When Pattern-by-Pattern Works: Theoretical and Empirical Insights for Logistic Models with Missing Values

Predicting a response with partially missing inputs remains a challenging task even in parametric models, since parameter estimation in itself is not sufficient to predict on partially observed inputs. Several works study prediction in linear models. In this paper, we focus on logistic models, which present their own difficulties. From a theoretical perspective, we prove that a Pattern-by-Pattern strategy (PbP), which learns one logistic model per missingness pattern, accurately approximates Bayes probabilities in various missing data scenarios (MCAR, MAR and MNAR). Empirically, we thoroughly compare various methods (constant and iterative imputations, complete case analysis, PbP, and an EM algorithm) across classification, probability estimation, calibration, and parameter inference. Our analysis provides a comprehensive view on the logistic regression with missing values. It reveals that mean imputation can be used as baseline for low sample sizes, and improved performance is obtained via nonlinear multiple iterative imputation techniques with the labels (MICE.RF.Y). For large sample sizes, PbP is the best method for Gaussian mixtures, and we recommend MICE.RF.Y in presence of nonlinear features.

Model Agnostic Differentially Private Causal Inference

Estimating causal effects from observational data is essential in fields such as medicine, economics and social sciences, where privacy concerns are paramount. We propose a general, model-agnostic framework for differentially private estimation of average treatment effects (ATE) that avoids strong structural assumptions on the data-generating process or the models used to estimate propensity scores and conditional outcomes. In contrast to prior work, which enforces differential privacy by directly privatizing these nuisance components and results in a privacy cost that scales with model complexity, our approach decouples nuisance estimation from privacy protection. This separation allows the use of flexible, state-of-the-art black-box models, while differential privacy is achieved by perturbing only predictions and aggregation steps within a fold-splitting scheme with ensemble techniques. We instantiate the framework for three classical estimators -- the G-formula, inverse propensity weighting (IPW), and augmented IPW (AIPW) -- and provide formal utility and privacy guarantees. Empirical results show that our methods maintain competitive performance under realistic privacy budgets. We further extend our framework to support meta-analysis of multiple private ATE estimates. Our results bridge a critical gap between causal inference and privacy-preserving data analysis.

Optimal Transport with Heterogeneously Missing Data

We consider the problem of solving the optimal transport problem between two empirical distributions with missing values. Our main assumption is that the data is missing completely at random (MCAR), but we allow for heterogeneous missingness probabilities across features and across the two distributions. As a first contribution, we show that the Wasserstein distance between empirical Gaussian distributions and linear Monge maps between arbitrary distributions can be debiased without significantly affecting the sample complexity. Secondly, we show that entropic regularized optimal transport can be estimated efficiently and consistently using iterative singular value thresholding (ISVT). We propose a validation set-free hyperparameter selection strategy for ISVT that leverages our estimator of the Bures-Wasserstein distance, which could be of independent interest in general matrix completion problems. Finally, we validate our findings on a wide range of numerical applications.

Causal mediation analysis with one or multiple mediators: a comparative study

Mediation analysis breaks down the causal effect of a treatment on an outcome into an indirect effect, acting through a third group of variables called mediators, and a direct effect, operating through other mechanisms. Mediation analysis is hard because confounders between treatment, mediators, and outcome blur effect estimates in observational studies. Many estimators have been proposed to adjust on those confounders and provide accurate causal estimates. We consider parametric and non-parametric implementations of classical estimators and provide a thorough evaluation for the estimation of the direct and indirect effects in the context of causal mediation analysis for binary, continuous, and multi-dimensional mediators. We assess several approaches in a comprehensive benchmark on simulated data. Our results show that advanced statistical approaches such as the multiply robust and the double machine learning estimators achieve good performances in most of the simulated settings and on real data. As an example of application, we propose a thorough analysis of factors known to influence cognitive functions to assess if the mechanism involves modifications in brain morphology using the UK Biobank brain imaging cohort. This analysis shows that for several physiological factors, such as hypertension and obesity, a substantial part of the effect is mediated by changes in the brain structure. This work provides guidance to the practitioner from the formulation of a valid causal mediation problem, including the verification of the identification assumptions, to the choice of an adequate estimator.

Expert Study on Interpretable Machine Learning Models with Missing Data

Inherently interpretable machine learning (IML) models provide valuable insights for clinical decision-making but face challenges when features have missing values. Classical solutions like imputation or excluding incomplete records are often unsuitable in applications where values are missing at test time. In this work, we conducted a survey with 71 clinicians from 29 trauma centers across France, including 20 complete responses to study the interaction between medical professionals and IML applied to data with missing values. This provided valuable insights into how missing data is interpreted in clinical machine learning. We used the prediction of hemorrhagic shock as a concrete example to gauge the willingness and readiness of the participants to adopt IML models from three classes of methods. Our findings show that, while clinicians value interpretability and are familiar with common IML methods, classical imputation techniques often misalign with their intuition, and that models that natively handle missing values are preferred. These results emphasize the need to integrate clinical intuition into future IML models for better human-computer interaction.

Federated Causal Inference: Multi-Centric ATE Estimation beyond Meta-Analysis

We study Federated Causal Inference, an approach to estimate treatment effects from decentralized data across centers. We compare three classes of Average Treatment Effect (ATE) estimators derived from the Plug-in G-Formula, ranging from simple meta-analysis to one-shot and multi-shot federated learning, the latter leveraging the full data to learn the outcome model (albeit requiring more communication). Focusing on Randomized Controlled Trials (RCTs), we derive the asymptotic variance of these estimators for linear models. Our results provide practical guidance on selecting the appropriate estimator for various scenarios, including heterogeneity in sample sizes, covariate distributions, treatment assignment schemes, and center effects. We validate these findings with a simulation study.

MMD-based Variable Importance for Distributional Random Forest

Distributional Random Forest (DRF) is a flexible forest-based method to estimate the full conditional distribution of a multivariate output of interest given input variables. In this article, we introduce a variable importance algorithm for DRFs, based on the well-established drop and relearn principle and MMD distance. While traditional importance measures only detect variables with an influence on the output mean, our algorithm detects variables impacting the output distribution more generally. We show that the introduced importance measure is consistent, exhibits high empirical performance on both real and simulated data, and outperforms competitors. In particular, our algorithm is highly efficient to select variables through recursive feature elimination, and can therefore provide small sets of variables to build accurate estimates of conditional output distributions.

Positivity-free Policy Learning with Observational Data

Policy learning utilizing observational data is pivotal across various domains, with the objective of learning the optimal treatment assignment policy while adhering to specific constraints such as fairness, budget, and simplicity. This study introduces a novel positivity-free (stochastic) policy learning framework designed to address the challenges posed by the impracticality of the positivity assumption in real-world scenarios. This framework leverages incremental propensity score policies to adjust propensity score values instead of assigning fixed values to treatments. We characterize these incremental propensity score policies and establish identification conditions, employing semiparametric efficiency theory to propose efficient estimators capable of achieving rapid convergence rates, even when integrated with advanced machine learning algorithms. This paper provides a thorough exploration of the theoretical guarantees associated with policy learning and validates the proposed framework's finite-sample performance through comprehensive numerical experiments, ensuring the identification of causal effects from observational data is both robust and reliable.

Variable importance for causal forests: breaking down the heterogeneity of treatment effects

Causal random forests provide efficient estimates of heterogeneous treatment effects. However, forest algorithms are also well-known for their black-box nature, and therefore, do not characterize how input variables are involved in treatment effect heterogeneity, which is a strong practical limitation. In this article, we develop a new importance variable algorithm for causal forests, to quantify the impact of each input on the heterogeneity of treatment effects. The proposed approach is inspired from the drop and relearn principle, widely used for regression problems. Importantly, we show how to handle the forest retrain without a confounding variable. If the confounder is not involved in the treatment effect heterogeneity, the local centering step enforces consistency of the importance measure. Otherwise, when a confounder also impacts heterogeneity, we introduce a corrective term in the retrained causal forest to recover consistency. Additionally, experiments on simulated, semi-synthetic, and real data show the good performance of our importance measure, which outperforms competitors on several test cases. Experiments also show that our approach can be efficiently extended to groups of variables, providing key insights in practice.

Conformal Prediction with Missing Values

Conformal prediction is a theoretically grounded framework for constructing predictive intervals. We study conformal prediction with missing values in the covariates -- a setting that brings new challenges to uncertainty quantification. We first show that the marginal coverage guarantee of conformal prediction holds on imputed data for any missingness distribution and almost all imputation functions. However, we emphasize that the average coverage varies depending on the pattern of missing values: conformal methods tend to construct prediction intervals that under-cover the response conditionally to some missing patterns. This motivates our novel generalized conformalized quantile regression framework, missing data augmentation, which yields prediction intervals that are valid conditionally to the patterns of missing values, despite their exponential number. We then show that a universally consistent quantile regression algorithm trained on the imputed data is Bayes optimal for the pinball risk, thus achieving valid coverage conditionally to any given data point. Moreover, we examine the case of a linear model, which demonstrates the importance of our proposal in overcoming the heteroskedasticity induced by missing values. Using synthetic and data from critical care, we corroborate our theory and report improved performance of our methods.