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High-dimensional networks play a key role in understanding complex relationships. These relationships are often dynamic in nature and can change with multiple external factors (e.g., time and groups). Methods for estimating graphical models are often restricted to static graphs or graphs that can change with a single covariate (e.g., time). We propose a novel class of graphical models, the covariate-varying network (CVN), that can change with multiple external covariates. In order to introduce sparsity, we apply a $L_1$-penalty to the precision matrices of $m \geq 2$ graphs we want to estimate. These graphs often show a level of similarity. In order to model this 'smoothness', we introduce the concept of a 'meta-graph' where each node in the meta-graph corresponds to an individual graph in the CVN. The (weighted) adjacency matrix of the meta-graph represents the strength with which similarity is enforced between the $m$ graphs. The resulting optimization problem is solved by employing an alternating direction method of multipliers. We test our method using a simulation study and we show its applicability by applying it to a real-world data set, the gene expression networks from the study 'German Cancer in childhood and molecular-epidemiology' (KiKme). An implementation of the algorithm in R is publicly available under https://github.com/bips-hb/cvn

We present an Alternating Direction Method of Multipliers (ADMM) algorithm designed to solve the Weighted Generalized Fused LASSO Signal Approximator (wFLSA). First, we show that wFLSAs can always be reformulated as a Generalized LASSO problem. With the availability of algorithms tailored to the Generalized LASSO, the issue appears to be, in principle, resolved. However, the computational complexity of these algorithms is high, with a time complexity of $O(p^4)$ for a single iteration, where $p$ represents the number of coefficients. To overcome this limitation, we propose an ADMM algorithm specifically tailored for wFLSA-equivalent problems, significantly reducing the complexity to $O(p^2)$. Our algorithm is publicly accessible through the R package wflsa.

Methods of causal discovery aim to identify causal structures in a data driven way. Existing algorithms are known to be unstable and sensitive to statistical errors, and are therefore rarely used with biomedical or epidemiological data. We present an algorithm that efficiently exploits temporal structure, so-called tiered background knowledge, for estimating causal structures. Tiered background knowledge is readily available from, e.g., cohort or registry data. When used efficiently it renders the algorithm more robust to statistical errors and ultimately increases accuracy in finite samples. We describe the algorithm and illustrate how it proceeds. Moreover, we offer formal proofs as well as examples of desirable properties of the algorithm, which we demonstrate empirically in an extensive simulation study. To illustrate its usefulness in practice, we apply the algorithm to data from a children's cohort study investigating the interplay of diet, physical activity and other lifestyle factors for health outcomes.

Despite extensive safety assessments of drugs prior to their introduction to the market, certain adverse drug reactions (ADRs) remain undetected. The primary objective of pharmacovigilance is to identify these ADRs (i.e., signals). In addition to traditional spontaneous reporting systems (SRSs), electronic health (EHC) data is being used for signal detection as well. Unlike SRS, EHC data is longitudinal and thus requires assumptions about the patient's drug exposure history and its impact on ADR occurrences over time, which many current methods do implicitly. We propose an exposure model framework that explicitly models the longitudinal relationship between the drug and the ADR. By considering multiple such models simultaneously, we can detect signals that might be missed by other approaches. The parameters of these models are estimated using maximum likelihood, and the Bayesian Information Criterion (BIC) is employed to select the most suitable model. Since BIC is connected to the posterior distribution, it servers the dual purpose of identifying the best-fitting model and determining the presence of a signal by evaluating the posterior probability of the null model. We evaluate the effectiveness of this framework through a simulation study, for which we develop an EHC data simulator. Additionally, we conduct a case study applying our approach to four drug-ADR pairs using an EHC dataset comprising over 1.2 million insured individuals. Both the method and the EHC data simulator code are publicly accessible as part of the R package https://github.com/bips-hb/expard., Comment: 40 pages, 7 figures

Variable selection in linear regression settings is a much discussed problem. Best subset selection (BSS) is often considered the intuitive 'gold standard', with its use being restricted only by its NP-hard nature. Alternatives such as the least absolute shrinkage and selection operator (Lasso) or the elastic net (Enet) have become methods of choice in high-dimensional settings. A recent proposal represents BSS as a mixed integer optimization problem so that much larger problems have become feasible in reasonable computation time. We present an extensive neutral comparison assessing the variable selection performance, in linear regressions, of BSS compared to forward stepwise selection (FSS), Lasso and Enet. The simulation study considers a wide range of settings that are challenging with regard to dimensionality (with respect to the number of observations and variables), signal-to-noise ratios and correlations between predictors. As main measure of performance, we used the best possible F1-score for each method to ensure a fair comparison irrespective of any criterion for choosing the tuning parameters, and results were confirmed by alternative performance measures. Somewhat surprisingly, it was only in settings where the signal-to-noise ratio was high and the variables were (nearly) uncorrelated that BSS reliably outperformed the other methods, even in low-dimensional settings. Further, the FSS's performance was nearly identical to BSS. Our results shed new light on the usual presumption of BSS being, in principle, the best choice for variable selection. Especially for correlated variables, alternatives like Enet are faster and appear to perform better in practical settings., Comment: 28 pages, 10 figures. Results can be explored interactively at https://bestsubset.bips.eu

In this guide, we present how to perform constraint-based causal discovery using three popular software packages: pcalg (with add-ons tpc and micd), bnlearn, and TETRAD. We focus on how these packages can be used with observational data and in the presence of mixed data (i.e., data where some variables are continuous, while others are categorical), a known time ordering between variables, and missing data. Throughout, we point out the relative strengths and limitations of each package, as well as give practical recommendations. We hope this guide helps anyone who is interested in performing constraint-based causal discovery on their data., Comment: 27 pages, 20 figures

Causal discovery algorithms estimate causal graphs from observational data. This can provide a valuable complement to analyses focussing on the causal relation between individual treatment-outcome pairs. Constraint-based causal discovery algorithms rely on conditional independence testing when building the graph. Until recently, these algorithms have been unable to handle missing values. In this paper, we investigate two alternative solutions: Test-wise deletion and multiple imputation. We establish necessary and sufficient conditions for the recoverability of causal structures under test-wise deletion, and argue that multiple imputation is more challenging in the context of causal discovery than for estimation. We conduct an extensive comparison by simulating from benchmark causal graphs: As one might expect, we find that test-wise deletion and multiple imputation both clearly outperform list-wise deletion and single imputation. Crucially, our results further suggest that multiple imputation is especially useful in settings with a small number of either Gaussian or discrete variables, but when the dataset contains a mix of both neither method is uniformly best. The methods we compare include random forest imputation and a hybrid procedure combining test-wise deletion and multiple imputation. An application to data from the IDEFICS cohort study on diet- and lifestyle-related diseases in European children serves as an illustrating example., Comment: 38 pages, 11 figures