MOTIVATION: This work presents the first large-scale neutral benchmark experiment focused on single-event, right-censored, low-dimensional survival data. Benchmark experiments are essential in methodological research to scientifically compare new and existing model classes through proper empirical evaluation. Existing benchmarks in the survival literature are smaller in scale regarding the number of used datasets and extent of empirical evaluation. They often lack appropriate tuning or evaluation procedures, while other comparison studies focus on qualitative reviews rather than quantitative comparisons. This comprehensive study aims to fill the gap by neutrally evaluating a broad range of methods and providing generalizable guidelines for practitioners. RESULTS: We benchmark 21 models, ranging from classical statistical approaches to many common machine learning methods, on 34 publicly available datasets. The benchmark tunes models using both a discrimination measure (Harrell's C-index) and a scoring rule (Integrated Survival Brier Score), and evaluates them across six metrics covering discrimination, calibration, and overall predictive performance. Despite superior average ranks in overall predictive performance from individual learners like oblique random survival forests and likelihood-based boosting, and better discrimination rankings from multiple boosting- and tree-based methods as well as parametric survival models, no method statistically significantly outperforms the commonly used Cox proportional hazards model for either tuning measure. We conclude that while the Cox Proportional Hazards model remains a robust default for low-dimensional, right-censored survival data, more flexible methods may be preferable for specific dataset characteristics. AVAILABILITY AND IMPLEMENTATION: All code, data, and results are publicly available on GitHub https://github.com/slds-lmu/paper_2023_survival_benchmark and archived on Zenodo https://doi.org/10.5281/zenodo.19075310.

This paper proposes a method for measuring conditional feature importance via generative modeling. In explainable artificial intelligence (XAI), conditional feature importance assesses the impact of a feature on a prediction model's performance given the information of other features. Model-agnostic post hoc methods to do so typically evaluate changes in the predictive performance under on-manifold feature value manipulations. Such procedures require creating feature values that respect conditional feature distributions, which can be challenging in practice. Recent advancements in generative modeling can facilitate this. For tabular data, which may consist of both categorical and continuous features, the adversarial random forest (ARF) stands out as a generative model that can generate on-manifold data points without requiring intensive tuning efforts or computational resources, making it a promising candidate model for subroutines in XAI methods. This paper proposes cARFi (conditional ARF feature importance), a method for measuring conditional feature importance through feature values sampled from ARF-estimated conditional distributions. cARFi requires only little tuning to yield robust importance scores that can flexibly adapt for conditional or marginal notions of feature importance, including straightforward extensions to condition on feature subsets and allows for inferring the significance of feature importances through statistical tests.

Deep learning survival models often outperform classical methods in time-to-event predictions, particularly in personalized medicine, but their "black box" nature hinders broader adoption. We propose a framework for gradient-based explanation methods tailored to survival neural networks, extending their use beyond regression and classification. We analyze the implications of their theoretical assumptions for time-dependent explanations in the survival setting and propose effective visualizations incorporating the temporal dimension. Experiments on synthetic data show that gradient-based methods capture the magnitude and direction of local and global feature effects, including time dependencies. We introduce GradSHAP(t), a gradient-based counterpart to SurvSHAP(t), which outperforms SurvSHAP(t) and SurvLIME in a computational speed vs. accuracy trade-off. Finally, we apply these methods to medical data with multi-modal inputs, revealing relevant tabular features and visual patterns, as well as their temporal dynamics.

Many existing interpretation methods are based on Partial Dependence (PD) functions that, for a pre-trained machine learning model, capture how a subset of the features affects the predictions by averaging over the remaining features. Notable methods include Shapley additive explanations (SHAP) which computes feature contributions based on a game theoretical interpretation and PD plots (i.e., 1-dim PD functions) that capture average marginal main effects. Recent work has connected these approaches using a functional decomposition and argues that SHAP values can be misleading since they merge main and interaction effects into a single local effect. However, a major advantage of SHAP compared to other PD-based interpretations has been the availability of fast estimation techniques, such as `TreeSHAP`. In this paper, we propose a new tree-based estimator, `FastPD`, which efficiently estimates arbitrary PD functions. We show that `FastPD` consistently estimates the desired population quantity -- in contrast to path-dependent `TreeSHAP` which is inconsistent when features are correlated. For moderately deep trees, `FastPD` improves the complexity of existing methods from quadratic to linear in the number of observations. By estimating PD functions for arbitrary feature subsets, `FastPD` can be used to extract PD-based interpretations such as SHAP, PD plots and higher-order interaction effects.

We propose a principled method for autoencoding with random forests. Our strategy builds on foundational results from nonparametric statistics and spectral graph theory to learn a low-dimensional embedding of the model that optimally represents relationships in the data. We provide exact and approximate solutions to the decoding problem via constrained optimization, split relabeling, and nearest neighbors regression. These methods effectively invert the compression pipeline, establishing a map from the embedding space back to the input space using splits learned by the ensemble's constituent trees. The resulting decoders are universally consistent under common regularity assumptions. The procedure works with supervised or unsupervised models, providing a window into conditional or joint distributions. We demonstrate various applications of this autoencoder, including powerful new tools for visualization, compression, clustering, and denoising. Experiments illustrate the ease and utility of our method in a wide range of settings, including tabular, image, and genomic data.

Global feature effect methods, such as partial dependence plots, provide an intelligible visualization of the expected marginal feature effect. However, such global feature effect methods can be misleading, as they do not represent local feature effects of single observations well when feature interactions are present. We formally introduce generalized additive decomposition of global effects (GADGET), which is a new framework based on recursive partitioning to find interpretable regions in the feature space such that the interaction-related heterogeneity of local feature effects is minimized. We provide a mathematical foundation of the framework and show that it is applicable to the most popular methods to visualize marginal feature effects, namely partial dependence, accumulated local effects, and Shapley additive explanations (SHAP) dependence. Furthermore, we introduce and validate a new permutation-based interaction detection procedure that is applicable to any feature effect method that fits into our proposed framework. We empirically evaluate the theoretical characteristics of the proposed methods based on various feature effect methods in different experimental settings. Moreover, we apply our introduced methodology to three real-world examples to showcase their usefulness.

We consider a global representation of a regression or classification function by decomposing it into the sum of main and interaction components of arbitrary order. We propose a new identification constraint that allows for the extraction of interventional SHAP values and partial dependence plots, thereby unifying local and global explanations. With our proposed identification, a feature’s partial dependence plot corresponds to the main effect term plus the intercept. The interventional SHAP value of feature $k$ is a weighted sum of the main component and all interaction components that include $k$, with the weights given by the reciprocal of the component’s dimension. This brings a new perspective to local explanations such as SHAP values which were previously motivated by game theory only. We show that the decomposition can be used to reduce direct and indirect bias by removing all components that include a protected feature. Lastly, we motivate a new measure of feature importance. In principle, our proposed functional decomposition can be applied to any machine learning model, but exact calculation is only feasible for low-dimensional structures or ensembles of those. We provide an algorithm and efficient implementation for gradient-boosted trees (xgboost) and random planted forest. Conducted experiments suggest that our method provides meaningful explanations and reveals interactions of higher orders. The proposed methods are implemented in an R package, available at https://github.com/PlantedML/glex.

We propose methods for density estimation and data synthesis using a novel form of unsupervised random forests. Inspired by generative adversarial networks, we implement a recursive procedure in which trees gradually learn structural properties of the data through alternating rounds of generation and discrimination. The method is provably consistent under minimal assumptions. Unlike classic tree-based alternatives, our approach provides smooth (un)conditional densities and allows for fully synthetic data generation. We achieve comparable or superior performance to state-of-the-art probabilistic circuits and deep learning models on various tabular data benchmarks while executing about two orders of magnitude faster on average. An accompanying $R$ package, $arf$, is available on $CRAN$.

SUMMARY: Due to their flexibility and superior performance, machine learning models frequently complement and outperform traditional statistical survival models. However, their widespread adoption is hindered by a lack of user-friendly tools to explain their internal operations and prediction rationales. To tackle this issue, we introduce the survex R package, which provides a cohesive framework for explaining any survival model by applying explainable artificial intelligence techniques. The capabilities of the proposed software encompass understanding and diagnosing survival models, which can lead to their improvement. By revealing insights into the decision-making process, such as variable effects and importances, survex enables the assessment of model reliability and the detection of biases. Thus, transparency and responsibility may be promoted in sensitive areas, such as biomedical research and healthcare applications. AVAILABILITY AND IMPLEMENTATION: survex is available under the GPL3 public license at https://github.com/modeloriented/survex and on CRAN with documentation available at https://modeloriented.github.io/survex.

Abstract We propose the conditional predictive impact (CPI), a consistent and unbiased estimator of the association between one or several features and a given outcome, conditional on a reduced feature set. Building on the knockoff framework of Candès et al. (J R Stat Soc Ser B 80:551–577, 2018), we develop a novel testing procedure that works in conjunction with any valid knockoff sampler, supervised learning algorithm, and loss function. The CPI can be efficiently computed for high-dimensional data without any sparsity constraints. We demonstrate convergence criteria for the CPI and develop statistical inference procedures for evaluating its magnitude, significance, and precision. These tests aid in feature and model selection, extending traditional frequentist and Bayesian techniques to general supervised learning tasks. The CPI may also be applied in causal discovery to identify underlying multivariate graph structures. We test our method using various algorithms, including linear regression, neural networks, random forests, and support vector machines. Empirical results show that the CPI compares favorably to alternative variable importance measures and other nonparametric tests of conditional independence on a diverse array of real and synthetic datasets. Simulations confirm that our inference procedures successfully control Type I error with competitive power in a range of settings. Our method has been implemented in an package, , which can be downloaded from https://github.com/dswatson/cpi .

Abstract Random survival forests (RSF) are a powerful nonparametric method for building prediction models with a time-to-event outcome. RSF do not rely on the proportional hazards assumption and can be readily applied to both low- and higher-dimensional data. A remaining limitation of RSF, however, arises from the fact that the method is almost entirely focussed on continuously measured event times. This issue may become problematic in studies where time is measured on a discrete scale $$t = 1, 2, ...$$ t=1,2,... , referring to time intervals $$[0,a_1), [a_1,a_2), \ldots $$ [0,a1),[a1,a2),… . In this situation, the application of methods designed for continuous time-to-event data may lead to biased estimators and inaccurate predictions if discreteness is ignored. To address this issue, we develop a RSF algorithm that is specifically designed for the analysis of (possibly right-censored) discrete event times. The algorithm is based on an ensemble of discrete-time survival trees that operate on transformed versions of the original time-to-event data using tree methods for binary classification. As the outcome variable in these trees is typically highly imbalanced, our algorithm implements a node splitting strategy based on Hellinger’s distance, which is a skew-insensitive alternative to classical split criteria such as the Gini impurity. The new algorithm thus provides flexible nonparametric predictions of individual-specific discrete hazard and survival functions. Our numerical results suggest that node splitting by Hellinger’s distance improves predictive performance when compared to the Gini impurity. Furthermore, discrete-time RSF improve prediction accuracy when compared to RSF approaches treating discrete event times as continuous in situations where the number of time intervals is small.

BACKGROUND: In the last years more and more multi-omics data are becoming available, that is, data featuring measurements of several types of omics data for each patient. Using multi-omics data as covariate data in outcome prediction is both promising and challenging due to the complex structure of such data. Random forest is a prediction method known for its ability to render complex dependency patterns between the outcome and the covariates. Against this background we developed five candidate random forest variants tailored to multi-omics covariate data. These variants modify the split point selection of random forest to incorporate the block structure of multi-omics data and can be applied to any outcome type for which a random forest variant exists, such as categorical, continuous and survival outcomes. Using 20 publicly available multi-omics data sets with survival outcome we compared the prediction performances of the block forest variants with alternatives. We also considered the common special case of having clinical covariates and measurements of a single omics data type available. RESULTS: We identify one variant termed "block forest" that outperformed all other approaches in the comparison study. In particular, it performed significantly better than standard random survival forest (adjusted p-value: 0.027). The two best performing variants have in common that the block choice is randomized in the split point selection procedure. In the case of having clinical covariates and a single omics data type available, the improvements of the variants over random survival forest were larger than in the case of the multi-omics data. The degrees of improvements over random survival forest varied strongly across data sets. Moreover, considering all clinical covariates mandatorily improved the performance. This result should however be interpreted with caution, because the level of predictive information contained in clinical covariates depends on the specific application. CONCLUSIONS: The new prediction method block forest for multi-omics data can significantly improve the prediction performance of random forest and outperformed alternatives in the comparison. Block forest is particularly effective for the special case of using clinical covariates in combination with measurements of a single omics data type.

Motivation: Random forests are fast, flexible and represent a robust approach to analyze high dimensional data. A key advantage over alternative machine learning algorithms are variable importance measures, which can be used to identify relevant features or perform variable selection. Measures based on the impurity reduction of splits, such as the Gini importance, are popular because they are simple and fast to compute. However, they are biased in favor of variables with many possible split points and high minor allele frequency. Results: We set up a fast approach to debias impurity-based variable importance measures for classification, regression and survival forests. We show that it creates a variable importance measure which is unbiased with regard to the number of categories and minor allele frequency and almost as fast as the standard impurity importance. As a result, it is now possible to compute reliable importance estimates without the extra computing cost of permutations. Further, we combine the importance measure with a fast testing procedure, producing p-values for variable importance with almost no computational overhead to the creation of the random forest. Applications to gene expression and genome-wide association data show that the proposed method is powerful and computationally efficient. Availability and implementation: The procedure is included in the ranger package, available at https://cran.r-project.org/package=ranger and https://github.com/imbs-hl/ranger. Supplementary information: Supplementary data are available at Bioinformatics online.

BACKGROUND: Random forests have often been claimed to uncover interaction effects. However, if and how interaction effects can be differentiated from marginal effects remains unclear. In extensive simulation studies, we investigate whether random forest variable importance measures capture or detect gene-gene interactions. With capturing interactions, we define the ability to identify a variable that acts through an interaction with another one, while detection is the ability to identify an interaction effect as such. RESULTS: Of the single importance measures, the Gini importance captured interaction effects in most of the simulated scenarios, however, they were masked by marginal effects in other variables. With the permutation importance, the proportion of captured interactions was lower in all cases. Pairwise importance measures performed about equal, with a slight advantage for the joint variable importance method. However, the overall fraction of detected interactions was low. In almost all scenarios the detection fraction in a model with only marginal effects was larger than in a model with an interaction effect only. CONCLUSIONS: Random forests are generally capable of capturing gene-gene interactions, but current variable importance measures are unable to detect them as interactions. In most of the cases, interactions are masked by marginal effects and interactions cannot be differentiated from marginal effects. Consequently, caution is warranted when claiming that random forests uncover interactions.