De-paradox Tree: Breaking Down Simpson's Paradox via A Kernel-Based Partition Algorithm

TL;DR

De-paradox Tree is introduced, an interpretable algorithm designed to uncover hidden subgroup patterns behind paradoxical associations under assumed causal structures involving confounders and effect heterogeneity, enabling more reliable and informed decision-making in complex observational data environments.

Abstract

Real-world observational datasets and machine learning have revolutionized data-driven decision-making, yet many models rely on empirical associations that may be misleading due to confounding and subgroup heterogeneity. Simpson's paradox exemplifies this challenge, where aggregated and subgroup-level associations contradict each other, leading to misleading conclusions. Existing methods provide limited support for detecting and interpreting such paradoxical associations, especially for practitioners without deep causal expertise. We introduce De-paradox Tree, an interpretable algorithm designed to uncover hidden subgroup patterns behind paradoxical associations under assumed causal structures involving confounders and effect heterogeneity. It employs novel split criteria and balancing-based procedures to adjust for confounders and homogenize heterogeneous effects through recursive partitioning. Compared to state-of-the-art methods, De-paradox Tree builds simpler, more interpretable trees, selects relevant covariates, and identifies nested opposite effects while ensuring robust estimation of causal effects when causally admissible variables are provided. Our approach addresses the limitations of traditional causal inference and machine learning methods by introducing an interpretable framework that supports non-expert practitioners while explicitly acknowledging causal assumptions and scope limitations, enabling more reliable and informed decision-making in complex observational data environments.

Figures (13)

• Figure 1: This work provides a De-paradox Tree method to address empirical spurious and paradoxical associations that might lead to misleading interpretations of causal effects. (A) Simpson's paradox, are prevalent in observational studies. E.g., in the study that investigates the effect of a job training program (ref. Section \ref{['sec:introduction']}), the cause (i.e., job training program) and outcome (i.e., earnings) can be distorted by a third variable (i.e., ethnicity), leading to a misleading interpretation of the causal effect. (B) Suppose that a tree produced by De-paradox Tree (hypothetical for illustration) by automatically decomposing cause-outcome associations in data, recursively splitting the data through (1) minimizing the imbalance of pretreatment variables over the treated and control arms, and (2) homogenizing inconsistent effects within subgroups. (C) The proposed De-paradox Tree helps to explain why paradoxical associations may appear in data. This can be due to (1) imbalance, where ethnicity plays as a confounding variable, causing different ethnic groups have different probabilities of participating in job training program, and (2) heterogeneity, where age plays as an effect modifier and the job training has different effects within the same ethnic group when decomposing by age.
• Figure 2: Two causal structures for Simpson's paradox considered in this study. (A) A confounder $Z$ influences both cause $X$ and outcome $Y$, causing the marginal and conditional associations diverge. (B) A third variable $Z$ does not confound $X$ and $Y$, but the conditional associations differ across levels of $Z$ due to effect heterogeneity, and from each other when conditioning on $Z$. The marginal association is also different from conditional associations. Other causal structures, such as collider-based structures, are intentionally not shown, as conditioning on colliders may induce selection bias and invalidate causal interpretations in balancing-based analyses---an important consideration when the method is used by non-expert practitioners.
• Figure 3: Kernel mean embeddings and kernel distance. Probability distributions are uniquely mapped into data points---i.e., kernel mean embeddings---in a reproducing kernel Hilbert space (RKHS) through an expectation operation, so that a kernel distance of two distributions could be computed via the distance of these two data embeddings.
• Figure 4: An illustration of policy learning to maximize expected outcomes. Suppose $\pi, \pi^{\prime}$ are two different policy trees, if the average outcome $W(\pi^{\prime}) > W(\pi)$, the second tree will be chosen and a root node will be further decomposed into two child subgroups with opposite treatments.
• Figure 5: The two-step data simulation process, where $h(\mathbf{W})$, defined in Equation (\ref{['eq:propensityscore']}) based on $f(\mathbf{W})$, determines the injected imbalance, as well as $g(\mathbf{V})$, seen in Equation (\ref{['eq:outcomeregression']}), determines the injected heterogeneity. (1) We design a function $f(\mathbf{W})$ corresponding to $f$-Design 0, $f$-Design 1, and $f$-Design 2 to create three variants of imbalance; (2) We design a function $g(\mathbf{W})$ corresponding to $g$-Design 0, $g$-Design 1, and $g$-Design 2 to create three variants of heterogeneity.