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,
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.
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.
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.
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
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Preston, S.H., Heuveline, P., Guillot, M.: Demography: Measuring and Modeling Population Processes. Blackwell Malden, MA (2000)
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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Moore, D.F. (2016). Basic Principles of Survival Analysis. In: Applied Survival Analysis Using R. Use R!. Springer, Cham. https://doi.org/10.1007/978-3-319-31245-3_2
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