Eric V. Strobl

Eric V. Strobl, Thomas A. Lasko

Complex diseases are caused by a multitude of factors that may differ between patients. As a result, hypothesis tests comparing all patients to all healthy controls can detect many significant variables with inconsequential effect sizes. A few highly predictive root causes may nevertheless generate disease within each patient. In this paper, we define patient-specific root causes as variables subject to exogenous "shocks" which go on to perturb an otherwise healthy system and induce disease. In other words, the variables are associated with the exogenous errors of a structural equation model (SEM), and these errors predict a downstream diagnostic label. We quantify predictivity using sample-specific Shapley values. This derivation allows us to develop a fast algorithm called Root Causal Inference for identifying patient-specific root causes by extracting the error terms of a linear SEM and then computing the Shapley value associated with each error. Experiments highlight considerable improvements in accuracy because the method uncovers root causes that may have large effect sizes at the individual level but clinically insignificant effect sizes at the group level. An R implementation is available at github.com/ericstrobl/RCI.

Eric V. Strobl, Thomas A. Lasko

Complex diseases are caused by a multitude of factors that may differ between patients even within the same diagnostic category. A few underlying root causes may nevertheless initiate the development of disease within each patient. We therefore focus on identifying patient-specific root causes of disease, which we equate to the sample-specific predictivity of the exogenous error terms in a structural equation model. We generalize from the linear setting to the heteroscedastic noise model where $Y = m(X) + \varepsilonσ(X)$ with non-linear functions $m(X)$ and $σ(X)$ representing the conditional mean and mean absolute deviation, respectively. This model preserves identifiability but introduces non-trivial challenges that require a customized algorithm called Generalized Root Causal Inference (GRCI) to extract the error terms correctly. GRCI recovers patient-specific root causes more accurately than existing alternatives.

Thomas A. Lasko, Eric V. Strobl, William W. Stead

The rising popularity of artificial intelligence in healthcare is highlighting the problem that a computational model achieving super-human clinical performance at its training sites may perform substantially worse at new sites. In this perspective, we present common sources for this failure to transport, which we divide into sources under the control of the experimenter and sources inherent to the clinical data-generating process. Of the inherent sources we look a little deeper into site-specific clinical practices that can affect the data distribution, and propose a potential solution intended to isolate the imprint of those practices on the data from the patterns of disease cause and effect that are the usual target of probabilistic clinical models.

Eric V. Strobl, Thomas A. Lasko

Root causal analysis seeks to identify the set of initial perturbations that induce an unwanted outcome. In prior work, we defined sample-specific root causes of disease using exogenous error terms that predict a diagnosis in a structural equation model. We rigorously quantified predictivity using Shapley values. However, the associated algorithms for inferring root causes assume no latent confounding. We relax this assumption by permitting confounding among the predictors. We then introduce a corresponding procedure called Extract Errors with Latents (EEL) for recovering the error terms up to contamination by vertices on certain paths under the linear non-Gaussian acyclic model. EEL also identifies the smallest sets of dependent errors for fast computation of the Shapley values. The algorithm bypasses the hard problem of estimating the underlying causal graph in both cases. Experiments highlight the superior accuracy and robustness of EEL relative to its predecessors.

Eric V. Strobl, Shyam Visweswaran

Personalized medicine seeks to identify the causal effect of treatment for a particular patient as opposed to a clinical population at large. Most investigators estimate such personalized treatment effects by regressing the outcome of a randomized clinical trial (RCT) on patient covariates. The realized value of the outcome may however lie far from the conditional expectation. We therefore introduce a method called Dirac Delta Regression (DDR) that estimates the entire conditional density from RCT data in order to visualize the probabilities across all possible outcome values. DDR transforms the outcome into a set of asymptotically Dirac delta distributions and then estimates the density using non-linear regression. The algorithm can identify significant differences in patient-specific outcomes even when no population level effect exists. Moreover, DDR outperforms state-of-the-art algorithms in conditional density estimation by a large margin even in the small sample regime. An R package is available at https://github.com/ericstrobl/DDR.

Eric V. Strobl

Neuroimaging studies of psychiatric disorders often correlate imaging patterns with diagnostic labels or composite symptom scores, yielding diffuse associations that obscure underlying mechanisms. We instead seek to identify root-causal maps -- localized BOLD disturbances that initiate pathological cascades -- and to link them selectively to symptom dimensions. We introduce a bilevel structural causal model that connects between-subject symptom structure to within-subject resting-state fMRI via independent latent sources with localized direct effects. Based on this model, we develop SOURCE (Symptom-Oriented Uncovering of Root-Causal Elements), a procedure that links interpretable symptom axes to a parsimonious set of localized drivers. Experiments show that SOURCE recovers localized maps consistent with root-causal BOLD drivers and increases interpretability and anatomical specificity relative to existing comparators.

Eric V. Strobl

Psychiatric questionnaires are highly context sensitive and often only weakly predict subsequent symptom severity, which makes the prognostic relationship difficult to learn. Although flexible nonlinear models can improve predictive accuracy, their limited interpretability can erode clinical trust. In fields such as imaging and omics, investigators commonly address visit- and instrument-specific artifacts by extracting stable signal through preprocessing and then fitting an interpretable linear model. We adopt the same strategy for questionnaire data by decoupling preprocessing from prediction: we restrict nonlinear capacity to a baseline preprocessing module that estimates stable item values, and then learn a linear mapping from these stabilized baseline items to future severity. We refer to this two-stage method as REFINE (Redundancy-Exploiting Follow-up-Informed Nonlinear Enhancement), which concentrates nonlinearity in preprocessing while keeping the prognostic relationship transparently linear and therefore globally interpretable through a coefficient matrix, rather than through post hoc local attributions. In experiments, REFINE outperforms other interpretable approaches while preserving clear global attribution of prognostic factors across psychiatric and non-psychiatric longitudinal prediction tasks.

Marco Barbero Mota, John M. Still, Jorge L. Gamboa et al.

Systemic lupus erythematosus (SLE) is a complex heterogeneous disease with many manifestational facets. We propose a data-driven approach to discover probabilistic independent sources from multimodal imperfect EHR data. These sources represent exogenous variables in the data generation process causal graph that estimate latent root causes of the presence of SLE in the health record. We objectively evaluated the sources against the original variables from which they were discovered by training supervised models to discriminate SLE from negative health records using a reduced set of labelled instances. We found 19 predictive sources with high clinical validity and whose EHR signatures define independent factors of SLE heterogeneity. Using the sources as input patient data representation enables models to provide with rich explanations that better capture the clinical reasons why a particular record is (not) an SLE case. Providers may be willing to trade patient-level interpretability for discrimination especially in challenging cases.

Marco Barbero-Mota, Eric V. Strobl, John M. Still et al.

We provide an accessible description of a peer-reviewed generalizable causal machine learning pipeline to (i) discover latent causal sources of large-scale electronic health records observations, and (ii) quantify the source causal effects on clinical outcomes. We illustrate how imperfect multimodal clinical data can be processed, decomposed into probabilistic independent latent sources, and used to train taskspecific causal models from which individual causal effects can be estimated. We summarize the findings of the two real-world applications of the approach to date as a demonstration of its versatility and utility for medical discovery at scale.

Eric V. Strobl, Thomas A. Lasko

Randomized clinical trials eliminate confounding but impose strict exclusion criteria that limit recruitment to a subset of the population. Observational datasets are more inclusive but suffer from confounding -- often providing overly optimistic estimates of treatment response over time due to partially optimized physician prescribing patterns. We therefore assume that the unconfounded treatment response lies somewhere in-between the observational estimate before and the observational estimate after treatment assignment. This assumption allows us to extrapolate results from exclusive trials to the broader population by analyzing observational and trial data simultaneously using an algorithm called Optimum in Convex Hulls (OCH). OCH represents the treatment effect either in terms of convex hulls of conditional expectations or convex hulls (also known as mixtures) of conditional densities. The algorithm first learns the component expectations or densities using the observational data and then learns the linear mixing coefficients using trial data in order to approximate the true treatment effect; theory importantly explains why this linear combination should hold. OCH estimates the treatment effect in terms both expectations and densities with state of the art accuracy.

Eric V. Strobl, Thomas A. Lasko

We consider estimating the conditional average treatment effect for everyone by eliminating confounding and selection bias. Unfortunately, randomized clinical trials (RCTs) eliminate confounding but impose strict exclusion criteria that prevent sampling of the entire clinical population. Observational datasets are more inclusive but suffer from confounding. We therefore analyze RCT and observational data simultaneously in order to extract the strengths of each. Our solution builds upon Difference in Differences (DD), an algorithm that eliminates confounding from observational data by comparing outcomes before and after treatment administration. DD requires a parallel slopes assumption that may not apply in practice when confounding shifts across time. We instead propose Synthesized Difference in Differences (SDD) that infers the correct (possibly non-parallel) slopes by linearly adjusting a conditional version of DD using additional RCT data. The algorithm achieves state of the art performance across multiple synthetic and real datasets even when the RCT excludes the majority of patients.

Eric V. Strobl

The PC algorithm infers causal relations using conditional independence tests that require a pre-specified Type I $α$ level. PC is however unsupervised, so we cannot tune $α$ using traditional cross-validation. We therefore propose AutoPC, a fast procedure that optimizes $α$ directly for a user chosen metric. We in particular force PC to double check its output by executing a second run on the recovered graph. We choose the final output as the one which maximizes stability between the two runs. AutoPC consistently outperforms the state of the art across multiple metrics.

Eric V. Strobl

Causal processes in biomedicine may contain cycles, evolve over time or differ between populations. However, many graphical models cannot accommodate these conditions. We propose to model causation using a mixture of directed cyclic graphs (DAGs), where the joint distribution in a population follows a DAG at any single point in time but potentially different DAGs across time. We also introduce an algorithm called Causal Inference over Mixtures that uses longitudinal data to infer a graph summarizing the causal relations generated from a mixture of DAGs. Experiments demonstrate improved performance compared to prior approaches.

Eric V. Strobl

Causal processes in nature may contain cycles, and real datasets may violate causal sufficiency as well as contain selection bias. No constraint-based causal discovery algorithm can currently handle cycles, latent variables and selection bias (CLS) simultaneously. I therefore introduce an algorithm called Cyclic Causal Inference (CCI) that makes sound inferences with a conditional independence oracle under CLS, provided that we can represent the cyclic causal process as a non-recursive linear structural equation model with independent errors. Empirical results show that CCI outperforms CCD in the cyclic case as well as rivals FCI and RFCI in the acyclic case.

Eric V. Strobl, Shyam Visweswaran, Peter L. Spirtes

Many real datasets contain values missing not at random (MNAR). In this scenario, investigators often perform list-wise deletion, or delete samples with any missing values, before applying causal discovery algorithms. List-wise deletion is a sound and general strategy when paired with algorithms such as FCI and RFCI, but the deletion procedure also eliminates otherwise good samples that contain only a few missing values. In this report, we show that we can more efficiently utilize the observed values with test-wise deletion while still maintaining algorithmic soundness. Here, test-wise deletion refers to the process of list-wise deleting samples only among the variables required for each conditional independence (CI) test used in constraint-based searches. Test-wise deletion therefore often saves more samples than list-wise deletion for each CI test, especially when we have a sparse underlying graph. Our theoretical results show that test-wise deletion is sound under the justifiable assumption that none of the missingness mechanisms causally affect each other in the underlying causal graph. We also find that FCI and RFCI with test-wise deletion outperform their list-wise deletion and imputation counterparts on average when MNAR holds in both synthetic and real data.

Eric V. Strobl, Kun Zhang, Shyam Visweswaran

Constraint-based causal discovery (CCD) algorithms require fast and accurate conditional independence (CI) testing. The Kernel Conditional Independence Test (KCIT) is currently one of the most popular CI tests in the non-parametric setting, but many investigators cannot use KCIT with large datasets because the test scales cubicly with sample size. We therefore devise two relaxations called the Randomized Conditional Independence Test (RCIT) and the Randomized conditional Correlation Test (RCoT) which both approximate KCIT by utilizing random Fourier features. In practice, both of the proposed tests scale linearly with sample size and return accurate p-values much faster than KCIT in the large sample size context. CCD algorithms run with RCIT or RCoT also return graphs at least as accurate as the same algorithms run with KCIT but with large reductions in run time.

Eric V. Strobl, Peter L. Spirtes, Shyam Visweswaran

The PC algorithm allows investigators to estimate a complete partially directed acyclic graph (CPDAG) from a finite dataset, but few groups have investigated strategies for estimating and controlling the false discovery rate (FDR) of the edges in the CPDAG. In this paper, we introduce PC with p-values (PC-p), a fast algorithm which robustly computes edge-specific p-values and then estimates and controls the FDR across the edges. PC-p specifically uses the p-values returned by many conditional independence tests to upper bound the p-values of more complex edge-specific hypothesis tests. The algorithm then estimates and controls the FDR using the bounded p-values and the Benjamini-Yekutieli FDR procedure. Modifications to the original PC algorithm also help PC-p accurately compute the upper bounds despite non-zero Type II error rates. Experiments show that PC-p yields more accurate FDR estimation and control across the edges in a variety of CPDAGs compared to alternative methods.

Eric V. Strobl, Shyam Visweswaran

Developing feature selection algorithms that move beyond a pure correlational to a more causal analysis of observational data is an important problem in the sciences. Several algorithms attempt to do so by discovering the Markov blanket of a target, but they all contain a forward selection step which variables must pass in order to be included in the conditioning set. As a result, these algorithms may not consider all possible conditional multivariate combinations. We improve on this limitation by proposing a backward elimination method that uses a kernel-based conditional dependence measure to identify the Markov blanket in a fully multivariate fashion. The algorithm is easy to implement and compares favorably to other methods on synthetic and real datasets.

Eric Strobl, Shyam Visweswaran

Deep learning methods have predominantly been applied to large artificial neural networks. Despite their state-of-the-art performance, these large networks typically do not generalize well to datasets with limited sample sizes. In this paper, we take a different approach by learning multiple layers of kernels. We combine kernels at each layer and then optimize over an estimate of the support vector machine leave-one-out error rather than the dual objective function. Our experiments on a variety of datasets show that each layer successively increases performance with only a few base kernels.