# GenGamma: Generalized gamma distribution In flexsurv: Flexible Parametric Survival and Multi-State Models

 GenGamma R Documentation

## Generalized gamma distribution

### Description

Density, distribution function, hazards, quantile function and random generation for the generalized gamma distribution, using the parameterisation originating from Prentice (1974). Also known as the (generalized) log-gamma distribution.

### Usage

```dgengamma(x, mu = 0, sigma = 1, Q, log = FALSE)

pgengamma(q, mu = 0, sigma = 1, Q, lower.tail = TRUE, log.p = FALSE)

Hgengamma(x, mu = 0, sigma = 1, Q)

hgengamma(x, mu = 0, sigma = 1, Q)

qgengamma(p, mu = 0, sigma = 1, Q, lower.tail = TRUE, log.p = FALSE)

rgengamma(n, mu = 0, sigma = 1, Q)
```

### Arguments

 `x, q` vector of quantiles. `mu` Vector of “location” parameters. `sigma` Vector of “scale” parameters. Note the inconsistent meanings of the term “scale” - this parameter is analogous to the (log-scale) standard deviation of the log-normal distribution, “sdlog” in `dlnorm`, rather than the “scale” parameter of the gamma distribution `dgamma`. Constrained to be positive. `Q` Vector of shape parameters. `log, log.p` logical; if TRUE, probabilities p are given as log(p). `lower.tail` logical; if TRUE (default), probabilities are P(X <= x), otherwise, P(X > x). `p` vector of probabilities. `n` number of observations. If `length(n) > 1`, the length is taken to be the number required.

### Details

If g ~ Gamma(Q^{-2}, 1) , and w = log(Q^2*g) / Q, then x = exp(mu + sigma w) follows the generalized gamma distribution with probability density function

f(x | mu, sigma, Q) = |Q| (Q^{-2})^{Q^{-2}} / (sigma * x * Gamma(Q^{-2})) exp(Q^{-2}*(Q*w - exp(Q*w)))

This parameterisation is preferred to the original parameterisation of the generalized gamma by Stacy (1962) since it is more numerically stable near to Q=0 (the log-normal distribution), and allows Q<=0. The original is available in this package as `dgengamma.orig`, for the sake of completion and compatibility with other software - this is implicitly restricted to `Q`>0 (or `k`>0 in the original notation). The parameters of `dgengamma` and `dgengamma.orig` are related as follows.

`dgengamma.orig(x, shape=shape, scale=scale, k=k) = `

```dgengamma(x, mu=log(scale) + log(k)/shape, sigma=1/(shape*sqrt(k)), Q=1/sqrt(k))```

The generalized gamma distribution simplifies to the gamma, log-normal and Weibull distributions with the following parameterisations:

 `dgengamma(x, mu, sigma, Q=0)` `=` `dlnorm(x, mu, sigma)` `dgengamma(x, mu, sigma, Q=1)` `=` `dweibull(x, shape=1/sigma, scale=exp(mu))` `dgengamma(x, mu, sigma, Q=sigma)` `=` ```dgamma(x, shape=1/sigma^2, rate=exp(-mu) / sigma^2)```

The properties of the generalized gamma and its applications to survival analysis are discussed in detail by Cox (2007).

The generalized F distribution `GenF` extends the generalized gamma to four parameters.

### Value

`dgengamma` gives the density, `pgengamma` gives the distribution function, `qgengamma` gives the quantile function, `rgengamma` generates random deviates, `Hgengamma` retuns the cumulative hazard and `hgengamma` the hazard.

### Author(s)

Christopher Jackson <chris.jackson@mrc-bsu.cam.ac.uk>

### References

Prentice, R. L. (1974). A log gamma model and its maximum likelihood estimation. Biometrika 61(3):539-544.

Farewell, V. T. and Prentice, R. L. (1977). A study of distributional shape in life testing. Technometrics 19(1):69-75.

Lawless, J. F. (1980). Inference in the generalized gamma and log gamma distributions. Technometrics 22(3):409-419.

Cox, C., Chu, H., Schneider, M. F. and Muñoz, A. (2007). Parametric survival analysis and taxonomy of hazard functions for the generalized gamma distribution. Statistics in Medicine 26:4252-4374

Stacy, E. W. (1962). A generalization of the gamma distribution. Annals of Mathematical Statistics 33:1187-92

`GenGamma.orig`, `GenF`, `Lognormal`, `GammaDist`, `Weibull`.