Standard errors via bootstrap for an mjoint object

This function takes a model fit from an mjoint object and calculates standard errors and confidence intervals for the main longitudinal and survival coefficient parameters, including the latent association parameters, using bootstrapping (Efron and Tibshirani, 2000).

Usage

bootSE(
  object,
  nboot = 100,
  ci = 0.95,
  use.mle = TRUE,
  verbose = FALSE,
  control = list(),
  progress = TRUE,
  ncores = 1L,
  safe.boot = FALSE,
  ...
)

Arguments

object

an object inheriting from class mjoint for a joint model of time-to-event and multivariate longitudinal data.

nboot

the number of bootstrap samples. Default is nboot=100.

ci

the confidence interval to be estimated using the percentile-method. Default is ci=0.95 for a 95% confidence interval.

use.mle

logical: should the algorithm use the maximizer from the converged model in object as initial values for coefficients in each bootstrap iteration. Default is use.mle=TRUE.

verbose

logical: if TRUE, the parameter estimates and other convergence statistics are value are printed at each iteration of the MCEM algorithm. Default is FALSE.

control

a list of control values with components:

nMC

integer: the initial number of Monte Carlo samples to be used for integration in the burn-in phase of the MCEM. Default is nMC=100K.

nMCscale

integer: the scale factor for the increase in Monte Carlo size when Monte Carlo has not reduced from the previous iteration. Default is nMCscale=5.

nMCmax

integer: the maximum number of Monte Carlo samples that the algorithm is allowed to reach. Default is nMCmax=20000.

burnin

integer: the number of iterations for 'burn-in' phase of the optimization algorithm. It is computationally inefficient to use a large number of Monte Carlo samples early on until one is approximately near the maximum likelihood estimate. Default is burnin=100K for type='antithetic' or type='montecarlo' and burnin=5 for type='sobol' or type='halton'. For standard methods, such a large burn-in will generally be unnecessary and can be reduced on an application-specific basis.

mcmaxIter

integer: the maximum number of MCEM algorithm iterations allowed. Default is mcmaxIter=burnin+200.

convCrit

character string: the convergence criterion to be used. See Details.

gammaOpt

character string: by default (gammaOpt='NR'), \(\gamma\) is updated using a one-step Newton-Raphson iteration, with the Hessian matrix calculated exactly. If gammaOpt='GN', a Gauss-Newton algorithm-type iteration is implemented, where the Hessian matrix is approximated based on calculations similar to those used for calculating the empirical information matrix? If it is used, then the step-length is adjusted by a nominal scaling parameter of 0.5 in order to reduce the chance of over-shooting the maximizer.

tol0

numeric: tolerance value for convergence in the parameters; see Details. Default is tol0=1e-03.

tol1

numeric: tolerance value for convergence in the parameters; see Details. Default is tol1=1e-03.

tol2

numeric: tolerance value for convergence in the parameters; see Details. Default is tol2=5e-03 for type='antithetic' or type='montecarlo' and tol2=1e-03 for type='sobol' or type='halton'.

tol.em

numeric: tolerance value for convergence in the multivariate linear mixed model (MV-LMM). When \(K > 1\), the optimal initial parameters are those from the MV-LMM, which is estimated using a separate EM algorithm. Since both the E- and M-steps are available in closed-form, this algorithm convergences relatively rapidly with a high precision. Default is min(1e-04, tol2).

rav

numeric: threshold when using convCrit='sas' that applies absolute change (when \(<\)rav) or relative change (when \(\ge\)rav) criterion; see Details. Default is 0.1, which is an order of magnitude higher than the SAS implementation.

type

character: type of Monte Carlo integration method to use. Options are

type='montecarlo': Vanilla Monte Carlo sampling.
type='antithetic': Variance reduction method using antithetic simulation. This is the default option.
type='sobol': Quasi-Monte Carlo with a low deterministic Sobol sequence with Owen-type scrambling.
type='halton': Quasi-Monte Carlo with a low deterministic Halton sequence.

progress

logical: should a progress bar be shown on the console to indicate the percentage of bootstrap iterations completed? Default is progress=TRUE.

ncores

integer: if more than one core is available, then the bootSE function can run in parallel via the foreach function. By default, ncores=1, which defaults to serial mode. Note that if ncores>1, then progress is set to FALSE by default, as it is not possible to display progress bars for parallel processes at the current time.

safe.boot

logical: should each bootstrap replication be wrapped in a tryCatch statement to catch errors (e.g. during the optimisation progress)? When model fitting throws errors, a new bootstrap sample is drawn for the current iteration and the model is re-fit; this process continues until a model fits successfully. Default is FALSE.

...

options passed to the control argument.

Value

An object of class bootSE.

Details

Standard errors and confidence intervals are obtained by repeated fitting of the requisite joint model to bootstrap samples of the original longitudinal and time-to-event data. Note that bootstrap is done by sampling subjects, not individual records.

Note

This function is computationally intensive. A dynamic progress bar is displayed showing the percentage of bootstrap models fitted. On computer systems with more than one core available, computational time can be reduced by passing the argument ncores (with integer value >1) to bootSE, which implements parallel processing via the foreach package. Note: if parallel processing is implemented, then the progress bar is not displayed.

Due to random sampling, an mjoint model fitted to some bootstrap samples may not converge within the specified control parameter settings. The bootSE code discards any models that failed to converge when calculating the standard errors and confidence intervals. If a large proportion of models have failed to converge, it is likely that it will need to be refitted with changes to the control arguments.

References

Efron B, Tibshirani R. An Introduction to the Bootstrap. 2000; Boca Raton, FL: Chapman & Hall/CRC.

Author

Graeme L. Hickey (graemeleehickey@gmail.com)

Examples

if (FALSE) { # \dontrun{
# Fit a joint model with bivariate longitudinal outcomes

data(heart.valve)
hvd <- heart.valve[!is.na(heart.valve$log.grad) & !is.na(heart.valve$log.lvmi), ]

fit <- mjoint(
    formLongFixed = list("grad" = log.grad ~ time + sex + hs,
                         "lvmi" = log.lvmi ~ time + sex),
    formLongRandom = list("grad" = ~ 1 | num,
                          "lvmi" = ~ time | num),
    formSurv = Surv(fuyrs, status) ~ age,
    data = list(hvd, hvd),
    inits = list("gamma" = c(0.11, 1.51, 0.80)),
    timeVar = "time",
    verbose = TRUE)

fit.boot <- bootSE(fit, 50, use.mle = TRUE, control = list(
    burnin = 25, convCrit = "either",
    tol0 = 6e-03, tol2 = 6e-03, mcmaxIter = 60))
} # }

Standard errors via bootstrap for an `mjoint` object