Flexible parametric models for right-truncated, uncensored data defined by times of initial and final events.

This function estimates the distribution of the time between an initial and final event, in situations where individuals are only observed if they have experienced both events before a certain time, thus they are right-truncated at this time. The time of the initial event provides information about the time from initial to final event, given the truncated observation scheme, and initial events are assumed to occur with an exponential growth rate.

Usage

flexsurvrtrunc(
  t,
  tinit,
  rtrunc,
  tmax,
  data = NULL,
  method = "joint",
  dist,
  theta = NULL,
  fixed.theta = TRUE,
  inits = NULL,
  fixedpars = NULL,
  dfns = NULL,
  integ.opts = NULL,
  cl = 0.95,
  optim_control = list()
)

Arguments

t

Vector of time differences between an initial and final event for a set of individuals.

tinit

Absolute time of the initial event for each individual.

rtrunc

Individual-specific right truncation points on the same scale as t, so that each individual's survival time t would not have been observed if it was greater than the corresponding element of rtrunc. Only used in method="joint". In method="final", the right-truncation is implicit.

tmax

Maximum possible time between initial and final events that could have been observed. This is only used in method="joint", and is ignored in method="final".

data

Data frame containing t, rtrunc and tinit.

method

If "joint" then the "joint-conditional" method is used. If "final" then the "conditional-on-final" method is used. The "conditional-on-initial" method can be implemented by using flexsurvreg with a rtrunc argument. These methods are all described in Seaman et al. (2020).

dist

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.

theta

Initial value (or fixed value) for the exponential growth rate theta. Defaults to 1.

fixed.theta

Should theta be fixed at its initial value or estimated. This only applies to method="joint". For method="final", theta must be fixed.

inits

Initial values for the parameters of the parametric survival distributon. If not supplied, a heuristic is used. as is done in flexsurvreg.

fixedpars

Integer indices of the parameters of the survival distribution that should be fixed to their values supplied in inits. Same length as inits.

dfns

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.

integ.opts

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)

cl

Width of symmetric confidence intervals for maximum likelihood estimates, by default 0.95.

optim_control

List to supply as the control argument to optim to control the likelihood maximisation.

Details

Covariates are not currently supported.

Note that flexsurvreg, with an rtrunc argument, can fit models for a similar kind of data, but those models ignore the information provided by the time of the initial event.

A nonparametric estimator of survival under right-truncation is also provided in survrtrunc. See Seaman et al. (2020) for a full comparison of the alternative models.

References

Seaman, S., Presanis, A. and Jackson, C. (2020) Estimating a Time-to-Event Distribution from Right-Truncated Data in an Epidemic: a Review of Methods

Examples

set.seed(1) 
## simulate time to initial event
X <- rexp(1000, 0.2)
## simulate time between initial and final event
T <- rgamma(1000, 2, 10) 

## right-truncate to keep only those with final event time
## before tmax
tmax <- 40
obs <- X + T < tmax 
rtrunc <- tmax - X
dat <- data.frame(X, T, rtrunc)[obs,]

flexsurvrtrunc(t=T, rtrunc=rtrunc, tinit=X, tmax=40, data=dat,
                dist="gamma", theta=0.2)
#> Call:
#> flexsurvrtrunc(t = T, tinit = X, rtrunc = rtrunc, tmax = 40, 
#>     data = dat, dist = "gamma", theta = 0.2)
#> 
#> Estimates: 
#>         pars   est   lcl   ucl  estlog   selog
#> shape  shape  1.93  1.78   2.1   0.659  0.0414
#> rate    rate  9.22  8.39  10.1   2.222  0.0483
#> theta  theta  1.22    NA    NA   0.200      NA
#> 
#> Log-likelihood = -7846.25, df = 2
#> AIC = 15696.5
#> 

flexsurvrtrunc(t=T, rtrunc=rtrunc, tinit=X, tmax=40, data=dat,
                dist="gamma", theta=0.2, fixed.theta=FALSE)
#> Call:
#> flexsurvrtrunc(t = T, tinit = X, rtrunc = rtrunc, tmax = 40, 
#>     data = dat, dist = "gamma", theta = 0.2, fixed.theta = FALSE)
#> 
#> Estimates: 
#>         pars    est    lcl     ucl  estlog    selog
#> shape  shape  1.932  1.781   2.095   0.658  0.04144
#> rate    rate  9.419  8.586  10.334   2.243  0.04728
#> theta  theta  0.824  0.814   0.834  -0.193  0.00621
#> 
#> Log-likelihood = -1949.284, df = 3
#> AIC = 3904.568
#> 

flexsurvrtrunc(t=T, rtrunc=rtrunc, tinit=X, tmax=40, data=dat,
                dist="gamma", theta=0.2, inits=c(1, 8))
#> Call:
#> flexsurvrtrunc(t = T, tinit = X, rtrunc = rtrunc, tmax = 40, 
#>     data = dat, dist = "gamma", theta = 0.2, inits = c(1, 8))
#> 
#> Estimates: 
#>         pars   est   lcl   ucl  estlog   selog
#> shape  shape  1.93  1.78   2.1   0.659  0.0414
#> rate    rate  9.22  8.39  10.1   2.222  0.0483
#> theta  theta  1.22    NA    NA   0.200      NA
#> 
#> Log-likelihood = -7846.25, df = 2
#> AIC = 15696.5
#> 

flexsurvrtrunc(t=T, rtrunc=rtrunc, tinit=X, tmax=40, data=dat,
                dist="gamma", theta=0.2, method="final")
#> Call:
#> flexsurvrtrunc(t = T, tinit = X, rtrunc = rtrunc, tmax = 40, 
#>     data = dat, method = "final", dist = "gamma", theta = 0.2)
#> 
#> Estimates: 
#>         pars   est   lcl   ucl  estlog   selog
#> shape  shape  1.94  1.79  2.11   0.663  0.0415
#> rate    rate  9.05  8.22  9.97   2.203  0.0493
#> theta  theta  1.22    NA    NA   0.200      NA
#> 
#> Log-likelihood = 715.5208, df = 2
#> AIC = -1427.042
#> 

flexsurvrtrunc(t=T, rtrunc=rtrunc, tinit=X, tmax=40, data=dat,
                dist="gamma", fixed.theta=TRUE)
#> Call:
#> flexsurvrtrunc(t = T, tinit = X, rtrunc = rtrunc, tmax = 40, 
#>     data = dat, dist = "gamma", fixed.theta = TRUE)
#> 
#> Estimates: 
#>         pars   est   lcl   ucl  estlog   selog
#> shape  shape  1.93  1.78  2.09   0.658  0.0415
#> rate    rate  8.41  7.58  9.33   2.130  0.0529
#> theta  theta  2.72    NA    NA   1.000      NA
#> 
#> Log-likelihood = -33947.91, df = 2
#> AIC = 67899.82
#> 

flexsurvrtrunc(t=T, rtrunc=rtrunc, tinit=X, tmax=40, data=dat,
                dist="weibull", fixed.theta=TRUE)
#> Call:
#> flexsurvrtrunc(t = T, tinit = X, rtrunc = rtrunc, tmax = 40, 
#>     data = dat, dist = "weibull", fixed.theta = TRUE)
#> 
#> Estimates: 
#>         pars    est    lcl    ucl  estlog   selog
#> shape  shape  1.501  1.430  1.577   0.406  0.0249
#> scale  scale  0.252  0.241  0.264  -1.378  0.0240
#> theta  theta  2.718     NA     NA   1.000      NA
#> 
#> Log-likelihood = -33950.21, df = 2
#> AIC = 67904.42
#> 

flexsurvrtrunc(t=T, rtrunc=rtrunc, tinit=X, tmax=40, data=dat,
                dist="lnorm", fixed.theta=TRUE)
#> Call:
#> flexsurvrtrunc(t = T, tinit = X, rtrunc = rtrunc, tmax = 40, 
#>     data = dat, dist = "lnorm", fixed.theta = TRUE)
#> 
#> Estimates: 
#>             pars     est     lcl    ucl  estlog   selog
#> meanlog  meanlog  -1.694  -1.758  -1.63  -1.694  0.0325
#> sdlog      sdlog   0.893   0.848   0.94  -0.113  0.0264
#> theta      theta   2.718      NA     NA   1.000      NA
#> 
#> Log-likelihood = -33991.77, df = 2
#> AIC = 67987.55
#> 

flexsurvrtrunc(t=T, rtrunc=rtrunc, tinit=X, tmax=40, data=dat,
                dist="gengamma", fixed.theta=TRUE)
#> Call:
#> flexsurvrtrunc(t = T, tinit = X, rtrunc = rtrunc, tmax = 40, 
#>     data = dat, dist = "gengamma", fixed.theta = TRUE)
#> 
#> Estimates: 
#>         pars     est     lcl     ucl  estlog   selog
#> mu        mu  -1.446  -1.520  -1.371  -1.446  0.0379
#> sigma  sigma   0.702   0.657   0.750  -0.354  0.0335
#> Q          Q   0.798   0.631   0.964   0.798  0.0851
#> theta  theta   2.718      NA      NA   1.000      NA
#> 
#> Log-likelihood = -33947.48, df = 3
#> AIC = 67900.97
#> 

flexsurvrtrunc(t=T, rtrunc=rtrunc, tinit=X, tmax=40, data=dat,
                dist="gompertz", fixed.theta=TRUE)
#> Call:
#> flexsurvrtrunc(t = T, tinit = X, rtrunc = rtrunc, tmax = 40, 
#>     data = dat, dist = "gompertz", fixed.theta = TRUE)
#> 
#> Estimates: 
#>         pars   est   lcl   ucl  estlog   selog
#> shape  shape  2.58  2.18  2.97   2.576  0.2009
#> rate    rate  2.63  2.37  2.92   0.968  0.0532
#> theta  theta  2.72    NA    NA   1.000      NA
#> 
#> Log-likelihood = -33990.12, df = 2
#> AIC = 67984.25
#>