0. Information on data source

This is an exemplary illustration of the Pseudo-value regression trees (PRT) method applied on data from the Rotterdam tumor bank (Royston and Altman (2013)) provided in the survival package in R. The outcome of interest considered for this illustrative example is overall survival. Information on R the version this code was written on can be found below.

# version info
version

##                _                           
## platform       x86_64-w64-mingw32          
## arch           x86_64                      
## os             mingw32                     
## system         x86_64, mingw32             
## status                                     
## major          4                           
## minor          1.0                         
## year           2021                        
## month          05                          
## day            18                          
## svn rev        80317                       
## language       R                           
## version.string R version 4.1.0 (2021-05-18)
## nickname       Camp Pontanezen

1. Source required functions and load necessary packages

The file Functions.R contains the implementation of PRT as well as functions related for analyzing the PRT method output.

rm(list = ls())

source("R/Functions.R")

library(batchtools)
library(tidyverse)
library(magrittr)
library(pseudo)
library(mboost)
library(partykit)
library(stringr)
library(VGAM)
library(Matrix)
library(ggparty)
library(cowplot)
library(ggtext)
library(grid)
library(gridExtra)
library(pander)
library(survival)
library(qdapRegex)

2. Load rotterdam data from the survival package

# load example data and functions
data(cancer, package = "survival")
data <- rotterdam %>%
  filter(nodes > 0) %>% # filter for nodes > 0 (as in Royston and Altman, 2013)
  mutate(meno   = as.factor(meno),
         grade  = as.factor(grade),
         hormon = as.factor(hormon),
         chemo  = as.factor(chemo))

The data frame data contains \(n = 1546\) female patients with primary breast cancer. Further, data contains the columns age at surgery (age), measured in years), menopausal status (meno, two categories, pre-(=0)/ post-(=1) menopausal), tumor size (size, three categories, <=20, 20-50, >50), differentiation grade (grade, two categories, 2/2), number of positive lymph nodes (nodes), progesterone receptor (pgr, measured in \(fmol/l\)), estrogen receptors (er, measured in \(fmol/l\)), hormonal treatment (hormon, two categories, yes(1)/no(0)), chemotherapy (chemo, two categories, yes(1)/no(0)) (see Royston and Altman (2013) for details on the data). Further, time (in days) and event status (0 = censored, 1 = event) for overall survival are contained in the columns dtime and death respectively.

3. Define covariates included into analysis

We include all aforementioned available covariates into the analysis.

# define covariates for tree building and node-wise gradient boosting
covariates    <- c("age", "meno", "size", "grade", "nodes", 
                   "pgr",  "er", "hormon", "chemo")

4. Define time points to calculate pseudo-values on

We chose five time points (\(t_1, \ldots , t_5\)) corresponding to 1 to 5 years of follow-up. Note that the maximal cutoff value should not exceed the maximal observation time (the maximal value in the dtime column).

# check distribution of observed times in the data 
summary(data$dtime)

##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##      45    1175    2166    2300    3275    7027

# define time points
cutoffs_pseudo <- c(1:5)*365.25

5. Model building with PRT

The main function PRT performs the following steps

1 Input
- Calculate pseudo-values \(\hat{\theta}_i(t_k)\), \(k = 1,\ldots, K\).
2 Determination of the tuning parameter \(m_{\text{stop}}(1)\)
- Create augmented data matrix with observations \(\tilde{\imath}=1,\ldots, n\cdot K\).
- Carry out five-fold cross-validation. For \(f = 1, \ldots, 5\):
  - Build a multivariate regression tree with pseudo-value outcome on the \(f\)-th training data (using the multivariate conditional inference framework).
  - Fit node-wise gradient boosting models to the tree (with time-dependent offset values).
- Determine the optimal \(m_{\text{stop}}(1)\).
3 Model building
- Grow the regression tree on the complete data set.
- Given the optimal \(m_{\text{stop}}(1)\), calculate the node-wise stopping iterations \(m_{\text{stop}}(\mathcal{N})\).
- Fit node-wise gradient boosting models to the tree (with time-dependent offset values).

# model building
prt <- PRT(data = data, covariates = covariates, cutoffs_pseudo = cutoffs_pseudo, 
           time_col = "dtime", status_col = "death", id_col = "pid", 
           D = 3, fold = 5, center = FALSE)

## [1] "Calculate pseudo-values..."
## [1] "Optimize mstop via cross-validation..."
## [1] "Fit PRT with optimal mstop on all data..."

The function PRT takes the following arguments:

The function PRT returns an object containing the following elements:

6. Estimated survival probabilities

The function predict.PRT returns estimated survival probabilities (column pred) in a long data format attached to the original data for each observation. Additionally, the terminal node the observations falls into is attached (column end_node).

# predictions
pred_PRT <- predict.PRT(prt)

pred_PRT %>% filter(pid %in% c(10, 12, 19)) %>% # show parts of the results for end nodes 4,7,8
  arrange(end_node) %>% 
  dplyr::select(c("nodes", "size", "meno", "er", "pgr", "time", "pred", "end_node"))

## # A tibble: 15 x 8
##    nodes size  meno     er   pgr  time  pred end_node
##    <int> <fct> <fct> <int> <int> <dbl> <dbl>    <dbl>
##  1     1 <=20  0        67   245   365 0.997        4
##  2     1 <=20  0        67   245   730 0.978        4
##  3     1 <=20  0        67   245  1095 0.930        4
##  4     1 <=20  0        67   245  1461 0.882        4
##  5     1 <=20  0        67   245  1826 0.829        4
##  6     1 20-50 0         0    14   365 0.983        7
##  7     1 20-50 0         0    14   730 0.928        7
##  8     1 20-50 0         0    14  1095 0.838        7
##  9     1 20-50 0         0    14  1461 0.770        7
## 10     1 20-50 0         0    14  1826 0.702        7
## 11     5 20-50 0       976   316   365 0.961        8
## 12     5 20-50 0       976   316   730 0.876        8
## 13     5 20-50 0       976   316  1095 0.764        8
## 14     5 20-50 0       976   316  1461 0.684        8
## 15     5 20-50 0       976   316  1826 0.610        8

7. Extract selected base-learner from PRT object

The function get.baselearner applied on the object returned by the PRT function returns a list containing (i) a data frame indicating which base-learner was selected (1 = yes, 0 = no) in which node-wise boosting model (stored in selected_bl) and (ii) a list containing the estimates for the base-learner in each node (stored in estimates_bl).

# extract selected base-learner 
baselearner <- get.baselearner(prt)
baselearner$selected_bl   # data frame indicating which base-learner was selected

##         age meno size grade nodes pgr er hormon chemo time x0
## Node 1    0    0    1     1     1   1  1      1     1    0  0
## Node 2    0    0    1     1     0   0  1      1     1    0  0
## Node 3    0    1    0     1     0   0  1      0     1    0  0
## Node 4    0    0    0     1     0   0  1      0     0    0  0
## Node 5    0    0    0     0     0   0  1      0     0    0  0
## Node 6    0    0    0     1     1   0  1      1     1    0  0
## Node 7    0    1    0     0     0   0  1      1     1    0  0
## Node 8    0    0    0     1     0   0  1      0     1    0  0
## Node 9    0    0    1     1     0   1  1      0     0    0  0
## Node 10   0    0    1     1     0   0  1      1     0    0  0
## Node 11   0    0    0     1     0   0  1      0     0    0  0
## Node 12   0    0    1     0     0   0  1      0     1    0  0
## Node 13   0    0    1     0     0   0  1      0     0    0  0
## Node 14   0    0    0     0     0   0  1      1     0    0  0
## Node 15   0    1    0     0     0   0  1      0     0    0  0

#baselearner$estimates_bl # list containing the estimates for the base-learner in each node

8. Extract split variables from PRT object

The function analyze.splits applied on the object returned by the PRT function returns a list containing (i) a data frame indicating which split variables were selected (1 = yes, 0 = no) in the path to an end node (stored in columns of splits_per_endnode) and (ii) a data frame containing the split variables and levels for each node in the regression tree (stored in splits_list).

# extract split variables
splits   <- analyze.splits(prt)
splits$splits_per_endnode # data frame indicating which split variables were selected in each path

##        4 5 7 8 11 12 14 15
## age    0 0 0 0  0  0  0  0
## meno   1 1 0 0  0  0  0  0
## size   1 1 1 1  0  0  1  1
## grade  0 0 0 0  0  0  0  0
## nodes  1 1 1 1  1  1  1  1
## pgr    0 0 0 0  1  1  1  1
## er     0 0 0 0  1  1  0  0
## hormon 0 0 0 0  0  0  0  0
## chemo  0 0 0 0  0  0  0  0

splits$splits_list        # data frame containing the split variables and levels for each node in the regression tree

##   node split_variable level
## 1    1          nodes     1
## 2    2           size     2
## 3    3           meno     3
## 4    6          nodes     3
## 5    9            pgr     2
## 6   10             er     3
## 7   13           size     3

9. Plot PRT results

The function plot.PRT plots the tree structure with selected base-learners at each node and estimated survival probabilities in the terminal nodes.

# define covariate labels in the plot
lbls         <- covariates
names(lbls)  <- covariates


# plot results
plot.PRT(PRT_object = prt, covariate_labels = lbls, pred = pred_PRT, 
         baselearner = baselearner$selected_bl, 
         covariate_labels_bl = lbls, work_data = data,
         plot.title = "Illustration of PRT", label_size_bl = 8, 
         left_bl  = -0.08, right_bl = 0.08, 
         y_bl = -0.015, y_surv = -0.07, y_bl_terminal = -0.17, 
         height_terminal = 0.5, width_terminal = 0.7, height_bl = 0.5, 
         width_bl = 0.7, ylim_surv = c(40,100))

The arguments of the plot.PRT function are given in the following table.

References

Hothorn, T., P. Bühlmann, T. Kneib, M. Schmid, and B. Hofner. 2022. mboost: Model-Based Boosting. https://cran.r-project.org/web/packages/mboost.

Hothorn, T., H. Seibold, and A. Zeileis. 2022. Partykit: A Toolkit for Recursive Partytioning. https://cran.r-project.org/web/packages/partykit.

Pohar Perme, M., and M. Gerster. 2017. Pseudo: Computes Pseudo-Observations for Modeling. https://cran.r-project.org/web/packages/pseudo.

Royston, P., and D. G. Altman. 2013. “External Validation of a Cox Prognostic Model: Principles and Methods.” BMC Medical Research Methodology 13.

Therneau, T. M. 2021. A Package for Survival Analysis in R. https://CRAN.R-project.org/package=survival.

Institute of Medical Biometry, Informatics and Epidemiology, University of Bonn, schenk@imbie.uni-bonn.de ↩︎

Pseudo-value Regression Trees

Alina Schenk¹, Moritz Berger¹, Matthias Schmid¹

2023-01-27

0. Information on data source

1. Source required functions and load necessary packages

2. Load rotterdam data from the survival package

3. Define covariates included into analysis

4. Define time points to calculate pseudo-values on

5. Model building with PRT

6. Estimated survival probabilities

7. Extract selected base-learner from PRT object

8. Extract split variables from PRT object

9. Plot PRT results

References

Pseudo-value Regression Trees

Alina Schenk1, Moritz Berger1, Matthias Schmid1

2023-01-27

0. Information on data source

1. Source required functions and load necessary packages

2. Load rotterdam data from the survival package

3. Define covariates included into analysis

4. Define time points to calculate pseudo-values on

5. Model building with PRT

6. Estimated survival probabilities

7. Extract selected base-learner from PRT object

8. Extract split variables from PRT object

9. Plot PRT results

References

Alina Schenk¹, Moritz Berger¹, Matthias Schmid¹