(Multivariate) random effects, normally distributed and acting as in the linear mixed model, normally distributed, can be included to adjust for clusters.
The error density of the regression model is specified as a mixture of Bayesian G-splines (normal densities with equidistant means and constant variances). This function performs an MCMC sampling from the posterior distribution of unknown quantities.
For details, see Komárek (2006), and Komárek, Lesaffre and Legrand (2007).
We explain first in more detail a model without doubly censoring. Let $T_{i,l},\; i=1,\dots, N,\; l=1,\dots, n_i$ be event times for $i$th cluster and the units within that cluster The following regression model is assumed: $$\log(T_{i,l}) = \beta'x_{i,l} + b_i'z_{i,l} + \varepsilon_{i,l},\quad i=1,\dots, N,\;l=1,\dots, n_i$$ where $\beta$ is unknown regression parameter vector, $x_{i,l}$ is a vector of covariates. $b_i$ is a (multivariate) cluster-specific random effect vector and $z_{i,l}$ is a vector of covariates for random effects.
The random effect vectors $b_i,\;i=1,\dots, N$ are assumed to be i.i.d. with a (multivariate) normal distribution with the mean $\beta_b$ and a~covariance matrix $D$. Hierarchical centring (see Gelfand, Sahu, Carlin, 1995) is used. I.e. $\beta_b$ expresses the average effect of the covariates included in $z_{i,l}$. Note that covariates included in $z_{i,l}$ may not be included in the covariate vector $x_{i,l}$. The covariance matrix $D$ is assigned an inverse Wishart prior distribution in the next level of hierarchy. The error terms $\varepsilon_{i,l},\;i=1,\dots, N, l=1,\dots, n_i$ are assumed to be i.i.d. with a~univariate density $g_{\varepsilon}(e)$. This density is expressed as a~mixture of Bayesian G-splines (normal densities with equidistant means and constant variances). We distinguish two, theoretically equivalent, specifications.
[object Object],[object Object] Personally, I found Specification 2 performing better. In the paper Komárek, Lesaffre and Legrand (2007) only Specification 2 is described.
The mixture weights $w_{j},\;j=-K,\dots, K$ are not estimated directly. To avoid the constraints $0 < w_{j} < 1$ and $\sum_{j=-K}^{K}\,w_j = 1$ transformed weights $a_{j},\;j=-K,\dots, K$ related to the original weights by the logistic transformation: $$a_{j} = \frac{\exp(w_{j})}{\sum_{m}\exp(w_{m})}$$ are estimated instead.
A~Bayesian model is set up for all unknown parameters. For more details I refer to Komárek (2006) and to Komárek, Lesafre, and Legrand (2007). If there are doubly-censored data the model of the same type as above can be specified for both the onset time and the time-to-event.
bayessurvreg2(formula, random, formula2, random2,
data = parent.frame(),
na.action = na.fail, onlyX = FALSE,
nsimul = list(niter = 10, nthin = 1, nburn = 0, nwrite = 10),
prior, prior.beta, prior.b, init = list(iter = 0),
mcmc.par = list(type.update.a = "slice", k.overrelax.a = 1,
k.overrelax.sigma = 1, k.overrelax.scale = 1),
prior2, prior.beta2, prior.b2, init2,
mcmc.par2 = list(type.update.a = "slice", k.overrelax.a = 1,
k.overrelax.sigma = 1, k.overrelax.scale = 1),
store = list(a = FALSE, a2 = FALSE, y = FALSE, y2 = FALSE,
r = FALSE, r2 = FALSE, b = FALSE, b2 = FALSE),
dir = getwd()) The left-hand side of the formula must be an~object created
using
random formula for
the onset time.If omitted, no random part is included in
formula applies here.random applies here.formula, formula2, random,
random2 statements.na.fail.TRUE no MCMC sampling is performed and only the
design matrix (matrices) are returned. This can be useful to set up
correctly priors for regression parameters in the presence of
factor covariates.formula and random. See prior argument of
formula and random
are to be updated and the specification
of priors for parameters related to these random effects. The list
is assumed to have the formula and
random.
Note that the beta vecformula and random. The list can have the following components:
[object Object],[object Object],[object Object],[object Object],[object Objecformula
are to be updated. See bayesBisurvreg for more
details.
formula2 and random2. See prior argument of
formula2 and random2. This should
be a~list with the same structure as prior.b.formula2 and random2).
This should be a~list with the same structure as prior.beta.formula2 and random2.
The list has the same structure as init.formula2 are to be updated. The list has the same
structure as mcmc.par.bayessurvreg2 containing an information
concerning the initial values and prior choices.Komárek, A. (2006). Accelerated Failure Time Models for Multivariate Interval-Censored Data with Flexible Distributional Assumptions. PhD. Thesis, Katholieke Universiteit Leuven, Faculteit Wetenschappen.
Komárek, A., Lesaffre, E., and Legrand, C. (2007). Baseline and treatment effect heterogeneity for survival times between centers using a random effects accelerated failure time model with flexible error distribution. Statistics in Medicine, 26, 5457-5472.
## See the description of R commands for
## the model with EORTC data,
## analysis described in Komarek, Lesaffre and Legrand (2007).
##
## R commands available in the documentation
## directory of this package
## as ex-eortc.R and
## http://www.karlin.mff.cuni.cz/~komarek/software/bayesSurv/ex-eortc.pdf
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