Parametric Survival Distributions

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Description

List of distributions for accelerated failure models. These are location-scale families for some transformation of time. The entry describes the cdf F and density f of a canonical member of the family.

Usage

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survreg.distributions

Format

There are two basic formats, the first defines a distribution de novo, the second defines a new distribution in terms of an old one.

name: name of distribution
variance: function(parms) returning the variance (currently unused)
init(x,weights,...): Function returning an initial
estimate of the mean and variance
(used for initial values in the iteration)
density(x,parms): Function returning a matrix with columns F,
1-F,f,f'/f,f''/f
quantile(p,parms): Quantile function
scale: Optional fixed value for the scale parameter
parms: Vector of default values and names for any additional parameters
deviance(y,scale,parms): Function returning the deviance for a
saturated model; used only for deviance residuals.

and to define one distribution in terms of another

name: name of distribution
dist: name of parent distribution
trans: transformation (eg log)
dtrans: derivative of transformation
itrans: inverse of transformation
scale: Optional fixed value for scale parameter

Details

There are four basic distributions:extreme, gaussian, logistic and t. The last three are parametrised in the same way as the distributions already present in R. The extreme value cdf is

F=1-e^{-e^t}.

When the logarithm of survival time has one of the first three distributions we obtain respectively weibull, lognormal, and loglogistic. The location-scale parameterizaion of a Weibull distribution found in survreg is not the same as the parameterization of rweibull.

The other predefined distributions are defined in terms of these. The exponential and rayleigh distributions are Weibull distributions with fixed scale of 1 and 0.5 respectively, and loggaussian is a synonym for lognormal.

For speed parts of the three most commonly used distributions are hardcoded in C; for this reason the elements of survreg.distributions with names of "Extreme value", "Logisitic" and "Gaussian" should not be modified. (The order of these in the list is not important, recognition is by name.) As an alternative to modifying survreg.distributions a new distribution can be specified as a separate list. This is the preferred method of addition and is illustrated below.

See Also

survreg, pweibull, pnorm,plogis, pt, survregDtest

Examples

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# time transformation
survreg(Surv(time, status) ~ ph.ecog + sex, dist='weibull', data=lung)
# change the transformation to work in years
# intercept changes by log(365), everything else stays the same
my.weibull <- survreg.distributions$weibull
my.weibull$trans <- function(y) log(y/365)
my.weibull$itrans <- function(y) 365*exp(y)
survreg(Surv(time, status) ~ ph.ecog + sex, lung, dist=my.weibull)

# Weibull parametrisation
y<-rweibull(1000, shape=2, scale=5)
survreg(Surv(y)~1, dist="weibull")
# survreg scale parameter maps to 1/shape, linear predictor to log(scale)

# Cauchy fit
mycauchy <- list(name='Cauchy',
                 init= function(x, weights, ...) 
                      c(median(x), mad(x)),
                 density= function(x, parms) {
                      temp <- 1/(1 + x^2)
                      cbind(.5 + atan(x)/pi, .5+ atan(-x)/pi,
                            temp/pi, -2 *x*temp, 2*temp*(4*x^2*temp -1))
                      },
                 quantile= function(p, parms) tan((p-.5)*pi),
                 deviance= function(...) stop('deviance residuals not defined')
                 )
survreg(Surv(log(time), status) ~ ph.ecog + sex, lung, dist=mycauchy)

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