CausalLab

Structural Causal Model

Build a DAG

Click canvas to add nodes. Drag from one node to another to add directed edges. Right-click to remove.

Preset Models

Actions

d-Separation Query

⊥ |

Structural Equations

Add nodes to define equations

Graph Properties

Nodes0

Edges0

Is DAG—

Topological Order—

Node Roles

Select a preset to see roles

Paths Between Selected

Select two nodes

Pearl's Causal Hierarchy

Rung 1: Association

P(Y|X) — What is? Seeing X tells us about Y.

Rung 2: Intervention

P(Y|do(X)) — What if? Forcing X and observing Y.

Rung 3: Counterfactual

P(Y_x|X=x',Y=y') — What if I had acted differently?

do-Calculus Engine

Graph Surgery

The do-operator do(X=x) simulates an intervention by removing all arrows INTO X, fixing X=x, and computing the resulting distribution.

Select Model

Causal Query

Treatment

Outcome

Identification Criteria

Select a model and run identification

Derivation Steps

Run identification to see steps

do-Calculus Rules

Rule 1: Insertion/deletion of observations

P(y|do(x),z,w) = P(y|do(x),w) if (Y ⊥ Z | X,W) in Ḡ_X

Rule 2: Action/observation exchange

P(y|do(x),do(z),w) = P(y|do(x),z,w) if (Y ⊥ Z | X,W) in Ḡ_X,G_Z̲

Rule 3: Insertion/deletion of actions

P(y|do(x),do(z),w) = P(y|do(x),w) if (Y ⊥ Z | X,W) in Ḡ_X,G_{Z̲(S)}

Result

Causal Discovery

PC Algorithm

Constraint-based causal discovery: starts with a complete graph, removes edges using conditional independence tests, then orients v-structures and applies Meek's rules.

True Model

Parameters

Sample Size500

Significance (α)0.05

Noise Level1.0

Algorithm Phase

Click "Run" to start

Discovery Metrics

SHD (Structural Hamming)—

Edge Precision—

Edge Recall—

Orientation Accuracy—

CI Tests Performed—

Discovered Graph

Run algorithm to see results

Algorithm Log

Simpson's Paradox

The Paradox

A trend that appears in several groups REVERSES when the groups are combined. Only causal reasoning (not statistics alone) can resolve which analysis is correct.

Dataset

Confounding Strength

Confounder → Treatment0.7

Confounder → Outcome0.8

True Treatment Effect0.1

Sample Size2000

Analysis Results

Marginal (Naive) Analysis

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Conditional (Stratified) Analysis

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The Causal DAG

Z → X → Y ← Z

Z (confounder) causes both X (treatment) and Y (outcome). The back-door path X ← Z → Y creates spurious correlation.

Resolution

Condition on Z (the confounder)

The stratified analysis gives the correct causal effect because conditioning on Z blocks the back-door path.

Key Insight

Statistics alone cannot resolve this

Whether to condition on Z depends on the causal structure, not the data. If Z were a collider instead, conditioning would be WRONG.

Instrumental Variables

When Experiments Fail

IVs let us estimate causal effects even with unobserved confounders. An instrument Z must: (1) affect X, (2) affect Y only through X, (3) be independent of confounders U.

Parameters

True Causal Effect (β)0.5

Instrument Strength (π)0.8

Confounding Strength1.0

Sample Size500

Estimation Comparison

True β0.5

OLS (biased)—

2SLS (IV)—

OLS Bias—

2SLS Bias—

First-Stage F—

2SLS Walkthrough

Stage 1: Regress X on Z
X̂ = π·Z + e
Get predicted values X̂ that are "clean" of confounding

Stage 2: Regress Y on X̂
Y = β·X̂ + ε
β̂_IV = Cov(Z,Y)/Cov(Z,X) — the Wald estimator

Instrument Diagnostics

Run 2SLS to see diagnostics

Monte Carlo Results

Click Monte Carlo to see distribution

Counterfactual Reasoning

Rung 3 of Pearl's Ladder

"Would the patient have recovered if given the treatment, given that they weren't treated and didn't recover?"

This requires the twin network model — two copies of the SCM sharing the same exogenous noise.

Scenario

Model Parameters

P(X=1) Treatment Rate0.5

P(Y=1|X=1) Success w/ Treatment0.8

P(Y=1|X=0) Success w/o Treatment0.4

Monotonicity

Probabilities of Causation

PN (Probability of Necessity)—

PS (Probability of Sufficiency)—

PNS (Necessity & Sufficiency)—

Interpretation

Select a scenario

Choose a preset to explore counterfactual reasoning.

Response Types

Helped

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Hurt

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Always

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Never

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Twin Network

How it works

1. Abduction: Use evidence to infer exogenous noise U
2. Action: Modify the model (set X to counterfactual value)
3. Prediction: Compute Y under the modified model with inferred U