Survival Models based on Distributions

Kinds

Exponential Distribution

Exponential Distribution assumes that the Hazard Function is a constant regardless of time. The distribution is memoryless $P(T>t+s|T>s)=P(T>t),0<s<t<\infty$.

The PDF of survival time follows Exponential Distribution $f(x)=\lambda e^{-\lambda t}$ The Hazard Function $\lambda (t)=\lambda$ The Survival Function $S(t)=e^{-\lambda t}$

Gamma Distribution

The PDF of survival time follows Gamma Distribution $f(t)=\frac{{\lambda }^{\alpha }}{\Gamma (\alpha )}{t}^{\alpha -1}{e}^{-\lambda t}$ The Hazard Function and Survival Function don’t have closed form expression.

Weibull Distribution

Weibull Distribution uses additional parameter $\alpha$ than Exponential Distribution to control the shape of the Hazard Function. If $\alpha 1$ then the hazard function is increasing, if $\alpha =1$ then it is constant, and if $0<\alpha <1$ then it is decreasing.

The PDF of survival time follows Weibull Distribution $f(t)=\alpha \lambda (\lambda t{)}^{\alpha -1}\exp [-(\lambda t{)}^{\alpha }]$ where $\lambda$ is a scale parameter and $\alpha$ is a shape parameter.

The Hazard Function $\lambda (t)=\alpha \lambda (\lambda t{)}^{\alpha -1}$ The Survival Function $S(t)=\exp [-(\lambda t{)}^{\alpha }]$

Rayleigh Distribution

The PDF of survival time follows Rayleigh distribution $f(x)=({\lambda }_0+{\lambda }_{1}t)\exp (-{\lambda }_{0}t-\frac{1}{2}{\lambda }_1{t}^2)$ The Hazard Function $\lambda (t)={\lambda }_0+{\lambda }_{1}t$ The Survival Function $S(t)=\exp (-{\lambda }_{0}t-\frac{1}{2}{\lambda }_1{t}^2)$

Log-Normal Distribution

We assume that the survival time follows a Log-Normal Distribution $T\sim Log-normal(\mu ,{\sigma }^2)$ or $\log T\sim N(\mu ,{\sigma }^2)$. The Hazard Function of the distribution is hump-shaped.

The PDF of survival time follows Log-Normal Distribution $f(t)=\frac{1}{\sqrt{2\pi \sigma }}\frac{1}{t}\exp \left(-\frac{(\log t-\mu {)}^2}{2{\sigma }^2}\right)$ The Survival Function $S(t)=1-\Phi \left(\frac{\log t-\mu }{\sigma }\right)$ where $\Phi$ is the CDF of standard normal distribution

The Hazard Function doesn’t have closed form expression.

Gompertz Distribution

The Hazard Function $\ln \lambda (t)={\lambda }_0+{\lambda }_{1}t\iff \lambda (t)=\lambda e^{\gamma t}$ where $\lambda :=e^{{\lambda }_0}$ and $\gamma ={\lambda }_1$

Gompertz-Makeham Distribution

The Hazard Function $\lambda (t)=\alpha +\lambda (t)=\lambda e^{\gamma t}$