Fast OpenMP parallel computing of random forests (Breiman 2001) for regression, classification, survival analysis (Ishwaran et al. 2008), competing risks (Ishwaran et al. 2012), multivariate (Segal and Xiao 2011), unsupervised (Mantero and Ishwaran 2020), quantile regression (Meinhausen 2006, Zhang et al. 2019, Greenwald-Khanna 2001), and class imbalanced q-classification (O'Brien and Ishwaran 2019). Different splitting rules invoked under deterministic or random splitting (Geurts et al. 2006, Ishwaran 2015) are available for all families. Different types of variable importance (VIMP), holdout VIMP, as well as confidence regions (Ishwaran and Lu 2019) can be calculated for single and grouped variables. Minimal depth variable selection (Ishwaran et al. 2010, 2011). Fast interface for missing data imputation using a variety of different random forest methods (Tang and Ishwaran 2017).

New items to be aware of:

1. For computational speed, the default VIMP is no longer "permute" (Breiman-Cutler permutation importance) and has been switched to "anti" (importance="anti", importance=TRUE; see below for details). Be aware in some situations, such as highly imbalanced classification, that permutation VIMP may perform better. Permutation VIMP is obtained using importance="permute".

2. save.memory can be used for big data to save memory; especially useful for survival and competing risks.

3. Mahalanobis splitting for multivariate regression with correlated y-outcomes (splitrule="mahalanobis"). Now allows for a user specified covariance matrix.

4. Visualize trees on your Safari or Google Chrome browser (works for all families). See get.tree.

This is the main entry point to the randomForestSRC package. For more information about this package and OpenMP parallel processing, use the command package?randomForestSRC.

rfsrc(formula, data, ntree = 500,
mtry = NULL, ytry = NULL,
nodesize = NULL, nodedepth = NULL,
splitrule = NULL, nsplit = NULL,
importance = c(FALSE, TRUE, "none", "anti", "permute", "random"),
block.size = if (any(is.element(as.character(importance),
c("none", "FALSE")))) NULL else 10,
bootstrap = c("by.root", "none", "by.user"),
samptype = c("swor", "swr"), samp = NULL, membership = FALSE,
sampsize = if (samptype == "swor") function(x){x * .632} else function(x){x},
na.action = c("na.omit", "na.impute"), nimpute = 1,
ntime = 150, cause,
perf.type = NULL,
proximity = FALSE, distance = FALSE, forest.wt = FALSE,
xvar.wt = NULL, yvar.wt = NULL, split.wt = NULL, case.wt  = NULL,
forest = TRUE,
save.memory = FALSE,
var.used = c(FALSE, "all.trees", "by.tree"),
split.depth = c(FALSE, "all.trees", "by.tree"),
seed = NULL,
do.trace = FALSE,
statistics = FALSE,
...)

## convenient interface for growing a CART tree
rfsrc.cart(formula, data, ntree = 1, mtry = ncol(data), bootstrap = "none", ...)

## Arguments

formula

Object of class 'formula' describing the model to fit. Interaction terms are not supported. If missing, unsupervised splitting is implemented.

data

Data frame containing the y-outcome and x-variables.

ntree

Number of trees.

mtry

Number of variables to possibly split at each node. Default is number of variables divided by 3 for regression. For all other families (including unsupervised settings), the square root of number of variables. Values are rounded up.

ytry

The number of randomly selected pseudo-outcomes for unsupervised families (see details below). Default is ytry=1.

nodesize

Minumum size of terminal node. The defaults are: survival (15), competing risk (15), regression (5), classification (1), mixed outcomes (3), unsupervised (3). It is recommended to experiment with different nodesize values.

nodedepth

Maximum depth to which a tree should be grown. Parameter is ignored by default.

splitrule

Splitting rule (see below).

nsplit

Non-negative integer specifying number of random splits for splitting a variable. When zero, all split values are used (deterministic splitting), which can be slower. By default 10 is used.

importance

Method for computing variable importance (VIMP); see below. Default action is importance="none" but VIMP can be recovered later using vimp or predict.

block.size

Determines how cumulative error rate is calculated. When NULL, only done once for entire forest; thus plot of the cumulative error rate will result in a flat line. To view the cumulative error rate on every nth tree, set the value to an integer between 1 and ntree. As an intended side effect, if importance is requested, VIMP is calculated in "blocks" of size equal to block.size, thus resulting in a compromise between ensemble and tree VIMP. The default action in that case is to use 10 trees.

bootstrap

Bootstrap protocol. Default is by.root which bootstraps the data by sampling with or without replacement (without replacement is the default; see the option samptype below). If none, the data is not bootstrapped (it is not possible to return OOB ensembles or prediction error in this case). If by.user, the bootstrap specified by samp is used.

samptype

Type of bootstrap used when by.root is in effect. Choices are swor (sampling without replacement; the default) and swr (sampling with replacement).

samp

Bootstrap specification when by.user is in effect. Array of dim n x ntree specifying how many times each record appears inbag in the bootstrap for each tree.

membership

Should terminal node membership and inbag information be returned?

sampsize

Function specifying bootstrap size when by.root is in effect. For sampling without replacement, it is the requested size of the sample, which by default is .632 times the sample size. For sampling with replacement, it is the sample size. Can also be specified using a number.

na.action

Action taken if the data contains NA's. Possible values are na.omit or na.impute. The default na.omit removes the entire record if any entry is NA. Selecting na.impute imputes the data (see below for details). Also see the function impute for fast imputation.

nimpute

Number of iterations of the missing data algorithm. Performance measures such as out-of-bag (OOB) error rates are optimistic if nimpute is greater than 1.

ntime

Integer value used for survival to constrain ensemble calculations to an ntime grid of time points over the observed event times. Alternatively if a vector of values of length greater than one is supplied, it is assumed these are the time points to be used (these will be adjusted to match closest observed event times). Setting ntime to zero (or NULL) uses all observed event times.

cause

Integer value between 1 and J indicating the event of interest for splitting a node for competing risks, where J is the number of event types. If not specified, the default is to use a composite splitting rule that averages over all event types. Can also be a vector of non-negative weights of length J specifying weights for each event (for example, a vector of ones reverts to the default composite split statistic). Regardless of how cause is specified, estimates for all event types are returned.

perf.type

Optional character value specifying metric used for predicted value, variable importance (VIMP), and error rate. Reverts to the family default metric if not specified. perf.type="none" turns off performance entirely which is a useful way to turn off C-index calculations for big survival data (which can be expensive). Values allowed for univariate/multivariate classification are: perf.type="misclass" (default), perf.type="brier" and perf.type="gmean".

proximity

Proximity of cases as measured by the frequency of sharing the same terminal node. This is an nxn matrix, which can be large. Choices are inbag, oob, all, TRUE, or FALSE. Setting proximity = TRUE is equivalent to proximity = "inbag".

distance

Distance between cases as measured by the ratio of the sum of the count of edges from each case to their immediate common ancestor node to the sum of the count of edges from each case to the root node. If the cases are co-terminal for a tree, this measure is zero and reduces to 1 - the proximity measure. This is an nxn matrix, which can be large. Choices are inbag, oob, all, TRUE, or FALSE. Setting distance = TRUE is equivalent to distance = "inbag".

forest.wt

Calculate the forest weight matrix? Creates an nxn matrix which can be used for prediction and constructing customized estimators. Choices are similar to proximity: inbag, oob, all, TRUE, or FALSE. The default is TRUE which is equivalent to inbag.

xvar.wt

Vector of non-negative weights (does not have to sum to 1) representing the probability of selecting a variable for splitting. Default is uniform weights.

yvar.wt

Used for sending in features with custom splitting. For expert use only.

split.wt

Vector of non-negative weights used for multiplying the split statistic for a variable. A large value encourages the node to split on a specific variable. Default is uniform weights.

case.wt

Vector of non-negative weights (does not have to sum to 1) for sampling cases. Observations with larger weights will be selected with higher probability in the bootstrap (or subsampled) samples. It is generally better to use real weights rather than integers. See the breast data example below illustrating its use for class imbalanced data.

forest

Save key forest values? Used for prediction on new data and required by many of the package functions. Turn this off if you are only interested in training a forest.

save.memory

Save memory? Default is to store terminal node quantities used for prediction on test data. This yields rapid prediction but can be memory intensive for big data, especially competing risks and survival models. Turn this flag off in those cases.

var.used

Return statistics on number of times a variable split? Default is FALSE. Possible values are all.trees which returns total number of splits of each variable, and by.tree which returns a matrix of number a splits for each variable for each tree.

split.depth

Records the minimal depth for each variable. Default is FALSE. Possible values are all.trees which returns a matrix of the average minimal depth for a variable (columns) for a specific case (rows), and by.tree which returns a three-dimensional array recording minimal depth for a specific case (first dimension) for a variable (second dimension) for a specific tree (third dimension).

seed

Negative integer specifying seed for the random number generator.

do.trace

Number of seconds between updates to the user on approximate time to completion.

statistics

Should split statistics be returned? Values can be parsed using stat.split.

...

Further arguments passed to or from other methods.

## Details

1. Types of forests

There is no need to set the type of forest as the package automagically determines the underlying random forest requested from the type of outcome and the formula supplied. There are several possible scenarios:

1. Regression forests for continuous outcomes.

2. Classification forests for factor outcomes.

3. Multivariate forests for continuous and/or factor outcomes and for mixed (both type) of outcomes.

4. Unsupervised forests when there is no outcome.

5. Survival forests for right-censored survival.

6. Competing risk survival forests for competing risk.

2. Splitting

1. Splitting rules are specified by the option splitrule.

2. For all families, pure random splitting can be invoked by setting splitrule="random".

3. For all families, computational speed can be increased using randomized splitting invoked by the option nsplit. See Improving Computational Speed.

3. Available splitting rules

• Regression analysis:

1. splitrule="mse" (default split rule): weighted mean-squared error splitting (Breiman et al. 1984, Chapter 8.4).

2. splitrule="quantile.regr": quantile regression splitting via the "check-loss" function. Requires specifying the target quantiles. See quantreg.rfsrc for further details.

3. la.quantile.regr: local adaptive quantile regression splitting. See quantreg.rfsrc.

• Classification analysis:

1. splitrule="gini" (default splitrule): Gini index splitting (Breiman et al. 1984, Chapter 4.3).

2. splitrule="auc": AUC (area under the ROC curve) splitting for both two-class and multiclass setttings. AUC splitting is appropriate for imbalanced data. See imbalanced for more information.

3. splitrule="entropy": entropy splitting (Breiman et al. 1984, Chapter 2.5, 4.3).

• Survival analysis:

1. splitrule="logrank" (default splitrule): log-rank splitting (Segal, 1988; Leblanc and Crowley, 1993).

2. splitrule="bs.gradient": gradient-based (global non-quantile) brier score splitting. The time horizon used for the Brier score is set to the 90th percentile of the observed event times. This can be over-ridden by the option prob, which must be a value between 0 and 1 (set to .90 by default).

3. splitrule="logrankscore": log-rank score splitting (Hothorn and Lausen, 2003).

• Competing risk analysis (for details see Ishwaran et al., 2014):

1. splitrule="logrankCR" (default splitrule): modified weighted log-rank splitting rule modeled after Gray's test (Gray, 1988). Use this to find *all* variables that are informative and when the goal is long term prediction.

2. splitrule="logrank": weighted log-rank splitting where each event type is treated as the event of interest and all other events are treated as censored. The split rule is the weighted value of each of log-rank statistics, standardized by the variance. Use this to find variables that affect a *specific* cause of interest and when the goal is a targeted analysis of a specific cause. However in order for this to be effective, remember to set the cause option to the targeted cause of interest. See examples below.

• Multivariate analysis:

1. Default is the multivariate normalized composite split rule using mean-squared error and Gini index (Tang and Ishwaran, 2017).

2. splitrule="mahalanobis": Mahalanobis splitting that adjusts for correlation (also allows for a user specified covariance matrix, see example below). Only works for multivariate regression (all outcomes must be real).

• Unsupervised analysis: In settings where there is no outcome, unsupervised splitting that uses pseudo-outcomes is applied using the default multivariate splitting rule (see below for details) Also see sidClustering for a more sophisticated method for unsupervised analysis (Mantero and Ishwaran, 2020).

• Custom splitting: All families except unsupervised are available for user defined custom splitting. Some basic C-programming skills are required. The harness for defining these rules is in splitCustom.c. In this file we give examples of how to code rules for regression, classification, survival, and competing risk. Each family can support up to sixteen custom split rules. Specifying splitrule="custom" or splitrule="custom1" will trigger the first split rule for the family defined by the training data set. Multivariate families will need a custom split rule for both regression and classification. In the examples, we demonstrate how the user is presented with the node specific membership. The task is then to define a split statistic based on that membership. Take note of the instructions in splitCustom.c on how to register the custom split rules. It is suggested that the existing custom split rules be kept in place for reference and that the user proceed to develop splitrule="custom2" and so on. The package must be recompiled and installed for the custom split rules to become available.

4. Improving computational speed

See the function rfsrc.fast for a fast implementation of rfsrc. Key methods for increasing speed are as follows:

• Nodesize

Increasing nodesize has the greatest effect in speeding calculations. In some big data settings this can also lead to better prediction performance.

• Save memory

Use option save.memory="TRUE" for big data competing risk and survival models. By default the package stores terminal node quantities to be used in prediction for test data but this can be memory intensive for big data.

• Block size

Make sure block.size="NULL" (or set to number of trees) so that the cumulative error is calculated only once.

• Turn off performace

The C-index error rate calculation can be very expensive for big survival data. Set perf.type="none" to turn this off and all other performance calculations (then consider using the function get.brier.survival as a fast way to get survival performance).

perf.type="none" applies to all other families as well.

• Randomized splitting rules

Set nsplit to a small non-zero integer value. Then a maximum of nsplit split points are chosen randomly for each of the candidate splitting variables when splitting a tree node, thus significantly reducing computational costs.

For more details about randomized splitting see Loh and Shih (1997), Dietterich (2000), and Lin and Jeon (2006). Geurts et al. (2006) introduced extremely randomized trees using the extra-trees algorithm. This algorithm corresponds to nsplit=1. In our experience however this may be too low for general use (Ishwaran, 2015).

For completely randomized (pure random) splitting use splitrule="random". In pure splitting, nodes are split by randomly selecting a variable and randomly selecting its split point (Cutler and Zhao, 2001).

• Subsampling

Reduce the size of the bootstrap using sampsize and samptype. See rfsrc.fast for a fast forest implementation using subsampling.

• Unique time points

Setting ntime to a reasonably small value such as 50 constrains survival ensemble calculations to a restricted grid of time points and significantly improves computational times.

• Large number of variables

Try filtering variables ahead of time. Make sure not to request VIMP (variable importance can always be recovered later using vimp or predict). Also if variable selection is desired, but is too slow, consider using max.subtree which calculates minimal depth, a measure of the depth that a variable splits, and yields fast variable selection (Ishwaran, 2010).

5. Prediction Error

Prediction error is calculated using OOB data. The metric used is mean-squared-error for regression, and misclassification error for classification. A normalized Brier score (relative to a coin-toss) and the AUC (area under the ROC curve) is also provided upon printing a classification forest. Performance for Brier score can be specified using perf.type="brier". G-mean performance is also available, see the function imbalanced for more details.

For survival, prediction error is measured by 1-C, where C is Harrell's (Harrell et al., 1982) concordance index. Prediction error is between 0 and 1, and measures how well the predictor correctly ranks (classifies) two random individuals in terms of survival. A value of 0.5 is no better than random guessing. A value of 0 is perfect.

When bootstrapping is by none, a coherent OOB subset is not available to assess prediction error. Thus, all outputs dependent on this are suppressed. In such cases, prediction error is only available via classical cross-validation (the user will need to use the predict.rfsrc function).

6. Variable Importance (VIMP)

VIMP is calculated using OOB data in several ways. importance="permute" yields permutation VIMP (Breiman-Cutler importance) by permuting OOB cases. importance="random" uses random left/right assignments whenever a split is encountered for the target variable. The default importance="anti" (equivalent to importance=TRUE) assigns cases to the anti (opposite) split.

VIMP depends upon block.size, an integer value between 1 and ntree, specifying number of trees in a block used for VIMP. When block.size=1, VIMP is calculated for each tree. When block.size="ntree", VIMP is calculated for the entire forest by comparing the perturbed OOB forest ensemble (using all trees) to the unperturbed OOB forest ensemble (using all trees). This yields ensemble VIMP, which does not measure the tree average effect of a variable, but rather its overall forest effect.

A useful compromise between tree VIMP and ensemble VIMP can be obtained by setting block.size to a value between 1 and ntree. Smaller values generally gives better accuracy, however computational times will be higher because VIMP is calculated over more blocks. However, see imbalanced for imbalanced classification data where larger block.size often works better (O'Brien and Ishwaran, 2019).

See vimp for a user interface for extracting VIMP and subsampling for calculating confidence intervals for VIMP.

Also see holdout.vimp for holdout VIMP, which calculates importance by holding out variables. This is more conservative, but with good false discovery properties.

For classification, VIMP is returned as a matrix with J+1 columns where J is the number of classes. The first column "all" is the unconditional VIMP, while the remaining columns are conditional VIMP calculated using only OOB cases with the class label.

7. Multivariate Forests

Multivariate forests can be specified in two ways:

rfsrc(Multivar(y1, y2, ..., yd) ~ . , my.data, ...)

rfsrc(cbind(y1, y2, ..., yd) ~ . , my.data, ...)

By default, a multivariate normalized composite splitting rule is used to split nodes (for multivariate regression, users have the option to use Mahalanobis splitting).

The nature of the outcomes informs the code as to what type of multivariate forest is grown; i.e. whether it is real-valued, categorical, or a combination of both (mixed). Performance measures (when requested) are returned for all outcomes.

Helper functions get.mv.formula, get.mv.predicted, get.mv.error can be used for defining the multivariate forest formula and extracting predicted values (all outcomes) and VIMP (all variables, all outcomes; assuming importance was requested in the call). The latter two functions also work for univariate (regular) forests. Both functions return standardized values (dividing by the variance for regression, or multiplying by 100, otherwise) using option standardize="TRUE".

8. Unsupervised Forests and sidClustering

See sidClustering sidClustering for a more sophisticated method for unsupervised analysis.

Otherwise a more direct (but naive) way to proceed is to use the unsupervised splitting rule. The following are equivalent ways to grow an unsupervised forest via unsupervised splitting:

rfsrc(data = my.data)

rfsrc(Unsupervised() ~ ., data = my.data)

In unsupervised mode, features take turns acting as target y-outcomes and x-variables for splitting. Specifically, mtry x-variables are randomly selected for splitting the node. Then for each mtry feature, ytry variables are selected from the remaining features to act as the target pseduo-outcomes. Splitting uses the multivariate normalized composite splitting rule.

The default value of ytry is 1 but can be increased. As illustration, the following equivalent unsupervised calls set mtry=10 and ytry=5:

rfsrc(data = my.data, ytry = 5, mtry = 10)

rfsrc(Unsupervised(5) ~ ., my.data, mtry = 10)

Note that all performance values (error rates, VIMP, prediction) are turned off in unsupervised mode.

9. Survival, Competing Risks

1. Survival settings require a time and censoring variable which should be identifed in the formula as the outcome using the standard Surv formula specification. A typical formula call looks like:

Surv(my.time, my.status) ~ .

where my.time and my.status are the variables names for the event time and status variable in the users data set.

2. For survival forests (Ishwaran et al. 2008), the censoring variable must be coded as a non-negative integer with 0 reserved for censoring and (usually) 1=death (event).

3. For competing risk forests (Ishwaran et al., 2013), the implementation is similar to survival, but with the following caveats:

• Censoring must be coded as a non-negative integer, where 0 indicates right-censoring, and non-zero values indicate different event types. While 0,1,2,..,J is standard, and recommended, events can be coded non-sequentially, although 0 must always be used for censoring.

• Setting the splitting rule to logrankscore will result in a survival analysis in which all events are treated as if they are the same type (indeed, they will coerced as such).

• Generally, competing risks requires a larger nodesize than survival settings.

10. Missing data imputation

na.action="na.impute" imputes missing data (both x and y-variables) using the missing data algorithm of Ishwaran et al. (2008). But also see the impute for an alternate way to do fast and accurate imputation.

The missing data algorithm can be iterated by setting nimpute to a positive integer greater than 1. When iterated, at the completion of each iteration, missing data is imputed using OOB non-missing terminal node data which is then used as input to grow a new forest. A side effect of iteration is that missing values in the returned objects xvar, yvar are replaced by imputed values. In other words the incoming data is overlaid with the missing data. Also, performance measures such as error rates and VIMP become optimistically biased.

Records in which all outcome and x-variable information are missing are removed from the forest analysis. Variables having all missing values are also removed.

11. Allowable data types and factors

Data types must be real valued, integer, factor or logical -- however all except factors are coerced and treated as if real valued. For ordered x-variable factors, splits are similar to real valued variables. For unordered factors, 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 number of splits attempted is not greater than number of cases in a node or the value of nsplit.

For coherence, an immutable map is applied to each factor that ensures factor levels in the training data are consistent with the factor levels in any subsequent test data. This map is applied to each factor before and after the native C library is executed. Because of this, if all x-variables all factors, then computational time will be long in high dimensional problems. Consider converting factors to real if this is the case.

## Value

An object of class (rfsrc, grow) with the following components:

call

The original call to rfsrc.

family

The family used in the analysis.

n

Sample size of the data (depends upon NA's, see na.action).

ntree

Number of trees grown.

mtry

Number of variables randomly selected for splitting at each node.

nodesize

Minimum size of terminal nodes.

nodedepth

Maximum depth allowed for a tree.

splitrule

Splitting rule used.

nsplit

Number of randomly selected split points.

yvar

y-outcome values.

yvar.names

A character vector of the y-outcome names.

xvar

Data frame of x-variables.

xvar.names

A character vector of the x-variable names.

xvar.wt

Vector of non-negative weights specifying the probability used to select a variable for splitting a node.

split.wt

Vector of non-negative weights specifying multiplier by which the split statistic for a covariate is adjusted.

cause.wt

Vector of weights used for the composite competing risk splitting rule.

leaf.count

Number of terminal nodes for each tree in the forest. Vector of length ntree. A value of zero indicates a rejected tree (can occur when imputing missing data). Values of one indicate tree stumps.

proximity

Proximity matrix recording the frequency of pairs of data points occur within the same terminal node.

forest

If forest=TRUE, the forest object is returned. This object is used for prediction with new test data sets and is required for other R-wrappers.

forest.wt

Forest weight matrix.

membership

Matrix recording terminal node membership where each column records node mebership for a case for a tree (rows).

splitrule

Splitting rule used.

inbag

Matrix recording inbag membership where each column contains the number of times that a case appears in the bootstrap sample for a tree (rows).

var.used

Count of the number of times a variable is used in growing the forest.

imputed.indv

Vector of indices for cases with missing values.

imputed.data

Data frame of the imputed data. The first column(s) are reserved for the y-outcomes, after which the x-variables are listed.

split.depth

Matrix (i,j) or array (i,j,k) recording the minimal depth for variable j for case i, either averaged over the forest, or by tree k.

node.stats

Split statistics returned when statistics=TRUE which can be parsed using stat.split.

err.rate

Tree cumulative OOB error rate.

err.block.rate

When importance=TRUE, vector of the cumulative error rate for each ensemble block comprised of block.size trees. So with block.size = 10, entries are the cumulative error rate for the first 10 trees, the first 20 trees, 30 trees, and so on. As another exmple, if block.size = 1, entries are the error rate for each tree.

importance

Variable importance (VIMP) for each x-variable.

predicted

In-bag predicted value.

predicted.oob

OOB predicted value.

++++++++

class

In-bag predicted class labels.

class.oob

OOB predicted class labels.

++++++++

regrOutput

List containing performance values for multivariate regression outcomes (applies only in multivariate settings).

clasOutput

List containing performance values for multivariate categorical (factor) outcomes (applies only in multivariate settings).

++++++++

survival

In-bag survival function.

survival.oob

OOB survival function.

chf

In-bag cumulative hazard function (CHF).

chf.oob

OOB CHF.

time.interest

Ordered unique death times.

Number of deaths.

++++++++

chf

In-bag cause-specific cumulative hazard function (CSCHF) for each event.

chf.oob

OOB CSCHF.

cif

In-bag cumulative incidence function (CIF) for each event.

cif.oob

OOB CIF.

## Note

Values returned depend heavily on the family. In particular, predicted values from the forest (predicted and predicted.oob) are as follows:

1. For regression, a vector of predicted y-outcomes.

2. For classification, a matrix with columns containing the estimated class probability for each class. Performance values and VIMP for classification are reported as a matrix with J+1 columns where J is the number of classes. The first column "all" is the unconditional value for performance (VIMP), while the remaining columns are performance (VIMP) conditioned on cases corresponding to that class label.

3. For survival, a vector of mortality values (Ishwaran et al., 2008) representing estimated risk for each individual calibrated to the scale of the number of events (as a specific example, if i has a mortality value of 100, then if all individuals had the same x-values as i, we would expect an average of 100 events). Also returned are matrices containing the CHF and survival function. Each row corresponds to an individual's ensemble CHF or survival function evaluated at each time point in time.interest.

4. For competing risks, a matrix with one column for each event recording the expected number of life years lost due to the event specific cause up to the maximum follow up (Ishwaran et al., 2013). Also returned are the cause-specific cumulative hazard function (CSCHF) and the cumulative incidence function (CIF) for each event type. These are encoded as a three-dimensional array, with the third dimension used for the event type, each time point in time.interest making up the second dimension (columns), and the case (individual) being the first dimension (rows).

5. For multivariate families, predicted values (and other performance values such as VIMP and error rates) are stored in the lists regrOutput and clasOutput which can be extracted using functions get.mv.error, get.mv.predicted and get.mv.vimp.

## Author

Hemant Ishwaran and Udaya B. Kogalur

## References

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

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Dietterich, T. G. (2000). An experimental comparison of three methods for constructing ensembles of decision trees: bagging, boosting, and randomization. Machine Learning, 40, 139-157.

Gray R.J. (1988). A class of k-sample tests for comparing the cumulative incidence of a competing risk, Ann. Statist., 16: 1141-1154.

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find.interaction.rfsrc,

get.tree.rfsrc,

holdout.vimp.rfsrc,

imbalanced.rfsrc, impute.rfsrc,

max.subtree.rfsrc,

partial.rfsrc, plot.competing.risk.rfsrc, plot.rfsrc, plot.survival.rfsrc, plot.variable.rfsrc, predict.rfsrc, print.rfsrc,

quantreg.rfsrc,

rfsrc, rfsrc.cart, rfsrc.fast,

sidClustering.rfsrc,

stat.split.rfsrc, subsample.rfsrc, synthetic.rfsrc,

tune.rfsrc,

var.select.rfsrc, vimp.rfsrc

## Examples

##------------------------------------------------------------
## survival analysis
##------------------------------------------------------------

## veteran data
## randomized trial of two treatment regimens for lung cancer
data(veteran, package = "randomForestSRC")
v.obj <- rfsrc(Surv(time, status) ~ ., data = veteran,
ntree = 100, block.size = 1)

## plot tree number 3
plot(get.tree(v.obj, 3))

## print results of trained forest
print(v.obj)

## plot results of trained forest
plot(v.obj)

## plot survival curves for first 10 individuals -- direct way
matplot(v.obj$time.interest, 100 * t(v.obj$survival.oob[1:10, ]),
xlab = "Time", ylab = "Survival", type = "l", lty = 1)

## plot survival curves for first 10 individuals
## using function "plot.survival"
plot.survival(v.obj, subset = 1:10)

# \donttest{
## obtain Brier score using KM and RSF censoring distribution estimators
bs.km <- get.brier.survival(v.obj, cens.model = "km")$brier.score bs.rsf <- get.brier.survival(v.obj, cens.model = "rfsrc")$brier.score

## plot the brier score
plot(bs.km, type = "s", col = 2)
lines(bs.rsf, type ="s", col = 4)
legend("topright", legend = c("cens.model = km", "cens.model = rfsrc"), fill = c(2,4))

## plot CRPS (continuous rank probability score) as function of time
## here's how to calculate the CRPS for every time point
trapz <- randomForestSRC:::trapz
time <- v.obj$time.interest crps.km <- sapply(1:length(time), function(j) { trapz(time[1:j], bs.km[1:j, 2] / diff(range(time[1:j]))) }) crps.rsf <- sapply(1:length(time), function(j) { trapz(time[1:j], bs.rsf[1:j, 2] / diff(range(time[1:j]))) }) plot(time, crps.km, ylab = "CRPS", type = "s", col = 2) lines(time, crps.rsf, type ="s", col = 4) legend("bottomright", legend=c("cens.model = km", "cens.model = rfsrc"), fill=c(2,4)) ## fast nodesize optimization for veteran data ## optimal nodesize in survival is larger than other families ## see the function "tune" for more examples tune.nodesize(Surv(time,status) ~ ., veteran) ## Primary biliary cirrhosis (PBC) of the liver data(pbc, package = "randomForestSRC") pbc.obj <- rfsrc(Surv(days, status) ~ ., pbc) print(pbc.obj) ## save.memory example for survival ## growing many deep trees creates memory issue without this option! data(pbc, package = "randomForestSRC") print(rfsrc(Surv(days, status) ~ ., pbc, splitrule = "random", ntree = 25000, nodesize = 1, save.memory = TRUE)) ##------------------------------------------------------------ ## trees can be plotted for any family ## see get.tree for details and more examples ##------------------------------------------------------------ ## survival where factors have many levels data(veteran, package = "randomForestSRC") vd <- veteran vd$celltype=factor(vd$celltype) vd$diagtime=factor(vd$diagtime) vd.obj <- rfsrc(Surv(time,status)~., vd, ntree = 100, nodesize = 5) plot(get.tree(vd.obj, 3)) ## classification iris.obj <- rfsrc(Species ~., data = iris) plot(get.tree(iris.obj, 25, class.type = "bayes")) plot(get.tree(iris.obj, 25, target = "setosa")) plot(get.tree(iris.obj, 25, target = "versicolor")) plot(get.tree(iris.obj, 25, target = "virginica")) ## ------------------------------------------------------------ ## simple example of VIMP using iris classification ## ------------------------------------------------------------ ## directly from trained forest print(rfsrc(Species~.,iris,importance=TRUE)$importance)

## VIMP (and performance) use misclassification error by default
## but brier prediction error can be requested
print(rfsrc(Species~.,iris,importance=TRUE,perf.type="brier")$importance) ## example using vimp function (see vimp help file for details) iris.obj <- rfsrc(Species ~., data = iris) print(vimp(iris.obj)$importance)
print(vimp(iris.obj,perf.type="brier")$importance) ## example using hold out vimp (see holdout.vimp help file for details) print(holdout.vimp(Species~.,iris)$importance)
print(holdout.vimp(Species~.,iris,perf.type="brier")$importance) ## ------------------------------------------------------------ ## confidence interval for vimp using subsampling ## compare with holdout vimp ## ------------------------------------------------------------ ## new York air quality measurements o <- rfsrc(Ozone ~ ., data = airquality) so <- subsample(o) plot(so) ## compare with holdout vimp print(holdout.vimp(Ozone ~ ., data = airquality)$importance)

##------------------------------------------------------------
## example of imputation in survival analysis
##------------------------------------------------------------

data(pbc, package = "randomForestSRC")
pbc.obj2 <- rfsrc(Surv(days, status) ~ ., pbc, na.action = "na.impute")

## same as above but iterate the missing data algorithm
pbc.obj3 <- rfsrc(Surv(days, status) ~ ., pbc,
na.action = "na.impute", nimpute = 3)

## fast way to impute data (no inference is done)
## see impute for more details
pbc.imp <- impute(Surv(days, status) ~ ., pbc, splitrule = "random")

##------------------------------------------------------------
## compare RF-SRC to Cox regression
## Illustrates C-index and Brier score measures of performance
## assumes "pec" and "survival" libraries are loaded
##------------------------------------------------------------

if (library("survival", logical.return = TRUE)
& library("pec", logical.return = TRUE)
& library("prodlim", logical.return = TRUE))

{
##prediction function required for pec
predictSurvProb.rfsrc <- function(object, newdata, times, ...){
ptemp <- predict(object,newdata=newdata,...)$survival pos <- sindex(jump.times = object$time.interest, eval.times = times)
p <- cbind(1,ptemp)[, pos + 1]
if (NROW(p) != NROW(newdata) || NCOL(p) != length(times))
stop("Prediction failed")
p
}

## data, formula specifications
data(pbc, package = "randomForestSRC")
pbc.na <- na.omit(pbc)  ##remove NA's
surv.f <- as.formula(Surv(days, status) ~ .)
pec.f <- as.formula(Hist(days,status) ~ 1)

## run cox/rfsrc models
## for illustration we use a small number of trees
cox.obj <- coxph(surv.f, data = pbc.na, x = TRUE)
rfsrc.obj <- rfsrc(surv.f, pbc.na, ntree = 150)

## compute bootstrap cross-validation estimate of expected Brier score
## see Mogensen, Ishwaran and Gerds (2012) Journal of Statistical Software
set.seed(17743)
prederror.pbc <- pec(list(cox.obj,rfsrc.obj), data = pbc.na, formula = pec.f,
splitMethod = "bootcv", B = 50)
print(prederror.pbc)
plot(prederror.pbc)

## compute out-of-bag C-index for cox regression and compare to rfsrc
rfsrc.obj <- rfsrc(surv.f, pbc.na)
cat("out-of-bag Cox Analysis ...", "\n")
cox.err <- sapply(1:100, function(b) {
if (b%%10 == 0) cat("cox bootstrap:", b, "\n")
train <- sample(1:nrow(pbc.na), nrow(pbc.na), replace = TRUE)
cox.obj <- tryCatch({coxph(surv.f, pbc.na[train, ])}, error=function(ex){NULL})
if (!is.null(cox.obj)) {
get.cindex(pbc.na$days[-train], pbc.na$status[-train], predict(cox.obj, pbc.na[-train, ]))
} else NA
})
cat("\n\tOOB error rates\n\n")
cat("\tRSF            : ", rfsrc.obj$err.rate[rfsrc.obj$ntree], "\n")
cat("\tCox regression : ", mean(cox.err, na.rm = TRUE), "\n")
}

##------------------------------------------------------------
## competing risks
##------------------------------------------------------------

## WIHS analysis
## cumulative incidence function (CIF) for HAART and AIDS stratified by IDU

data(wihs, package = "randomForestSRC")
wihs.obj <- rfsrc(Surv(time, status) ~ ., wihs, nsplit = 3, ntree = 100)
plot.competing.risk(wihs.obj)
cif <- wihs.obj$cif.oob Time <- wihs.obj$time.interest
idu <- wihs$idu cif.haart <- cbind(apply(cif[,,1][idu == 0,], 2, mean), apply(cif[,,1][idu == 1,], 2, mean)) cif.aids <- cbind(apply(cif[,,2][idu == 0,], 2, mean), apply(cif[,,2][idu == 1,], 2, mean)) matplot(Time, cbind(cif.haart, cif.aids), type = "l", lty = c(1,2,1,2), col = c(4, 4, 2, 2), lwd = 3, ylab = "Cumulative Incidence") legend("topleft", legend = c("HAART (Non-IDU)", "HAART (IDU)", "AIDS (Non-IDU)", "AIDS (IDU)"), lty = c(1,2,1,2), col = c(4, 4, 2, 2), lwd = 3, cex = 1.5) ## illustrates the various splitting rules ## illustrates event specific and non-event specific variable selection if (library("survival", logical.return = TRUE)) { ## use the pbc data from the survival package ## events are transplant (1) and death (2) data(pbc, package = "survival") pbc$id <- NULL

## modified Gray's weighted log-rank splitting
## (equivalent to cause=c(1,1) and splitrule="logrankCR")
pbc.cr <- rfsrc(Surv(time, status) ~ ., pbc)

## log-rank cause-1 specific splitting and targeted VIMP for cause 1
pbc.log1 <- rfsrc(Surv(time, status) ~ ., pbc,
splitrule = "logrankCR", cause = c(1,0), importance = TRUE)

## log-rank cause-2 specific splitting and targeted VIMP for cause 2
pbc.log2 <- rfsrc(Surv(time, status) ~ ., pbc,
splitrule = "logrankCR", cause = c(0,1), importance = TRUE)

## extract VIMP from the log-rank forests: event-specific
## extract minimal depth from the Gray log-rank forest: non-event specific
var.perf <- data.frame(md = max.subtree(pbc.cr)$order[, 1], vimp1 = 100 * pbc.log1$importance[ ,1],
vimp2 = 100 * pbc.log2$importance[ ,2]) print(var.perf[order(var.perf$md), ], digits = 2)

}

## ------------------------------------------------------------
## regression analysis
## ------------------------------------------------------------

## new York air quality measurements
airq.obj <- rfsrc(Ozone ~ ., data = airquality, na.action = "na.impute")

# partial plot of variables (see plot.variable for more details)
plot.variable(airq.obj, partial = TRUE, smooth.lines = TRUE)

## motor trend cars
mtcars.obj <- rfsrc(mpg ~ ., data = mtcars)

## ------------------------------------------------------------
## regression with custom bootstrap
## ------------------------------------------------------------

ntree <- 25
n <- nrow(mtcars)
s.size <- n / 2
swr <- TRUE
samp <- randomForestSRC:::make.sample(ntree, n, s.size, swr)
o <- rfsrc(mpg ~ ., mtcars, bootstrap = "by.user", samp = samp)

## ------------------------------------------------------------
## classification analysis
## ------------------------------------------------------------

## iris data
iris.obj <- rfsrc(Species ~., data = iris)

## wisconsin prognostic breast cancer data
data(breast, package = "randomForestSRC")
breast.obj <- rfsrc(status ~ ., data = breast, block.size=1)
plot(breast.obj)

## ------------------------------------------------------------
## big data set, reduce number of variables using simple method
## ------------------------------------------------------------

## use Iowa housing data set
data(housing, package = "randomForestSRC")

## original data contains lots of missing data, use fast imputation
## however see impute for other methods
housing2 <- impute(data = housing, fast = TRUE)

## run shallow trees to find variables that split any tree
xvar.used <- rfsrc(SalePrice ~., housing2, ntree = 250, nodedepth = 4,
var.used="all.trees", mtry = Inf, nsplit = 100)$var.used ## now fit forest using filtered variables xvar.keep <- names(xvar.used)[xvar.used >= 1] o <- rfsrc(SalePrice~., housing2[, c("SalePrice", xvar.keep)]) print(o) ## ------------------------------------------------------------ ## imbalanced classification data ## see the "imbalanced" function for further details ## ## a) use balanced random forests with undersampling of the majority class ## Specifically let n0, n1 be sample sizes for majority, minority ## cases. We sample 2 x n1 cases with majority, minority cases chosen ## with probabilities n1/n, n0/n where n=n0+n1 ## ## b) balanced random forests using "imbalanced" ## ## c) q-classifier (RFQ) using "imbalanced" ## ## ------------------------------------------------------------ ## Wisconsin breast cancer example data(breast, package = "randomForestSRC") breast <- na.omit(breast) ## balanced random forests - brute force y <- breast$status
obdirect <- rfsrc(status ~ ., data = breast, nsplit = 10,
case.wt = randomForestSRC:::make.wt(y),
sampsize = randomForestSRC:::make.size(y))
print(obdirect)
print(get.imbalanced.performance(obdirect))

## balanced random forests - using "imbalanced"
ob <- imbalanced(status ~ ., data = breast, method = "brf")
print(ob)
print(get.imbalanced.performance(ob))

## q-classifier (RFQ) - using "imbalanced"
oq <- imbalanced(status ~ ., data = breast)
print(oq)
print(get.imbalanced.performance(oq))

## q-classifier (RFQ) - with auc splitting
oqauc <- imbalanced(status ~ ., data = breast, splitrule = "auc")
print(oqauc)
print(get.imbalanced.performance(oqauc))

## ------------------------------------------------------------
## unsupervised analysis
## ------------------------------------------------------------

## two equivalent ways to implement unsupervised forests
mtcars.unspv <- rfsrc(Unsupervised() ~., data = mtcars)
mtcars2.unspv <- rfsrc(data = mtcars)

## illustration of sidClustering for the mtcars data
## see sidClustering for more details
mtcars.sid <- sidClustering(mtcars, k = 1:10)
print(split(mtcars, mtcars.sid$cl[, 3])) print(split(mtcars, mtcars.sid$cl[, 10]))

## ------------------------------------------------------------
## bivariate regression using Mahalanobis splitting
## also illustrates user specified covariance matrix
## ------------------------------------------------------------

if (library("mlbench", logical.return = TRUE)) {

## load boston housing data, specify the bivariate regression
data(BostonHousing)
f <- formula("Multivar(lstat, nox) ~.")

## Mahalanobis splitting
bh.mreg <- rfsrc(f, BostonHousing, importance = TRUE, splitrule = "mahal")

## performance error and vimp
vmp <- get.mv.vimp(bh.mreg)
pred <- get.mv.predicted(bh.mreg)

## standardized error and vimp
err.std <- get.mv.error(bh.mreg, standardize = TRUE)
vmp.std <- get.mv.vimp(bh.mreg, standardize = TRUE)

## same analysis, but with user specified covariance matrix
sigma <- cov(BostonHousing[, c("lstat","nox")])
bh.mreg2 <- rfsrc(f, BostonHousing, splitrule = "mahal", sigma = sigma)

}

## ------------------------------------------------------------
## multivariate mixed forests (nutrigenomic study)
## study effects of diet, lipids and gene expression for mice
## diet, genotype and lipids used as the multivariate y
## genes used for the x features
## ------------------------------------------------------------

## load the data (data is a list)
data(nutrigenomic, package = "randomForestSRC")

## assemble the multivariate y data
ydta <- data.frame(diet = nutrigenomic$diet, genotype = nutrigenomic$genotype,
nutrigenomic$lipids) ## multivariate mixed forest call ## uses "get.mv.formula" for conveniently setting formula mv.obj <- rfsrc(get.mv.formula(colnames(ydta)), data.frame(ydta, nutrigenomic$genes),
importance=TRUE, nsplit = 10)

## print results for diet and genotype y values
print(mv.obj, outcome.target = "diet")
print(mv.obj, outcome.target = "genotype")

## extract standardized VIMP
svimp <- get.mv.vimp(mv.obj, standardize = TRUE)

## plot standardized VIMP for diet, genotype and lipid for each gene
boxplot(t(svimp), col = "bisque", cex.axis = .7, las = 2,
outline = FALSE,
ylab = "standardized VIMP",
main = "diet/genotype/lipid VIMP for each gene")

## ------------------------------------------------------------
## custom splitting using the pre-coded examples
## ------------------------------------------------------------

## motor trend cars
mtcars.obj <- rfsrc(mpg ~ ., data = mtcars, splitrule = "custom")

## iris analysis
iris.obj <- rfsrc(Species ~., data = iris, splitrule = "custom1")

## WIHS analysis
wihs.obj <- rfsrc(Surv(time, status) ~ ., wihs, nsplit = 3,
ntree = 100, splitrule = "custom1")

# }