crisk2.bart: BART for competing risks
In BART: Bayesian Additive Regression Trees
crisk2.bart R Documentation
BART for competing risks
Description
Here we have implemented another approach to utilize BART
for competing risks that is very flexible, and is akin to discrete-time
survival analysis. Following the capabilities of BART, we allow for
maximum flexibility in modeling the dependence of competing failure times on
covariates. In particular, we do not impose proportional hazards.
Similar to crisk.bart, we utilize two BART models, yet they are
two different BART models than previously considered. First, given an
event of either cause occurred, we employ a typical binary BART model to
discriminate between cause 1 and 2. Next, we proceed as if it were a typical
survival analysis with BART for an absorbing event from either cause.
To elaborate, consider data in the form: (s_i, \delta_i,
{x}_i) where s_i is the event time;
\delta_i is an indicator distinguishing events,
\delta_i=h due to cause h in {1, 2}, from
right-censoring, \delta_i=0; {x}_i is a vector of
covariates; and i=1, ..., N indexes subjects.
We denote the K distinct event/censoring times by
0<t_{(1)}<...<t_{(K)}<\infty thus
taking t_{(j)} to be the j^{th} order statistic
among distinct observation times and, for convenience,
t_{(0)}=0.
First, consider event indicators for an event from either cause:
y_{1ij} for each subject i at each distinct time
t_{(j)} up to and including the subject's last observation
time s_i=t_{(n_i)} with n_i=\arg \max_j [t_{(j)}\leq
s_i]. We denote by p_{1ij} the
probability of an event at time t_{(j)} conditional on no
previous event. We now write the model for y_{1ij} as a
nonparametric probit (or logistic) regression of y_{1ij} on
the time t_{(j)} and the covariates {x}_{1i},
and then utilize BART for binary responses. Specifically,
y_{1ij}\ =\ I[\delta_i>0] I[s_i=t_{(j)}],\ j=1, ..., n_i. Therefore, we have p_{1ij} =
F(mu_{1ij}),\ mu_{1ij} = mu_1+f_1(t_{(j)}, {x}_{1i}) where F denotes the Normal (or
Logistic) cdf.
Next, we denote by p_{2i} the probability of a cause 1
event at time s_i conditional on an event having
occurred. We now write the model for y_{2i} as a
nonparametric probit (or logistic) regression of y_{2i} on
the time s_i and the covariates {x}_{2i},
via BART for binary responses. Specifically,
y_{2i}\ =\ I[\delta_i=1]. Therefore, we
have p_{2i} = F(mu_{2i}),\ mu_{2i} = mu_2+f_2(s_i,
{x}_{2i}) where
F denotes the Normal (or Logistic) cdf. Although, we modeled
p_{2i} at the time of an event, s_i, we can
estimate this probability at any other time points on the grid via
p(t_{(j)}, x_2)=F( mu_2+f_2(t_{(j)}, {x}_2)).
Finally, based on these probabilities,
p_{hij}, we can construct targets of inference such as the
cumulative incidence functions.
Usage
crisk2.bart(x.train=matrix(0,0,0), y.train=NULL,
x.train2=x.train, y.train2=NULL,
times=NULL, delta=NULL, K=NULL,
x.test=matrix(0,0,0), x.test2=x.test,
sparse=FALSE, theta=0, omega=1,
a=0.5, b=1, augment=FALSE,
rho=NULL, rho2=NULL,
xinfo=matrix(0,0,0), xinfo2=matrix(0,0,0),
usequants=FALSE,
rm.const=TRUE, type='pbart',
ntype=as.integer(
factor(type, levels=c('wbart', 'pbart', 'lbart'))),
k=2, power=2, base=0.95,
offset=NULL, offset2=NULL,
tau.num=c(NA, 3, 6)[ntype],
ntree=50, numcut=100, ndpost=1000, nskip=250,
keepevery = 10L,
printevery=100L,
id=NULL, ## crisk2.bart only
seed=99, ## mc.crisk2.bart only
mc.cores=2, ## mc.crisk2.bart only
nice=19L ## mc.crisk2.bart only
)
mc.crisk2.bart(x.train=matrix(0,0,0), y.train=NULL,
x.train2=x.train, y.train2=NULL,
times=NULL, delta=NULL, K=NULL,
x.test=matrix(0,0,0), x.test2=x.test,
sparse=FALSE, theta=0, omega=1,
a=0.5, b=1, augment=FALSE,
rho=NULL, rho2=NULL,
xinfo=matrix(0,0,0), xinfo2=matrix(0,0,0),
usequants=FALSE,
rm.const=TRUE, type='pbart',
ntype=as.integer(
factor(type, levels=c('wbart', 'pbart', 'lbart'))),
k=2, power=2, base=0.95,
offset=NULL, offset2=NULL,
tau.num=c(NA, 3, 6)[ntype],
ntree=50, numcut=100, ndpost=1000, nskip=250,
keepevery = 10L,
printevery=100L,
id=NULL, ## crisk2.bart only
seed=99, ## mc.crisk2.bart only
mc.cores=2, ## mc.crisk2.bart only
nice=19L ## mc.crisk2.bart only
)
Arguments
x.train
Covariates for training (in sample) data for an event.
Must be a data.frame or a matrix with rows corresponding to
observations and columns to variables.
crisk2.bart will
generate draws of f_1(t, x) for each x which is
a row of x.train (note that the definition of x.train is
dependent on whether y.train has been specified; see below).
y.train
Event binary response for training (in sample)
data.
If y.train is NULL, then y.train
(x.train and x.test, if specified) are generated by a
call to surv.pre.bart (which require that times and
delta be provided: see below); otherwise, y.train
(x.train and x.test, if specified) are utilized as
given assuming that the data construction has already been performed.
x.train2
Covariates for training (in sample)
data of for a cause 1 event. Similar to x.train above.
y.train2
Cause 1 event binary response for training (in sample) data.
Similar to y.train above.
times
The time of event or right-censoring, s_i.
If y.train is NULL, then times (and delta)
must be provided.
delta
The event indicator: 1 for cause 1, 2 for cause 2 and 0 is censored.
If y.train is NULL, then delta (and times)
must be provided.
K
If provided, then coarsen times per the quantiles
1/K, 2/K, ..., K/K.
x.test
Covariates for test (out of sample) data of an event.
Must be a data.frame or a matrix and have the same structure as
x.train.
crisk2.bart will generate draws of
f_1(t, x) for each x which is a row of
x.test.
x.test2
Covariates for test (out of sample) data of a cause 1 event.
Similar to x.test above.
sparse
Whether to perform variable selection based on a
sparse Dirichlet prior; see Linero 2016.
theta
Set theta parameter; zero means random.
omega
Set omega parameter; zero means random.
a
Sparse parameter for Beta(a, b) prior:
0.5<=a<=1 where lower values inducing more sparsity.
b
Sparse parameter for Beta(a, b) prior; typically,
b=1.
rho
Sparse parameter: typically rho=p where p is the
number of covariates in x.train.
rho2
Sparse parameter: typically rho2=p where p is the
number of covariates in x.train2.
augment
Whether data augmentation is to be performed in sparse
variable selection.
xinfo
You can provide the cutpoints to BART or let BART
choose them for you. To provide them, use the xinfo
argument to specify a list (matrix) where the items (rows) are the
covariates and the contents of the items (columns) are the
cutpoints.
xinfo2
Cause 2 cutpoints.
usequants
If usequants=FALSE, then the
cutpoints in xinfo are generated uniformly; otherwise,
if TRUE, uniform quantiles are used for the cutpoints.
rm.const
Whether or not to remove constant variables.
type
Whether to employ probit BART via Albert-Chib,
'pbart', or logistic BART by Holmes-Held, 'lbart'.
ntype
The integer equivalent of type where
'wbart' is 1, 'pbart' is 2 and
'lbart' is 3.
k
k is the number of prior standard deviations
f_h(t, x)
is away from +/-3. The bigger k is, the more conservative the
fitting will be.
power
Power parameter for tree prior.
base
Base parameter for tree prior.
offset
Cause 1 binary offset.
offset2
Cause 2 binary offset.
tau.num
The numerator in the tau definition.
ntree
The number of trees in the sum.
numcut
The number of possible values of cutpoints (see
usequants). If a single number if given, this is used for all
variables. Otherwise a vector with length equal to
ncol(x.train) is required, where the i^{th}
element gives the number of cutpoints used for the i^{th}
variable in x.train. If usequants is FALSE,
numcut equally spaced cutoffs are used covering the range of
values in the corresponding column of x.train. If
usequants is TRUE, then min(numcut, the number of
unique values in the corresponding columns of x.train - 1) cutpoint
values are used.
ndpost
The number of posterior draws returned.
nskip
Number of MCMC iterations to be treated as burn in.
keepevery
Every keepevery draw is kept to be returned to the user.
printevery
As the MCMC runs, a message is printed every printevery draws.
id
crisk2.bart only: unique identifier added to returned list.
seed
mc.crisk2.bart only: seed required for reproducible MCMC.
mc.cores
mc.crisk2.bart only: number of cores to employ in parallel.
nice
mc.crisk2.bart only: set the job niceness. The
default niceness is 19: niceness goes from 0 (highest priority) to
19 (lowest priority).
Value
crisk2.bart returns an object of type crisk2bart which is
essentially a list. Besides the items listed
below, the list has offset, offset2,
times which are the unique times, K
which is the number of unique times, tx.train and
tx.test, if any.
yhat.train
A matrix with ndpost rows and
nrow(x.train) columns. Each row corresponds to a draw
f^*_1 from the posterior of f_1 and each column
corresponds to a row of x.train. The (i,j) value is
f^*_1(t, x) for the i^{th} kept draw of
f_1 and the j^{th} row of x.train.
Burn-in is dropped.
yhat.test
Same as yhat.train but now the x's are the rows of the test data.
surv.test
test data fits for the survival function, S(t, x).
surv.test.mean
mean of surv.test over the posterior samples.
prob.test
The probability of suffering an event.
prob.test2
The probability of suffering a cause 1 event.
cif.test
The cumulative incidence function of cause 1,
F_1(t, x).
cif.test2
The cumulative incidence function of cause 2,
F_2(t, x).
cif.test.mean
mean of cif.test columns for cause 1.
cif.test2.mean
mean of cif.test2 columns for cause 2.
varcount
a matrix with ndpost rows and
nrow(x.train) columns.
Each row is for a draw. For each variable (corresponding to the columns),
the total count of the number of times
this variable is used for an event in a tree decision rule (over all trees) is given.
varcount2
For each variable the total count of the number of times
this variable is used for a cause 1 event in a tree decision rule is given.
See Also
surv.pre.bart, predict.crisk2bart,
mc.crisk2.pwbart, crisk.bart
Examples
data(transplant)
pfit <- survfit(Surv(futime, event) ~ abo, transplant)
# competing risks for type O
plot(pfit[4,], xscale=7, xmax=735, col=1:3, lwd=2, ylim=c(0, 1),
xlab='t (weeks)', ylab='Aalen-Johansen (AJ) CI(t)')
legend(450, .4, c("Death", "Transplant", "Withdrawal"), col=1:3, lwd=2)
## plot(pfit[4,], xscale=30.5, xmax=735, col=1:3, lwd=2, ylim=c(0, 1),
## xlab='t (months)', ylab='Aalen-Johansen (AJ) CI(t)')
## legend(450, .4, c("Death", "Transplant", "Withdrawal"), col=1:3, lwd=2)
delta <- (as.numeric(transplant$event)-1)
## recode so that delta=1 is cause of interest; delta=2 otherwise
delta[delta==1] <- 4
delta[delta==2] <- 1
delta[delta>1] <- 2
table(delta, transplant$event)
times <- pmax(1, ceiling(transplant$futime/7)) ## weeks
##times <- pmax(1, ceiling(transplant$futime/30.5)) ## months
table(times)
typeO <- 1*(transplant$abo=='O')
typeA <- 1*(transplant$abo=='A')
typeB <- 1*(transplant$abo=='B')
typeAB <- 1*(transplant$abo=='AB')
table(typeA, typeO)
x.train <- cbind(typeO, typeA, typeB, typeAB)
x.test <- cbind(1, 0, 0, 0)
dimnames(x.test)[[2]] <- dimnames(x.train)[[2]]
##test BART with token run to ensure installation works
set.seed(99)
post <- crisk2.bart(x.train=x.train, times=times, delta=delta,
x.test=x.test, nskip=1, ndpost=1, keepevery=1)
## Not run:
## run one long MCMC chain in one process
## set.seed(99)
## post <- crisk2.bart(x.train=x.train, times=times, delta=delta, x.test=x.test)
## in the interest of time, consider speeding it up by parallel processing
## run "mc.cores" number of shorter MCMC chains in parallel processes
post <- mc.crisk2.bart(x.train=x.train, times=times, delta=delta,
x.test=x.test, seed=99, mc.cores=8)
K <- post$K
typeO.cif.mean <- apply(post$cif.test, 2, mean)
typeO.cif.025 <- apply(post$cif.test, 2, quantile, probs=0.025)
typeO.cif.975 <- apply(post$cif.test, 2, quantile, probs=0.975)
plot(pfit[4,], xscale=7, xmax=735, col=1:3, lwd=2, ylim=c(0, 0.8),
xlab='t (weeks)', ylab='CI(t)')
points(c(0, post$times)*7, c(0, typeO.cif.mean), col=4, type='s', lwd=2)
points(c(0, post$times)*7, c(0, typeO.cif.025), col=4, type='s', lwd=2, lty=2)
points(c(0, post$times)*7, c(0, typeO.cif.975), col=4, type='s', lwd=2, lty=2)
legend(450, .4, c("Transplant(BART)", "Transplant(AJ)",
"Death(AJ)", "Withdrawal(AJ)"),
col=c(4, 2, 1, 3), lwd=2)
##dev.copy2pdf(file='../vignettes/figures/liver-BART.pdf')
## plot(pfit[4,], xscale=30.5, xmax=735, col=1:3, lwd=2, ylim=c(0, 0.8),
## xlab='t (months)', ylab='CI(t)')
## points(c(0, post$times)*30.5, c(0, typeO.cif.mean), col=4, type='s', lwd=2)
## points(c(0, post$times)*30.5, c(0, typeO.cif.025), col=4, type='s', lwd=2, lty=2)
## points(c(0, post$times)*30.5, c(0, typeO.cif.975), col=4, type='s', lwd=2, lty=2)
## legend(450, .4, c("Transplant(BART)", "Transplant(AJ)",
## "Death(AJ)", "Withdrawal(AJ)"),
## col=c(4, 2, 1, 3), lwd=2)
## End(Not run)
BART documentation built on June 22, 2024, 11:33 a.m.
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| crisk2.bart | R Documentation |
BART for competing risks
Description
Here we have implemented another approach to utilize BART for competing risks that is very flexible, and is akin to discrete-time survival analysis. Following the capabilities of BART, we allow for maximum flexibility in modeling the dependence of competing failure times on covariates. In particular, we do not impose proportional hazards.
Similar to crisk.bart, we utilize two BART models, yet they are
two different BART models than previously considered. First, given an
event of either cause occurred, we employ a typical binary BART model to
discriminate between cause 1 and 2. Next, we proceed as if it were a typical
survival analysis with BART for an absorbing event from either cause.
To elaborate, consider data in the form: (s_i, \delta_i,
{x}_i) where s_i is the event time;
\delta_i is an indicator distinguishing events,
\delta_i=h due to cause h in {1, 2}, from
right-censoring, \delta_i=0; {x}_i is a vector of
covariates; and i=1, ..., N indexes subjects.
We denote the K distinct event/censoring times by
0<t_{(1)}<...<t_{(K)}<\infty thus
taking t_{(j)} to be the j^{th} order statistic
among distinct observation times and, for convenience,
t_{(0)}=0.
First, consider event indicators for an event from either cause:
y_{1ij} for each subject i at each distinct time
t_{(j)} up to and including the subject's last observation
time s_i=t_{(n_i)} with n_i=\arg \max_j [t_{(j)}\leq
s_i]. We denote by p_{1ij} the
probability of an event at time t_{(j)} conditional on no
previous event. We now write the model for y_{1ij} as a
nonparametric probit (or logistic) regression of y_{1ij} on
the time t_{(j)} and the covariates {x}_{1i},
and then utilize BART for binary responses. Specifically,
y_{1ij}\ =\ I[\delta_i>0] I[s_i=t_{(j)}],\ j=1, ..., n_i. Therefore, we have p_{1ij} =
F(mu_{1ij}),\ mu_{1ij} = mu_1+f_1(t_{(j)}, {x}_{1i}) where F denotes the Normal (or
Logistic) cdf.
Next, we denote by p_{2i} the probability of a cause 1
event at time s_i conditional on an event having
occurred. We now write the model for y_{2i} as a
nonparametric probit (or logistic) regression of y_{2i} on
the time s_i and the covariates {x}_{2i},
via BART for binary responses. Specifically,
y_{2i}\ =\ I[\delta_i=1]. Therefore, we
have p_{2i} = F(mu_{2i}),\ mu_{2i} = mu_2+f_2(s_i,
{x}_{2i}) where
F denotes the Normal (or Logistic) cdf. Although, we modeled
p_{2i} at the time of an event, s_i, we can
estimate this probability at any other time points on the grid via
p(t_{(j)}, x_2)=F( mu_2+f_2(t_{(j)}, {x}_2)).
Finally, based on these probabilities,
p_{hij}, we can construct targets of inference such as the
cumulative incidence functions.
Usage
crisk2.bart(x.train=matrix(0,0,0), y.train=NULL,
x.train2=x.train, y.train2=NULL,
times=NULL, delta=NULL, K=NULL,
x.test=matrix(0,0,0), x.test2=x.test,
sparse=FALSE, theta=0, omega=1,
a=0.5, b=1, augment=FALSE,
rho=NULL, rho2=NULL,
xinfo=matrix(0,0,0), xinfo2=matrix(0,0,0),
usequants=FALSE,
rm.const=TRUE, type='pbart',
ntype=as.integer(
factor(type, levels=c('wbart', 'pbart', 'lbart'))),
k=2, power=2, base=0.95,
offset=NULL, offset2=NULL,
tau.num=c(NA, 3, 6)[ntype],
ntree=50, numcut=100, ndpost=1000, nskip=250,
keepevery = 10L,
printevery=100L,
id=NULL, ## crisk2.bart only
seed=99, ## mc.crisk2.bart only
mc.cores=2, ## mc.crisk2.bart only
nice=19L ## mc.crisk2.bart only
)
mc.crisk2.bart(x.train=matrix(0,0,0), y.train=NULL,
x.train2=x.train, y.train2=NULL,
times=NULL, delta=NULL, K=NULL,
x.test=matrix(0,0,0), x.test2=x.test,
sparse=FALSE, theta=0, omega=1,
a=0.5, b=1, augment=FALSE,
rho=NULL, rho2=NULL,
xinfo=matrix(0,0,0), xinfo2=matrix(0,0,0),
usequants=FALSE,
rm.const=TRUE, type='pbart',
ntype=as.integer(
factor(type, levels=c('wbart', 'pbart', 'lbart'))),
k=2, power=2, base=0.95,
offset=NULL, offset2=NULL,
tau.num=c(NA, 3, 6)[ntype],
ntree=50, numcut=100, ndpost=1000, nskip=250,
keepevery = 10L,
printevery=100L,
id=NULL, ## crisk2.bart only
seed=99, ## mc.crisk2.bart only
mc.cores=2, ## mc.crisk2.bart only
nice=19L ## mc.crisk2.bart only
)
Arguments
x.train |
Covariates for training (in sample) data for an event. |
y.train |
Event binary response for training (in sample)
data. |
x.train2 |
Covariates for training (in sample)
data of for a cause 1 event. Similar to |
y.train2 |
Cause 1 event binary response for training (in sample) data.
Similar to |
times |
The time of event or right-censoring, |
delta |
The event indicator: 1 for cause 1, 2 for cause 2 and 0 is censored. |
K |
If provided, then coarsen |
x.test |
Covariates for test (out of sample) data of an event. |
x.test2 |
Covariates for test (out of sample) data of a cause 1 event.
Similar to |
sparse |
Whether to perform variable selection based on a sparse Dirichlet prior; see Linero 2016. |
theta |
Set |
omega |
Set |
a |
Sparse parameter for |
b |
Sparse parameter for |
rho |
Sparse parameter: typically |
rho2 |
Sparse parameter: typically |
augment |
Whether data augmentation is to be performed in sparse variable selection. |
xinfo |
You can provide the cutpoints to BART or let BART
choose them for you. To provide them, use the |
xinfo2 |
Cause 2 cutpoints. |
usequants |
If |
rm.const |
Whether or not to remove constant variables. |
type |
Whether to employ probit BART via Albert-Chib,
|
ntype |
The integer equivalent of |
k |
k is the number of prior standard deviations
|
power |
Power parameter for tree prior. |
base |
Base parameter for tree prior. |
offset |
Cause 1 binary offset. |
offset2 |
Cause 2 binary offset. |
tau.num |
The numerator in the |
ntree |
The number of trees in the sum. |
numcut |
The number of possible values of cutpoints (see
|
ndpost |
The number of posterior draws returned. |
nskip |
Number of MCMC iterations to be treated as burn in. |
keepevery |
Every |
printevery |
As the MCMC runs, a message is printed every |
id |
|
seed |
|
mc.cores |
|
nice |
|
Value
crisk2.bart returns an object of type crisk2bart which is
essentially a list. Besides the items listed
below, the list has offset, offset2,
times which are the unique times, K
which is the number of unique times, tx.train and
tx.test, if any.
yhat.train |
A matrix with |
yhat.test |
Same as |
surv.test |
test data fits for the survival function, |
surv.test.mean |
mean of |
prob.test |
The probability of suffering an event. |
prob.test2 |
The probability of suffering a cause 1 event. |
cif.test |
The cumulative incidence function of cause 1,
|
cif.test2 |
The cumulative incidence function of cause 2,
|
cif.test.mean |
mean of |
cif.test2.mean |
mean of |
varcount |
a matrix with |
varcount2 |
For each variable the total count of the number of times this variable is used for a cause 1 event in a tree decision rule is given. |
See Also
surv.pre.bart, predict.crisk2bart,
mc.crisk2.pwbart, crisk.bart
Examples
data(transplant)
pfit <- survfit(Surv(futime, event) ~ abo, transplant)
# competing risks for type O
plot(pfit[4,], xscale=7, xmax=735, col=1:3, lwd=2, ylim=c(0, 1),
xlab='t (weeks)', ylab='Aalen-Johansen (AJ) CI(t)')
legend(450, .4, c("Death", "Transplant", "Withdrawal"), col=1:3, lwd=2)
## plot(pfit[4,], xscale=30.5, xmax=735, col=1:3, lwd=2, ylim=c(0, 1),
## xlab='t (months)', ylab='Aalen-Johansen (AJ) CI(t)')
## legend(450, .4, c("Death", "Transplant", "Withdrawal"), col=1:3, lwd=2)
delta <- (as.numeric(transplant$event)-1)
## recode so that delta=1 is cause of interest; delta=2 otherwise
delta[delta==1] <- 4
delta[delta==2] <- 1
delta[delta>1] <- 2
table(delta, transplant$event)
times <- pmax(1, ceiling(transplant$futime/7)) ## weeks
##times <- pmax(1, ceiling(transplant$futime/30.5)) ## months
table(times)
typeO <- 1*(transplant$abo=='O')
typeA <- 1*(transplant$abo=='A')
typeB <- 1*(transplant$abo=='B')
typeAB <- 1*(transplant$abo=='AB')
table(typeA, typeO)
x.train <- cbind(typeO, typeA, typeB, typeAB)
x.test <- cbind(1, 0, 0, 0)
dimnames(x.test)[[2]] <- dimnames(x.train)[[2]]
##test BART with token run to ensure installation works
set.seed(99)
post <- crisk2.bart(x.train=x.train, times=times, delta=delta,
x.test=x.test, nskip=1, ndpost=1, keepevery=1)
## Not run:
## run one long MCMC chain in one process
## set.seed(99)
## post <- crisk2.bart(x.train=x.train, times=times, delta=delta, x.test=x.test)
## in the interest of time, consider speeding it up by parallel processing
## run "mc.cores" number of shorter MCMC chains in parallel processes
post <- mc.crisk2.bart(x.train=x.train, times=times, delta=delta,
x.test=x.test, seed=99, mc.cores=8)
K <- post$K
typeO.cif.mean <- apply(post$cif.test, 2, mean)
typeO.cif.025 <- apply(post$cif.test, 2, quantile, probs=0.025)
typeO.cif.975 <- apply(post$cif.test, 2, quantile, probs=0.975)
plot(pfit[4,], xscale=7, xmax=735, col=1:3, lwd=2, ylim=c(0, 0.8),
xlab='t (weeks)', ylab='CI(t)')
points(c(0, post$times)*7, c(0, typeO.cif.mean), col=4, type='s', lwd=2)
points(c(0, post$times)*7, c(0, typeO.cif.025), col=4, type='s', lwd=2, lty=2)
points(c(0, post$times)*7, c(0, typeO.cif.975), col=4, type='s', lwd=2, lty=2)
legend(450, .4, c("Transplant(BART)", "Transplant(AJ)",
"Death(AJ)", "Withdrawal(AJ)"),
col=c(4, 2, 1, 3), lwd=2)
##dev.copy2pdf(file='../vignettes/figures/liver-BART.pdf')
## plot(pfit[4,], xscale=30.5, xmax=735, col=1:3, lwd=2, ylim=c(0, 0.8),
## xlab='t (months)', ylab='CI(t)')
## points(c(0, post$times)*30.5, c(0, typeO.cif.mean), col=4, type='s', lwd=2)
## points(c(0, post$times)*30.5, c(0, typeO.cif.025), col=4, type='s', lwd=2, lty=2)
## points(c(0, post$times)*30.5, c(0, typeO.cif.975), col=4, type='s', lwd=2, lty=2)
## legend(450, .4, c("Transplant(BART)", "Transplant(AJ)",
## "Death(AJ)", "Withdrawal(AJ)"),
## col=c(4, 2, 1, 3), lwd=2)
## End(Not run)
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