bayesBisurvreg {bayesSurv} | R Documentation |
A function to estimate a regression model with bivariate (possibly right-, left-, interval- or doubly-interval-censored) data. In the case of doubly interval censoring, different regression models can be specified for the onset and event times.
The error density of the regression model is specified as a mixture of Bayesian G-splines (normal densities with equidistant means and constant variance matrices). This function performs an MCMC sampling from the posterior distribution of unknown quantities.
For details, see Komarek (2006) and Komarek and Lesaffre (2006).
We explain first in more detail a model without doubly censoring. Let T[i,l], i=1,..., N, l=1, 2 be event times for ith cluster and the first and the second unit. The following regression model is assumed:
log(T[i,l]) = beta'x[i,l] + epsilon[i,l], i=1,..., N, l=1,2
where beta is unknown regression parameter vector and x[i,l] is a vector of covariates. The bivariate error terms epsilon[i] = (epsilon[i,1], epsilon[i,2])', i=1,..., N are assumed to be i.i.d. with a~bivariate density g[epsilon](e[1], e[2]). This density is expressed as a~mixture of Bayesian G-splines (normal densities with equidistant means and constant variance matrices). We distinguish two, theoretically equivalent, specifications.
(epsilon[1],,epsilon[2])' is distributed as sum[j[1]=-K[1]][K[1]] sum[j[2]=-K[2]][K[2]] w[j[1],j[2]] N(mu[(j[1],j[2])], diag(sigma[1]^2, sigma[2]^2))
where sigma[1]^2, sigma[2]^2 are unknown basis variances and mu[(j[1],j[2])] = (mu[1,j[1]], mu[2,j[2]])' is an~equidistant grid of knots symmetric around the unknown point (gamma[1], gamma[2])' and related to the unknown basis variances through the relationship
mu[1,j[1]] = gamma[1] + j[1]*delta[1]*sigma[1], j[1]=-K[1],..., K[1]
mu[2,j[2]] = gamma[2] + j[2]*delta[2]*sigma[2], j[2]=-K[2],..., K[2]
where delta[1], delta[2] are fixed constants, e.g. delta[1]=delta[2]=2/3 (which has a~justification of being close to cubic B-splines).
(epsilon[1],,epsilon[2])' is distributed as (alpha[1], alpha[2])' + S (V[1], V[2])'
where (alpha[1], alpha[2])' is an unknown intercept term and S is a diagonal matrix with tau[1] and tau[2] on a diagonal, i.e. tau[1], tau[2] are unknown scale parameters. (V[1], V[2])' is then standardized bivariate error term which is distributed according to the bivariate normal mixture, i.e.
(V[1], V[2])' is distributed as sum[j[1]=-K[1]][K[1]] sum[j[2]=-K[2]][K[2]] w[j[1],j[2]] N(mu[(j[1],j[2])], diag(sigma[1]^2, sigma[2]^2))
where mu[(j[1],j[2])] = (mu[1,j[1]], mu[2,j[2]])' is an~equidistant grid of fixed knots (means), usually symmetric about the fixed point (gamma[1], gamma[2])' = (0, 0)' and sigma[1]^2, sigma[2]^2 are fixed basis variances. Reasonable values for the numbers of grid points K[1] and K[2] are K[1]=K[2]=15 with the distance between the two knots equal to delta=0.3 and for the basis variances sigma[1]^2=sigma[2]^2=0.2^2.
Personally, I found Specification 2 performing better. In the paper Komarek and Lesaffre (2006) only Specification 2 is described.
The mixture weights w[j[1],j[2]], j[1]=-K[1],..., K[1], j[2]=-K[2],..., K[2] are not estimated directly. To avoid the constraints 0 < w[j[1],j[2]] < 1 and sum[j[1]=-K[1]][K[1]]sum[j[2]=-K[2]][K[2]]w[j[1],j[2]]=1 transformed weights a[j[1],j[2]], j[1]=-K[1],..., K[1], j[2]=-K[2],..., K[2] related to the original weights by the logistic transformation:
a[j[1],j[2]] = exp(w[j[1],j[2]])/sum[m[1]]sum[m[2]] exp(w[m[1],m[2]])
are estimated instead.
A~Bayesian model is set up for all unknown parameters. For more details I refer to Komarek and Lesaffre (2006) and to Komarek (2006).
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.
bayesBisurvreg(formula, formula2, data = parent.frame(), na.action = na.fail, onlyX = FALSE, nsimul = list(niter = 10, nthin = 1, nburn = 0, nwrite = 10), prior, prior.beta, 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, 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), dir = getwd())
formula |
model formula for the regression. In the case of
doubly-censored data, this is the model formula for the onset
time. Data are assumed to be sorted according to subjects and within
subjects according to the types of the events that determine the
bivariate survival distribution, i.e. the response vector must be
t[1,1],t[1,2],t[2,1],t[2,2],t[3,1],t[3,2],...,t[n,1],t[n,2].
The rows of the design matrix with covariates must be sorted
analogically.
The left-hand side of the formula must be an object created using Surv .
|
formula2 |
model formula for the regression of the time-to-event in
the case of doubly-censored data. Ignored otherwise. The same remark as
for formula concerning the sort order applies here.
|
data |
optional data frame in which to interpret the variables occuring in the formulas. |
na.action |
the user is discouraged from changing the default
value na.fail . |
onlyX |
if 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. |
nsimul |
a list giving the number of iterations of the MCMC and
other parameters of the simulation.
|
prior |
a~list specifying the prior distribution of the G-spline
defining the distribution of the error term in the regression model
given by formula . See prior argument of
bayesHistogram function for more detail. In this list
also ‘Specification’ as described above is specified.
|
prior.beta |
prior specification for the regression parameters,
in the case of doubly censored data for the regression parameters of
the onset time. I.e. it is related to formula .
This should be a~list with the following components:
onlyX set to TRUE
to find out how the betas are sorted. They must correspond to a
design matrix X taken from formula .
|
init |
an~optional list with initial values for the MCMC related
to the model given by formula . The list can have the following components:
init of
the function bayesHistogram for more details. |
mcmc.par |
a~list specifying how some of the G-spline parameters
related to formula are to be updated. The list can have the following components (all
of them have their default values):
|
prior2 |
a~list specifying the prior distribution of the G-spline
defining the distribution of the error term in the regression model
given by formula2 . See prior argument of
bayesHistogram function for more detail. |
prior.beta2 |
prior specification for the regression parameters
of time-to-event in the case of doubly censored data (related to
formula2 ).
This should be a~list with the same structure as prior.beta .
|
init2 |
an~optional list with initial values for the MCMC related
to the model given by formula2 . The list has the same
structure as init .
|
mcmc.par2 |
a~list specifying how some of the G-spline parameters
related to formula2 are to be updated. The list has the same
structure as mcmc.par .
|
store |
a~list of logical values specifying which chains that are
not stored by default are to be stored. The list can have the
following components.
|
dir |
a string that specifies a directory where all sampled values are to be stored. |
A list of class bayesBisurvreg
containing an information
concerning the initial values and prior choices.
Additionally, the following files with sampled values
are stored in a directory specified by dir
argument of this
function (some of them are created only on request, see store
parameter of this function).
Headers are written to all files created by default and to files asked
by the user via the argument store
. During the burn-in, only
every nsimul$nwrite
value is written. After the burn-in, all
sampled values are written in files created by default and to files
asked by the user via the argument store
. In the files for
which the corresponding store
component is FALSE
, every
nsimul$nwrite
value is written during the whole MCMC (this
might be useful to restart the MCMC from some specific point).
The following files are created:
iteration
with
indeces of MCMC iterations to which the stored sampled values
correspond.
k
, Mean.1
, Mean.2
,
D.1.1
, D.2.1
, D.2.2
, where
k = number of mixture components that had probability numerically higher than zero;
Mean.1 = E(epsilon[i,1]);
Mean.2 = E(epsilon[i,2]);
D.1.1 = var(epsilon[i,1]);
D.2.1 = cov(epsilon[i,1], epsilon[i,2]);
D.2.2 = var(epsilon[i,2]);
all related to the distribution of the error term from the model given by formula
.
mixmoment.sim
, however related to the model
given by formula2
.
formula
.
mweight.sim
, however related to the model
given by formula2
.
mweight.sim
. Related to the model given by formula
.
mmean.sim
, however related to the model
given by formula2
.
formula
. This file together with mixmoment.sim
,
mweight.sim
and mmean.sim
can be used to reconstruct
the G-spline in each MCMC iteration.
The file has columns labeled gamma1
,
gamma2
, sigma1
, sigma2
, delta1
,
delta2
, intercept1
, intercept2
,
scale1
, scale2
. The meaning of the values in these
columns is the following:
gamma1 = the middle knot gamma[1] in the first dimension. If ‘Specification’ is 2, this column usually contains zeros;
gamma2 = the middle knot gamma[2] in the second dimension. If ‘Specification’ is 2, this column usually contains zeros;
sigma1 = basis standard deviation sigma[1] of the G-spline in the first dimension. This column contains a~fixed value if ‘Specification’ is 2;
sigma2 = basis standard deviation sigma[2] of the G-spline in the second dimension. This column contains a~fixed value if ‘Specification’ is 2;
delta1 = distance delta[1] between the two knots of the G-spline in the first dimension. This column contains a~fixed value if ‘Specification’ is 2;
delta2 = distance delta[2] between the two knots of the G-spline in the second dimension. This column contains a~fixed value if ‘Specification’ is 2;
intercept1 = the intercept term alpha[1] of the G-spline in the first dimension. If ‘Specification’ is 1, this column usually contains zeros;
intercept2 = the intercept term alpha[2] of the G-spline in the second dimension. If ‘Specification’ is 1, this column usually contains zeros;
scale1 = the scale parameter tau[1] of the G-spline in the first dimension. If ‘Specification’ is 1, this column usually contains ones;
scale2 = the scale parameter tau[2] of the G-spline in the second dimension. ‘Specification’ is 1, this column usually contains ones.
gspline.sim
, however related to the model
given by formula2
.
store$a = TRUE
. The
file contains the transformed weights
a[k[1],k[2]],
k[1]=-K[1],..., K[1],
k[2]=-K[2],..., K[2] of all mixture
components, i.e. also of components that had numerically zero
probabilities.
This file is related to the model given by formula
.
store$a2 =
TRUE
and in the case of doubly-censored data, the same
structure as mlogweight.sim
, however related to the model
given by formula2
.
store$r = TRUE
. The file
contains the labels of the mixture components into which the
residuals are intrinsically assigned. Instead of double indeces
(k[1], k[2]), values from 1 to (2*K[1]+1)*(2*K[2]+1) are stored here. Function
vecr2matr
can be used to transform it back to double
indeces.
store$r2 =
TRUE
and in the case of doubly-censored data, the same
structure as r.sim
, however related to the model
given by formula2
.
lambda
or two
columns labeled lambda1
and lambda2
. These are the
values of the smoothing parameter(s) lambda
(hyperparameters of the prior distribution of the transformed
mixture weights a[k[1],k[2]]). This file is
related to the model given by formula
.
lambda.sim
, however related to the model
given by formula2
.
formula
. The columns are labeled according to the
colnames
of the design matrix.
beta.sim
, however related to the model
given by formula2
.
store$y = TRUE
. It
contains sampled (augmented) log-event times for all observations
in the data set.
store$y2 =
TRUE
and in the case of doubly-censored data, the same
structure as Y.sim
, however related to the model
given by formula2
.
loglik
, penalty
or penalty1
and
penalty2
, logprw
. This file is related to the model
given by formula
. The columns have the following meaning.
loglik = -N(log(2*pi) + log(sigma[1]) + log(sigma[2])) -0.5*sum[i=1][N]( (sigma[1]^2*tau[1]^2)^(-1) * (y[i,1] - x[i,1]'beta - alpha[1] - tau[1]*mu[1,r[i,1]])^2 + (sigma[2]^2*tau[2]^2)^(-1) * (y[i,2] - x[i,2]'beta - alpha[2] - tau[2]*mu[2,r[i,2]])^2 )
where y[i,l] denotes (augmented) (i,l)th
true log-event time. In other words, loglik
is equal to the
conditional log-density
sum[i=1][N] log(p((y[i,1], y[i,2]) | r[i], beta, G-spline));
penalty1: If prior$neighbor.system
= "uniCAR"
:
the penalty term for the first dimension not multiplied by
lambda1
;
penalty2: If prior$neighbor.system
= "uniCAR"
:
the penalty term for the second dimension not multiplied by
lambda2
;
penalty: If prior$neighbor.system
is different from "uniCAR"
:
the penalty term not multiplied by \lambda
;
logprw = -2*N*log(sum[k[1]]sum[k[2]] exp(a[k[1],k[2]])) + sum[k[1]]sum[k[2]] N[k[1],k[2]]*a[k[1],k[2]], where N[k[1],k[2]] is the number of residuals assigned intrinsincally to the (k[1], k[2])th mixture component.
In other words, logprw
is equal to the conditional
log-density
sum[i=1][N] log(p(r[i] | G-spline weights)).
lambda.sim
, however related to the model
given by formula2
.
Arnošt Komárek arnost.komarek[AT]mff.cuni.cz
Gilks, W. R. and Wild, P. (1992). Adaptive rejection sampling for Gibbs sampling. Applied Statistics, 41, 337 - 348.
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. and Lesaffre, E. (2006). Bayesian semi-parametric accelerated failure time model for paired doubly interval-censored data. Statistical Modelling, 6, 3–22.
Neal, R. M. (2003). Slice sampling (with Discussion). The Annals of Statistics, 31, 705 - 767.
## See the description of R commands for ## the population averaged AFT model ## with the Signal Tandmobiel data, ## analysis described in Komarek and Lesaffre (2006), ## ## R commands available in the documentation ## directory of this package as ## tandmobPA.pdf, tandmobPA.R.