surv.bart {BART} R Documentation

## Survival analysis with BART

### Description

Here we have implemented a simple and direct approach to utilize BART in survival analysis 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 survival times on covariates. In particular, we do not impose proportional hazards.

To elaborate, consider data in the usual form: (t_i, \delta_i, {x}_i) where t_i is the event time, \delta_i is an indicator distinguishing events (\delta=1) from right-censoring (\delta=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. Now consider event indicators y_{ij} for each subject i at each distinct time t_{(j)} up to and including the subject's observation time t_i=t_{(n_i)} with n_i=\sum_j I[t_{(j)}\leq t_i]. This means y_{ij}=0 if j<n_i and y_{in_i}=\delta_i.

We then denote by p_{ij} the probability of an event at time t_{(j)} conditional on no previous event. We now write the model for y_{ij} as a nonparametric probit regression of y_{ij} on the time t_{(j)} and the covariates {x}_i, and then utilize BART for binary responses. Specifically,  y_{ij}\ =\ \delta_i I[t_i=t_{(j)}],\ j=1, ..., n_i ; we have p_{ij} = F(\mu_{ij}),\ \mu_{ij} = \mu_0+f(t_{(j)}, {x}_i) where F denotes the standard normal cdf (probit link). As in the binary response case, f is the sum of many tree models.

### Usage


surv.bart( x.train=matrix(0,0,0),
y.train=NULL, times=NULL, delta=NULL,
x.test=matrix(0,0,0),
K=NULL, events=NULL, ztimes=NULL, zdelta=NULL,
sparse=FALSE, theta=0, omega=1,
a=0.5, b=1, augment=FALSE, rho=NULL,
xinfo=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=.95,
offset=NULL, tau.num=c(NA, 3, 6)[ntype],
ntree=50, numcut=100, ndpost=1000, nskip=250,
keepevery = 10L,

printevery=100L,

id=NULL,    ## surv.bart only
seed=99,    ## mc.surv.bart only
mc.cores=2, ## mc.surv.bart only
nice=19L    ## mc.surv.bart only
)

mc.surv.bart( x.train=matrix(0,0,0),
y.train=NULL, times=NULL, delta=NULL,
x.test=matrix(0,0,0),
K=NULL, events=NULL, ztimes=NULL, zdelta=NULL,
sparse=FALSE, theta=0, omega=1,
a=0.5, b=1, augment=FALSE, rho=NULL,
xinfo=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=.95,
offset=NULL, tau.num=c(NA, 3, 6)[ntype],
ntree=50, numcut=100, ndpost=1000, nskip=250,
keepevery = 10L,

printevery=100L,

id=NULL,    ## surv.bart only
seed=99,    ## mc.surv.bart only
mc.cores=2, ## mc.surv.bart only
nice=19L    ## mc.surv.bart only
)


### Arguments

 x.train Explanatory variables for training (in sample) data. Must be a matrix with (as usual) rows corresponding to observations and columns to variables. surv.bart will generate draws of f(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 Binary response dependent variable 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. times The time of event or right-censoring. If y.train is NULL, then times (and delta) must be provided. delta The event indicator: 1 is an event while 0 is censored. If y.train is NULL, then delta (and times) must be provided. x.test Explanatory variables for test (out of sample) data. Must be a matrix and have the same structure as x.train. surv.bart will generate draws of f(t, x) for each x which is a row of x.test. K If provided, then coarsen times per the quantiles 1/K, 2/K, ..., K/K. events If provided, then use for the grid of time points. ztimes If provided, then these columns of x.train (and x.test if any) are the times for time-dependent covariates. They will be transformed into time-dependent covariate sojourn times. zdelta If provided, then these columns of x.train (and x.test if any) are the delta for time-dependent covariates. They will be transformed into time-dependent covariate binary events. sparse Whether to perform variable selection based on a sparse Dirichlet prior rather than simply uniform; 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 under consideration. 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. 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 Albert-Chib, 'pbart', or 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(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 With binary BART, the centering is P(Y=1 | x) = F(f(x) + offset) where offset defaults to F^{-1}(mean(y.train)). You can use the offset parameter to over-ride these defaults. tau.num The numerator in the tau definition, i.e., tau=tau.num/(k*sqrt(ntree)). ntree The number of trees in the sum. ndpost The number of posterior draws returned. nskip Number of MCMC iterations to be treated as burn in. printevery As the MCMC runs, a message is printed every printevery draws. keepevery Every keepevery draw is kept to be returned to the user. A “draw” will consist of values f^*(t, x) at x = rows from the train(optionally) and test data, where f^* denotes the current draw of f. numcut The number of possible values of c (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 c 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) c values are used. id surv.bart only: unique identifier added to returned list. seed mc.surv.bart only: seed required for reproducible MCMC. mc.cores mc.surv.bart only: number of cores to employ in parallel. nice mc.surv.bart only: set the job niceness. The default niceness is 19: niceness goes from 0 (highest) to 19 (lowest).

### Value

surv.bart returns an object of type survbart which is essentially a list. Besides the items listed below, the list has a binaryOffset component giving the value used, a times component giving 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^* from the posterior of f and each column corresponds to a row of x.train. The (i,j) value is f^*(t, x) for the i^{th} kept draw of f 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 The survival function, S(t|x), where x's are the rows of the test data. yhat.train.mean train data fits = mean of yhat.train columns. yhat.test.mean test data fits = mean of yhat.test columns. surv.test.mean mean of surv.test columns. 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 that variable is used in a tree decision rule (over all trees) is given.

Note that yhat.train and yhat.test are f(t, x) + binaryOffset. If you want draws of the probability P(Y=1 | t, x) you need to apply the normal cdf (pnorm) to these values.

surv.pre.bart

### Examples


data(lung)

N <- length(lung$status) table(lung$ph.karno, lung$pat.karno) ## if physician's KPS unavailable, then use the patient's h <- which(is.na(lung$ph.karno))
lung$ph.karno[h] <- lung$pat.karno[h]

times <- lung$time delta <- lung$status-1 ##lung$status: 1=censored, 2=dead ##delta: 0=censored, 1=dead ## this study reports time in days rather than weeks or months ## coarsening from days to weeks or months will reduce the computational burden ##times <- ceiling(times/30) times <- ceiling(times/7) ## weeks table(times) table(delta) ## matrix of observed covariates x.train <- cbind(lung$sex, lung$age, lung$ph.karno)

## lung$sex: Male=1 Female=2 ## lung$age:        Age in years
## lung$ph.karno: Karnofsky performance score (dead=0:normal=100:by=10) ## rated by physician dimnames(x.train)[[2]] <- c('M(1):F(2)', 'age(39:82)', 'ph.karno(50:100:10)') table(x.train[ , 1]) summary(x.train[ , 2]) table(x.train[ , 3]) ##test BART with token run to ensure installation works set.seed(99) post <- surv.bart(x.train=x.train, times=times, delta=delta, nskip=1, ndpost=1, keepevery=1) ## Not run: ## run one long MCMC chain in one process ## set.seed(99) ## post <- surv.bart(x.train=x.train, times=times, delta=delta, x.test=x.test) ## in the interest of time, consider speeding it up by parallel processing ## run "mc.cores" number of shorter MCMC chains in parallel processes post <- mc.surv.bart(x.train=x.train, times=times, delta=delta, mc.cores=8, seed=99) pre <- surv.pre.bart(times=times, delta=delta, x.train=x.train, x.test=x.train) K <- pre$K
M <- nrow(post$yhat.train) pre$tx.test <- rbind(pre$tx.test, pre$tx.test)
pre$tx.test[ , 2] <- c(rep(1, N*K), rep(2, N*K)) ## sex pushed to col 2, since time is always in col 1 pred <- predict(post, newdata=pre$tx.test, mc.cores=8)

pd <- matrix(nrow=M, ncol=2*K)

for(j in 1:K) {
h <- seq(j, N*K, by=K)
pd[ , j] <- apply(pred$surv.test[ , h], 1, mean) pd[ , j+K] <- apply(pred$surv.test[ , h+N*K], 1, mean)
}

pd.mu  <- apply(pd, 2, mean)
pd.025 <- apply(pd, 2, quantile, probs=0.025)
pd.975 <- apply(pd, 2, quantile, probs=0.975)

males <- 1:K
females <- males+K

plot(c(0, pre$times), c(1, pd.mu[males]), type='s', col='blue', ylim=0:1, ylab='S(t, x)', xlab='t (weeks)', main=paste('Advanced Lung Cancer ex. (BART::lung)', "Friedman's partial dependence function", 'Male (blue) vs. Female (red)', sep='\n')) lines(c(0, pre$times), c(1, pd.025[males]), col='blue', type='s', lty=2)
lines(c(0, pre$times), c(1, pd.975[males]), col='blue', type='s', lty=2) lines(c(0, pre$times), c(1, pd.mu[females]), col='red', type='s')
lines(c(0, pre$times), c(1, pd.025[females]), col='red', type='s', lty=2) lines(c(0, pre$times), c(1, pd.975[females]), col='red', type='s', lty=2)

## End(Not run)


[Package BART version 2.9.7 Index]