simsurv: Simulate survival data

Description Usage Arguments Details Value Note Author(s) References Examples

Description

Simulate survival times from standard parametric survival distributions, 2-component mixture distributions, or a user-defined hazard or log hazard function.

Usage

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simsurv(dist = c("weibull", "exponential", "gompertz"), lambdas, gammas, x,
  betas, tde, tdefunction = NULL, mixture = FALSE, pmix = 0.5, hazard,
  loghazard, cumhazard, logcumhazard, idvar = NULL, ids = NULL,
  nodes = 15, maxt = NULL, interval = c(1e-08, 500),
  rootsolver = c("uniroot", "dfsane"), rootfun = log,
  seed = sample.int(.Machine$integer.max, 1), ...)

Arguments

dist

Character string specifying the parametric survival distribution. Can be "weibull" (the default), "exponential", or "gompertz". Note that this is ignored if a user-defined baseline hazard function is provided via the hazard or loghazard argument.

lambdas

A numeric vector corresponding to the scale parameters for the exponential, Weibull or Gompertz distributions. This vector should be length one for any of the standard parametric distributions, or length two for a 2-component mixture distribution when mixture = TRUE.

gammas

A numeric vector corresponding to the shape parameters for the Weibull or Gompertz distributions. This vector should be length one for the standard Weibull or Gompertz distributions, or length two for a 2-component mixture distribution when mixture = TRUE.

x

A data frame containing the covariate values for each individual. Each row of the data frame should supply covariate data for one individual. The column names should correspond to the named elements in betas. If no covariates are being used to simulate the event times, then this data frame should just include an ID value for each individual; this is necessary since the number of individuals to simulate event times for is taken to be the number of rows in x (unless idvar and ids are specified, in which case the number of individuals to simulate event times for is taken to be length(ids)).

betas

A named vector, a data frame, or a list of data frames, containing the "true" parameters (e.g. log hazard ratios). If a standard exponential, Weibull, or Gompertz distribution, or a 2-component mixture distribution is being used then betas should provide the log hazard ratios for each of the baseline covariates that is to be included in the linear predictor of the proportional hazards model. This is most easily specified as a named vector (see the Examples). Alternatively, if a user-defined baseline hazard (or log hazard) function is provided via the hazard (or loghazard) argument, then betas can be specified as a vector, a data frame, or a list of data frames, and the of user-defined hazard (or log hazard) function should extract named elements betas however necessary in order to calculate the hazard (or log hazard) for each individual. See the Examples.

tde

A named vector, containing the "true" parameters that will be used to create time dependent effects (i.e. non-proportional hazards). The values specified in tde are used as coefficients (in the linear predictor of the proportional hazards model) on an interaction term between the corresponding covariate and time (or some function of time, for example log time, if tdefunction is not NULL).

tdefunction

An optional function of time to which covariates specified in tde will be interacted, in order to generate time dependent effects (i.e. non-proportional hazards). If NULL then the covariates specified in tde will be interacted with linear time. This can be the character string corresponding to a standard function (such as "log") or it can be a user-defined function with a single argument (for example function(x) x ^ 2).

mixture

Logical specifying whether to use a 2-component mixture model for the survival distribution. If TRUE, then the distribution of the mixture components is determined by the dist argument.

pmix

Scalar between 0 and 1 defining the mixing parameter when mixture = TRUE. The baseline survival at time t is taken to be S(t) = p S_1(t) + (1 - p) S_2(t) where S_1(t) and S_2(t) are the baseline survival under each component of the mixture distribution.

hazard

Optionally, a user-defined hazard function, with arguments t, x, and betas. This function should return the hazard at time t for an individual with covariates supplied via x and parameters supplied via betas. See the Examples.

loghazard

Optionally, a user-defined log hazard function, with arguments t, x, and betas. This function should return the log hazard at time t for an individual with covariates supplied via x and parameters supplied via betas. See the Examples.

cumhazard

Optionally, a user-defined cumulative hazard function, with arguments t, x, and betas. This function should return the cumulative hazard at time t for an individual with covariates supplied via x and parameters supplied via betas. See the Examples.

logcumhazard

Optionally, a user-defined log cumulative hazard function, with arguments t, x, and betas. This function should return the log cumulative hazard at time t for an individual with covariates supplied via x and parameters supplied via betas. See the Examples.

idvar

The name of the ID variable identifying individuals. This is only required when x (and betas if it is supplied as a data frame) contains multiple rows per individual. Otherwise, if idvar = NULL then each row of x (and betas if it is supplied as a data frame) is assumed to correspond to a different individual.

ids

A vector containing the unique values of idvar (i.e. the unique individual IDs). This is only required when x (and betas if it is supplied as a data frame) contain multiple rows per individual. Otherwise, if idvar = NULL then each row of x (and betas if it is supplied as a data frame) is assumed to correspond to a different individual.

nodes

Integer specifying the number of quadrature nodes to use for the Gauss-Kronrod quadrature. Can be 7, 11, or 15.

maxt

The maximum event time. For simulated event times greater than maxt, the event time ("eventtime") returned in the data frame will be truncated at maxt and the event indicator ("status") will be set to zero indicating that the individual was right-censored.

interval

The interval over which to search for the uniroot corresponding to each simulated event time.

rootsolver

Character string specifying the function to use for univariate root finding when required. This can currently be "uniroot" for using the uniroot function, or "dfsane" for using the dfsane function.

rootfun

A function to apply to each side of the root finding equation when numerical root finding is used to solve for the simulated event time. An appropriate function helps to improve numerical stability. The default is to use a log transformation; that is, to solve log(S(T)) - log(U) = 0 where S(T) is the survival probability at the event time and U is a uniform random variate. Suitable alternatives might be to specify rootfun = NULL, which corresponds to S(T) - U = 0, or rootfun = sqrt, which corresponds to sqrt(S(T)) - sqrt(U) = 0. It is unexpected that the user should need to change this argument from its default value, except perhaps in the extreme case that the numerical root finding fails.

seed

The seed to use.

...

Other arguments passed to hazard, loghazard, cumhazard, or logcumhazard.

Details

The simsurv function simulates survival times from standard parametric survival distributions (exponential, Weibull, Gompertz), 2-component mixture distributions, or a user-defined hazard or log hazard function. Baseline covariates can be included under a proportional hazards assumption. Time dependent effects (i.e. non-proportional hazards) can be included by interacting covariates with time (by specifying them in the tde argument); the default behaviour is to interact the covariates with linear time, however, they can be interacted with some other function of time simply by using the tdefunction argument.

Under the 2-component mixture distributions (obtained by setting mixture = TRUE) the baseline survival at time t is taken to be S(t) = p * S_1(t) + (1 - p) * S_2(t) where S_1(t) and S_2(t) are the baseline survival under each component of the mixture distribution and p is the mixing parameter specified via the argument pmix. Each component of the mixture distribution is assumed to be either exponential, Weibull or Gompertz. The 2-component mixture distributions can allow for a variety of flexible baseline hazard functions (see Crowther and Lambert (2013) for some examples).

If the user wishes to provide a user-defined hazard or log hazard function (instead of using one of the standard parametric survival distributions) then this is also possible via the hazard or loghazard argument. If a user-defined hazard or log hazard function is specified, then this is allowed to be time-dependent, and the resulting cumulative hazard function does not need to have a closed-form solution. The survival times are generated using the approach described in Crowther and Lambert (2013), whereby the cumulative hazard is evaluated using numerical quadrature and survival times are generated using an iterative algorithm which nests the quadrature-based evaluation of the cumulative hazard inside Brent's (1973) univariate root finder (for the latter the uniroot function is used). Not requiring a closed form solution to the cumulative hazard function has the benefit that survival times can be generated for complex models such as joint longitudinal and survival models; the Examples section provides an example of this.

Parameterisation for the exponential distribution

For the exponential distribution, with scale parameter λ > 0, the baseline hazard and survival functions used by simsurv are: h(t) = λ and S(t) = \exp(-λ).

Our parameterisation is equivalent to the one used by Wikipedia, the dexp function, the eha package, and the flexsurv package, except what we call the scale parameter they call the rate parameter.

Parameterisation for the Weibull distribution

For the Weibull distribution, with shape parameter γ > 0 and scale parameter λ > 0, the baseline hazard and survival functions used by simsurv are: h(t) = γ λ t ^ {γ - 1} and S(t) = \exp(-λ t ^ {γ}). Setting γ equal to 1 leads to the exponential distribution as a special case.

Our parameterisation differs from the one used by Wikipedia, dweibull, the phreg modelling function in the eha package, and the flexsurvreg modelling function in the flexsurv package. The parameterisation used in those functions can be achieved by transforming the scale parameter via the relationship b = λ ^ {\frac{-1}{γ}}, or equivalently λ = b ^ {-γ} where b is the scale parameter under their parameterisation of the Weibull distribution.

Parameterisation for the Gompertz distribution

For the Gompertz distribution, with and shape parameter γ > 0 and scale parameter λ > 0, the baseline hazard and survival functions used by simsurv are: h(t) = λ \exp(γ t) and S(t) = \exp(\frac{-λ (\exp(γ t) - 1)}{γ}). Setting γ equal to 0 leads to the exponential distribution as a special case.

Our parameterisation is equivalent to the one used by the dgompertz and flexsurvreg functions in the flexsurv package, except they use slightly different terminology. Their parameterisation can be achieved via the relationship a = γ and b = λ where a and b are their shape and rate parameters, respectively.

Our parameterisation differs from the one used in the dgompertz and phreg functions in the eha package. Their parameterisation can be achieved via the relationship a = λ and b = \frac{1}{γ} where a and b are their shape and scale parameters, respectively.

Our parameterisation differs from the one used by Wikipedia. Their parameterisation can be achieved via the relationship a = \frac{λ}{γ} and b = γ where a and b are their shape and scale parameters, respectively.

Value

A data frame with a row for each individual, and the following three columns:

Note

This package is modelled on the user-written survsim package available in the Stata software (see Crowther and Lambert (2012)).

Author(s)

Sam Brilleman ([email protected])

References

Crowther MJ, and Lambert PC. (2013) Simulating biologically plausible complex survival data. Statistics in Medicine 32, 4118–4134. doi: 10.1002/sim.5823

Bender R, Augustin T, and Blettner M. (2005) Generating survival times to simulate Cox proportional hazards models. Statistics in Medicine 24(11), 1713–1723.

Brent R. (1973) Algorithms for Minimization without Derivatives. Englewood Cliffs, NJ: Prentice-Hall.

Crowther MJ, and Lambert PC. (2012) Simulating complex survival data. The Stata Journal 12(4), 674–687. http://www.stata-journal.com/sjpdf.html?articlenum=st0275

Examples

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  #-------------- Simpler examples

  # Generate times from a Weibull model including a binary
  # treatment variable, with log(hazard ratio) = -0.5, and censoring
  # after 5 years:
  set.seed(9911)
  covs <- data.frame(id = 1:100, trt = stats::rbinom(100, 1L, 0.5))
  s1 <- simsurv(lambdas = 0.1, gammas = 1.5, betas = c(trt = -0.5),
                x = covs, maxt = 5)
  head(s1)

  # Generate times from a Gompertz model:
  s2 <- simsurv(dist = "gompertz", lambdas = 0.1, gammas = 0.05, x = covs)

  # Generate times from a 2-component mixture Weibull model:
  s3 <- simsurv(lambdas = c(0.1, 0.05), gammas = c(1, 1.5),
                mixture = TRUE, pmix = 0.5, x = covs, maxt = 5)

  # Generate times from user-defined log hazard function:
  fn <- function(t, x, betas, ...)
    (-1 + 0.02 * t - 0.03 * t ^ 2 + 0.005 * t ^ 3)
  s4 <- simsurv(loghazard = fn, x = covs, maxt = 1.5)

  # Generate times from a Weibull model with diminishing treatment effect:
  s5 <- simsurv(lambdas = 0.1, gammas = 1.5, betas = c(trt = -0.5),
                x = covs, tde = c(trt = 0.05), tdefunction = "log")

  #-------------- Complex examples

  # Here we present an example of simulating survival times
  # based on a joint longitudinal and survival model

  # First we define the hazard function to pass to simsurv
  # (NB this is a Weibull proportional hazards regression submodel
  # from a joint longitudinal and survival model with a "current
  # value" association structure).
  haz <- function(t, x, betas, ...) {
      betas[["shape"]] * (t ^ (betas[["shape"]] - 1)) * exp(
        betas[["betaEvent_intercept"]] +
        betas[["betaEvent_binary"]] * x[["Z1"]] +
        betas[["betaEvent_continuous"]] * x[["Z2"]] +
        betas[["betaEvent_assoc"]] * (
          betas[["betaLong_intercept"]] +
          betas[["betaLong_slope"]] * t +
          betas[["betaLong_binary"]] * x[["Z1"]] +
          betas[["betaLong_continuous"]] * x[["Z2"]]
        )
      )
  }

  # Then we construct data frames with the true parameter
  # values and the covariate data for each individual
  set.seed(5454) # set seed before simulating data
  N <- 20        # number of individuals

  # Population (fixed effect) parameters
  betas <- data.frame(
    shape                = rep(2,    N),
    betaEvent_intercept  = rep(-11.9,N),
    betaEvent_binary     = rep(0.6,  N),
    betaEvent_continuous = rep(0.08, N),
    betaEvent_assoc      = rep(0.03, N),
    betaLong_binary      = rep(-1.5, N),
    betaLong_continuous  = rep(1,    N),
    betaLong_intercept   = rep(90,   N),
    betaLong_slope       = rep(2.5,  N)
  )

  # Individual-specific (random effect) parameters
  b_corrmat <- matrix(c(1, 0.5, 0.5, 1), 2, 2)
  b_sds     <- c(20, 3)
  b_means   <- rep(0, 2)
  b_z       <- MASS::mvrnorm(n = N, mu = b_means, Sigma = b_corrmat)
  b         <- sapply(1:length(b_sds), function(x) b_sds[x] * b_z[,x])
  betas$betaLong_intercept <- betas$betaLong_intercept + b[,1]
  betas$betaLong_slope     <- betas$betaLong_slope     + b[,2]

  # Covariate data
  covdat <- data.frame(
    Z1 = stats::rbinom(N, 1, 0.45), # a binary covariate
    Z2 = stats::rnorm(N, 44, 8.5)   # a continuous covariate
  )

  # Then we simulate the survival times based on the
  # hazard function, covariates, and true parameter values
  times <- simsurv(hazard = haz, x = covdat, betas = betas, maxt = 10)
  head(times)

sambrilleman/simsurv documentation built on May 20, 2019, 4:13 p.m.