rocTree:Receiver Operating Characteristic (ROC)-Guided Classification Survival Tree and Ensemble.

Description

The rocTree package uses a Receiver Operating Characteristic (ROC) guided classification algorithm to grow prune survival trees and ensemble.

Introduction

The rocTree package provides implementations to a unified framework for tree-structured analysis with censored survival outcomes. Different from many existing tree building algorithms, the rocTree package incorporates time-dependent covariates by constructing a time-invariant partition scheme on the survivor population. The partition-based risk prediction function is constructed using an algorithm guided by the Receiver Operating Characteristic (ROC) curve. The generalized time-dependent ROC curves for survival trees show that the target hazard function yields the highest ROC curve. The optimality of the target hazard function motivates us to use a weighted average of the time-dependent area under the curve on a set of time points to evaluate the prediction performance of survival trees and to guide splitting and pruning. Moreover, the rocTree package also offers a novel ensemble algorithm, where the ensemble is on unbiased martingale estimating equations.

Methods

The package contains functions to construct ROC-guided survival trees and ensemble through the main function rocTree.

Author(s)

Maintainer: Sy Han Chiou schiou@utdallas.edu

Authors:

• Yifei Sun ys3072@cumc.columbia.edu

• Mei-Cheng Wang mcwang@jhu.edu

See Also

rocTree

Surv function imported from survival

Description

This function is imported from the survival package. See Surv.

Plotting an rocTree object

Description

Plots an rocTree object. The function returns a dgr_graph object that is rendered in the RStudio Viewer or survival/hazard estimate at terminal nodes.

Usage

## S3 method for class 'rocTree'
plot(
  x,
  output = c("graph", "visNetwork"),
  digits = 4,
  tree = 1L,
  rankdir = c("TB", "BT", "LR", "RL"),
  shape = "ellipse",
  nodeOnly = FALSE,
  savePlot = FALSE,
  file_name = "pic.pdf",
  file_type = "pdf",
  type = c("tree", "survival", "hazard"),
  ...
)

Arguments

See Also

See rocTree for creating rocTree objects.

Examples

## Not run: 
data(simDat)
fit <- rocTree(Surv(Time, death) ~ z1 + z2, id = id, data = simDat, ensemble = FALSE)
## Plot tree
plot(fit)
## Plot survival estimates at terminal nodes
plot(fit, type = "survival")
## Plot hazard estimates at terminal nodes
plot(fit, type = "haz")

## End(Not run)

Predicting based on a rocTree model.

Description

The function gives predicted values with a rocTree fit.

Usage

## S3 method for class 'rocTree'
predict(object, newdata, type = c("survival", "hazard"), control = list(), ...)

Arguments

Value

Returns a data.frame of the predicted survival probabilities or cumulative hazard.

Examples

data(simDat)
fit <- rocTree(Surv(Time, death) ~ z1 + z2, id = id, data = simDat, ensemble = FALSE)

## testing data
newdat <- data.frame(Time = sort(unique(simDat$Time)), 
                     z2 = runif(1))
newdat$z1 <- 1 * (newdat$Time < median(newdat$Time))
head(newdat)

## Predict survival 
pred <- predict(fit, newdat)
plot(pred)

## Predict hazard
pred <- predict(fit, newdat, type = "hazard")
plot(pred)

Printing an rocTree object

Description

The function prints an rocTree object. It is a method for the generic function print of class "rocTree".

Usage

## S3 method for class 'rocTree'
print(x, digits = 5, tree = NULL, ...)

Arguments

Examples

data(simDat)

## Fitting a pruned survival tree
rocTree(Surv(Time, death) ~ z1 + z2, id = id, data = simDat, ensemble = FALSE)

## Fitting a unpruned survival tree
rocTree(Surv(Time, death) ~ z1 + z2, id = id, data = simDat, ensemble = FALSE,
        control = list(numFold = 0))

## Not run: 
## Fitting the ensemble algorithm (default)
rocTree(Surv(Time, death) ~ z1 + z2, id = id, data = simDat, ensemble = TRUE)

## End(Not run)

Roc-guided survival trees

Description

Fits a "rocTree" model.

Usage

rocTree(
  formula,
  data,
  id,
  subset,
  ensemble = TRUE,
  splitBy = c("dCON", "CON"),
  control = list()
)

Arguments

Details

The argument "control" defaults to a list with the following values:

tau

is the maximum follow-up time; default value is the 90th percentile of the unique observed survival times.

maxTree

is the number of survival trees to be used in the ensemble method (when ensemble = TRUE).

maxNode

is the maximum node number allowed to be in the tree; the default value is 500.

numFold

is the number of folds used in the cross-validation. When numFold > 0, the survival tree will be pruned; when numFold = 0, the unpruned survival tree will be presented. The default value is 10.

h

is the smoothing parameter used in the Kernel; the default value is tau / 20.

minSplitTerm

is the minimum number of baseline observations in each terminal node; the default value is 15.

minSplitNode

is the minimum number of baseline observations in each splitable node; the default value is 30.

disc

is a logical vector specifying whether the covariates in formula are discrete (TRUE) or continuous (FALSE). The length of disc should be the same as the number of covariates in formula. When not specified, the rocTree() function assumes continuous covariates for all.

K

is the number of time points on which the concordance measure is computed. A less refined time grids (smaller K) generally yields faster speed but a very small K is not recommended. The default value is 20.

Value

An object of S4 class "rocTree" representig the fit, with the following components:

References

Sun Y. and Wang, M.C. (2018+). ROC-guided classification and survival trees. Technical report.

See Also

See print.rocTree and plot.rocTree for printing and plotting an rocTree, respectively.

Examples

data(simDat)

## Fitting a pruned survival tree
rocTree(Surv(Time, death) ~ z1 + z2, id = id, data = simDat, ensemble = FALSE)

## Fitting a unpruned survival tree
rocTree(Surv(Time, death) ~ z1 + z2, id = id, data = simDat, ensemble = FALSE,
        control = list(numFold = 0))

## Not run: 
## Fitting the ensemble algorithm (default)
rocTree(Surv(Time, death) ~ z1 + z2, id = id, data = simDat, ensemble = TRUE)

## End(Not run)

Simulated dataset for demonstration

Description

A simulated data frame with variables

id

subjects identification

Time

observation times

death

event/death indicator; 1 = an event (death) is recorded

z1

baseline covariate generated from a standard uniform distribution

z2

baseline covariate generated from a standard uniform distribution (independent from z1

Format

A data frame with 5050 rows and 5 variables.

Details

The sample dataset is generated by set.seed(1); simu(100, 0, 1.3). See simu for details of the simu function.

Function to generate simulated data used in the manuscript.

Description

This function is used to generate simulated data under various settings. Let Z be a p-dimensional vector of possible time-dependent covariates and \beta be the vector of regression coefficient. The survival times (T) are generated from the hazard function specified as follow:

Scenario 1.1

Proportional hazards model:

\lambda(t|Z) = \lambda_0(t) e^{-0.5 Z_1 + 0.5 Z_2 - 0.5 Z_3 ... + 0.5 Z_{10}},

where \lambda_0(t) = 2t.

Scenario 1.2

Proportional hazards model with noise variable:

\lambda(t|Z) = \lambda_0(t) e^{2Z_1 + 2Z_2 + 0Z_3 + ... + 0Z_{10}},

where \lambda_0(t) = 2t.

Scenario 1.3

Proportional hazards model with nonlinear covariate effects:

\lambda(t|Z) = \lambda_0(t) e^{[2\sin(2\pi Z_1) + 2|Z_2 - 0.5|]},

where \lambda_0(t) = 2t.

Scenario 1.4

Accelerated failure time model:

\log(T) = -2 + 2Z_1 + 2Z_2 + \epsilon,

where \epsilon follows N(0, 0.5^2).

Scenario 1.5

Generalized gamma family:

T = e^{\sigma\omega},

where \omega = \log(Q^2 g) / Q, g follows Gamma(Q^{-2}, 1), \sigma = 2Z_1, Q = 2Z_2.

Scenario 2.1

Dichotomous time dependent covariate with at most one change in value:

\lambda(t|Z(t)) = \lambda_0(t)e^{2Z_1(t) + 2Z_2},

where Z_1(t) is the time-dependent covariate: Z_1(t) = \theta I(t \ge U_0) + (1 - \theta) I(t < U_0), ,\theta is a Bernoulli variable with equal probability, and U_0 follows a uniform distribution over [0, 1].

Scenario 2.2

Dichotomous time dependent covariate with multiple changes:

\lambda(t|Z(t)) = e^{2Z_1(t) + 2Z_2},

where Z_1(t) = \theta[I(U_1\le t < U_2) + I(U_3 \le t)] + (1 - \theta)[I(t < U_1) + I(U_2\le t < U_3)], \theta is a Bernoulli variable with equal probability, and U_1\le U_2\le U_3 are the first three terms of a stationary Poisson process with rate 10.

Scenario 2.3

Proportional hazard model with a continuous time dependent covariate:

\lambda(t|Z(t)) = 0.1 e^{Z_1(t) + Z_2},

where Z_1(t) = kt + b, k and b are independent uniform random variables over [1, 2].

Scenario 2.4

Non-proportional hazards model with a continuous time dependent covariate:

\lambda(t|Z(t)) = 0.1 \cdot[1 + \sin\{Z_1(t) + Z_2\}],

where Z_1(t) = kt + b, k and b follow independent uniform distributions over [1, 2].

Scenario 2.5

Non-proportional hazards model with a nonlinear time dependent covariate:

\lambda(t|Z(t)) = 0.1 \cdot[1 + \sin\{Z_1(t) + Z_2\}],

where Z_1(t) = 2kt\cdot \{I(t > 5) - 1\} + b, k and b follow independent uniform distributions over [1, 2].

The censoring times are generated from an independent uniform distribution over [0, c], where c was tuned to yield censoring percentages of 25

Usage

simu(n, cen, scenario, summary = FALSE)

trueHaz(dat)

trueSurv(dat)

Arguments

Value

simu returns a data.frame. The returned data.frame consists of columns:

id

is the subject id.

Y

is the observed follow-up time.

death

is the death indicator; death = 0 if censored.

z1–z10

is the possible time-independent covariate.

k, b, U

are the latent variables used to generate $Z_1(t)$ in Scenario 2.1 – 2.5.

The returned data.frame can be supply to trueHaz and trueSurv to generate the true cumulative hazard function and the survival function, respectively.

Examples

set.seed(1)
simu(10, 0.25, 1.2, TRUE)

set.seed(1)
simu(10, 0.50, 2.2, TRUE)