# GenF: Generalized F distribution In flexsurv: Flexible Parametric Survival and Multi-State Models

## Description

Density, distribution function, hazards, quantile function and random generation for the generalized F distribution, using the reparameterisation by Prentice (1975).

## Usage

 ``` 1 2 3 4 5 6 7 8 9 10 11``` ```dgenf(x, mu = 0, sigma = 1, Q, P, log = FALSE) pgenf(q, mu = 0, sigma = 1, Q, P, lower.tail = TRUE, log.p = FALSE) Hgenf(x, mu = 0, sigma = 1, Q, P) hgenf(x, mu = 0, sigma = 1, Q, P) qgenf(p, mu = 0, sigma = 1, Q, P, lower.tail = TRUE, log.p = FALSE) rgenf(n, mu = 0, sigma = 1, Q, P) ```

## Arguments

 `x, q` Vector of quantiles. `mu` Vector of location parameters. `sigma` Vector of scale parameters. `Q` Vector of first shape parameters. `P` Vector of second 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 y ~ F(2*s1, 2*s2), and w = log(y)w = log(y) then x = exp(w*sigma + mu) has the original generalized F distribution with location parameter mu, scale parameter sigma>0 and shape parameters s1,s2.

In this more stable version described by Prentice (1975), s1,s2 are replaced by shape parameters Q,P, with P>0, and

s1 = 2 / (Q^2 + 2P + Q*delta), s2 = 2 / (Q^2 + 2P - Q*delta)

equivalently

Q = (1/s1 - 1/s2) / (1/s1 + 1/s2)^{-1/2}, P = 2 / (s1 + s2)

Define delta = (Q^2 + 2P)^{1/2}, and w = (log(x) - mu)delta / sigma, then the probability density function of x is

f(x) = (delta (s1/s2)^{s1} e^{s1 w}) / (sigma t (1 + s1 e^w/s2) ^ {(s1 + s2)} B(s1, s2))

f(x) = (delta (s1/s2)^{s1} e^{s1 w}) / (sigma t (1 + s1 e^w/s2) ^ {(s1 + s2)} B(s1, s2))

The original parameterisation is available in this package as `dgenf.orig`, for the sake of completion / compatibility. With the above definitions,

```dgenf(x, mu=mu, sigma=sigma, Q=Q, P=P) = dgenf.orig(x, mu=mu, sigma=sigma/delta, s1=s1, s2=s2)```

The generalized F distribution with `P=0` is equivalent to the generalized gamma distribution `dgengamma`, so that `dgenf(x, mu, sigma, Q, P=0)` equals ```dgengamma(x, mu, sigma, Q)```. The generalized gamma reduces further to several common distributions, as described in the `GenGamma` help page.

The generalized F distribution includes the log-logistic distribution (see `Llogis`) as a further special case:

```dgenf(x, mu=mu, sigma=sigma, Q=0, P=1) = dllogis(x, shape=sqrt(2)/sigma, scale=exp(mu))```

The range of hazard trajectories available under this distribution are discussed in detail by Cox (2008). Jackson et al. (2010) give an application to modelling oral cancer survival for use in a health economic evaluation of screening.

## Value

`dgenf` gives the density, `pgenf` gives the distribution function, `qgenf` gives the quantile function, `rgenf` generates random deviates, `Hgenf` retuns the cumulative hazard and `hgenf` the hazard.

## Note

The parameters `Q` and `P` are usually called q and p in the literature - they were made upper-case in these R functions to avoid clashing with the conventional arguments `q` in the probability function and `p` in the quantile function.

## Author(s)

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

## References

R. L. Prentice (1975). Discrimination among some parametric models. Biometrika 62(3):607-614.

Cox, C. (2008). The generalized F distribution: An umbrella for parametric survival analysis. Statistics in Medicine 27:4301-4312.

Jackson, C. H. and Sharples, L. D. and Thompson, S. G. (2010). Survival models in health economic evaluations: balancing fit and parsimony to improve prediction. International Journal of Biostatistics 6(1):Article 34.

`GenF.orig`, `GenGamma`