Scalable and Versatile Identification for Hierarchical Structural Causal Models: A New Look at Project STAR

Janis Aiad; Aghiles Drali; Aymen El Ouadrhiri; Anass Ettahiri; Yasser Oufqir; Simon Patry; David Cortes; Marianne Clausel

stat.MLAug 25, 2026

Scalable and Versatile Identification for Hierarchical Structural Causal Models: A New Look at Project STAR

Authors: Janis Aiad, Aghiles Drali, Aymen El Ouadrhiri, Anass Ettahiri, Yasser Oufqir, Simon Patry, David Cortes, Marianne Clausel,

Abstract

The STAR (Student-Teacher Achievement Ratio) experiment (1985, Tennessee, USA) is a landmark hierarchical dataset designed to assess the impact of class size on student outcomes, with observations nested within classes. To encode class-level interventions in such hierarchical settings, we develop a complete, scalable, open-source pipeline for Hierarchical Structural Causal Models (HSCM) that bridges symbolic identification and practical estimation. Our approach integrates graph transformations, pyAgrum's do-calculus for automatic identification of causal effects, adaptation of symbolic expression into closed-form HSCM formulas, and numerical estimation from fitted local probability models. A key innovation is our adapted Abstract Syntax Tree (AST), which decomposes pyAgrum's identified formulas into independent density, expectation, and marginalization tasks, enabling parallel and scalable computation. We validate the pipeline on canonical HSCM motifs and benchmark scenarios with known ground truth, then apply it to STAR kindergarten mathematics outcomes. The results show that flat baselines (ignoring hierarchy) recover associations but fail to encode class-level interventions, and that symbolic identification alone is not enough for practical Hierarchical Structural Causal inference; scalable estimation and numerical stability checks are central parts of the scientific object.

Figures & tables

Panel of Figure 1 — Figure 1: An example of hierarchical causal model and its transformation. (a) an HSCM with $n$ units, each of which containing $m$ subunits, (b) HCGM with explicit HCM’s latent $Q$ variables, (c) HCM’s matching collapsed model and (d) its augmented/marginalized version. We apply do-calculus to this final model to perform identification in the original HCM.

Panel of Figure 2 — Figure 2: Pipeline for causal identification in Hierarchical Structural Causal Models (HSCMs).

Panel of Figure 3 — Figure 3: An example of hierarchical causal model and its transformation. (a) an HSCM with $n$ units, each of which containing $m$ subunits, (b) HCGM with explicit HCM’s latent $Q$ variables, (c) HCM’s matching collapsed model and (d) its augmented/marginalized version. We apply do-calculus to this final model to perform identification in the original HCM.

Step	Sequential cost	Parallelization structure	Effective cost with $P$ workers
Collapse/augmentation	$O(e+v)$ graph operations	graph-level (negligible)	$O(e+v)$
Build pyAgrum model	$O(e+v)$ (copy plus identification search)	symbolic backend	$O(e+v)$
AST translation	$O(K)$ expression traversal	factor-level traversal	$O(K)$
Per-unit $Q$ fitting	$O(nmd)$ for simple parametric factors	independent accross units and factors	$\sim O(nmd/P)$ + scheduling overhead
Monte Carlo evaluation	$O(BK)$ per intervention	Independent across samples, units, and branches	$\sim O(BK/P)$ when branches are balanced

Table 1: Algorithmic cost and parallelization structure of the pipeline. The effective cost assumes ideal parallelization across

P

workers. We denote

n

is the number of units,

m

is the average number of sub-units per unit,

d

is the maximum parent count,

v

and

e

are the numbers of transformed graph nodes and edges,

K

is the number of density factors in the identified formula,

B

is the number of Monte Carlo samples or quadrature points.

Panel of Figure 6 — Figure 6: Class-level STAR diagnostics. Left: gender heterogeneity across classes, showing why an additive individual gender coefficient does not represent the full distribution of class composition. Right: reading and mathematics class-level score means, together with a standardized Q-Q diagnostic for Mathematics. In the Q-Q panel, blue points are empirical quantiles and the dashed black line is the Gaussian reference. The green band marks the central range where the Gaussian approximation is most relevant, while the red bands mark lower and upper score extremes. The rightmost extreme is kept as an empirical check rather than fitted separately because it contains only a few repeated score values.

Model	Specification	Coef.	SE	$p$ -value	$R^{2}$
$\mathcal{M}^{1}$	No controls	$\hat{\beta}_{A}$ =8.096	1.597	$4.0\times 10^{-7}$	0.006
$\mathcal{M}^{2}$	School fixed effects	$\hat{\beta}_{A}$ =9.499	1.463	$8.5\times 10^{-11}$	0.219
$\mathcal{M}^{3}$	Student controls	$\hat{\beta}_{A}$ =9.380	1.415	$3.4\times 10^{-11}$	0.268
$\mathcal{M}^{4}$	Full OLS replication	$\hat{\beta}_{A}$ =9.432	1.419	$3.0\times 10^{-11}$	0.269
$\mathcal{M}^{5}$	IV class-size contrast	$\hat{\theta}$ =-1.219	0.163	$7.0\times 10^{-14}$	–
$\mathcal{M}^{6}$	Random class intercept	$\hat{\beta}_{A}$ =9.099	2.263	$5.8\times 10^{-5}$	–

Table 3: Baseline regression estimates for kindergarten mathematics. For M1–M4 and M6, the reported coefficient is

\hat{\beta}_{A}

, the coefficient of the small-class assignment indicator

A_{ij}

. For M5, the reported coefficient is the IV class-size coefficient

\hat{\theta}

Figure 7: ExactBIC-derived graphs for the STAR HSCM analysis. The left panel shows the directed ExactBIC graph over the flat STAR variables, after adding the unit-level context used by the HSCM analysis. The right panel shows the corresponding collapsed and augmented

Q

-variable graph for the Mathematics outcome. Blue nodes are unit-level context variables, orange nodes are observed variables or fitted

Q

-mechanisms, and the green node is the target class-level Mathematics outcome distribution

Q^{y}

. Solid arrows are retained directed dependencies; dashed red arrows mark factors normalized in the stabilized numerical evaluation.

Symbol	Type	Interpretation
$Q^{a}$	Bernoulli/proportion	Small-class assignment distribution within a class
$Q^{g}$	Bernoulli/proportion	Gender composition within a class
$Q^{e}$	Bernoulli/proportion	Ethnicity composition within a class
$Q^{l}$	Bernoulli/proportion	Free-lunch composition within a class
$Q^{b}$	Gaussian score mechanism	Reading score distribution for the class
$Q^{y}$	Gaussian score mechanism	Mathematics score distribution for the class

Table 4: Class-level

Q

-variables used by the STAR HSCM evaluator. These are distributional summaries or mechanisms fitted at the class level from the student-level variables in Table 2 .

Run	Target	$E[\cdot\|\mathrm{do}(1)]$	$E[\cdot\|\mathrm{do}(0)]$	ATE
ExactBIC	Mathematics outcome ( $Y$ )	2402.75	2365.95	36.80
ExactBIC, normalized factors	Reading ( $B$ ) diagnostic	1788.10	1766.47	21.63

Table 5: STAR HSCM estimates and normalized-factor diagnostic. The Mathematics row is the main outcome analysis, where the outcome is denoted

Y

; the Reading row is a separate diagnostic run used to inspect the normalized

P(Q^{b\,|\,a,g}\,|\,S)

factor.

Figure 8: Estimated class-level HSCM response parameters and means in STAR. Reading ( $Q^{b}$ ) and the Mathematics outcome ( $Q^{y}$ ) are represented by Gaussian summaries $(\hat{\mu}_{i},\hat{\sigma}_{i}^{2})$ , and the right panel reports the joint class means used to inspect the relationship between the two score mechanisms. In the right panel, colors encode the class-level urbanicity category $S$ ; the panel is a descriptive diagnostic rather than an oriented causal graph.

Figure 9: Estimated class-level HSCM probability quantities used in the STAR analysis. Binary sub-unit variables are represented by class-level Bernoulli probabilities for Small-class assignment (

Q^{a}

), Gender (

Q^{g}

), Ethnicity (

Q^{e}

), and free lunch (

Q^{l}

);

Q^{l\,|\,e}

summarizes the free-lunch mechanism conditional on Ethnicity; and

S

is the observed class-level Urbanicity variable. In the conditional free-lunch scatter, point colors encode the urbanicity category

S

; in the

S

panel itself, the bar color is only a visual grouping choice.

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sHAIL-Causal: A Sequential Staircase Procedure for Invariant Causal Predictor Discovery

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School of Mathematics and Statistics, Rochester Institute of Technology

Causal Discovery Causal Inference Online Learning

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We introduce sHAIL-Causal, the causal specialization of the Saturated Hierarchical Atomic Incremental Learning (sHAIL) paradigm: a sequential staircase procedure that ascends a nested hierarchy of hypothesis classes H_0 < H_1 < ... < H_K once a saturation signal indicates that mastery of the current stage has plateaued. Where general sHAIL leaves the saturation criterion open, sHAIL-Causal instantiates it with a joint criterion of goodness-of-fit saturation and cross-environment invariance, replacing the complexity control of Structural Risk Minimization. We show, theoretically and by simulation, that complexity-only staircases are seduced by confounded predictors that lower empirical risk without reflecting stable causal structure, whereas an invariance-gated staircase provably halts at the true causal predictor set under a per-variable Richness condition. We show that naive greedy search fails to recover the causal set even under Richness, trace the failure to non-monotonicity of the invariance statistic along single-variable paths, and validate a fix combining bounded-exhaustive block-seeding with a calibrated acceptance threshold. We then extend the guarantee to environments arriving sequentially, yielding a confidence guarantee that stays valid at every arrival, which one-shot exhaustive search cannot offer without repeating its full combinatorial search. We close by formalizing the intervention of a wise teacher who lifts a saturated learner off a plateau of boredom.

Abstract

Causal Effect Estimation

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