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

Xian Teng; Yu-Ru Lin

cs.LGMar 2, 2026

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

Authors: Xian Teng, Yu-Ru Lin

Organizations: University of Pittsburgh, Pittsburgh, Pennsylvania, USA

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 & tables

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 1 ), 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.

Symbols	Explanations
$\mathcal{D}\triangleq\{X_{i},Y_{i},\mathbf{Z}_{i}\}$	The observational data consisting of i.i.d. triplets $\{X_{i},Y_{i},\mathbf{Z}_{i}\}$
$p,q$	The feature distributions for the treated/control arms
$D[p,q],D_{\mathcal{H}_{1}}[p,q]$	A generic distance as well as kernel distance of two distributions $p,q$
$\mathcal{F}$	A function class
$\mathcal{H}_{1}$	The unit ball of a reproducing kernel Hilbert space (RKHS)
$\langle\cdot,\cdot\rangle_{\mathcal{H}}$	Inner product of a RKHS

Table 1. A table of math symbols used in the proposed De-paradox Tree algorithm.

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 ( 10 ) based on $f(\mathbf{W})$ , determines the injected imbalance, as well as $g(\mathbf{V})$ , seen in Equation ( 11 ), 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.

Figure 6. Comparing De-paradox Tree and baseline methods on simulation data with injected confounder imbalance— $f$ -Design 0, $f$ -Design 1, $f$ -Design 2 . Three rows represent three designs of confounder imbalance. We measure how balanced the subgroups are through subgroup kernel balance, and average Kolmogorov-Smirnov (KS) test, how accurate the subgroup-level effect estimates are through coverage rate and MSE of causal effects, and how many subgroups generated using the number of leaves. In each design we report the average and 95% CI from 100 simulation trials with $N=2000$ data units. Tree configuration parameters are $n_{\mathrm{min}}=30,n^{1}_{\mathrm{min}}=n^{0}_{\mathrm{min}}=15,d_{1}=4,d_{2}=4$ .

Figure 7. Comparing De-paradox Tree and baseline methods on simulation data with injected heterogeneous effects. Three rows suggest three designs of heterogeneous effects— $g$ -Design 0, $g$ -Design 1, $g$ -Design 2 . We measure how accurate the assigned treatments are through error rate of treatment assignments and the regret between optimal outcome and the outcome achieved by tested methods. The smaller values the better. The dimensions of $\mathbf{W},\mathbf{V}$ are both 2. In each design we report the average and 95% CI from 100 simulation trials with $N=2000$ data units. Tree configuration parameters are $n_{\mathrm{min}}=30,n^{1}_{\mathrm{min}}=n^{0}_{\mathrm{min}}=15,d_{1}=4,d_{2}=4$ .

Figure 8. Comparing trees from baseline (A, B) on hybrid voting data and our De-paradox Tree (C, D). For (A, C), $p_{{X_{\mathrm{flip}}}}=0.5$ and $p_{{Y_{\mathrm{flip}}}}=0.5$ , while for (B, D), $p_{{X_{\mathrm{flip}}}}=0.9$ and $p_{{Y_{\mathrm{flip}}}}=0.9$ . Node boxes include ID, split criterion, kernel balance, effect size with 95% CI, and sample size for treated (NT) and control (NC) groups. Square nodes are to mitigate confounding bias, circular nodes are to search mixed effects. Marked nodes match the ground-truth variable for subgroup specification. Same tree configurations are used for both De-paradox Tree and baseline $n_{\mathrm{min}}=30,n^{1}_{\mathrm{min}}=n^{0}_{\mathrm{min}}=15,d_{1}=4,d_{2}=2$ . Explanation: The comparison shows that De-paradox Tree outperforms baseline methods by correctly identifying key splitting variables to reduce imbalance and reveal heterogeneous treatment effects. It selects household size and year of birth for balancing, then uncovers nested subgroups using ground-truth features. In contrast, baselines often split on incorrect variables, missing key confounders and effect modifiers. Even with stronger treatment imbalance (flip probability 0.9), baselines still overlook critical features, demonstrating the robustness of De-paradox Tree in complex observational data.

Figure 9. The tree learned from the De-paradox Tree algorithm when applied on the Lalonde dataset. The left tree shows detailed statistics, the right one is for paradox investigation. Slopes represent effect size, percentages are participation rates. Tree parameters are $d_{1}=4,d_{2}=2,n_{\mathrm{min}}=30,n^{1}_{\mathrm{min}}=n^{0}_{\mathrm{min}}=15$ . Explanation: The results show that De-paradox Tree effectively uncovers subgroup-level treatment effects in this dataset, revealing meaningful heterogeneity that traditional analyses overlook. While the overall effect appears spurious (-0.142, 95% CI: [-0.797, 0.514]), subgroup analysis identifies positive treatment effects across all subgroups , with the non-Black subgroup benefiting the most. Additionally, kernel balance improves after partitioning, reinforcing the reliability of subgroup-level estimates. These findings highlight the method’s interpretability and potential for uncovering hidden disparities in treatment response.

Figure 10. The tree learned from the De-paradox Tree algorithm when applied on the Python Programming dataset. The left tree shows detailed statistics, the right one is for paradox investigation. Slopes represent effect size, circle nodes mean it’s an opposite-effects policy tree by splitting on two effect modifiers ${\tt first\_attempts\_performance},{\tt EXAMPLE\_speed}$ Tree parameters are $d_{1}=4,d_{2}=2,n_{\mathrm{min}}=30,n^{1}_{\mathrm{min}}=n^{0}_{\mathrm{min}}=15$ . Explanation: The results show that De-paradox Tree effectively uncovers subgroup-level treatment effects in this dataset, revealing meaningful heterogeneity that traditional analyses overlook. While the overall effect is close to zero (0.008, 95% CI: [-0.07, 0.08]), subgroup analysis identifies positive treatment effects for low-entry-score students , with the low-entry-score yet slow steady learners benefiting the most (0.23, 95% CI: [0.07, 0.40]).

Appendix figures & tables3 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 11. Comparing trees from our De-paradox Tree method (A) and baseline (B) on the hybrid voting data, when $p_{X_{\mathrm{flip}}}=0.1$ and $p_{Y_{\mathrm{flip}}}=0.1$ . Node boxes include ID, split criterion, kernel balance, effect size with 95% CI, and sample size for treated (NT) and control (NC) groups. Square nodes are to mitigate confounding bias, circular nodes are to search mixed effects. Marked nodes match the ground-truth variable for subgroup specification. Same tree configurations are used for both De-paradox Tree and baseline $n_{\mathrm{min}}=30,n^{1}_{\mathrm{min}}=n^{0}_{\mathrm{min}}=15,d_{1}=4,d_{2}=2$ .

Figure 12. A large version of Figure 8 . Comparing De-paradox Tree vs baseline method on the hybrid voting datasets.

Figure 13. Applying a non-decision-tree baseline ( Alipourfard et al., 2018b ) to the hybrid voting dataset where $p_{{X_{\mathrm{flip}}}}=0.9$ and $p_{{Y_{\mathrm{flip}}}}=0.9$ . The left plot shows the aggregated trend between g2000 and outcome voted . The right plot shows the disaggregated trends detected by the method conditioning on hh_size .

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Splitting the Difference: Interpretable Causal Forests for Treatment Effect Heterogeneity and Bias

Nicolas Alexander Ihlo, Merle Behr

Faculty of Informatics and Data Science University of Regensburg, Germany

Causal Effect Estimation Conditional Average Treatment Effect Estimation Causal Interpretability Individualized Treatment Effect Estimation Random Forest

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In various fields, such as medicine and marketing, accurately predicting individual treatment effects holds significant promise. However, achieving reliable predictions alone is often insufficient for making informed decisions; it is equally important to understand why the treatment effect is higher for some individuals than for others. To address this two-fold challenge of prediction and interpretation, we introduce an algorithm based on decision trees and random forests for estimating individual treatment effects. Our algorithm is simple: it operates exactly like a standard random forest, but with a different splitting criterion, and requires no additional workarounds such as double machine learning or orthogonalization as used in Generalized random forests. It handles observational studies with varying treatment propensities without requiring separate estimation of the full propensity function. This is achieved by combining two splitting criteria---one targeting heterogeneity in the treatment effect, the other targeting bias correction for the average treatment effect---which together improve split point selection and automatically distinguish confounders from features responsible for heterogeneity. As a result, interpretation follows directly from the fitted tree structure itself, that is, from which features the trees split on and with which split statistics, without requiring separate post-hoc analysis. For the theoretical analysis of this algorithm, we consider a change point model with step functions for potential outcomes and treatment propensity and provide insights into the theoretical underpinnings of our approach. Simulation studies show that our simple algorithm achieves comparable, and often better, prediction accuracy than existing methods, while substantially improving interpretability.

Abstract

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