rsf.default {randomSurvivalForest}R Documentation

Random Survival Forest Entry Point

Description

Random Survival Forests (RSF) for right censored survival data (Ishwaran, Kogalur, Blackstone and Lauer, 2007). RSF is an extension of Breiman's Random Forests (Breiman, 2001) to survival analysis settings. Algorithm uses a binary recursive tree growing procedure with different splitting rules for growing an ensemble cumulative hazard function. An “out-of-bag” estimate of Harrell's concordance index (Harrell, 1982) is provided for assessing prediction. Importance values for variables can be computed. Prediction on test data is also available. Missing data (x-variables, survival times, censoring indicators) can be imputed on both training and test data. Note this is the default generic method for the package.

Usage

## Default S3 method:
rsf(formula,
    data = NULL,
    ntree = 1000,
    mtry = NULL,
    nodesize = NULL,
    splitrule = c("logrank", "conserve", "logrankscore", "logrankapprox")[1],
    importance = TRUE,
    big.data = FALSE,
    na.action = c("na.omit", "na.impute")[1],
    predictorWt = NULL,
    forest = FALSE,
    proximity = FALSE, 
    seed = NULL,
    ntime = NULL,
    add.noise = FALSE,
    do.trace = FALSE,
    ...)

Arguments

formula A symbolic description of the model to be fit. Details for model specification are given below.
data Data frame containing the data used in the formula. Missing values allowed. See na.action for details.
ntree Number of trees to grow. This should not be set to a number too small, in order to ensure that every input row gets predicted at least a few times.
mtry Number of variables randomly sampled at each split. The default is sqrt(p), where p equals the number of variables.
nodesize Minimum number of deaths with unique survival times required for a terminal node. Default is roughly min(3,round(0.632*ndead)). Larger values cause smaller trees to be grown.
splitrule Splitting rule used for splitting nodes in growing the survival tree. Possible values are logrank, conserve, logrankscore and logrankapprox. Default value is logrank. See details below.
importance Logical. Should importance of variables be estimated?
big.data Logical. Set this value to true when the number of variables p is very large, or the data is very large. See details below.
na.action Action taken if the data contain NA's. Possible values are na.omit and na.impute. Default is na.omit, which removes the entire record if even one of its entries is NA (applies only to entries specifically called in 'formula'). The action na.impute implements a sophisticated tree imputation technique. See details below.
predictorWt Vector of non-negative weights where entry k represents the likehood of selecting variable k as a candidate for splitting. Default is to use uniform weights. Vector must be of dimension p, where p equals the number of variables.
forest Logical. Should the forest object be returned? Used for prediction on new data. Default is FALSE.
proximity Logical. Should proximity measure between observations be calculated? Creates an nxn matrix (which can be huge). Default is FALSE.
seed Seed for random number generator. Must be a negative integer (the R wrapper handles incorrectly set seed values).
ntime Maximum number of desired distinct time points considered for evaluating ensemble. Default equals number of distinct events.
add.noise Logical. Should noise variable be added?
do.trace Logical. Should trace output be enabled? Default is FALSE. Integer values can also be passed. A positive value causes output to be printed each do.trace iteration.
... Further arguments passed to or from other methods.

Details

The default rule, the logrank splitting rule, grows trees by splitting nodes by maximization of the log-rank test statistic (Segal, 1988; Leblanc and Crowley, 1993). The conserve splitting rule splits nodes by finding daughters closest to the conservation of events principle (see Naftel, Blackstone and Turner, 1985). The logrankscore splitting rule uses a standardized log-rank statistic (Hothorn and Lausen, 2003). The logrankapprox splitting rule splits nodes using an approximation to the log-rank test (suggested by Michael Leblanc; also see Cox and Oakes, page 105).

All four rules often yield roughly the same prediction error performance, but users are encouraged to try all methods in any given example. The logrankapprox splitting rule is almost always fastest, especially with large data sets. After that, conserve is often second fastest. For very large data sets, discretizing continuous variables and/or the observed survival times can greatly speed up computational times. Discretization does not have to be overly granular for substantial gains to be seen.

A typical formula has the form Survrsf(time, censoring) ~ terms, where time is survival time and censoring is a binary censoring indicator. Note that censoring must be coded as 0=censored and 1=death (event) and time must be strictly positive.

Variables which are encoded as factors will be coerced into dummy variables. These dummy variables will be automatically labelled using the original variable name. For example, if marital status is a variable named “marital” encoded as a factor with levels “S”, “M” and “D”, two new dummy variables will be created labeled “maritalM” and “maritalS”.

Importance values for variables are computed as outlined in Breiman (2001). After each tree is grown, a given variable is randomly permuted in the out-of-bag (OOB) data and this data is dropped down the in-bag tree. An OOB ensemble cumulative hazard function (CHF) is computed from all such perturbed trees and its OOB error rate calculated. The difference between this and the OOB error rate (without permuting) is the importance value for the predictor. Error rates are measured by 1-C, where C is Harrell's concordance index. Error rates are between 0 and 1, with 0.5 representing the benchmark value of a procedure based on random guessing. A value of 0 is perfect. Thus, the importance value indicates how much misclassification increases, or decreases, for a new test case if the given variable were not available for that case, adjusting for all other variables used in growing the forest.

For very large data sets, or data with a large number of variables, users should consider setting the logical flag big.data to TRUE. This bypasses the large overhead needed by R in creating design matrices and parsing formula. Be aware, however, that variables are not processed and are interpreted as is when this option is turned on. Think of the data frame as containing time and censoring information and the rest of the data as the pre-processed design matrix when this option is on. Side effects are that factors are not transformed to dummy values (in fact they are coerced to NAs), and transformations used in the formula (such as logs etc.) are ignored.

Setting na.action to na.impute implements a tree imputation method whereby missing data (x-variables or outcomes) are imputed dynamically as a tree is grown by randomly sampling from the distribution within the current node (Ishwaran et al. 2007). OOB data is not used in imputation to avoid biasing prediction error and importance value estimates. Final imputation for integer valued variables and censoring indicators use a maximal class rule, whereas continuous variables and survival time use a mean rule. Records in which all outcome and x-variable information are missing are removed. Variables having all missing values are removed.

Value

An object of class (rsf, grow), which is a list with the following components:

call The original call to rsf.
formula The formula used in the call.
n Sample size of the data (depends upon NA's, see na.action).
ndead Number of deaths.
ntree Number of trees grown.
mtry Number of variables randomly selected for splitting at each node.
nodesize Minimum size of terminal nodes.
splitrule Splitting rule used.
time Vector of length n of survival times.
cens Vector of length n of censoring information (0=censored, 1=death).
timeInterest Sorted unique event times. Ensemble values are given for these time points only.
predictorNames A character vector of the variable names used in growing the forest.
predictorWt Vector of non-negative weights used for randomly sampling variables for splitting.
predictors Matrix of x-variables used to grow the forest.
ensemble A matrix of the bootstrap ensemble CHF with each row corresponding to an individual's CHF evaluated at each of the time points in timeInterest.
oob.ensemble Same as ensemble, but based on the OOB CHF.
mortality A vector of length n, with each value containing the bootstrap ensemble mortality for an individual in the data. Ensemble mortality values should be interpreted in terms of total number of deaths.
oob.mortality Same as mortality, but based on oob.ensemble.
err.rate Vector of length ntree containing OOB error rates for the ensemble, with the b-th element being the error rate for the ensemble formed using the first b trees. Error rates are measured using 1-C, where C is Harrell's concordance index.
leaf.count Number of terminal nodes for each tree in the forest. Vector of length ntree. A value of zero indicates a rejected tree (sometimes occurs when imputing missing data). Values of one indicate tree stumps.
importance Importance measure of each variable. For each variable this is the difference in the OOB error rate when the variable is randomly permuted compared to the OOB error rate without any permutation (i.e. the B-th component of err.rate). Large positive values indicate informative variables, whereas small values, or negative values, indicate variables unlikely to be informative.
forest If forest=TRUE, the forest object is returned. This object can then be used for prediction with new test data sets.
proximity If proximity=TRUE, a matrix of dimension nxn recording the frequency pairs of data points occur within the same terminal node. Value returned is a vector of the lower diagonal of the matrix. Use plot.proximity() to extract this information.
imputedIndv Vector of indices for cases with missing values. Can be NULL.
imputedData Matrix of imputed data. First two columns are censoring and survival time, respectively. Remaining columns are the x-variables. Row i contains imputed outcomes and x-variables for row imputedIndv[i] of predictors. Can be NULL.

Note

The key deliverable is the matrix ensemble containing the bootstrap ensemble CHF function for each individual evaluated at a set of distinct time points (an OOB ensemble, oob.ensemble, is also returned). The vector mortality (likewise oob.mortality) is a weighted sum over the columns of ensemble, weighted by the number of individuals at risk at the different time points. Entry i of the vector represents the estimated total mortality of individual i in terms of total number of deaths. In other words, if i has a mortality value of 100, then if all individuals had the same x-values as i, there would be on average 100 deaths in the dataset.

Different R wrappers are provided with the package to aid in interpreting the ensemble.

Author(s)

Hemant Ishwaran hemant.ishwaran@gmail.com and Udaya B. Kogalur ubk2101@columbia.edu

References

H. Ishwaran, U.B. Kogalur, E.H. Blackstone and M.S. Lauer (2007). Random Survival Forests. Cleveland Clinic Technical Report.

H. Ishwaran (2007). Variable importance in binary trees. Cleveland Clinic Technical Report.

L. Breiman (2001). Random forests, Machine Learning, 45:5-32.

F.E. Harrell et al. (1982). Evaluating the yield of medical tests, J. Amer. Med. Assoc., 247:2543-2546.

M. R. Segal. (1988). Regression trees for censored data, Biometrics, 44:35-47.

M. LeBlanc and J. Crowley (1993). Survival trees by goodness of split, J. Amer. Stat. Assoc., 88:457-467.

D.C. Naftel, E.H. Blackstone and M.E. Turner (1985). Conservation of events, unpublished notes.

T. Hothorn and B. Lausen (2003). On the exact distribution of maximally selected rank statistics, Computational Statistics & Data Analysis, 43:121-137.

D.R. Cox and D. Oakes (1988). Analysis of Survival Data, Chapman and Hall.

A. Liaw and M. Wiener (2002). Classification and regression by randomForest, R News, 2:18-22.

See Also

plot.ensemble, plot.variable, plot.error, plot.proximity, predict.rsf, print.rsf, find.interaction, pmml_to_rsf, rsf_to_pmml, Survrsf.

Examples

# Example 1:  Veteran's Administration lung cancer trial from
# Kalbfleisch & Prentice.  Randomized trial of two treatment
# regimens for lung cancer.  Minimal argument call.  Print
# results, then plot error rate and importance values.

data(veteran, package = "randomSurvivalForest")
veteran.out <- rsf(Survrsf(time, status)~., data = veteran)
print(veteran.out)
plot(veteran.out)

# Example 2:  Richer argument call.
# Note that forest option is set to true to illustrate 
# how one might use 'rsf' for prediction (see 'rsf.predict'
# for more details).

data(veteran, package = "randomSurvivalForest")
veteran.f <- as.formula(Survrsf(time, status)~.)
ntree <- 200
mtry <- 2
nodesize <- 3
splitrule <- "logrank"
proximity <- TRUE
forest <- TRUE
seed <- -1
ntime <- NULL
do.trace <- 25
veteran2.out <- rsf(veteran.f, veteran, ntree, mtry, nodesize, splitrule,
                    proximity = proximity,
                    forest = forest,
                    seed = seed,
                    ntime = ntime,
                    do.trace = do.trace)
print(veteran2.out)
plot.proximity(veteran2.out)

# Take a peek at the forest ...
head(veteran2.out$forest$nativeArray)

# Partial plot of top variable.
plot.variable(veteran2.out, partial = TRUE, n.pred=1)

## Not run: 
# Example 3:  Veteran data again. Look specifically at
# Karnofsky performance score.  Compare to Kaplan-Meier.
# Assumes "survival" library is loaded.

if (library("survival", logical.return = TRUE))
{
        data(veteran, package = "randomSurvivalForest")
        veteran3.out <- rsf(Survrsf(time, status)~karno,
                       veteran,
                       ntree = 1000)
        plot.ensemble(veteran3.out)
        par(mfrow = c(1,1))
        plot(survfit(Surv(time, status)~karno, data = veteran))
}

# Example 4:  Primary biliary cirrhosis (PBC) of the liver.
# Data found in Appendix D.1 of Fleming and Harrington, Counting
# Processes and Survival Analysis, Wiley, 1991 (only differences
# are that age is in days and sex and stage variables are not
# missing for observations 313-418).  

data(pbc, package = "randomSurvivalForest") 
pbc.out <- rsf(Survrsf(days,status)~., pbc, ntree = 1000)
print(pbc.out)

# Example 5:  Same as Example 4, but with imputation for missing values.

data(pbc, package = "randomSurvivalForest") 
pbc2.out <- rsf(Survrsf(days,status)~., pbc, ntree = 1000, na.action="na.impute")
# summary of analysis
print(pbc2.out)
# combine original data + imputed data 
pbc.imputed.data <- cbind(status=pbc2.out$cens, days=pbc2.out$time, pbc2.out$predictors)
pbc.imputed.data[pbc2.out$imputedIndv,] <- pbc2.out$imputedData
tail(pbc)
tail(pbc.imputed.data)

# Example 6:  Compare Cox regression to Random Survival Forests
# for PBC data.  Compute OOB estimate of Harrell's concordance 
# index for Cox regression using B = 100 bootstrap draws.
# Assumes "Hmisc" and "survival" libraries are loaded. 

if (library("survival", logical.return = TRUE) 
    & library("Hmisc", logical.return = TRUE))
{
    data(pbc, package = "randomSurvivalForest")
    pbc3.out <- rsf(Survrsf(days,status)~., pbc, mtry = 2, ntree = 1000)
    B <- 100 
    cox.err <- rep(NA, B) 
    cox.f <- as.formula(Surv(days,status)~.)  
    pbc.data <- pbc[apply(is.na(pbc), 1, sum) == 0,] ##remove NA's 
    cat("Out-of-bag Cox Analysis ...", "\n")
    for (b in 1:B) {
        cat("Cox bootstrap", b, "\n") 
        bag.sample <- sample(1:dim(pbc.data)[1],
                             dim(pbc.data)[1],
                             replace = TRUE) 
        oob.sample <- setdiff(1:dim(pbc.data)[1], bag.sample)
        train <- pbc.data[bag.sample,]
        test <- pbc.data[oob.sample,]
        cox.out <- coxph(cox.f, train)
        cox.out <- tryCatch({coxph(cox.f, train)}, error=function(ex){NULL})
        if (is.list(cox.out)) {
          cox.predict <- predict(cox.out, test)
          cox.err[b] <- rcorr.cens(cox.predict, 
                 Surv(pbc.data$days[oob.sample], pbc.data$status[oob.sample]))[1]
        }
     }
     cat("Error rates:", "\n")
     cat("Random Survival Forests:", pbc3.out$err.rate[pbc3.out$ntree], "\n")
     cat("         Cox Regression:", mean(cox.err, na.rm = TRUE), "\n")
}

# Example 7:  Using an external data set.

file.in <- "other.data"
other.data <- read.table(file.in, header = TRUE)
rsf.f <- as.formula(Survrsf(time, status)~.)
rsf.out <- rsf(formula = rsf.f, data = other.data) 

## End(Not run)

[Package randomSurvivalForest version 3.0.1 Index]