# Survival Distributions in R

### Overview

This page summarizes common parametric distributions in R, based on the R functions shown in the table below.

Parametric survival distributions in R
DistributionDensityCDFHazardCumulative hazardRandom sample
Exponentialdexppexpflexsurv::hexpflexsurv::Hexprexp
Weibull (AFT) dweibullpweibullflexsurv::hweibullflexsurv::Hweibullrweibull
Lognormal dlnormplnormflexsurv::hlnormflexsurv::Hlnormrlnorm
Gompertz flexsurv::dgompertzflexsurv::pgompertzflexsurv::hgompertzflexsurv::Hgompertzflexsurv::rgompertz
Log-logistic flexsurv::dllogisflexsurv::pllogisflexsurv::hllogisflexsurv::Hllogisflexsurv::rllogis
Generalized gamma (Prentice 1975) flexsurv::dgengammaflexsurv::pgengammaflexsurv::hgengammaflexsurv::Hgengammaflexsurv::rgengamma

### General Survival Distributions

Survival function: \begin{aligned} S(t) = Pr(T > t) = \exp\left(-H(t)\right) \end{aligned}

Hazard function: \begin{aligned} h(t) = f(t)/S(t) \end{aligned}

Cumulative hazard function: \begin{aligned} H(t) = \int_0^t h(z)dz = -log S(t) \end{aligned}

### Exponential Distribution

Notation: $\lambda > 0$ (rate)

Density: $f(t) = \lambda e^{-\lambda t}$

Survival: $S(t) = e^{-\lambda t}$

Hazard: $h(t) = \lambda$

Cumulative hazard: $h(t) = \lambda t$

Mean: $1/\lambda$

Median: $\ln(2)/\lambda$

### Weibull Distribution

Notation: $\kappa > 0$ (shape), $\eta > 0$ (scale), $\Gamma(x)$ = gamma function

Density: $f(t) = \frac{\kappa}{\eta}\left(\frac{t}{\eta}\right)^{\kappa -1}e^{-(x/\eta)^\kappa}$

Survival: $S(t) = e^{-(t/\eta)^\kappa}$

Hazard: $h(t) = \frac{\kappa}{\eta}\left(\frac{t}{\eta}\right)^{\kappa -1}$

Cumulative hazard: $H(t) = \left(\frac{t}{\eta}\right)^\kappa$

Mean: $\eta \Gamma(1 + 1/\kappa)$

Median: $\eta (\ln(2))^{1/\kappa}$

Notes: The exponential distribution is a special case of the Weibull with $\kappa = 1$ and $\lambda = 1/\eta$

### Gamma Distribution

Notation: $a > 0$ (shape), $b > 0$ (rate), $\gamma(k, x) = \int_0^x z^{k-1}e^{-z}dz$ is the lower incomplete gamma function

Density: $f(t) = \frac{b^a}{\Gamma(a)}t^{a -1}e^{-bt}$

Survival: $S(t) = 1 - \frac{\gamma(a, bt)}{\Gamma(a)}$

Mean: $a/b$

Notes: When $a=1$, the gamma distribution simplifies to the exponential distribution with rate parameter $b$.

### Lognormal Distribution

Notation: $\mu \in (-\infty, \infty)$ (mean), $\sigma > 0$ (standard deviation), $\Phi(t)$ is the CDF of the standard normal distribution

Density: $f(t) = \frac{1}{t\sigma\sqrt{2\pi}}e^{-\frac{(\ln t - \mu)^2}{2\sigma^2}}$

Survival: $1- \Phi\left(\frac{\ln t - \mu}{\sigma}\right)$

Mean: $e^{\mu + \sigma^2/2}$

Median: $e^\mu$

### Gompertz Distribution

Notation: $a \in (-\infty, \infty)$ (shape), $b > 0$ (rate)

Density: $f(t) = be^{at}\exp\left[-\frac{b}{a}(e^{at}-1)\right]$

Survival: $S(t) = \exp\left[-\frac{b}{a}(e^{at}-1)\right]$

Hazard: $h(t) = be^{at}$

Cumulative Hazard: $H(t) = \frac{b}{a}\left(e^{at}-1\right)$

Median: $\frac{1}{b}\ln\left[-(1/a)\ln(1/2) + 1\right]$

Notes: When $a=0$ the Gompertz distribution is equivalent to the exponential with constant hazard and rate $b$.

### Log-logistic Distribution

Notation: $\kappa>0$ (shape), $\eta > 0$ (scale)

Density: \begin{aligned} f(t) =\frac{(\kappa/\eta)(t/\eta)^{\kappa-1}}{\left(1 + (t/\eta)^\kappa\right)^2} \end{aligned}

Survival: \begin{aligned} S(t) = \frac{1}{(1+(t/\eta)^\kappa)} \end{aligned}

Hazard: \begin{aligned} h(t) =\frac{(\kappa/\eta)(t/\eta)^{\kappa-1}}{\left(1 + (t/\eta)^\kappa\right)} \end{aligned}

Median: $\eta$

Mean: If $\kappa > 1$, \begin{aligned} \frac{\eta (\pi/\kappa)}{\sin(\pi/\kappa)}; \end{aligned} else undefined

### Generalized Gamma Distribution

Notation: $\mu \in (-\infty, \infty)$ (location parameter), $\sigma > 0$ (scale parameter), $Q \in (-\infty, \infty)$ (shape parameter), $w = (log(t) - \mu)/\sigma$, and $u = Q^{-2}e^{Qw}$

Density: \begin{aligned} f(t) = \frac{|Q|(Q^{-2})^{Q^{-2}}}{\sigma t \Gamma(Q^{-2})} \exp\left[Q^{-2}\left(Qw-e^{Qw}\right)\right] \end{aligned}

Survival: \begin{aligned} S(t) = \begin{cases} 1 - \frac{\gamma(Q^{-2}, u)}{\Gamma(Q^{-2})} \text{ if } Q \neq 0 \\ 1 - \Phi(w) \text{ if } Q = 0 \end{cases} \end{aligned}

Notes: Simplifies to lognormal when $Q=0$, Weibull when $Q=1$, exponential when $Q=\sigma=1$, and gamma when $Q = \sigma$