surv.bart: Survival analysis with BART

In BART: Bayesian Additive Regression Trees

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.

See Also

surv.pre.bart

Examples

## load survival package for the advanced lung cancer example
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)

BART documentation built on June 22, 2024, 11:33 a.m.