Multi-Method Causal Evidence Synthesis: Ranking Candidate Drivers by Convergent Cross-Method Evidence from Observational Data

Manish Gupta; Dipanjan De

stat.MEAug 20, 2026

Multi-Method Causal Evidence Synthesis: Ranking Candidate Drivers by Convergent Cross-Method Evidence from Observational Data

Authors: Manish Gupta, Dipanjan De

Organizations: Tricon Infotech

Abstract

Practitioners inferring causality from observational data usually rely on a single method and treat its output as causal truth. Recent tools select an optimal method for a dataset, and recent ensembles aggregate multiple causal-discovery algorithms into one graph, but little work pools evidence across different mathematical traditions, including non-causal ones. We present Multi-Method Causal Evidence Synthesis (MCES), a framework that ranks which candidate drivers in an observational system are most likely relevant to a set of outcomes, and with what strength of evidence. MCES runs eleven methods across eight mathematical traditions on observational panel data and pools their outputs into a Convergent Evidence Score (CES), a linear opinion pool. CES quantifies convergence of evidence across analytical lenses: the degree to which methods with different assumptions point to the same driver-outcome relationship. It does not claim causal identification in the interventionist sense; it supports hypothesis prioritization, not a transferable probability of causation. MCES first applies Structural-Behavioral Decomposition to remove definitional (algebraic) relationships, then runs all methods, normalizes outputs to [0,1], and pools them. We distinguish MCES from method selection, structural ensembles, prediction ensembles, and literature synthesis. Using synthetic data with embedded ground truth, the Sachs protein-signaling benchmark, six Bayesian-network structure benchmarks, and two further synthetic domains, we show MCES ranks true edges near the top (Precision@5 = 1.0, Precision@10 = 0.96 on the primary scenario), with a low empirical rate of null pairs reaching Moderate-or-higher convergence. Our central point is not that the pool beats every individual method, but that no single method is uniformly best across the evaluated scenarios, so MCES offers a method-agnostic default.

Figures & tables

Domain	Drivers ( $X$ )	Outcomes ( $Y$ )	Units	Time
Enterprise Software	Product perf., training, usability, staffing	Revenue, throughput, efficiency	Sites	Weeks
Healthcare	Treatments, protocols, staffing, equipment	Mortality, readmission, satisfaction	Hospitals	Months
Manufacturing	Machine settings, materials, operator skill	Defect rate, yield, throughput	Lines	Shifts
Education	Teaching methods, class size, technology	Test scores, graduation rate	Schools	Semesters
SaaS / Product	Features, UX changes, pricing, onboarding	Retention, NPS, revenue, churn	Cohorts	Months

Table 1: The causal diagnostic problem across domains.

Method	Captures	Misses
Regression	Linear associations	Non-linear effects, causal direction
SHAP	Non-linear importance	Causal direction, temporal dynamics
Granger Causality	Temporal predictability	Non-linear relationships
Bayesian Networks	DAG structure	Hidden confounders
Causal Forest	Heterogeneous effects	Median binarization (impl. choice)

Table 2: Complementary blind spots of component analytical methods.

Method	Key Papers	Limitation
Partial Correlation	Pearson (1895)	Cannot determine direction
Lasso Regression	Tibshirani (1996)	Assumes linearity
Distance Correlation	Székely et al. (2007)	Unsigned; no direction
Predictive Power Score	Wetschoreck et al. (2020)	Univariate; no confounder control
Mixed-Effects Models	Laird and Ware (1982)	Assumes linearity
Random Forest + SHAP	Breiman (2001) ; Lundberg and Lee (2017)	Importance $\neq$ causation

Table 3: Foundational literature for MCES component methods.

Paradigm	Examples	What It Does	Limitation
Method Selection	CausalTune, Causal-Copilot	Picks best ONE method	Discards all other evidence
Same-Method Ensemble	TETRAD, E-CIT	Same algorithm on data splits	Shared assumptions & failure modes
Structural Ensemble	Vo et al. (2026) , Li et al. (2026)	Pools causal-discovery algorithms into one graph	Members share the graph-recovery task and causal-family assumptions
Literature Synthesis	Shi et al. (2025)	LLM extracts from papers	Requires existing studies
Cross-Tradition (MCES)	This paper	11 methods, 8 traditions, incl. non-causal; ranks a declared grid	No recovery guarantee; empirical FP control only

Table 4: The 4-way distinction: existing paradigms vs. MCES.

Panel of Figure 1 — Figure 1: The MCES pipeline. Definitional (identity) pairs are removed first; the surviving behavioral candidates are scored by eleven methods from eight mathematical traditions, normalized, and pooled into the Convergent Evidence Score.

Tier	Methods	What They Provide	$w_{k}$
1: Associational	Partial Corr., Lasso, Distance Corr.	Associations, no direction	0.06
2: Predictive	Mixed Effects, RF+SHAP, PPS	Importance, structure	0.05–0.08
3: Temporal	Granger, ITS, Transfer Entropy	Direction and timing	0.13–0.14
4: Structural and treatment-effect	Bayesian Net, Causal Forest	Causal structure, effects	0.10–0.11

Table 5: Analytical groupings and optional tiered weights. The default is uniform; the tiered column is one alternative studied in the weight sensitivity analysis.

Method	F1@10
Ensemble (MCES)	0.686 $\pm$ 0.035
Partial Corr.	0.711 $\pm$ 0.016
Lasso	0.714 $\pm$ 0.000
Distance Corr.	0.475 $\pm$ 0.088
PPS	0.129 $\pm$ 0.029
RF+SHAP	0.539 $\pm$ 0.070

Table 6: E1, F1@10 on primary_panel (mean

\pm

s.d. over 20 seeds). The ensemble is competitive with, but does not strictly dominate, the best individual method; its value is robustness across scenarios (Section 7.6 , 8 ).

Figure 2: E1, F1@10 of the ensemble (blue) against each individual method (gray) on primary_panel . Error bars are s.d. over 20 seeds. The pool sits at the top of the range without depending on any single member.

Scenario	MCES F1@10	Baseline F1@10
Primary panel (20 seeds)	0.69 $\pm$ 0.04	0.70 $\pm$ 0.04
Non-linear	0.74 $\pm$ 0.00	0.42 $\pm$ 0.00
Confounded	0.73 $\pm$ 0.00	0.82 $\pm$ 0.00
Healthcare	1.00 $\pm$ 0.00	0.80 $\pm$ 0.00
Manufacturing	0.74 $\pm$ 0.00	0.63 $\pm$ 0.00
Stacked logistic (rotating folds)	0.69 $\pm$ 0.04	0.71 $\pm$ 0.00 a

Table 7: External baselines, F1@10. The baseline in the scenario rows is NOTEARS-linear ( Zheng et al., 2018 ) , fit per scenario on the pooled panel; in the final row it is a supervised logistic combination of the eleven method scores (rotating folds; superscript a: trained on ground-truth labels, an upper reference available only when labels exist).

Mean pairwise rank corr. $\rho=0.945$ (min 0.814)
Weight scheme	F1@10
uniform	0.686 $\pm$ 0.035
tiered	0.646 $\pm$ 0.042
tier heavy	0.629 $\pm$ 0.058
associational heavy	0.707 $\pm$ 0.021
causal only	0.564 $\pm$ 0.081

Table 8: E2, F1@10 by weight scheme on primary_panel , with mean pairwise rank correlation between schemes.

$N$	$T$	Obs.	Methods	P@10	F1@10
23	5	115	11.0	0.83 $\pm$ 0.09	0.60 $\pm$ 0.07
23	10	230	11.0	0.87 $\pm$ 0.05	0.62 $\pm$ 0.03
23	20	460	11.0	0.97 $\pm$ 0.05	0.69 $\pm$ 0.03
23	40	920	11.0	0.97 $\pm$ 0.05	0.69 $\pm$ 0.03
6	20	120	11.0	0.47 $\pm$ 0.17	0.33 $\pm$ 0.12
12	20	240	11.0	0.83 $\pm$ 0.12	0.60 $\pm$ 0.09

Table 9: E3, sample-size sweep on primary_panel (mean

\pm

s.d. over 3 seeds). Top block: panel length

T

N=23

; bottom block: panel width

N

T=20

. “Methods” is the mean number of methods that ran; all eleven remain active in every cell, so the degradation is statistical power, not applicability.

	Without SBD	With SBD
Precision@3	0.333	1.000
Precision@5	0.600	0.600
Identity pairs in top 5	2.0	0.0

Table 10: E4, Structural–Behavioral Decomposition on a controlled identity scenario (mean over 3 seeds).

Method	Recall of non-linear edges
Ensemble (MCES)	0.667
Partial Corr.	0.667
Lasso	0.667
Distance Corr.	0.667
PPS	0.444
RF+SHAP	0.667

Table 11: E5, recall of non-linear edges (top-

n_{\text{true}}

) on the nonlinear scenario.

Figure 3: Held-out reliability diagram: raw CES (gray) and isotonic-calibrated CES (blue) against the empirical true-edge rate, over the pooled held-out predictions of all rotating folds (no fold’s calibrator sees its own test seeds). The dashed line is perfect calibration; the calibrated curve lies closer to it.

CES threshold	Null pairs $\geq$ t	True pairs $\geq$ t
$\geq 0.3$	0.0051 $\pm$ 0.0033	0.69 $\pm$ 0.09
$\geq 0.35$	0.0012 $\pm$ 0.0012	0.60 $\pm$ 0.06
$\geq 0.4$ (default)	0.0003 $\pm$ 0.0006	0.50 $\pm$ 0.05
$\geq 0.45$	0.0002 $\pm$ 0.0005	0.41 $\pm$ 0.06
$\geq 0.5$	0.0000 $\pm$ 0.0000	0.33 $\pm$ 0.07
$\geq 0.6$	0.0000 $\pm$ 0.0000	0.19 $\pm$ 0.04

Table 12: E8, CES-threshold sensitivity on primary_panel (mean

\pm

s.d. over 20 seeds): fraction of null pairs and of true pairs at or above each candidate threshold.

Figure 4: E9, out-of-sample predictive lift vs. CES on primary_panel (all behavioral pairs, 20 seeds). High-CES pairs concentrate at positive lift; null pairs at zero.

Domain	# true edges	Ensemble F1@10	Best individual F1@10
Healthcare (synthetic)	10	$1.000$	$1.000$
Manufacturing (synthetic)	9	$0.737$	$0.842$

Table 13: Domain applications (synthetic, full ensemble). F1@10 with the best individual method for comparison.

Network	Nodes	Pairs	P@5	Ens. F1@K	Best ind. F1@K
Child	20	380	0.20	0.32	0.40
Insurance	27	702	0.20	0.25	0.33
Alarm	37	1332	0.20	0.35	0.35
Hailfinder	56	3080	0.60	0.30	0.35
Hepar2	70	4830	0.40	0.28	0.31
Win95pts	76	5700	0.60	0.34	0.31

Table 14: E10b, orientation audit: the same networks with no gold-DAG role assignment (all ordered pairs; reverse orientations are errors). Compare Table 15 : removing the declared partition sharply reduces performance for the ensemble and the best individual method alike.

Network	Nodes	Scoreable	P@5	Ens. F1@K	Best ind. F1@K
Child	20	12/25	1.00	0.75	0.75
Insurance	27	28/52	1.00	0.68	0.79
Alarm	37	28/46	1.00	0.71	0.79
Hailfinder	56	40/66	0.60	0.40	0.47
Hepar2	70	95/123	1.00	0.49	0.47
Win95pts	76	82/112	1.00	0.63	0.65

Table 15: Bayesian-network structure benchmarks ( bnlearn ).

K

is the number of scoreable forward edges under the driver/outcome orientation. The ensemble tracks the best individual method at each scale.

Approach	When to Use Instead of MCES
A/B test	When randomization is possible
Single well-specified causal model	Strong theory + large $N$ + one specific question
CausalTune / method selection	Computational budget is limited, need one fast answer
Causal-Copilot	Need guided exploration, not rigorous synthesis

Table 16: When to use alternative approaches instead of MCES.

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

Method	Captures	Misses	Key Assumption	Optional tiered $w_{k}$
Partial Corr.	Conditional linear assoc. after adjustment	Non-linear, direction	Linearity	0.06
Lasso	Sparse selection	Non-linear, temporal	Linearity, sparsity	0.06
Distance Corr.	Arbitrary-form dependence	Direction, confounders	Exchangeability (perm. test)	0.06
Mixed Effects	Unit baselines, panel	Non-linear, direction	Linearity	0.07
RF + SHAP	Non-linear, interactions	Direction, temporal	Predictive validity, exchangeability	0.08
PPS	Univariate OOS skill, asymmetry	Confounders, temporal	Representative folds, exchangeability	0.05

Appendix

Table 17: Complete method comparison: what each method captures, misses, and assumes. The final column lists the optional tiered weight; the default weighting is uniform.

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