# Basic Principles of Survival Analysis

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## Abstract

Survival analysis methods depend on the survival distribution, and two key ways of specifying it are the survival function and the hazard function . The survival function defines the probability of surviving up to a point t. Formally,

$$\displaystyle{S(t) = pr(T > t),\begin{array}{cc} &0 < t < \infty \end{array} }$$

This function takes the value 1 at time 0, decreases (or remains constant) over time, and of course never drops below 0. As defined here it is right continuous.

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The original version of this chapter was revised. An erratum to this chapter can be found at DOI 10.1007/978-3-319-31245-3_13

An erratum to this chapter can be found at http://dx.doi.org/10.1007/978-3-319-31245-3_13

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## Notes

1. 1.

In some texts the survival function is defined as S(t) = Pr(T ≥ t), resulting in a left-continuous survival function. This issue arises with step function survival curves, e.g. the Kaplan-Meier estimate discussed in the next chapter.

2. 2.

To establish this formula, we also need the result that $$\mathop{\lim }\limits _{t\rightarrow \infty }\left (t \cdot S(t)\right ) = 0$$. This is easy to show for the exponential distribution, but it is non-trivial to prove in general.

## References

1. Cox, D.R., Oakes, D.: Analysis of Survival Data. Chapman and Hall/CRC, London; New York (1984)

2. Kalbfleisch, J.D., Prentice, R.L.: The Statistical Analysis of Failure Time Data, 2nd edn. Wiley, Hoboken (2002)

3. Klein, J.P., Moeschberger, M.L.: Survival Analysis: Techniques for Censored and Truncated Data, 2nd edn. Springer, New York (2005)

4. Preston, S.H., Heuveline, P., Guillot, M.: Demography: Measuring and Modeling Population Processes. Blackwell Malden, MA (2000)

5. Therneau, T.M., Offord, J.: Expected survival based on hazard rates (update). Technical Report 63, Mayo Clinic Department of Health Science Research (1999)

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