This page summarizes common parametric distributions in R, based on the R functions shown in the table below.
| Distribution | Density | CDF | Hazard | Cumulative hazard | Random sample |
|---|---|---|---|---|---|
| Exponential | dexp | pexp | flexsurv::hexp | flexsurv::Hexp | rexp |
| Weibull (AFT) | dweibull | pweibull | flexsurv::hweibull | flexsurv::Hweibull | rweibull |
| Gamma | dgamma | pgamma | flexsurv::hgamma | flexsurv::Hgamma | rgamma |
| Lognormal | dlnorm | plnorm | flexsurv::hlnorm | flexsurv::Hlnorm | rlnorm |
| Gompertz | flexsurv::dgompertz | flexsurv::pgompertz | flexsurv::hgompertz | flexsurv::Hgompertz | flexsurv::rgompertz |
| Log-logistic | flexsurv::dllogis | flexsurv::pllogis | flexsurv::hllogis | flexsurv::Hllogis | flexsurv::rllogis |
| Generalized gamma (Prentice 1975) | flexsurv::dgengamma | flexsurv::pgengamma | flexsurv::hgengamma | flexsurv::Hgengamma | flexsurv::rgengamma |
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: \(\begin{aligned} H(t) = \int_0^t h(z)dz = -log S(t) \end{aligned}\)
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: \(\ln(2)/\lambda\)
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 \(\lambda = 1/\eta\)
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 \(b\).
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: \(e^\mu\)
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 \(b\).
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
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 \(Q = \sigma\)