arXiv subjects

Lucas Kook

Publications and source records attributed to Lucas Kook.

17 recordsLinked to original sources

arXiv2026

Exact Graph Learning via Integer Programming

Learning the dependence structure among variables in complex systems is a central problem across medical, natural, and social sciences. These structures can be naturally represented by graphs, and the task of inferring such graphs from data is known as graph learning or causal discovery. Existing approaches typically rely on restrictive assumptions about the data-generating process, employ greedy oracle algorithms, or solve approximate formulations of the graph learning problem. Therefore, they are either sensitive to violations of central assumptions or fail to guarantee globally optimal solutions. We address these limitations by introducing a nonparametric graph learning framework based on conditional independence testing and integer programming. We reformulate the graph learning problem as a mixed-integer program and prove that solving this integer-programming problem provides a globally optimal solution to the original graph learning problem. Our method leverages efficient encodings of graphical separation criteria, enabling the exact recovery of larger graphs than was previously feasible. We provide an open-source R package 'glip' which supports learning (acyclic) directed (mixed) graphs and chain graphs. We demonstrate that our approach is often faster than existing exact graph learning procedures and achieves state-of-the-art performance on simulated and benchmark data across all aforementioned classes of graphs.

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arXiv2025

Comparison of generalised additive models and neural networks in applications: A systematic review

Neural networks have become a popular tool in predictive modelling, more commonly associated with machine learning and artificial intelligence than with statistics. Generalised Additive Models (GAMs) are flexible non-linear statistical models that retain interpretability. Both are state-of-the-art in their own right, with their respective advantages and disadvantages. This paper analyses how these two model classes have performed on real-world tabular data. Following PRISMA guidelines, we conducted a systematic review of papers that performed empirical comparisons of GAMs and neural networks. Eligible papers were identified, yielding 143 papers, with 430 datasets. Key attributes at both paper and dataset levels were extracted and reported. Beyond summarising comparisons, we analyse reported performance metrics using mixed-effects modelling to investigate potential characteristics that can explain and quantify observed differences, including application area, study year, sample size, number of predictors, and neural network complexity. Across datasets, no consistent evidence of superiority was found for either GAMs or neural networks when considering the most frequently reported metrics (RMSE, $R^2$, and AUC). Neural networks tended to outperform in larger datasets and in those with more predictors, but this advantage narrowed over time. Conversely, GAMs remained competitive, particularly in smaller data settings, while retaining interpretability. Reporting of dataset characteristics and neural network complexity was incomplete in much of the literature, limiting transparency and reproducibility. This review highlights that GAMs and neural networks should be viewed as complementary approaches rather than competitors. For many tabular applications, the performance trade-off is modest, and interpretability may favour GAMs.

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arXiv2025

Rethinking player evaluation in sports: Goals above expectation and beyond

A popular quantitative approach to evaluating player performance in sports involves comparing an observed outcome to the expected outcome ignoring player involvement, which is estimated using statistical or machine learning methods. In soccer, for instance, goals above expectation (GAX) of a player measure how often shots of this player led to a goal compared to the model-derived expected outcome of the shots. Typically, sports data analysts rely on flexible machine learning models, which are capable of handling complex nonlinear effects and feature interactions, but fail to provide valid statistical inference due to finite-sample bias and slow convergence rates. In this paper, we close this gap by presenting a framework for player evaluation with metrics derived from differences in actual and expected outcomes using flexible machine learning algorithms, which nonetheless allows for valid frequentist inference. We first show that the commonly used metrics are directly related to Rao's score test in parametric regression models for the expected outcome. Motivated by this finding and recent developments in double machine learning, we then propose the use of residualized versions of the original metrics. For GAX, the residualization step corresponds to an additional regression predicting whether a given player would take the shot under the circumstances described by the features. We further relate metrics in the proposed framework to player-specific effect estimates in interpretable semiparametric regression models, allowing us to infer directional effects, e.g., to determine players that have a positive impact on the outcome. Our primary use case are GAX in soccer. We further apply our framework to evaluate goal-stopping ability of goalkeepers, shooting skill in basketball, quarterback passing skill in American football, and injury-proneness of soccer players.

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arXiv2024

Instrumental Variable Estimation of Distributional Causal Effects

Estimating the causal effect of a treatment on the entire response distribution is an important yet challenging task. For instance, one might be interested in how a pension plan affects not only the average savings among all individuals but also how it affects the entire savings distribution. While sufficiently large randomized studies can be used to estimate such distributional causal effects, they are often either not feasible in practice or involve non-compliance. A well-established class of methods for estimating average causal effects from either observational studies with unmeasured confounding or randomized studies with non-compliance are instrumental variable (IV) methods. In this work, we develop an IV-based approach for identifying and estimating distributional causal effects. We introduce a distributional IV model with corresponding assumptions, which leads to a novel identification result for the interventional cumulative distribution function (CDF) under a binary treatment. We then use this identification to construct a nonparametric estimator, called DIVE, for estimating the interventional CDFs under both treatments. We empirically assess the performance of DIVE in a simulation experiment and illustrate the usefulness of distributional causal effects on two real-data applications.

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arXiv2024

How Inverse Conditional Flows Can Serve as a Substitute for Distributional Regression

Neural network representations of simple models, such as linear regression, are being studied increasingly to better understand the underlying principles of deep learning algorithms. However, neural representations of distributional regression models, such as the Cox model, have received little attention so far. We close this gap by proposing a framework for distributional regression using inverse flow transformations (DRIFT), which includes neural representations of the aforementioned models. We empirically demonstrate that the neural representations of models in DRIFT can serve as a substitute for their classical statistical counterparts in several applications involving continuous, ordered, time-series, and survival outcomes. We confirm that models in DRIFT empirically match the performance of several statistical methods in terms of estimation of partial effects, prediction, and aleatoric uncertainty quantification. DRIFT covers both interpretable statistical models and flexible neural networks opening up new avenues in both statistical modeling and deep learning.

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arXiv2024

Generalizing Orthogonalization for Models with Non-Linearities

The complexity of black-box algorithms can lead to various challenges, including the introduction of biases. These biases present immediate risks in the algorithms' application. It was, for instance, shown that neural networks can deduce racial information solely from a patient's X-ray scan, a task beyond the capability of medical experts. If this fact is not known to the medical expert, automatic decision-making based on this algorithm could lead to prescribing a treatment (purely) based on racial information. While current methodologies allow for the "orthogonalization" or "normalization" of neural networks with respect to such information, existing approaches are grounded in linear models. Our paper advances the discourse by introducing corrections for non-linearities such as ReLU activations. Our approach also encompasses scalar and tensor-valued predictions, facilitating its integration into neural network architectures. Through extensive experiments, we validate our method's effectiveness in safeguarding sensitive data in generalized linear models, normalizing convolutional neural networks for metadata, and rectifying pre-existing embeddings for undesired attributes.

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arXiv2024

Algorithm-agnostic significance testing in supervised learning with multimodal data

Valid statistical inference is crucial for decision-making but difficult to obtain in supervised learning with multimodal data, e.g., combinations of clinical features, genomic data, and medical images. Multimodal data often warrants the use of black-box algorithms, for instance, random forests or neural networks, which impede the use of traditional variable significance tests. We address this problem by proposing the use of COvariance Measure Tests (COMETs), which are calibrated and powerful tests that can be combined with any sufficiently predictive supervised learning algorithm. We apply COMETs to several high-dimensional, multimodal data sets to illustrate (i) variable significance testing for finding relevant mutations modulating drug-activity, (ii) modality selection for predicting survival in liver cancer patients with multiomics data, and (iii) modality selection with clinical features and medical imaging data. In all applications, COMETs yield results consistent with domain knowledge without requiring data-driven pre-processing which may invalidate type I error control. These novel applications with high-dimensional multimodal data corroborate prior results on the power and robustness of COMETs for significance testing. COMETs are implemented in the comets R package available on CRAN and pycomets Python library available on GitHub. Source code for reproducing all results is available at https://github.com/LucasKook/comets. All data sets used in this work are openly available.

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arXiv2023

Model-based causal feature selection for general response types

Discovering causal relationships from observational data is a fundamental yet challenging task. Invariant causal prediction (ICP, Peters et al., 2016) is a method for causal feature selection which requires data from heterogeneous settings and exploits that causal models are invariant. ICP has been extended to general additive noise models and to nonparametric settings using conditional independence tests. However, the latter often suffer from low power (or poor type I error control) and additive noise models are not suitable for applications in which the response is not measured on a continuous scale, but reflects categories or counts. Here, we develop transformation-model (TRAM) based ICP, allowing for continuous, categorical, count-type, and uninformatively censored responses (these model classes, generally, do not allow for identifiability when there is no exogenous heterogeneity). As an invariance test, we propose TRAM-GCM based on the expected conditional covariance between environments and score residuals with uniform asymptotic level guarantees. For the special case of linear shift TRAMs, we also consider TRAM-Wald, which tests invariance based on the Wald statistic. We provide an open-source R package 'tramicp' and evaluate our approach on simulated data and in a case study investigating causal features of survival in critically ill patients.

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arXiv2022

Estimating Conditional Distributions with Neural Networks using R package deeptrafo

Contemporary empirical applications frequently require flexible regression models for complex response types and large tabular or non-tabular, including image or text, data. Classical regression models either break down under the computational load of processing such data or require additional manual feature extraction to make these problems tractable. Here, we present deeptrafo, a package for fitting flexible regression models for conditional distributions using a tensorflow backend with numerous additional processors, such as neural networks, penalties, and smoothing splines. Package deeptrafo implements deep conditional transformation models (DCTMs) for binary, ordinal, count, survival, continuous, and time series responses, potentially with uninformative censoring. Unlike other available methods, DCTMs do not assume a parametric family of distributions for the response. Further, the data analyst may trade off interpretability and flexibility by supplying custom neural network architectures and smoothers for each term in an intuitive formula interface. We demonstrate how to set up, fit, and work with DCTMs for several response types. We further showcase how to construct ensembles of these models, evaluate models using inbuilt cross-validation, and use other convenience functions for DCTMs in several applications. Lastly, we discuss DCTMs in light of other approaches to regression with non-tabular data.

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arXiv2022

Deep conditional transformation models for survival analysis

An every increasing number of clinical trials features a time-to-event outcome and records non-tabular patient data, such as magnetic resonance imaging or text data in the form of electronic health records. Recently, several neural-network based solutions have been proposed, some of which are binary classifiers. Parametric, distribution-free approaches which make full use of survival time and censoring status have not received much attention. We present deep conditional transformation models (DCTMs) for survival outcomes as a unifying approach to parametric and semiparametric survival analysis. DCTMs allow the specification of non-linear and non-proportional hazards for both tabular and non-tabular data and extend to all types of censoring and truncation. On real and semi-synthetic data, we show that DCTMs compete with state-of-the-art DL approaches to survival analysis.

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arXiv2022

Distribution-Free Location-Scale Regression

We introduce a generalized additive model for location, scale, and shape (GAMLSS) next of kin aiming at distribution-free and parsimonious regression modelling for arbitrary outcomes. We replace the strict parametric distribution formulating such a model by a transformation function, which in turn is estimated from data. Doing so not only makes the model distribution-free but also allows to limit the number of linear or smooth model terms to a pair of location-scale predictor functions. We derive the likelihood for continuous, discrete, and randomly censored observations, along with corresponding score functions. A plethora of existing algorithms is leveraged for model estimation, including constrained maximum-likelihood, the original GAMLSS algorithm, and transformation trees. Parameter interpretability in the resulting models is closely connected to model selection. We propose the application of a novel best subset selection procedure to achieve especially simple ways of interpretation. All techniques are motivated and illustrated by a collection of applications from different domains, including crossing and partial proportional hazards, complex count regression, non-linear ordinal regression, and growth curves. All analyses are reproducible with the help of the "tram" add-on package to the R system for statistical computing and graphics.

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arXiv2022

Deep transformation models for functional outcome prediction after acute ischemic stroke

In many medical applications, interpretable models with high prediction performance are sought. Often, those models are required to handle semi-structured data like tabular and image data. We show how to apply deep transformation models (DTMs) for distributional regression which fulfill these requirements. DTMs allow the data analyst to specify (deep) neural networks for different input modalities making them applicable to various research questions. Like statistical models, DTMs can provide interpretable effect estimates while achieving the state-of-the-art prediction performance of deep neural networks. In addition, the construction of ensembles of DTMs that retain model structure and interpretability allows quantifying epistemic and aleatoric uncertainty. In this study, we compare several DTMs, including baseline-adjusted models, trained on a semi-structured data set of 407 stroke patients with the aim to predict ordinal functional outcome three months after stroke. We follow statistical principles of model-building to achieve an adequate trade-off between interpretability and flexibility while assessing the relative importance of the involved data modalities. We evaluate the models for an ordinal and dichotomized version of the outcome as used in clinical practice. We show that both, tabular clinical and brain imaging data, are useful for functional outcome prediction, while models based on tabular data only outperform those based on imaging data only. There is no substantial evidence for improved prediction when combining both data modalities. Overall, we highlight that DTMs provide a powerful, interpretable approach to analyzing semi-structured data and that they have the potential to support clinical decision making.

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arXiv2022

Deep interpretable ensembles

Ensembles improve prediction performance and allow uncertainty quantification by aggregating predictions from multiple models. In deep ensembling, the individual models are usually black box neural networks, or recently, partially interpretable semi-structured deep transformation models. However, interpretability of the ensemble members is generally lost upon aggregation. This is a crucial drawback of deep ensembles in high-stake decision fields, in which interpretable models are desired. We propose a novel transformation ensemble which aggregates probabilistic predictions with the guarantee to preserve interpretability and yield uniformly better predictions than the ensemble members on average. Transformation ensembles are tailored towards interpretable deep transformation models but are applicable to a wider range of probabilistic neural networks. In experiments on several publicly available data sets, we demonstrate that transformation ensembles perform on par with classical deep ensembles in terms of prediction performance, discrimination, and calibration. In addition, we demonstrate how transformation ensembles quantify both aleatoric and epistemic uncertainty, and produce minimax optimal predictions under certain conditions.

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arXiv2022

Pitfalls and potentials in simulation studies: Questionable research practices in comparative simulation studies allow for spurious claims of superiority of any method

Comparative simulation studies are workhorse tools for benchmarking statistical methods. As with other empirical studies, the success of simulation studies hinges on the quality of their design, execution and reporting. If not conducted carefully and transparently, their conclusions may be misleading. In this paper we discuss various questionable research practices which may impact the validity of simulation studies, some of which cannot be detected or prevented by the current publication process in statistics journals. To illustrate our point, we invent a novel prediction method with no expected performance gain and benchmark it in a pre-registered comparative simulation study. We show how easy it is to make the method appear superior over well-established competitor methods if questionable research practices are employed. Finally, we provide concrete suggestions for researchers, reviewers and other academic stakeholders for improving the methodological quality of comparative simulation studies, such as pre-registering simulation protocols, incentivizing neutral simulation studies and code and data sharing.

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arXiv2021

deepregression: a Flexible Neural Network Framework for Semi-Structured Deep Distributional Regression

In this paper we describe the implementation of semi-structured deep distributional regression, a flexible framework to learn conditional distributions based on the combination of additive regression models and deep networks. Our implementation encompasses (1) a modular neural network building system based on the deep learning library \pkg{TensorFlow} for the fusion of various statistical and deep learning approaches, (2) an orthogonalization cell to allow for an interpretable combination of different subnetworks, as well as (3) pre-processing steps necessary to set up such models. The software package allows to define models in a user-friendly manner via a formula interface that is inspired by classical statistical model frameworks such as \pkg{mgcv}. The packages' modular design and functionality provides a unique resource for both scalable estimation of complex statistical models and the combination of approaches from deep learning and statistics. This allows for state-of-the-art predictive performance while simultaneously retaining the indispensable interpretability of classical statistical models.

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arXiv2021

Distributional Anchor Regression

Prediction models often fail if train and test data do not stem from the same distribution. Out-of-distribution (OOD) generalization to unseen, perturbed test data is a desirable but difficult-to-achieve property for prediction models and in general requires strong assumptions on the data generating process (DGP). In a causally inspired perspective on OOD generalization, the test data arise from a specific class of interventions on exogenous random variables of the DGP, called anchors. Anchor regression models, introduced by Rothenhaeusler et al. (2021), protect against distributional shifts in the test data by employing causal regularization. However, so far anchor regression has only been used with a squared-error loss which is inapplicable to common responses such as censored continuous or ordinal data. Here, we propose a distributional version of anchor regression which generalizes the method to potentially censored responses with at least an ordered sample space. To this end, we combine a flexible class of parametric transformation models for distributional regression with an appropriate causal regularizer under a more general notion of residuals. In an exemplary application and several simulation scenarios we demonstrate the extent to which OOD generalization is possible.

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arXiv2020

Deep and interpretable regression models for ordinal outcomes

Outcomes with a natural order commonly occur in prediction tasks and often the available input data are a mixture of complex data like images and tabular predictors. Deep Learning (DL) models are state-of-the-art for image classification tasks but frequently treat ordinal outcomes as unordered and lack interpretability. In contrast, classical ordinal regression models consider the outcome's order and yield interpretable predictor effects but are limited to tabular data. We present ordinal neural network transformation models (ONTRAMs), which unite DL with classical ordinal regression approaches. ONTRAMs are a special case of transformation models and trade off flexibility and interpretability by additively decomposing the transformation function into terms for image and tabular data using jointly trained neural networks. The performance of the most flexible ONTRAM is by definition equivalent to a standard multi-class DL model trained with cross-entropy while being faster in training when facing ordinal outcomes. Lastly, we discuss how to interpret model components for both tabular and image data on two publicly available datasets.

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