Mark A. van de Wiel

Jeroen M. Goedhart, Thomas Klausch, Jurriaan Janssen et al.

Medical prediction applications often need to deal with small sample sizes compared to the number of covariates. Such data pose problems for prediction and variable selection, especially when the covariate-response relationship is complicated. To address these challenges, we propose to incorporate co-data, i.e. external information on the covariates, into Bayesian additive regression trees (BART), a sum-of-trees prediction model that utilizes priors on the tree parameters to prevent overfitting. To incorporate co-data, an empirical Bayes (EB) framework is developed that estimates, assisted by a co-data model, prior covariate weights in the BART model. The proposed method can handle multiple types of co-data simultaneously. Furthermore, the proposed EB framework enables the estimation of the other hyperparameters of BART as well, rendering an appealing alternative to cross-validation. We show that the method finds relevant covariates and that it improves prediction compared to default BART in simulations. If the covariate-response relationship is nonlinear, the method benefits from the flexibility of BART to outperform regression-based co-data learners. Finally, the use of co-data enhances prediction in an application to diffuse large B-cell lymphoma prognosis based on clinical covariates, gene mutations, DNA translocations, and DNA copy number data. Keywords: Bayesian additive regression trees; Empirical Bayes; Co-data; High-dimensional data; Omics; Prediction

Giorgio Spadaccini, Marjolein Fokkema, Mark A. van de Wiel

In epidemiological settings, Machine Learning (ML) is gaining popularity for hypothesis-free discovery of risk (or protective) factors. Although ML is strong at discovering non-linearities and interactions, this power is currently compromised by a lack of reliable inference. Although local measures of feature effect can be combined with tree ensembles, uncertainty quantifications for these measures remain only partially available and oftentimes unsatisfactory. We propose RuleSHAP, a framework for using rule-based, hypothesis-free discovery that combines sparse Bayesian regression, tree ensembles and Shapley values in a one-step procedure that both detects and tests complex patterns at the individual level. To ease computation, we derive a formula that computes marginal Shapley values more efficiently for our setting. We demonstrate the validity of our framework on simulated data. To illustrate, we apply our machinery to data from an epidemiological cohort to detect and infer several effects for high cholesterol and blood pressure, such as nonlinear interaction effects between features like age, sex, ethnicity, BMI and glucose level.

Mark A. van de Wiel, Jeroen Goedhart, Martin Jullum et al.

In clinical prediction settings the importance of a high-dimensional feature like genomics is often assessed by evaluating the change in predictive performance when adding it to a set of traditional clinical variables. This approach is questionable, because it does not account for collinearity nor known directionality of dependencies between variables. We suggest to use asymmetric Shapley values as a more suitable alternative to quantify feature importance in the context of a mixed-dimensional prediction model. We focus on a setting that is particularly relevant in clinical prediction: disease state as a mediating variable for genomic effects, with additional confounders for which the direction of effects may be unknown. We derive efficient algorithms to compute local and global asymmetric Shapley values for this setting. The former are shown to be very useful for inference, whereas the latter provide interpretation by decomposing any predictive performance metric into contributions of the features. Throughout, we illustrate our framework by a leading example: the prediction of progression-free survival for colorectal cancer patients.

Mingyang Cai, Thomas Klausch, Mark A. van de Wiel

Machine learning models used in medical applications often face challenges due to the covariate shift, which occurs when there are discrepancies between the distributions of training and target data. This can lead to decreased predictive accuracy, especially with unknown outcomes in the target data. This paper introduces a semi-supervised classification and regression tree (CART) that uses importance weighting to address these distribution discrepancies. Our method improves the predictive performance of the CART model by assigning greater weights to training samples that more accurately represent the target distribution, especially in cases of covariate shift without target outcomes. In addition to CART, we extend this weighted approach to generalized linear model trees and tree ensembles, creating a versatile framework for managing the covariate shift in complex datasets. Through simulation studies and applications to real-world medical data, we demonstrate significant improvements in predictive accuracy. These findings suggest that our weighted approach can enhance reliability in medical applications and other fields where the covariate shift poses challenges to model performance across various data distributions.

Mirrelijn M. van Nee, Tim van de Brug, Mark A. van de Wiel

Nowadays, clinical research routinely uses omics data, such as gene expression, for predicting clinical outcomes or selecting markers. Additionally, so-called co-data are often available, providing complementary information on the covariates, like p-values from previously published studies or groups of genes corresponding to pathways. Elastic net penalisation is widely used for prediction and covariate selection. Group-adaptive elastic net penalisation learns from co-data to improve the prediction and covariate selection, by penalising important groups of covariates less than other groups. Existing methods are, however, computationally expensive. Here we present a fast method for marginal likelihood estimation of group-adaptive elastic net penalties for generalised linear models. We first derive a low-dimensional representation of the Taylor approximation of the marginal likelihood and its first derivative for group-adaptive ridge penalties, to efficiently estimate these penalties. Then we show by using asymptotic normality of the linear predictors that the marginal likelihood for elastic net models may be approximated well by the marginal likelihood for ridge models. The ridge group penalties are then transformed to elastic net group penalties by using the variance function. The method allows for overlapping groups and unpenalised variables. We demonstrate the method in a model-based simulation study and an application to cancer genomics. The method substantially decreases computation time and outperforms or matches other methods by learning from co-data.

Mark A. van de Wiel, Mirrelijn M. van Nee, Armin Rauschenberger

High-dimensional prediction with multiple data types needs to account for potentially strong differences in predictive signal. Ridge regression is a simple model for high-dimensional data that has challenged the predictive performance of many more complex models and learners, and that allows inclusion of data type specific penalties. The largest challenge for multi-penalty ridge is to optimize these penalties efficiently in a cross-validation (CV) setting, in particular for GLM and Cox ridge regression, which require an additional estimation loop by iterative weighted least squares (IWLS). Our main contribution is a computationally very efficient formula for the multi-penalty, sample-weighted hat-matrix, as used in the IWLS algorithm. As a result, nearly all computations are in low-dimensional space, rendering a speed-up of several orders of magnitude. We developed a flexible framework that facilitates multiple types of response, unpenalized covariates, several performance criteria and repeated CV. Extensions to paired and preferential data types are included and illustrated on several cancer genomics survival prediction problems. Moreover, we present similar computational shortcuts for maximum marginal likelihood and Bayesian probit regression. The corresponding R-package, multiridge, serves as a versatile standalone tool, but also as a fast benchmark for other more complex models and multi-view learners.

Mirrelijn M. van Nee, Lodewyk F. A. Wessels, Mark A. van de Wiel

Clinical research often focuses on complex traits in which many variables play a role in mechanisms driving, or curing, diseases. Clinical prediction is hard when data is high-dimensional, but additional information, like domain knowledge and previously published studies, may be helpful to improve predictions. Such complementary data, or co-data, provide information on the covariates, such as genomic location or p-values from external studies. Our method enables exploiting multiple and various co-data sources to improve predictions. We use discrete or continuous co-data to define possibly overlapping or hierarchically structured groups of covariates. These are then used to estimate adaptive multi-group ridge penalties for generalised linear and Cox models. We combine empirical Bayes estimation of group penalty hyperparameters with an extra level of shrinkage. This renders a uniquely flexible framework as any type of shrinkage can be used on the group level. The hyperparameter shrinkage learns how relevant a specific co-data source is, counters overfitting of hyperparameters for many groups, and accounts for structured co-data. We describe various types of co-data and propose suitable forms of hypershrinkage. The method is very versatile, as it allows for integration and weighting of multiple co-data sets, inclusion of unpenalised covariates and posterior variable selection. We demonstrate it on two cancer genomics applications and show that it may improve the performance of other dense and parsimonious prognostic models substantially, and stabilises variable selection.

Carel F. W. Peeters, Caroline Übelhör, Steven W. Mes et al.

Motivation: Radiomics refers to the high-throughput mining of quantitative features from radiographic images. It is a promising field in that it may provide a non-invasive solution for screening and classification. Standard machine learning classification and feature selection techniques, however, tend to display inferior performance in terms of (the stability of) predictive performance. This is due to the heavy multicollinearity present in radiomic data. We set out to provide an easy-to-use approach that deals with this problem. Results: We developed a four-step approach that projects the original high-dimensional feature space onto a lower-dimensional latent-feature space, while retaining most of the covariation in the data. It consists of (i) penalized maximum likelihood estimation of a redundancy filtered correlation matrix. The resulting matrix (ii) is the input for a maximum likelihood factor analysis procedure. This two-stage maximum-likelihood approach can be used to (iii) produce a compact set of stable features that (iv) can be directly used in any (regression-based) classifier or predictor. It outperforms other classification (and feature selection) techniques in both external and internal validation settings regarding survival in squamous cell cancers.

Thomas Klausch, Peter van de Ven, Tim van de Brug et al.

Optimal treatment regimes (OTR) are individualised treatment assignment strategies that identify a medical treatment as optimal given all background information available on the individual. We discuss Bayes optimal treatment regimes estimated using a loss function defined on the bivariate distribution of dichotomous potential outcomes. The proposed approach allows considering more general objectives for the OTR than maximization of an expected outcome (e.g., survival probability) by taking into account, for example, unnecessary treatment burden. As a motivating example we consider the case of oropharynx cancer treatment where unnecessary burden due to chemotherapy is to be avoided while maximizing survival chances. Assuming ignorable treatment assignment we describe Bayesian inference about the OTR including a sensitivity analysis on the unobserved partial association of the potential outcomes. We evaluate the methodology by simulations that apply Bayesian parametric and more flexible non-parametric outcome models. The proposed OTR for oropharynx cancer reduces the frequency of the more burdensome chemotherapy assignment by approximately 75% without reducing the average survival probability. This regime thus offers a strong increase in expected quality of life of patients.

Carel F. W. Peeters, Mark A. van de Wiel, Wessel N. van Wieringen

Many modern statistical applications ask for the estimation of a covariance (or precision) matrix in settings where the number of variables is larger than the number of observations. There exists a broad class of ridge-type estimators that employs regularization to cope with the subsequent singularity of the sample covariance matrix. These estimators depend on a penalty parameter and choosing its value can be hard, in terms of being computationally unfeasible or tenable only for a restricted set of ridge-type estimators. Here we introduce a simple graphical tool, the spectral condition number plot, for informed heuristic penalty parameter selection. The proposed tool is computationally friendly and can be employed for the full class of ridge-type covariance (precision) estimators.