Parametric modelling or regression for time-to-event data. Several built-in distributions are available, and users may supply their own.
flexsurvreg(
formula,
anc = NULL,
data,
weights,
bhazard,
rtrunc,
subset,
na.action,
dist,
inits,
fixedpars = NULL,
dfns = NULL,
aux = NULL,
cl = 0.95,
integ.opts = NULL,
sr.control = survreg.control(),
hessian = TRUE,
hess.control = NULL,
...
)A list of class "flexsurvreg" containing information about
the fitted model. Components of interest to users may include:
A copy of the function call, for use in post-processing.
List defining the survival distribution used.
Matrix of maximum likelihood estimates and confidence limits, with parameters on their natural scales.
Matrix of maximum
likelihood estimates and confidence limits, with parameters all
transformed to the real line (using a log transform for all built-in
models where this is necessary). The
coef, vcov
and confint methods for flexsurvreg objects work on
this scale.
The transformed maximum likelihood
estimates, as in res.t. Calling coef() on a
flexsurvreg object simply returns this component.
Log-likelihood. This will differ from Stata, where the sum of the log uncensored survival times is added to the log-likelihood in survival models, to remove dependency on the time scale.
For relative survival models specified with bhazard, this is a partial
log-likelihood which omits a constant term defined by the sum of the
cumulative hazards over all event or censoring times.
Vector of individual contributions to the log-likelihood
Akaike's information criterion (-2*log likelihood + 2*number of estimated parameters)
Covariance matrix of the parameters, on
the real-line scale (e.g. log scale), which can be extracted with
vcov.
Data used in the model fit. To extract
this in the standard R formats, use use
model.frame.flexsurvreg or
model.matrix.flexsurvreg.
A formula expression in conventional R linear modelling
syntax. The response must be a survival object as returned by the
Surv function, and any covariates are given on the
right-hand side. For example,
Surv(time, dead) ~ age + sex
Surv objects of type="right","counting",
"interval1" or "interval2" are supported, corresponding to
right-censored, left-truncated or interval-censored observations.
If there are no covariates, specify 1 on the right hand side, for
example Surv(time, dead) ~ 1.
If the right hand side is specified as . all remaining variables are
included as covariates. For example, Surv(time, dead) ~ .
corresponds to Surv(time, dead) ~ age + sex if data contains
the variables time, dead, age, and sex.
By default, covariates are placed on the ``location'' parameter of the
distribution, typically the "scale" or "rate" parameter, through a linear
model, or a log-linear model if this parameter must be positive. This
gives an accelerated failure time model or a proportional hazards model
(see dist below) depending on how the distribution is
parameterised.
Covariates can be placed on other (``ancillary'') parameters by using the
name of the parameter as a ``function'' in the formula. For example, in a
Weibull model, the following expresses the scale parameter in terms of age
and a treatment variable treat, and the shape parameter in terms of
sex and treatment.
Surv(time, dead) ~ age + treat + shape(sex) + shape(treat)
However, if the names of the ancillary parameters clash with any real
functions that might be used in formulae (such as I(), or
factor()), then those functions will not work in the formula. A
safer way to model covariates on ancillary parameters is through the
anc argument to flexsurvreg.
survreg users should also note that the function
strata() is ignored, so that any covariates surrounded by
strata() are applied to the location parameter. Likewise the
function frailty() is not handled.
An alternative and safer way to model covariates on ancillary parameters, that is, parameters other than the main location parameter of the distribution. This is a named list of formulae, with the name of each component giving the parameter to be modelled. The model above can also be defined as:
Surv(time, dead) ~ age + treat, anc = list(shape = ~ sex +
treat)
A data frame in which to find variables supplied in
formula. If not given, the variables should be in the
working environment.
Optional numeric variable giving weights for each individual in the data. The fitted model is then defined by maximising the weighted sum of the individual-specific log-likelihoods.
Optional variable giving expected hazards for relative survival models. The model is described by Nelson et al. (2007).
bhazard should contain a vector of values for each person in
the data.
For people with observed events, bhazard refers to the
hazard at the observed event time.
For people whose event time is
left-censored or interval-censored, bhazard should contain the
probability of dying by the end of the corresponding interval,
conditionally on being alive at the start.
For people whose event time
is right-censored, the value of bhazard is ignored and does not
need to be specified.
If bhazard is supplied, then the parameter estimates returned by
flexsurvreg and the outputs returned by summary.flexsurvreg
describe the parametric model for relative survival.
For relative survival models, the log-likelihood returned by flexsurvreg is a partial
log-likelihood, which omits a constant term defined by the sum of the
cumulative hazards at the event or censoring time for each individual.
Hence this constant must be added if a full likelihood is needed.
Optional variable giving individual-specific right-truncation times. Used for analysing data with "retrospective ascertainment". For example, suppose we want to estimate the distribution of the time from onset of a disease to death, but have only observed cases known to have died by the current date. In this case, times from onset to death for individuals in the data are right-truncated by the current date minus the onset date. Predicted survival times for new cases can then be described by an un-truncated version of the fitted distribution.
These models can suffer from weakly identifiable parameters and
badly-behaved likelihood functions, and it is advised to compare
convergence for different initial values by supplying different
inits arguments to flexsurvreg.
Vector of integers or logicals specifying the subset of the observations to be used in the fit.
a missing-data filter function, applied after any 'subset'
argument has been used. Default is options()$na.action.
Typically, one of the strings in the first column of the following table, identifying a built-in distribution. This table also identifies the location parameters, and whether covariates on these parameters represent a proportional hazards (PH) or accelerated failure time (AFT) model. In an accelerated failure time model, the covariate speeds up or slows down the passage of time. So if the coefficient (presented on the log scale) is log(2), then doubling the covariate value would give half the expected survival time.
"gengamma" | Generalized gamma (stable) | mu | AFT |
"gengamma.orig" | Generalized gamma (original) | scale | AFT |
"genf" | Generalized F (stable) | mu | AFT |
"genf.orig" | Generalized F (original) | mu | AFT |
"weibull" | Weibull | scale | AFT |
"gamma" | Gamma | rate | AFT |
"exp" | Exponential | rate | PH |
"llogis" | Log-logistic | scale | AFT |
"lnorm" | Log-normal | meanlog | AFT |
"gompertz" | Gompertz | rate | PH |
"exponential" and "lognormal" can be used as aliases for
"exp" and "lnorm", for compatibility with
survreg.
Alternatively, dist can be a list specifying a custom distribution.
See section ``Custom distributions'' below for how to construct this list.
Very flexible spline-based distributions can also be fitted with
flexsurvspline.
The parameterisations of the built-in distributions used here are the same
as in their built-in distribution functions: dgengamma,
dgengamma.orig, dgenf,
dgenf.orig, dweibull, dgamma,
dexp, dlnorm, dgompertz,
respectively. The functions in base R are used where available,
otherwise, they are provided in this package.
A package vignette "Distributions reference" lists the survivor functions and covariate effect parameterisations used by each built-in distribution.
For the Weibull, exponential and log-normal distributions,
flexsurvreg simply works by calling survreg to
obtain the maximum likelihood estimates, then calling optim
to double-check convergence and obtain the covariance matrix for
flexsurvreg's preferred parameterisation.
The Weibull parameterisation is different from that in
survreg, instead it is consistent with
dweibull. The "scale" reported by
survreg is equivalent to 1/shape as defined
by dweibull and hence flexsurvreg. The first
coefficient (Intercept) reported by survreg
is equivalent to log(scale) in dweibull and
flexsurvreg.
Similarly in the exponential distribution, the rate, rather than the mean, is modelled on covariates.
The object flexsurv.dists lists the names of the built-in
distributions, their parameters, location parameter, functions used to
transform the parameter ranges to and from the real line, and the
functions used to generate initial values of each parameter for
estimation.
An optional numeric vector giving initial values for each unknown parameter. These are numbered in the order: baseline parameters (in the order they appear in the distribution function, e.g. shape before scale in the Weibull), covariate effects on the location parameter, covariate effects on the remaining parameters. This is the same order as the printed estimates in the fitted model.
If not specified, default initial values are chosen from a simple summary
of the survival or censoring times, for example the mean is often used to
initialize scale parameters. See the object flexsurv.dists for the
exact methods used. If the likelihood surface may be uneven, it is
advised to run the optimisation starting from various different initial
values to ensure convergence to the true global maximum.
Vector of indices of parameters whose values will be fixed
at their initial values during the optimisation. The indices are ordered
as in inits. For example, in a stable generalized Gamma model with
two covariates, to fix the third of three generalized gamma parameters
(the shape Q, see the help for GenGamma) and the
second covariate, specify fixedpars = c(3, 5)
An alternative way to define a custom survival distribution (see
section ``Custom distributions'' below). A list whose components may
include "d", "p", "h", or "H" containing the
probability density, cumulative distribution, hazard, or cumulative hazard
functions of the distribution. For example,
list(d=dllogis, p=pllogis).
If dfns is used, a custom dlist must still be provided, but
dllogis and pllogis need not be visible from the global
environment. This is useful if flexsurvreg is called within other
functions or environments where the distribution functions are also
defined dynamically.
A named list of other arguments to pass to custom distribution
functions. This is used, for example, by flexsurvspline to
supply the knot locations and modelling scale (e.g. hazard or odds). This
cannot be used to fix parameters of a distribution --- use
fixedpars for that.
Width of symmetric confidence intervals for maximum likelihood estimates, by default 0.95.
List of named arguments to pass to
integrate, if a custom density or hazard is provided without
its cumulative version. For example,
integ.opts = list(rel.tol=1e-12)
For the models which use survreg to find the
maximum likelihood estimates (Weibull, exponential, log-normal), this list
is passed as the control argument to survreg.
Calculate the covariances and confidence intervals for the
parameters. Defaults to TRUE.
List of options to control covariance matrix computation. Available options are:
numeric. If TRUE then numerical methods are used
to compute the Hessian for models where an analytic Hessian is
available. These models include the Weibull (both versions),
exponential, Gompertz and spline models with hazard or odds
scale. The default is to use the analytic Hessian for these
models. For all other models, numerical methods are always used
to compute the Hessian, whether or not this option is set.
tol.solve. The tolerance used for solve
when inverting the Hessian (default .Machine$double.eps)
tol.evalues The accepted tolerance for negative
eigenvalues in the covariance matrix (default 1e-05).
The Hessian is positive definite, thus invertible, at the maximum
likelihood. If the Hessian computed after optimisation convergence can't
be inverted, this is either because the converged result is not the
maximum likelihood (e.g. it could be a "saddle point"), or because the
numerical methods used to obtain the Hessian were inaccurate. If you
suspect that the Hessian was computed wrongly enough that it is not
invertible, but not wrongly enough that the nearest valid inverse would be
an inaccurate estimate of the covariance matrix, then these tolerance
values can be modified (reducing tol.solve or increasing
tol.evalues) to allow the inverse to be computed.
Optional arguments to the general-purpose optimisation routine
optim. For example, the BFGS optimisation algorithm is the
default in flexsurvreg, but this can be changed, for example
to method="Nelder-Mead" which can be more robust to poor initial
values. If the optimisation fails to converge, consider normalising the
problem using, for example, control=list(fnscale = 2500), for
example, replacing 2500 by a number of the order of magnitude of the
likelihood. If 'false' convergence is reported with a
non-positive-definite Hessian, then consider tightening the tolerance
criteria for convergence. If the optimisation takes a long time,
intermediate steps can be printed using the trace argument of the
control list. See optim for details.
flexsurvreg is intended to be
easy to extend to handle new distributions. To define a new distribution
for use in flexsurvreg, construct a list with the following
elements:
"name"A string naming the distribution. If this
is called "dist", for example, then there must be visible in the
working environment, at least, either
a) a function called ddist which defines the probability density,
or
b) a function called hdist which defines the hazard.
Ideally, in case a) there should also be a function called pdist
which defines the probability distribution or cumulative density, and in
case b) there should be a function called Hdist defining the
cumulative hazard. If these additional functions are not provided,
flexsurv attempts to automatically create them by numerically
integrating the density or hazard function. However, model fitting will
be much slower, or may not even work at all, if the analytic versions of
these functions are not available.
The functions must accept vector arguments (representing different times,
or alternative values for each parameter) and return the results as a
vector. The function Vectorize may be helpful for doing
this: see the example below.
These functions may be in an add-on package (see below for an example) or
may be user-written. If they are user-written they must be defined in the
global environment, or supplied explicitly through the dfns argument
to flexsurvreg. The latter may be useful if the functions are
created dynamically (as in the source of flexsurvspline) and thus
not visible through R's scoping rules.
Arguments other than parameters must be named in the conventional way --
for example x for the first argument of the density function or
hazard, as in dnorm(x, ...) and q for the first
argument of the probability function. Density functions should also have
an argument log, after the parameters, which when TRUE,
computes the log density, using a numerically stable additive formula if
possible.
Additional functions with names beginning with "DLd" and
"DLS" may be defined to calculate the derivatives of the log density
and log survival probability, with respect to the parameters of the
distribution. The parameters are expressed on the real line, for example
after log transformation if they are defined as positive. The first
argument must be named t, representing the time, and the remaining
arguments must be named as the parameters of the density function. The
function must return a matrix with rows corresponding to times, and columns
corresponding to the parameters of the distribution. The derivatives are
used, if available, to speed up the model fitting with optim.
"pars"Vector of strings naming the parameters of the distribution. These must be the same names as the arguments of the density and probability functions.
"location"Name of the main parameter governing the mean of
the distribution. This is the default parameter on which covariates are
placed in the formula supplied to flexsurvreg.
"transforms"List of R
functions which transform the range of values taken by each parameter onto
the real line. For example, c(log, log) for a distribution with two
positive parameters.
"inv.transforms"List of R functions defining the
corresponding inverse transformations. Note these must be lists, even for
single parameter distributions they should be supplied as, e.g.
c(exp) or list(exp).
"inits"A function of the
observed survival times t (including right-censoring times, and
using the halfway point for interval-censored times) which returns a vector
of reasonable initial values for maximum likelihood estimation of each
parameter. For example, function(t){ c(1, mean(t)) } will always
initialize the first of two parameters at 1, and the second (a scale
parameter, for instance) at the mean of t.
For example, suppose we want to use an extreme value survival distribution.
This is available in the CRAN package eha, which provides
conventionally-defined density and probability functions called
eha::dEV and eha::pEV. See the Examples below
for the custom list in this case, and the subsequent command to fit the
model.
Christopher Jackson <chris.jackson@mrc-bsu.cam.ac.uk>
Parameters are estimated by maximum likelihood using the algorithms
available in the standard R optim function. Parameters
defined to be positive are estimated on the log scale. Confidence intervals
are estimated from the Hessian at the maximum, and transformed back to the
original scale of the parameters.
The usage of flexsurvreg is intended to be similar to
survreg in the survival package.
Jackson, C. (2016). flexsurv: A Platform for Parametric Survival Modeling in R. Journal of Statistical Software, 70(8), 1-33. doi:10.18637/jss.v070.i08
Cox, C. (2008) The generalized \(F\) distribution: An umbrella for parametric survival analysis. Statistics in Medicine 27:4301-4312.
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
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.
Nelson, C. P., Lambert, P. C., Squire, I. B., & Jones, D. R. (2007). Flexible parametric models for relative survival, with application in coronary heart disease. Statistics in medicine, 26(30), 5486-5498.
flexsurvspline for flexible survival modelling using
the spline model of Royston and Parmar.
plot.flexsurvreg and lines.flexsurvreg to plot
fitted survival, hazards and cumulative hazards from models fitted by
flexsurvreg and flexsurvspline.
## Compare generalized gamma fit with Weibull
fitg <- flexsurvreg(formula = Surv(futime, fustat) ~ 1, data = ovarian, dist="gengamma")
fitg
fitw <- flexsurvreg(formula = Surv(futime, fustat) ~ 1, data = ovarian, dist="weibull")
fitw
plot(fitg)
lines(fitw, col="blue", lwd.ci=1, lty.ci=1)
## Identical AIC, probably not enough data in this simple example for a
## very flexible model to be worthwhile.
## Custom distribution
## make "dEV" and "pEV" from eha package (if installed)
## available to the working environment
if (require("eha")) {
custom.ev <- list(name="EV",
pars=c("shape","scale"),
location="scale",
transforms=c(log, log),
inv.transforms=c(exp, exp),
inits=function(t){ c(1, median(t)) })
fitev <- flexsurvreg(formula = Surv(futime, fustat) ~ 1, data = ovarian,
dist=custom.ev)
fitev
lines(fitev, col="purple", col.ci="purple")
}
## Custom distribution: supply the hazard function only
hexp2 <- function(x, rate=1){ rate } # exponential distribution
hexp2 <- Vectorize(hexp2)
custom.exp2 <- list(name="exp2", pars=c("rate"), location="rate",
transforms=c(log), inv.transforms=c(exp),
inits=function(t)1/mean(t))
flexsurvreg(Surv(futime, fustat) ~ 1, data = ovarian, dist=custom.exp2)
flexsurvreg(Surv(futime, fustat) ~ 1, data = ovarian, dist="exp")
## should give same answer
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