R: Fast Unified Random Forests for Survival, Regression, and...

rfsrc {randomForestSRC} | R Documentation

## Fast Unified Random Forests for Survival, Regression, and Classification (RF-SRC)

### Description

Fast OpenMP parallel computing of random forests (Breiman 2001) for regression, classification, survival analysis (Ishwaran 2008), competing risks (Ishwaran 2012), multivariate (Segal and Xiao 2011), unsupervised (Mantero and Ishwaran 2020), quantile regression (Meinhausen 2006, Zhang et al. 2019), 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. Variable importance (VIMP), and holdout VIMP, as well as confidence regions (Ishwaran and Lu 2019) can be calculated for single and grouped variables. Fast interface for missing data imputation using a variety of different random forest methods (Tang and Ishwaran 2017).

Key items users should be aware of:

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

2. sidClustering (see `sidClustering`) for unsupervised data analysis.

3. Fast hold out VIMP (see `holdout.vimp`). Can be conservative but has good false discovery properties unlike Breiman-Cutler VIMP (see `vimp`).

4. VIMP confidence intervals using subsampling (see `subsample`).

5. q-classifier for class imbalanced data, including G-mean VIMP which is superior to Breiman-Cutler (see `imbalanced`).

6. Fast imputation using on the fly imputation, missForest and multivariate missForest (see `impute`).

7. Fast random forests using subsampling (see `rfsrc.fast`).

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`.

### Usage

    rfsrc(formula, data, ntree = 1000,
      mtry = NULL, ytry = NULL,
      nodesize = NULL, nodedepth = NULL,
      splitrule = NULL, nsplit = 10,
      importance = c(FALSE, TRUE, "none", "permute", "random", "anti"),
      block.size = if (any(is.element(as.character(importance),
                         c("none", "FALSE")))) NULL else 10,
      ensemble = c("all", "oob", "inbag"),
      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, cause,
      proximity = FALSE, distance = FALSE, forest.wt = FALSE,
      xvar.wt = NULL, yvar.wt = NULL, split.wt = NULL, case.wt  = NULL,
      forest = TRUE,
      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` | A symbolic description of the model to be fit. If missing, unsupervised splitting is implemented.
`data` | Data frame containing the y-outcome and x-variables.
`ntree` | Number of trees.
`mtry` | Number of variables randomly selected as candidates for splitting a node. The default is `p`/3 for regression, where `p` equals the number of variables. For all other families (including unsupervised settings), the default is sqrt(`p`). Values are always rounded up.
`ytry` | For unsupervised forests, sets the number of randomly selected pseudo-responses (see below for more details). The default is `ytry`=1, which selects one pseudo-response.
`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. The default behaviour is that this parameter is ignored.
`splitrule` | Splitting rule (see below).
`nsplit` | Non-negative integer value for number of random splits to consider for each candidate splitting variable. This significantly increases speed. When zero or NULL, uses much slower deterministic splitting where all possible splits considered.
`importance` | Method for computing variable importance (VIMP); see below. Default action is codeimportance="none" but VIMP can always be recovered later using `vimp` or `predict`.
`block.size` | Should the cumulative error rate be calculated on every tree? When `NULL`, it will only be calculated on the last tree and the 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 useful compromise between ensemble and permutation VIMP. The default action is to use 10 trees.
`ensemble` | Specifies the type of ensemble. By default both out-of-bag (OOB) and inbag ensembles are returned. Always use OOB values for interfence on the training data.
`bootstrap` | Bootstrap protocol. The default is `by.root` which bootstraps the data by sampling with or without replacement (by default sampling is without replacement; see the option `samptype` below). If `none` is chosen, the data is not bootstrapped at all. If `by.user` is choosen, the bootstrap specified by `samp` is used. It is not possible to return OOB ensembles or prediction error if `none` is in effect.
`samptype` | Type of bootstrap when `by.root` is in effect. Choices are `swor` (sampling without replacement) and `swr` (sampling with replacement). Unlike Breiman's random forests, the default action here is sampling without replacement. Thus out-of-bag (OOB) technically means out-of-sample, but for legacy reasons we retain the term OOB.
`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 size of bootstrap data 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 by 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 even one of its entries is `NA` (for x-variables this applies only to those specifically listed in 'formula'). Selecting `na.impute` imputes the data. Also see the function `impute` for fast direct imputation.
`nimpute` | Number of iterations of the missing data algorithm. Performance measures such as out-of-bag (OOB) error rates tend to become optimistic if `nimpute` is greater than 1.
`ntime` | Integer value used for survival to constrain ensemble calculations to a grid of `ntime` time points. Alternatively if a vector of values of length greater than one is supplied, it is assumed these are the time points to be used to constrain the calculations (note that the constrained time points used will be the observed event times closest to the user supplied time points). If no value is specified, the default action is to use 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, passing a vector of ones reverts to the default composite split statistic). Finally, regardless of how `cause` is specified, the returned forest object always provides estimates for all event types.
`proximity` | Proximity of cases as measured by the frequency of sharing the same terminal node. This is an `n`x`n` 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 for these cases in a tree. This is an `n`x`n` matrix, which can be large. Choices are `inbag`, `oob`, `all`, `TRUE`, or `FALSE`. Setting `distance = TRUE` is equivalent to `distance = "inbag"`.
`forest.wt` | Should the forest weight matrix be calculated? Creates an `n`x`n` 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 to use uniform weights.
`yvar.wt` | NOT YET IMPLEMENTED: Vector of non-negative weights (does not have to sum to 1) representing the probability of selecting a response as a candidate for the split statistic in unsupervised settings. Default is to use uniform weights.
`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 to use 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 example below for the breast data set for an illustration of its use for class imbalanced data.
`forest` | Should the forest object be returned? Used for prediction on new data and required by many of the package functions.
`var.used` | Return variables used for splitting? Default is `FALSE`. Possible values are `all.trees` which returns a vector of the number of times a split occurred on a variable, and `by.tree` which returns a matrix recording the number of times a split occurred on a variable in a specific 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 response and the formula supplied. There are several possible scenarios:

   1. Regression forests for continuous responses.

   2. Classification forests for factor responses.

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

   4. Unsupervised forests when there is no response.

   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. `mse`: default split rule implements weighted mean-squared error splitting (Breiman et al. 1984, Chapter 8.4).

     2. `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. `gini`: default splitrule implements Gini index splitting (Breiman et al. 1984, Chapter 4.3).

     2. `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. `entropy`: entropy splitting (Breiman et al. 1984, Chapter 2.5, 4.3).

   - Survival analysis:

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

     2. `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. `logrankscore`: log-rank score splitting (Hothorn and Lausen, 2003).

   - Competing risk analysis:

     1. `logrankCR`: default splitrule implements a modified weighted log-rank splitting rule modeled after Gray's test (Gray, 1988).

     2. `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. For more details see Ishwaran et al. (2014).

   - Multivariate analysis: When one or both regression and classification responses are detected, a multivariate normalized composite split rule of mean-squared error and Gini index splitting is invoked. See Tang and Ishwaran (2017) for details on this splitting rule. This splitting rule is different than Segal and Xiao (2011) who use Mahalanobis distance splitting.

   - Unsupervised analysis: In settings where there is no outcome, unsupervised splitting is invoked. In this case, the mixed outcome splitting rule (above) is applied. 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`. In general, the key methods for increasing speed are as follows:

   - Randomized splitting rules

     Computational speed can be significantly improved with randomized splitting invoked by the option `nsplit`. When `nsplit` is set to a non-zero positive integer, a maximum of `nsplit` split points are chosen randomly for each of the candidate splitting variables when splitting a tree node, thus significantly reducing the cost from having to consider all possible split-values. To revert to traditional deterministic splitting (all split values considered) use `nsplit`=0.

     Randomized splitting for trees has a long history. See for example, Loh and Shih (1997), Dietterich (2000), and Lin and Jeon (2006). Geurts et al. (2006) recently introduced extremely randomized trees using what they call the extra-trees algorithm. We can see that this algorithm essentially corresponds to randomized splitting with `nsplit`=1. In our experience however this is not always the optimal value for `nsplit` and may often be too low. See Ishwaran (2015).

     Finally, 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

     Subsampling can be used to reduce the size of the in-sample data used to grow a tree and therefore can greatly reduce computational load. Subsampling is implemented using options `sampsize` and `samptype`. See `rfsrc.fast` for a fast forest implementation using subsampling.

   - Unique time points

     For large survival problems, users should consider setting `ntime` to a reasonably small value (such as 100 or 250). This constrains ensemble calculations such as survival functions to a restricted grid of time points.

   - Large number of variables

     The default setting `importance="none"` turns off variable importance (VIMP) calculations and considerably reduces computational times. Variable importance can always be recovered later using functions `vimp` or `predict`. Also 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. Performance is measured in terms of 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)

   To obtain VIMP use the option `importance`. Setting this to "permute" or "TRUE" returns permutation VIMP from permuting OOB cases. Choosing "random" replaces permutation with random assignment. Thus when an OOB case is dropped down a tree, the case goes left or right randomly whenever a split is encountered for the target variable. If "anti" is specified, the case is assigned to the opposite split. By default `importance`="FALSE" and VIMP is not requested.

   VIMP depends upon `block.size`, an integer value between 1 and `ntree`, specifying the number of trees in a block used to determine VIMP. When `block.size`=1, VIMP is calculated by tree. The difference between prediction error under the perturbed predictor and the original predictor is calculated for each tree and averaged over the forest. This yields Breiman-Cutler VIMP (Breiman 2001).

   When `block.size`="ntree", VIMP is calculated across the forest by comparing the perturbed OOB forest ensemble (using all trees) to the unperturbed OOB forest ensemble (using all trees). Unlike Breiman-Cutler VIMP, ensemble VIMP does not measure the tree average effect of x, but rather its overall forest effect. This is called Ishwaran-Kogalur VIMP (Ishwaran et al. 2008).

   A useful compromise between Breiman-Cutler (BC) and Ishwaran-Kogalur (IK) VIMP can be obtained by setting `block.size` to a value between 1 and `ntree`. Smaller values are closer to BC and larger values closer to IK. Smaller generally gives better accuracy, however computational times will be higher because VIMP is calculated over more blocks. Also see `imbalanced` for imbalanced classification data where a larger `block.size` 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 leads to a more conservative procedure but with good false discovery properties.

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, ...)

   A multivariate normalized composite splitting rule is used to split nodes. The nature of the outcomes will inform the code as to the type of multivariate forest to be grown; i.e. whether it is real-valued, categorical, or a combination of both (mixed). Note that performance measures (when requested) are returned for all outcomes.

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-responses. 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 response 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

    Set `na.action="na.impute"` to impute missing data (both x and y-variables) using the missing data algorithm of Ishwaran et al. (2008). However, see the function `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-responses, 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.
`importance` | Variable importance (VIMP) for each x-variable.
`predicted` | In-bag predicted value.
`predicted.oob` | OOB predicted value.

`++++++++` | for classification settings, additionally ++++++++

`class` | In-bag predicted class labels.
`class.oob` | OOB predicted class labels.

`++++++++` | for multivariate settings, additionally ++++++++

`regrOutput` | List containing performance values for multivariate regression responses (applies only in multivariate settings).
`clasOutput` | List containing performance values for multivariate categorical (factor) responses (applies only in multivariate settings).

`++++++++` | for survival settings, additionally ++++++++

`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.
`ndead` | Number of deaths.

`++++++++` | for competing risks, additionally ++++++++

`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.
`time.interest` | Ordered unique event times.
`ndead` | Number of events.

### 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-responses.

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 or VIMP, while the remaining columns are performance and 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(s)

Hemant Ishwaran and Udaya B. Kogalur

### References

Breiman L., Friedman J.H., Olshen R.A. and Stone C.J. (1984). Classification and Regression Trees, Belmont, California.

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

Cutler A. and Zhao G. (2001). PERT-Perfect random tree ensembles. Comp. Sci. Statist., 33: 490-497.

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.

Geurts, P., Ernst, D. and Wehenkel, L., (2006). Extremely randomized trees. Machine learning, 63(1):3-42.

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

Hothorn T. and Lausen B. (2003). On the exact distribution of maximally selected rank statistics, Comp. Statist. Data Anal., 43:121-137.

Ishwaran H. (2007). Variable importance in binary regression trees and forests, Electronic J. Statist., 1:519-537.

Ishwaran H. and Kogalur U.B. (2007). Random survival forests for R, Rnews, 7(2):25-31.

Ishwaran H., Kogalur U.B., Blackstone E.H. and Lauer M.S. (2008). Random survival forests, Ann. App. Statist., 2:841-860.

Ishwaran H., Kogalur U.B., Gorodeski E.Z, Minn A.J. and Lauer M.S. (2010). High-dimensional variable selection for survival data. J. Amer. Statist. Assoc., 105:205-217.

Ishwaran H., Kogalur U.B., Chen X. and Minn A.J. (2011). Random survival forests for high-dimensional data. Stat. Anal. Data Mining, 4:115-132

Ishwaran H., Gerds T.A., Kogalur U.B., Moore R.D., Gange S.J. and Lau B.M. (2014). Random survival forests for competing risks. Biostatistics, 15(4):757-773.

Ishwaran H. and Malley J.D. (2014). Synthetic learning machines. BioData Mining, 7:28.

Ishwaran H. (2015). The effect of splitting on random forests. Machine Learning, 99:75-118.

Lin, Y. and Jeon, Y. (2006). Random forests and adaptive nearest neighbors. J. Amer. Statist. Assoc., 101(474), 578-590.

Lu M., Sadiq S., Feaster D.J. and Ishwaran H. (2018). Estimating individual treatment effect in observational data using random forest methods. J. Comp. Graph. Statist, 27(1), 209-219

Ishwaran H. and Lu M. (2019). Standard errors and confidence intervals for variable importance in random forest regression, classification, and survival. Statistics in Medicine, 38, 558-582.

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

Loh W.-Y and Shih Y.-S (1997). Split selection methods for classification trees, Statist. Sinica, 7:815-840.

Mantero A. and Ishwaran H. (2020). Unsupervised random forests. To appear in Statistical Analysis and Data Mining.

Meinshausen N. (2006) Quantile regression forests, Journal of Machine Learning Research, 7:983-999.

Mogensen, U.B, Ishwaran H. and Gerds T.A. (2012). Evaluating random forests for survival analysis using prediction error curves, J. Statist. Software, 50(11): 1-23.

O'Brien R. and Ishwaran H. (2019). A random forests quantile classifier for class imbalanced data. Pattern Recognition, 90, 232-249

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

Segal M.R. and Xiao Y. Multivariate random forests. (2011). Wiley Interdisciplinary Reviews: Data Mining and Knowledge Discovery. 1(1):80-87.

Tang F. and Ishwaran H. (2017). Random forest missing data algorithms. Statistical Analysis and Data Mining, 10:363-377.

Zhang H., Zimmerman J., Nettleton D. and Nordman D.J. (2019). Random forest prediction intervals. The American Statistician. 4:1-5.

### See Also

`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.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)

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

    ## print the grow object and plot various useful details
    print(v.obj)
    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)

    ## 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)

    ##------------------------------------------------------------
    ## 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 grow call
    print(rfsrc(Species~.,iris,importance=TRUE)$importance)

    ## note by default VIMP (and performance) uses misclassification error,
    ## but brier prediction error can be specified
    print(rfsrc(Species~.,iris,importance=TRUE,perf.type="brier")$importance)

    ## 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)

    ## 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,
               nsplit = 10, na.action = "na.impute")

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

    ## fast way to impute the 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
      pbc.cr <- rfsrc(Surv(time, status) ~ ., pbc)

      ## log-rank event-one specific splitting
      pbc.log1 <- rfsrc(Surv(time, status) ~ ., pbc,
                  splitrule = "logrank", cause = c(1,0), importance = TRUE)

      ## log-rank event-two specific splitting
      pbc.log2 <- rfsrc(Surv(time, status) ~ ., pbc,
                  splitrule = "logrank", 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), ])

    }

    ## ------------------------------------------------------------
    ## 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)

    ## ------------------------------------------------------------
    ## 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)

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

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

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

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

    mtcars.mreg <- rfsrc(Multivar(mpg, cyl) ~., data = mtcars,
               block.size=1, importance = TRUE)

    ## extract error rates, vimp, and OOB predicted values for all targets
    err <- get.mv.error(mtcars.mreg)
    vmp <- get.mv.vimp(mtcars.mreg)
    pred <- get.mv.predicted(mtcars.mreg)

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

    ## ------------------------------------------------------------
    ## 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(do.call(cbind, nutrigenomic)),
                 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")

[Package randomForestSRC version 2.10.1 Index]