Table 1 Survival, hazard, and cumulative hazard functions with interpretations for various survival models.
Model | Survival Function S(t)S(t) | Hazard Function h(t)h(t) | Cumulative Hazard H(t)H(t) |
|---|---|---|---|
Exponential | \(\:{e}^{-\lambda\:t}\) | \(\:\lambda\:\) | \(\:\lambda\:t\) |
Weibull | \(\:{e}^{-(\lambda\:t{)}^{k}}\) | \(\:k{\lambda\:}^{k}{t}^{k-1}\) | \(\:(\lambda\:t{)}^{k}\) |
Gaussian (Normal) | \(\:\text{(}1-{\Phi\:}\left(\frac{t-\mu\:}{\sigma\:}\right)\) | \(\:\frac{f\left(t\right)}{S\left(t\right)}=\frac{\frac{1}{\sigma\:\sqrt{2\pi\:}}{e}^{-\frac{(t-\mu\:{)}^{2}}{2{\sigma\:}^{2}}}}{1-{\Phi\:}\left(\frac{t-\mu\:}{\sigma\:}\right)}\) | \(\:-\text{log}\left(1-{\Phi\:}\left(\frac{t-\mu\:}{\sigma\:}\right)\right)\) |
Log-Logistic | \(\:\frac{1}{1+(\lambda\:t{)}^{k}}\) | \(\:\frac{k{\lambda\:}^{k}{t}^{k-1}}{1+(\lambda\:t{)}^{k}}\) | \(\:\text{l}\text{o}\text{g}(1+(\lambda\:t{)}^{k})\) |
Logistic | \(\:\frac{1}{1+{e}^{\frac{t-\mu\:}{\sigma\:}}}\) | \(\:\frac{{e}^{\frac{t-\mu\:}{\sigma\:}}}{\sigma\:\left(1+{e}^{\frac{t-\mu\:}{\sigma\:}}\right)}\) | \(\:\text{log}\left(1+{e}^{\frac{t-\mu\:}{\sigma\:}}\right)\) |
Log-Gaussian (Log-Normal) | \(\:1-{\Phi\:}\left(\frac{\text{log}t-\mu\:}{\sigma\:}\right)\) | \(\:\frac{f\left(t\right)}{S\left(t\right)}=\frac{\frac{1}{t\sigma\:\sqrt{2\pi\:}}{e}^{-\frac{(\text{l}\text{o}\text{g}t-\mu\:{)}^{2}}{2{\sigma\:}^{2}}}}{1-{\Phi\:}\left(\frac{\text{log}t-\mu\:}{\sigma\:}\right)}\) | \(\:-\text{log}\left(1-{\Phi\:}\left(\frac{\text{log}t-\mu\:}{\sigma\:}\right)\right)\) |
