rsf.default {randomSurvivalForest} | R Documentation |
Random Survival Forests (RSF) (Ishwaran, Kogalur, Blackstone and Lauer, 2008) is an extension of Breiman's Random Forests (Breiman, 2001) to right-censored survival analysis settings. A forest of survival trees is grown and used to estimate an ensemble cumulative hazard function (CHF). Trees can be grown using different survival tree splitting rules. An “out-of-bag” estimate of Harrell's concordance index (Harrell, 1982) is provided for assessing prediction accuracy of the CHF. Variable importance (VIMP) can be computed for single, as well as grouped variables, as a means to filter variables and to assess variable predictiveness. RSF can be used to predict on test data. 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.
## Default S3 method: rsf(formula, data = NULL, ntree = 1000, mtry = NULL, nodesize = NULL, splitrule = c("logrank", "conserve", "logrankscore", "random")[1], nsplit = 0, importance = c("randomsplit", "permute", "none")[1], big.data = FALSE, na.action = c("na.omit", "na.impute")[1], nimpute = 1, predictorWt = NULL, forest = FALSE, proximity = FALSE, varUsed = NULL, seed = NULL, do.trace = FALSE, ...)
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 to grow trees. See details below. |
nsplit |
Non-negative integer value. If non-zero, a random version of the specified tree splitting rule is implemented. This can significantly increase speed. See details below. |
importance |
Method used to compute variable importance. See details below. |
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 to be 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 (for x-variables this applies only
to those specifically listed in 'formula'). The action
na.impute implements a sophisticated tree imputation
technique. See details below. |
nimpute |
Number of iterations of missing data algorithm. |
predictorWt |
Vector of non-negative weights where entry
k , after normalizing, is the probability 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 n xn
matrix (which can be huge). Default is FALSE. |
varUsed |
Analyzes which variables are used (split upon) in the topology
of the forest. Default is NULL. Possible values are all.trees ,
by.tree . See details below. |
seed |
Seed for random number generator. Must be a negative integer (the R wrapper handles incorrectly set seed values). |
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. |
Four primary splitting rules are available for growing a survival
forest. The default rule, logrank
, splits tree nodes by
maximization of the log-rank test statistic (Segal, 1988; Leblanc and
Crowley, 1993). A second rule, conserve
, splits nodes by
finding daughters closest to the conservation of events principle (see
Naftel, Blackstone and Turner, 1985). A third rule,
logrankscore
, uses a standardized log-rank statistic (Hothorn
and Lausen, 2003). A fourth rule, random
, implements pure
random splitting. For each node, a variable is randomly selected from
a random set of mtry
variables and the node is split using a
random split point (Lin and Jeon, 2006).
A random version of the logrank
, conserve
and
logrankscore
splitting rules can be invoked using
nsplit
. If nsplit
is set to a non-zero positive
integer, then a maximum of nsplit
split points are chosen
randomly for each of the mtry
variables within a node (this is
in contrast to deterministic splitting where all possible split points
for each of the mtry
variables are considered). The splitting
rule is applied to these random split points and the node is split on
that variable and random split point maximizing survival difference
(as measured by the splitting rule). Note that nsplit
has no
effect if the splitting rule is random
.
In terms of performance, a detailed study carried out by Ishwaran et
al. (2008) found logrank
and logrankscore
to be the most
accurate in terms of prediction error, followed by conserve
.
Setting nsplit
=1 and using logrank
splitting gave
performance close to logrank
, but with significantly shorter
computational times. Accuracy can be good further improved without
overly compromising speed by using larger values of nsplit
.
In addition to random splitting, computation times for very
large data sets can be improved by discretizing continuous variables
and/or the observed survival times. Discretization does not have to
be overly granular for substantial gains to be seen. Users may also
consider setting big.data
=TRUE for data with a large number of
variables. This bypasses the large overhead R needs to create design
matrices and parse 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. In particular, transformations used in
the formula (such as logs etc.) are ignored.
A typical RSF 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 encoded as factors are treated as such. If the factor is ordered, then splits are similar to real valued variables. If the factor is unordered, a split will move a subset of the levels in the parent node to the left daughter, and the complementary subset to the right daughter. All possible complementary pairs are considered and apply to factors with an unlimited number of levels. However, there is an optimization check to ensure that the number of splits attempted is not greater than the number of cases in a node (this internal check will override the nsplit value in random splitting mode if nsplit is large enough). Note that when predicting on test data involving factors, the factor labels in the test data must be the same as in the grow (training) data. Consider setting labels that are unique in the test data to missing.
Other than factors, all other x-variables are coerced and treated as being real valued.
Variable importance (VIMP) is computed similar to Breiman (2001),
although there are two ways to perturb a variable to determine its
VIMP: randomsplit
, permute
. The default method is
randomsplit
which works as follows. To calculate VIMP for a
variable x
, out-of-bag (OOB) cases are dropped down the
bootstrap (in-bag) survival tree. A case is assigned a daughter node
randomly whenever an x
-split is encountered. An OOB ensemble
cumulative hazard function (CHF) is computed from the forest of such
trees and its OOB error rate calculated. The VIMP for x
is the
difference between this and the OOB error rate for the original forest
(without random node assignment using x
). If permute
is
used, then x
is randomly permuted in OOB data and dropped down
the in-bag tree. See Ishwaran et al. (2008) for further details.
Prediction error is measured by 1-C, where C is Harrell's concordance index. Prediction error is between 0 and 1, and measures how well the ensemble correctly ranks (classifies) any two individuals in terms of survival. A value of 0.5 is no better than random guessing. A value of 0 is perfect. Because VIMP is based on the concordance index, VIMP indicates how much misclassification increases, or decreases, for a new test case if a given variable were not available for that case (given that the forest was grown using that variable).
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. 2008). OOB data
is not used in imputation to avoid biasing prediction error and VIMP
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. The algorithm can be iterated
by setting nimpute
to a positive integer greater than 1. A few
iterations should be used in heavy missing data settings to improve
accuracy of imputed values (see Ishwaran et al., 2008). Note if the
algorithm is iterated, a side effect is that missing values in
returned objects predictors
, time
and cens
are
replaced by imputed values. Further, imputed objects such as
imputedData
are set to NULL.
If varUsed
=all.trees
, a vector of size p
is
returned. Each element contains a count of the number of times a
split has occurred on this variable. If
varUsed
=by.tree
, a matrix of size ntree
xp
is returned. Each element [i][j] contains a count of the number of
times a split has occurred on variable [j] in tree [i].
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. |
nsplit |
Number of randomly selected split points. |
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 |
Data frame comprising 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 |
VIMP for each variable. |
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
n xn 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. |
varUsed |
Count of the number of times a variable is used in growing the forest. Can be a vector, matrix, or NULL. |
imputedIndv |
Vector of indices for cases with missing values. Can be NULL. |
imputedData |
Data frame comprising 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. |
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.
Hemant Ishwaran hemant.ishwaran@gmail.com and Udaya B. Kogalur ubk2101@columbia.edu
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.
T. Hothorn and B. Lausen (2003). On the exact distribution of maximally selected rank statistics, Comp. Stat. Data Anal., 43:121-137.
H. Ishwaran, U.B. Kogalur, E.H. Blackstone and M.S. Lauer (2008). Random survival forests, To appear in Ann. App. Statist..
H. Ishwaran, U.B. Kogalur (2007). Random survival forests for R, Rnews, 7/2:25-31.
H. Ishwaran (2007). Variable importance in binary regression trees and forests, Electronic J. Statist., 1:519-537.
M. LeBlanc and J. Crowley (1993). Survival trees by goodness of split, J. Amer. Stat. Assoc., 88:457-467.
A. Liaw and M. Wiener (2002). Classification and regression by randomForest, R News, 2:18-22.
Y. Lin and Y. Jeon (2006). Random forests and adaptive nearest neighbors, J. Amer. Stat. Assoc., 101:578-590.
D.C. Naftel, E.H. Blackstone and M.E. Turner (1985). Conservation of events, unpublished notes.
M. R. Segal. (1988). Regression trees for censored data, Biometrics, 44:35-47.
plot.ensemble
,
plot.variable
,
plot.error
,
plot.proximity
,
predict.rsf
,
print.rsf
,
find.interaction
,
pmml2rsf
,
rsf2pmml
,
Survrsf
.
#------------------------------------------------------------ # 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 (veteran data). # Forest is saved by setting 'forest' option to true # (see 'rsf.predict' for more details about prediction). # Coerce variable 'celltype' as a factor, and karnofsky score # as an ordered factor to illustrate factor useage in RSF. # Use random splitting with 'nsplit'. # Use 'varUsed' option. data(veteran, package = "randomSurvivalForest") veteran.f <- as.formula(Survrsf(time, status)~.) veteran$celltype <- factor(veteran$celltype, labels=c("squamous", "smallcell", "adeno", "large")) veteran$karno <- factor(veteran$karno, ordered = TRUE) ntree <- 200 mtry <- 2 nodesize <- 3 splitrule <- "logrank" nsplit <- 10 varUsed <- "by.tree" forest <- TRUE proximity <- TRUE do.trace <- 1 veteran2.out <- rsf(veteran.f, veteran, ntree, mtry, nodesize, splitrule, nsplit, varUsed = varUsed, forest = forest, proximity = proximity, do.trace = do.trace) print(veteran2.out) plot.proximity(veteran2.out) # Take a peek at the forest ... head(veteran2.out$forest$nativeArray) # Average number of times a variable was split on. apply(veteran2.out$varUsed,2,mean) # Partial plot of top variable. plot.variable(veteran2.out, partial = TRUE, npred=1) ## Not run: #------------------------------------------------------------ # Example 3: Veteran data (again). # Consider Karnofsky performance score. Compare to Kaplan-Meier. # Assumes "survival" library is loaded. if (library("survival", logical.return = TRUE)) { data(veteran, package = "randomSurvivalForest") veteran3.out <- rsf(Survrsf(time, status)~karno, veteran, ntree = 1000) plot.ensemble(veteran3.out) par(mfrow = c(1,1)) plot(survfit(Surv(time, status)~karno, data = veteran)) } #------------------------------------------------------------ # Example 4: Primary biliary cirrhosis (PBC) of the liver. # Data found in Appendix D.1 of Fleming and Harrington, Counting # Processes and Survival Analysis, Wiley, 1991 (modified so # that age is in days and sex and stage variables are not # missing for observations 313-418). data(pbc, package = "randomSurvivalForest") pbc.out <- rsf(Survrsf(days,status)~., pbc, ntree = 1000) print(pbc.out) #------------------------------------------------------------ # Example 5: Same as Example 4, but with imputation for # missing values. data(pbc, package = "randomSurvivalForest") pbc2.out <- rsf(Survrsf(days,status)~., pbc, ntree = 1000, na.action="na.impute") # summary of analysis print(pbc2.out) # Combine original data + imputed data. pbc.imputed.data <- cbind(status=pbc2.out$cens, days=pbc2.out$time, pbc2.out$predictors) pbc.imputed.data[pbc2.out$imputedIndv,] <- pbc2.out$imputedData tail(pbc) tail(pbc.imputed.data) # Iterate the missing data algorithm. # Use logrank random splitting (with nsplit=5) to increase speed. # Use trace to track algorithm in detail. # Note that a side effect of iterating is that the original data # are replaced by imputed values. pbc3.out <- rsf(Survrsf(days,status)~., pbc, ntree = 1000, nsplit=5, na.action="na.impute", nimpute=3, do.trace = TRUE) pbc.iterate.imputed.data <- cbind(status=pbc3.out$cens, days=pbc3.out$time, pbc3.out$predictors) #------------------------------------------------------------ # Example 6: Compare Cox regression to RSF (PBC data). # Compute OOB estimate of Harrell's concordance # index for Cox regression using B = 100 bootstrap draws. # Assumes "Hmisc" and "survival" libraries are loaded. if (library("survival", logical.return = TRUE) & library("Hmisc", logical.return = TRUE)) { data(pbc, package = "randomSurvivalForest") pbc3.out <- rsf(Survrsf(days,status)~., pbc, mtry = 2, ntree = 1000) B <- 100 cox.err <- rep(NA, B) cox.f <- as.formula(Surv(days,status)~.) pbc.data <- pbc[apply(is.na(pbc), 1, sum) == 0,] ##remove NA's cat("Out-of-bag Cox Analysis ...", "\n") for (b in 1:B) { cat("Cox bootstrap:", b, "\n") bag.sample <- sample(1:nrow(pbc.data), nrow(pbc.data), replace = TRUE) oob.sample <- setdiff(1:nrow(pbc.data), bag.sample) train <- pbc.data[bag.sample,] test <- pbc.data[oob.sample,] cox.out <- coxph(cox.f, train) cox.out <- tryCatch({coxph(cox.f, train)}, error=function(ex){NULL}) if (is.list(cox.out)) { cox.predict <- predict(cox.out, test) cox.err[b] <- rcorr.cens(cox.predict, Surv(pbc.data$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") } ## End(Not run)