Pseudo-value regression trees

The estimation of individual-specific survival probabilities is a common task in time-to-event analysis. A plethora of methods has been developed to address this issue, including, among many other approaches, group-wise Kaplan-Meier estimation, Cox regression (Cox 1972), parametric accelerated failure time models (Kalbfleisch and Prentice 2002), and inverse-probability-of-censoring-(IPC)-weighted regression models (Molinaro et al. 2004). Although these approaches are widely used in many disciplines, they often rely on restrictive assumptions limiting their utility. A notable example is the Cox regression model, which requires careful interpretation when the proportional hazards assumption is violated (e.g. Stensrud and Hernán 2020). Similarly, parametric accelerated failure time models may produce invalid results when the underlying distributional assumptions are not met, and IPC-based methods are biased if the working model for the censoring process is misspecified (van der Laan and Robins 2003). Invalid findings may also occur when the complexity of the data-generating process is not fully captured by the model, for instance when relevant covariates are excluded or when interactions between covariates remain undetected (e.g. Vatcheva et al. 2015). In some cases, model misspecification can be avoided by employing methods from the machine learning field (e.g. survival random forests, Ishwaran et al. 2008, or deep neural networks, Lee et al. 2018; Zhao and Feng 2020); however, application of these techniques is often infeasible due to small sample sizes or limitations in the interpretability of the estimated predictor-response relationships. For these reasons, it remains a challenging task to specify time-to-event models yielding accurate and interpretable estimates of individual-specific survival probabilities.

In this paper we propose a novel model building technique named pseudo-value regression trees (PRT). Our method is based on pseudo-value regression (Klein and Andersen 2005), which provides a direct modeling framework to estimate the survival function from a set of right-censored time-to-event data. Unlike Cox regression, pseudo-value regression is not based on a statistical model for the hazard function (from which the survival function can subsequently be derived by application of a suitable transformation); instead, it defines a direct link between the survival function and the covariate values on a grid of pre-specified time points \(t_1, \ldots , t_K\). Usually, K is set to a moderate number, e.g. \(K=5\) or \(K = 10\) (see Andersen and Pohar 2010). Given data from a set of n independent individuals with survival times \(T_i \in \mathbb {R}^+\) and time-independent baseline covariates \(X_i \in \mathbb {R}^p\), \(i=1,\ldots , n\), the key idea of pseudo-value regression is to approximate the survival probabilities \(S(t_k |X_i) = \text{ P }(T_i> t_k|X_i) = \text{ E }[\mathbbm {1}_{\lbrace {T_i > t_k\rbrace }}|X_i]\), \(k=1,\ldots , K\), by a set of jackknife pseudo-values. The latter are defined aswhere \(\hat{S}_{\text {KM}}(t_k)\) and \(\hat{S}_{\text {KM}}(t_k)^{-i}\) denote the Kaplan-Meier estimators based on the complete data and the reduced data (without individual i), respectively. Since it can be shown that \(\text{ E }[\hat{\theta }_i (t_k)|X_i] {\mathop {\longrightarrow }\limits ^{P}}\text{ E }[\mathbbm {1}_{\lbrace {T_i > t_k\rbrace }} |X_i]\) as \(n\rightarrow \infty\) (provided that the censoring mechanism is independent of the event times and the covariates, Graw et al. 2009; Overgaard et al. 2017), consistent estimates of \(S(t_k |X_i)\) can be obtained by fitting a statistical model that regresses the pseudo-values on the covariates (Andersen and Pohar 2010). Unlike IPC-based methods, which often discard the covariate information of censored individuals when used in combination with (weighted) regression techniques (Molinaro et al. 2004), pseudo-value regression is based on a “pseudo” complete data set that includes all available values \(X_1,\ldots , X_n\) in the estimation equation (Andersen and Pohar 2010).

The standard approach to fit a pseudo-value model is to specify a monotonically increasing link function \(g(\cdot )\) and to use \(\hat{\theta }_{i}(t_k)\) as outcome variable in the regression modelwhere \((\alpha _1, \ldots , \alpha _K, \gamma ^\top )^\top \in \mathbb {R}^{K+p}\) is a vector of unknown coefficients. Estimation of the coefficients is usually based on generalized estimating equations (GEE, Liang and Zeger 1986), setting \(g(\cdot )\) equal to the complementary log-log link function (Andersen et al. 2003; Andersen and Pohar 2010). While the GEE approach accounts for possible dependencies between the pseudo-values \(\hat{\theta }_{i}(t_1),\ldots , \hat{\theta }_{i}(t_K)\) obtained from the same individual, it is limited by the restrictive definition of the predictor \(\eta _{ik} = \alpha _k + \gamma ^T X_{i}\). In particular, \(\eta _{ik}\) does not allow for modeling time-dependent effects (since \(\gamma\) is assumed to be constant in time), and it is restricted to modeling main covariate effects only. Although more flexible effect terms (representing e.g. interactions with time or between the covariates) could be included in (2), we are not aware of any algorithm to identify these terms in a data-driven way. On the other hand, pre-specification of the interaction terms is often infeasible, as it would require detailed knowledge on the, usually hidden, interaction structure in the data-generating process. Another limitation of the standard regression model in (2) is that the intercept terms \(\alpha _1,\ldots , \alpha _K\) (representing the “baseline” risk function) are estimated in an unrestricted fashion. As a consequence, the fitted survival probabilities are not guaranteed to decrease with time.

To address these limitations, we extend the standard model in (2) by a semi-parametric approach for the estimation of survival probabilities via pseudo-value regression. Our proposed PRT method is inspired by logistic model trees (LMT, Landwehr et al. 2005), which is a popular classification method combining the strengths of tree learning and binary regression by fitting a series of regularized logistic models to the data in the nodes of a classification tree. In order to adapt LMT to pseudo-value regression, we propose to replace the classification tree by a multivariate conditional inference tree (Hothorn et al. 2006) and to use a novel GEE-type optimization criterion for modeling the pseudo-values in the nodes. The proposed PRT method does not require pre-specification of any main or interaction effects, neither among the covariates nor between the covariates and time.

Briefly, the PRT method is characterized by the following steps: First, in order to identify the most important interactions between the covariates, we build a multivariate conditional inference tree (Hothorn et al. 2006) using the pseudo-values as K-dimensional continuous response variable. In the second step, we apply a gradient boosting algorithm with linear base-learners (Bühlmann and Hothorn 2007; Hofner et al. 2014) to the data in each node of the tree. Our node-wise boosting algorithm is based on the aforementioned GEE-type optimization criterion, including a pre-specified link function to ensure that survival probability estimates are bounded between 0 and 1. Following the idea of LMT, the fitted values of boosting models in higher-level nodes are used as offset values to refine models in lower-level nodes, leading to the stabilization of estimates along single paths. In each node, the fitting of boosting models is stopped early, enabling the selection of relevant covariates. Furthermore, to model interactions between time and the covariates used to build the tree, we include a time-dependent monotonic base-learner (Hofner et al. 2011) in each node-wise model. This base-learner also ensures that survival probability estimates decrease with time.

The result of our model building technique is a set of pseudo-value regression models, each corresponding to a single path from the root node to a terminal node of the conditional inference tree. Due to the combination of tree learning and model-based boosting, the node-wise models include a mixture of interaction and time-dependent effects, all of which are identified in a data-driven way. Furthermore, PRT guarantees interpretability of the node-wise boosting fits by additively combining linear and monotonic base-learners (Hofner et al. 2014). Estimates of individual survival probabilities are obtained by dropping the covariate values down the tree and by evaluating the pseudo-value regression model in the respective terminal node.

The rest of the paper is organized as follows: In Sect. 2.1, we will start with the definition and properties of pseudo-values, including a description of the standard GEE approach for pseudo-value regression. Section 2.2 provides a brief introduction to logistic model trees (Landwehr et al. 2005). Section 3 contains a detailed description of the PRT method, including definitions of the multivariate recursive partitioning and model-based boosting techniques. In Sect. 4 we will present two simulation studies investigating the properties of the PRT method. Furthermore, we will present a comparison to established methods for survival probability estimation. In Sect. 5, we will apply the PRT method to data from the randomized phase III SUCCESS-A trial (de Gregorio et al. 2020), demonstrating that PRT are able to identify subgroups and predictors of disease-free survival in patients with non-metastatic breast cancer. The main findings of the paper are summarized and discussed in Sect. 6, along with a brief overview and discussion of related approaches. Further results and illustrations, as well as details on the implementation of the PRT method, are provided in the Supplementary Material.

Given the limitations of the standard GEE approach, and considering the flexibility of LMT in dealing with complex interaction structures, we propose to build pseudo-value regression models by extending the LMT methodology to the estimation of survival probabilities. Briefly, the idea of our pseudo-value regression trees (PRT) approach is to replace the binary classification tree by a conditional inference tree with multivariate pseudo-value outcome (accounting for possible dependencies between pseudo-values from the same individual, Sect. 3.1), to replace LogitBoost by a component-wise gradient boosting algorithm (including a time-dependent monotonic base-learner and a novel GEE-type optimization criterion, Sect. 3.2), and to use the successively refined boosting models for the estimation of individual-specific survival probabilities (Sect. 3.3).

To investigate the properties of the PRT method, we carried out two simulation studies. In the first study (Sect. 4.1), the data-generating process was characterized by a tree with lognormal survival models in the terminal nodes (reflecting the true structure of a PRT model). The aim of this study was to analyze whether PRT was able to identify relevant covariates and subgroups defined by the interaction effects. In the second study (Sect. 4.2), the data-generating process was based on a lognormal model with an additive mixture of main and interaction effects. Here, the aim was to analyze the performance of PRT in the presence of model misspecification (since the additive model did not have a tree structure). Both studies were based on 100 Monte Carlo replications and \(K=5\) time points (following the suggestions by Andersen and Pohar 2010). In each replication, we generated a training data set for model building and a test data set for model evaluation. The sample sizes of all data sets were set to \(n=1000\). The time points were chosen as the mean empirical \(10\%\), \(30\%\), \(50\%\), \(70\%\), and \(90\%\) quantiles of \(\tilde{T_i}, i = 1, \ldots , 1000\), computed from 1000 additional individuals (generated independently using the data-generating processes of the simulation studies). Five-fold cross-validation on the training data was used to optimize the stopping parameter \(m_{\text {stop}}(1)\).

Accuracy was measured by applying the fitted models to the test data set. We computed the mean squared error (MSE) defined bywhere \(\hat{S}_{\tilde{\imath }}(t_{\tilde{\imath }}| X_{\tilde{\imath }})\) and \(S_{\tilde{\imath }}(t_{\tilde{\imath }}| X_{\tilde{\imath }})\) denote the estimated and the true survival probabilities, respectively, of observation \(\tilde{\imath }\) at time \(t_{\tilde{\imath }}\). To evaluate the error on the scale of the survival probabilities, we further computed the root mean squared error (RMSE), defined as the square root of (13). Additionally, we calculated the bias of the estimated survival probabilities (defined as the average deviation of the estimated survival probabilities from their respective true survival probabilities) and the Brier score (Kvamme and Borgan 2023). Discrimination ability was measured using the concordance index (C-index, Gerds et al. 2013). Bias and Brier score values were averaged across the five time points. The time horizon for the C-index was set equal to the largest time point \(t_5\).

To illustrate the PRT method, we analyzed data from the SUCCESS-A trial (NCT02181101), which was a multicenter randomized phase III study that enrolled 3,754 patients with a primary invasive breast cancer between September 2005 and March 2007. All patients had a high recurrence risk, meaning that the SUCCESS-A study population did not constitute a random sample from the general population; for details on the inclusion/exclusion criteria and the design of the study see de Gregorio et al. (2020). The study had two treatment arms, with patients either receiving one of the standard chemotherapy regimens (control group) or standard chemotherapy with the addition of gemcitabine (experimental group). The randomization ratio was 1:1. The aims of SUCCESS-A were to compare the two groups with respect to disease-free survival (DFS) and overall survival (OS) within a five-year follow-up period. Here we focus on DFS, which, according to the STEEP system, was defined as the period from the date of randomization to the earliest date of disease progression (distant metastases, local and contralocal recurrence, and secondary primary tumors or death from any cause, de Gregorio et al. 2020). Since the definition of DFS included death from any cause, we did not consider death as a competing event.

Patients were censored at the last date on which they were known to be disease-free, resulting in an event rate of \(12.2\%\) (458 events in 3,754 patients). The maximum observation time was 5.5 years (6 months of chemotherapy followed by 5 years of follow-up; median 5.2 years, first quartile 3.7 years, third quartile 5.5 years). In addition to the survival times, the study collected data on several established prognostic factors, including age at randomization (age, measured in years), body mass index (BMI, measured in \(kg/m^2\)), tumor stage (stage, four categories, pT1/pT2/pT3/pT4), tumor grade (grade, three categories, G1/G2/G3), lymph node status (nodal status, two categories, pN0/pN+), tumor type (type, three categories, ductal/lobular/other), menopausal status (meno, two categories, pre-/post-menopausal), and receptor status for estrogen (ER), progesterone (PR), and HER2 (two categories each, negative/positive), see de Gregorio et al. (2020). A descriptive summary of the prognostic variables is given in Table S3 in Section S7 in the supplementary material.

A key issue in the development of treatment guidelines for breast cancer is the identification of patient subgroups with possibly different risks of disease progression (Coates et al. 2015; Senkus et al. 2015). To illustrate the PRT method, we investigated the existence of such subgroups in the SUCCESS-A data, noting that tree-based methods have a long tradition in medical risk assessment (including, among other techniques, univariate survival trees, LeBlanc and Crowley 1992; Bacchetti and Segal 1995, tree-structured classification and regression, Ciampi et al. 1995; Puth et al. 2020, and mixtures of survival trees, Jia et al. 2022). In addition to using the aforementioned prognostic factors as covariates, we included the group status (group, two categories, control/experimental) and time (monotonic P-spline base-learner) in our model. Pseudo-values for DFS were computed at 1, 2, 3, 4, and 5 years (\(K=5\)). The depth of the regression tree was fixed at \(D=3\). Our rationale for choosing this number was that it allowed for capturing interaction effects while, at the same time, resulting in a tree with a reasonably simple interpretation (analogous to the specification of the interaction order in linear regression). Patients with missing values in any of the variables (102 patients, \(2.7\%\)) were excluded from analysis, resulting in an analysis data set with n = 3,652 patients. The accuracy of the PRT model was evaluated by computing five-fold cross-validated values of the concordance index (C-index, Uno et al. 2011, with five-year time horizon) and the Brier score (Kvamme and Borgan 2023, averaged across the five time points).

As mentioned above, the aim of applying PRT to the SUCCESS-A data was to illustrate our method but not to optimize it with respect to prediction accuracy. For sensitivity analysis, we additionally present the Brier score and C-index values obtained from PRT with tree depths fixed at \(D \in \lbrace {2,4,5\rbrace }\), corresponding to maximum numbers of 4, 16 and 32 terminal nodes, respectively. We also compared PRT to the methods described in Sect. 4.2. Note that we had to fix the tree depth of the MOB method at \(D=1\), as larger values resulted in convergence issues.

The regression tree obtained from the PRT fit is shown in Fig. 7. Overall, the tree structure reflected several established prognostic factors and subgroups, which have been frequently reported in the literature and have also been included in treatment guidelines for breast cancer (Coates et al. 2015; Senkus et al. 2015). Specifically, the first split variable (selected in the root node) was ER, indicating the importance of this variable in adjuvant hormonal and chemotherapeutic treatment regimens. The survival advantage of estrogen receptor-positive patients (Goldhirsch et al. 2003) is reflected by the estimated five-year DFS probabilities, which were \(80.87\%\) on average in Nodes 4, 5, and 6 and \(89.89\%\) on average in Nodes 9, 10, and 11 (corresponding to estrogen receptor-negative patients and estrogen receptor-positive patients, respectively). The split variables in the second level of the tree were nodal status and age (threshold = 67 years), reflecting the higher risk of lymph node-positive patients (Senkus et al. 2015) and the increased risk of patients aged 67 or older, respectively (Chen et al. 2016). This result is confirmed by the average estimated five-year DFS probabilities, which were smaller in lymph node-positive patients than in lymph node-negative patients (\(72.46\%\), Nodes 4 and 5, vs. \(88.04\%\), Node 6), and were higher in patients aged \(\le 67\) years than in patients aged \(> 67\) years (\(90.34\%\), Nodes 9 and 10, vs. \(85.82\%\), Node 11). Patients with negative estrogen receptor status and positive lymph node status were further split into subgroups defined by PR. Of note, progesterone receptor-negative patients were estimated to have the lowest average five-year DFS probabilities (\(70.31\%\), Node 4). This group of patients also included the high-risk group of “triple negative” patients (negative ER, PR, and HER2, von Minckwitz et al. 2012), given through the base-learner for HER2 selected by the boosting model in Node 4. In line with the literature (von Minckwitz et al. 2012), the subgroup of triple negative patients had lower estimated five-year DFS probabilities on average than HER2 receptor-positive patients in Node 4 (red vs. green lines in the lower left panel of Fig. 7). Another prognostic variable is grade, which was selected as split variable in the group of patients \(\le 67\) years in Node 8. We note that the grouping of grade (G1/G2 vs. G3) reflects the grouping in current treatment guidelines for breast cancer (Coates et al. 2015; Senkus et al. 2015). As expected, patients with a low or intermediate grade had a higher estimated five-year DFS probability (G1/G2, \(92.03\%\), Node 9) than patients with a high grade (G3, \(86.97\%\), Node 10). Furthermore, tumor stage (selected by the boosting model) had a strong impact on survival in Node 9 (see colored lines in Fig. 7). Regarding the treatments investigated in the SUCCESS-A trial, we observed that the group was not selected in any of the nodes, neither for splitting nor as base-learner in the boosting models. This is in line with the findings in the original study report by de Gregorio et al. (2020), who concluded that the addition of gemcitabine to standard chemotherapy did not improve DFS.

The five-fold cross-validated Brier score and C-index values obtained from PRT and the alternative methods are shown in Table 3. It is seen that the Brier score and C-index values obtained from PRT were similar for all considered tree depths, suggesting that higher values of D did not increase predictive performance but only led to a more difficult interpretation of the models. Overall, PRT were very similar to the alternative methods in terms of prediction accuracy (except TreeOnly and KaplanMeier, which performed worse than PRT as expected, and BoostedTree and IPCW-LS, which also performed worse than PRT). These results support the plausibility of the above interpretations and the validity of the PRT model in Fig. 7.

This paper presents a semi-parametric approach for building time-to-event models with a pseudo-value outcome. Our method, entitled pseudo-value regression trees (PRT), results in a piecewise regression model for the survival function, where the “pieces” are obtained by recursively partitioning the covariate space. As described in Sect. 3, developing a model tree algorithm for pseudo-values involved, among other components, a method for multivariate tree building, a loss function for non-normal continuous outcomes, and an appropriately defined time base-learner to ensure monotonicity of the probability estimates. Our numerical experiments in Sects. 4 and 5 demonstrated that the PRT method was able to identify relevant covariates and interactions (Sect. 4.1), showed a favorable estimation accuracy (Sect. 4.2), and yielded highly plausible results in our application on primary invasive breast cancer (Sect. 5). Importantly, by restricting the tree depth to a moderate value (\(D \le 5\)), the fitted PRT models had an easily accessible interpretation (see e.g. Fig. 7). This is considered to be a major advantage when the focus is not solely on prediction accuracy, especially when compared to black-box methods like support vector machines, random forests, or deep neural networks (see e.g. Mogensen and Gerds 2013; van der Ploeg et al. 2014; Zhao and Feng 2020; Rahman et al. 2021).

Conceptually, the PRT method belongs to the class of “direct” modeling approaches, relating the covariates directly to the survival probabilities instead of relating them “indirectly” to \(S(t|X_i) = \exp (- \int _0^t \lambda (u|X_i) du )\) via the hazard function \(\lambda (t|X_i)\) (as done e.g. by the Cox model and Aalen’s additive hazard model). Prominent examples of direct models are the proportional odds model and the Cox-Aalen model, which can be fitted to a set of censored time-to-event data using inverse-probability-of-censoring-(IPC)-weighted binomial regression (Scheike et al. 2008, see also Grøn and Gerds 2014 and the references therein). Analogous to pseudo-value regression, these models provide estimates of \(S(t_k | X_i)\) on a pre-defined grid of time points \(t_k = t_1, \ldots , t_K\). The same is true for the hierarchical modeling approach by Garcia et al. (2019), which is a mixture of binomial regression and pseudo-value regression; instead of using IPC weights (effectively excluding censored individuals from the estimation equation), the authors replaced the binary values of censored individuals by pseudo-values and fitted the (pseudo-)binomial model within the generalized additive modeling framework. Hothorn et al. (2014) proposed the class of conditional transformation models, which is a general approach to model the distribution function \(F(t|X_i) = 1 - S(t|X_i)\) conditional on a set of covariates (including direct survival models as special cases). Of note, Hothorn et al. (2018) developed a likelihood-based approach for the modeling of \(F(t|X_i)\) that does not require pre-specification of a grid of time points (see also Hothorn 2019 and the references therein). Similar to Garcia et al. (2019), Hothorn et al. (2018) proposed to model the baseline risk and the covariate effects using basis functions. Despite the flexibility of the aforementioned approaches, we emphasize that the building of survival models (in particular, the specification of the model structure) remains a challenging task. Tree-based methods like PRT are useful in addressing this issue, providing tools for variable selection and the identification of interaction effects. On the other hand, the selection steps performed by PRT (and also by related tree methods) preclude the application of standard hypothesis tests in the nodes (Loh et al. 2019). As a consequence, tree-based methods like PRT should be handled with care if the model structure is fixed and if statistical inference is of major interest.

The PRT approach is also related to other methods for building model trees. In Sect. 4, for instance, we used the model-based recursive partitioning approach (MOB) as a comparator to PRT. Conceptually, PRT and MOB are of similar nature; however, they differ with respect to their tree building approaches: While PRT uses the generalized correlation coefficient in (5) for (multivariate) recursive partitioning, MOB applies a test for parameter instability (Zeileis and Hornik 2007) to determine the split variable in each node. In contrast to (5) (which is based on the bivariate relationships between the pseudo-values and the covariates), this instability test requires the node-wise fitting of a regression model including all covariates. As a consequence, MOB is usually more sensitive in detecting interaction effects in the models of interest (controlling for possible confounding instead of considering the “marginal” distributions as in (5)); on the other hand, the validity of the test results might be compromised by multicollinearity, especially when the number of covariates is large relative to the node size (see Sect. 4). To the best of our knowledge, there exists no regularized version of the MOB algorithm (performing e.g. variable selection like the boosting models in PRT). We further note that the current implementation of MOB in the R package partykit does not allow for modeling correlated observations (e.g. via mixed-effects models or a multivariate tree as in PRT). Similar arguments hold for the GUIDE algorithm, which has recently been extended by Loh et al. (2019) to build model trees within the proportional hazards framework.

As stated in Sect. 1, the consistency results by Graw et al. (2009) rely on the assumption that the censoring times \(C_i\) are independent of the survival times \(T_i\) (“random censoring”). Under this assumption, it can be shown that \(\text{ E }[\hat{\theta }_i (t_k) | X_i] \,{\mathop {\rightarrow }\limits ^{}} \text{ E }[\mathbbm {1}_{\{ T_i > t_k\}} |X_i] = S(t_k | X_i)\), so that using the pseudo-values as outcome variable in a statistical model (as done by PRT) is equivalent to substituting the outcome values of interest (here, the unobserved survival probabilities) by consistent estimates of these values. Later, Binder et al. (2014) showed that the random censoring assumption can be relaxed to allow for censoring times \(C_i\) that are only conditionally independent of \(T_i\) given the covariate values \(X_i\). More specifically, the authors considered a scenario where pseudo-values are based on the Aalen-Johansen estimator of the cumulative incidence function \(\text{ P }(T_i \le t_k)\). By fitting a regression model for the censoring survival function \(G(t_k|X_i):= \text{ P }(C_i > t_k|X_i)\) and incorporating the resulting IPC weights in the Aalen-Johansen estimator, they were able to eliminate the bias occurring from a “naive” calculation of the pseudo-values ignoring dependency on \(X_i\). Analogously, the PRT method could be extended to scenarios with covariate-dependent censoring. This could be done by replacing the Kaplan-Meier estimators in (1) by one minus the respective IPC-weighted Aalen-Johansen estimators. We point out that the results by Binder et al. (2014) rely on the correct specification of the regression model for \(G(t_k|X_i)\).

We further note that the pseudo-value methodology is not restricted to the estimation of survival probabilities from right-censored data. For example, Andersen and Pohar (2010) considered a general class of functionals of the form \(\text{ E}\left[ \psi (T_i) \vert X_i \right]\), suggesting that any of these functionals could be estimated by an appropriately defined pseudo-value regression model. In the same manner, the PRT approach could be adapted to a wider class of functionals, an obvious example being the cumulative incidence function in competing-risks analysis. Following the idea described in Zhao et al. (2020), PRT could also be applied for dynamic risk prediction. Furthermore, PRT can easily be extended to incorporate left-truncated survival times referring to individuals not yet at risk at time \(t=0\). Provided that the truncation times are independent of the survival times \(T_i\) (at least conditional on \(X_i\), see above), this could be done by an appropriate definition of the risk sets used in the calculation of the Kaplan-Meier estimators in (1). With regard to the latter, robustness can be increased by computing “stopped” pseudo-values \(\hat{\theta }_{i}(t_k)\) that are based on only those individuals who entered the sample before the respective time points \(t_k\). For details, see Grand et al. (2019).

Finally, we would like to note that PRT is, in general, applicable to high-dimensional scenarios with \(p > n\). Naturally, very large numbers of covariates may result in increased run-times (which are mainly due to the permutation tests conducted for the selection of the split variables). It should also be noted that both conditional inference trees and gradient boosting usually require some sort of pre-filtering in order to work well in “ultra-high”-dimensional scenarios with \(p \gg n\). The latter aspects are, of course, not specific to PRT but apply to many methods for modeling high-dimensional data (e.g. to penalized regression and even to random forests).

All computations were carried out using the R Language for Statistical Computing (version 4.1.2, R Core Team 2022). An implementation of the PRT method is available at https://​www.​imbie.​uni-bonn.​de/​cloud/​index.​php/​s/​5oZDBSJjW4pLjtb. Details are given in Section S1 in the supplementary material.