as well as MLE and Method of Moments methods.

Systems reliability is an off-shoot of this topic which gets involved with Bayesian methods, renewal theory, and accelerated life testing. Wayne Nelson and Bill Meeker have several good books on the topics.

• (+1) Censoring is the key idea; time is just a particular (and the most common) application. For instance, Dennis Helsel has applied survival analysis to chemical measurements having detection limits (left censoring). There's no time involved, but the model and the techniques are still useful.
– whuber
Commented Apr 4, 2011 at 14:40

In survival analysis, or time-to-event analysis, the variable or interest measures the time from a starting point to a point of interest like death due to some disease. So the response variable is a positive variable which is in most cases skewed. As a consequence the usual assumption of normality fails and, for instance, the classical regression techniques are not applicable. (Though, note that sometimes a transformation of the variable could make the situation better). But the main difference is censoring: a very common feature when dealing with time-to-event data. In its most common form (right censoring), you do not know the exact time for a given individual but you do know that it is larger than some value $t^{\star}$. For example, suppose you follow a patient up to death. At time $t=10$ days, he is alive. At time $t=30$ days he is still alive but then he is lost to follow-up. Then you do not know the exact time of death but you do know that $t > 30$. Ignoring censoring is clearly not the best think to do; instead, you can record $t^{\star} = 30$ as a censored observation. Techniques of survival analysis (e.g., Kaplan-Meier estimator, Cox regression, ...) are specially designed to deal with censoring.

To my point of view Modelling Survival Data in Medical Research is a very good choice as a first book in survival analysis... but there are many others.

5, 10, 12+, 14, 17, 18+, 20+


A first approximation description of survival analysis: Analysing data where the dependent variable has (1) precise values (the complete observations) and (2) values know to be above a given threshold (the censored observations). The above may be a survival data sample, values without + are precisely known; values with + are known to be more, but not how much more. (And there are many extensions.)

As schenectady pointed out, censoring is the key issue in survival analysis. Without censored observations in your dataset, your task has no difference with a simple regression task. This also suggests that the variable of interest $T$ is not restricted to time, i.e., it can be wage, price and so on.

Therefore, it may be more appropriate to call it censoring regression or censoring analysis, instead of survival analysis, I suppose.

Mathematically speaking, survival analysis (sometimes called reliability analysis) studies the stochastic behaviour of various kinds of quantities relating to non-negative random variables $$T_1,...,T_n$$ that we call "survival times" or "times-to-failure". The analysis usually involves observation of the survival status or survival times (sometimes censored) of a set of people/objects, and an attempt to make an inference to the underlying "hazard" operating on those objects.

Survival analysis is usually framed in terms of times for survival of people/objects in a hazardous process. However, mathematically speaking, it is just probabilistic analysis on a set of non-negative random variables; in principle, these could represent something other than time (e.g., spacial distance from an origin, etc.). Consequently, it is possible for the mathematics of survival analysis to be used to problems that have nothing to do with survival, but which involve the stochastic analysis of a set of random non-negative quantities.

A rough introduction to the setup and scope of survival analysis: The usual mathematical set-up of survival analysis goes like this. Suppose we have non-negative times-to-failure $$T_1,...,T_n \sim \text{IID Dist}$$ for a set of $$n$$ people/objects/components. Depending on which representation is convenient in a particular context, the unknown distribution of the time-to-failure can be represented equivalently by its CDF, its survival function, its density/mass function, or its hazard function. (Since we are dealing with non-negative random variables, the survival function and hazard function are particularly useful ways of expressing the distribution.)

Standard survival analysis problems involve an attempt to estimate the underlying distribution of the times-to-failure (or at least some probabilities or moments from this distribution) using various kinds of observations relating to the times. Standard situations examined in survival analysis are the following observational/inference problems:

• Observing failure/non-failure: We observe the times-to-failure $$t_1,...,t_n$$, and we use these to try to estimate the underlying distribution.

• Observing with right/left censorship: For some/all of the objects of interest, we observe a censored version of the time-to-failure, such as $$\max(T_i, t_0)$$ (left censoring) or $$\min(T_i, t_0)$$ (right censoring), and we use these to try to estimate the underlying distribution.

• Observing survival indicators at a fixed time: For some/all of the objects of interest, we observe the survival indicators $$\mathbb{I}(T_i > t)$$, and we use these to try to estimate some probabilities pertaining to the underlying distribution.

• Observations of combinations of people/objects/components: For some/all cases, we observe survival outcomes only through series or parallel systems (or more complicated systems) of individual people/components. For example, in a pure series system we observe the system time-to-failure $$\min(T_1,...,T_n)$$, or in a pure parallel system we observe the system time-to-failure $$\max(T_1,...,T_n)$$. (In more complex systems we might observe some more complex function of the times-to-failure.) Here we observe a time-to-failure from a "system" and we attempt to estimate the underlying distribution for the time-to-failure of the components.

• Combinations of the above: Some complex survival problems involve combinations of the above. For example, we might observe a complex system and also have censoring issues in our observations, or we might observe some times-to-failure directly and observe others only through indicators at fixed times. Some complex problems in survival analysis involve a complicated interplay between systems/components and direct observation, censored observation, or mere survival observation (i.e., observation of indicators for times-to-failure).