Association Is Not Always Causation and the Potential Outcome Framework

Construct the study

The population’s potential outcomes and the treatment-assignment mechanism can be changed separately.

Conceptual presets

Treatment assignment

Design

Selection by baseline CVD risk

high-risk less likelyhigh-risk more likely

Overlap quality

limited overlapgood overlap

Target smoking prevalence

15%65%

Strength of hidden prognostic factor U

inactivestrong U effect

Under observational assignment, U affects both treatment and outcome and is therefore an unmeasured confounder. Under complete randomization, U may still predict the outcome, but it does not affect treatment and is not a confounder.

Potential outcomes

Number of subjects

120600

Geometric-mean conditional causal OR

geometric-mean OR < 1geometric-mean OR > 1

Conditional-effect heterogeneity

common conditional OROR varies with X

The conditional odds ratio is a structural simulation parameter, not the marginal ATE or marginal risk ratio. The effect modifier is bounded and centered, so the selected OR is the finite-cohort geometric mean of the subject-specific conditional ORs. When heterogeneity is nonzero, a geometric-mean OR of 1 is not a sharp null because some subject-specific ORs can be above 1 and others below 1.

Population seed

Reveal simulation truth

Reveal simulation-oracle information: both potential outcomes, the finite-population ATE, true assignment probabilities, and structural effect parameters. Simulation only: real data do not reveal these quantities.

Turn this off for an observed-data view. Counterfactual outcomes, oracle assignment probabilities, structural effects, response types, and repeated-reassignment experiments are then hidden or locked.

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Mathematical formulation

Potential outcomes separate the causal estimand from the observed-data comparison. The tabs below connect individual counterfactuals, population effects, selection bias, and identification.

Graduate-level notation

Potential-outcome map

{Y_{i} (1), Y_{i} (0)} ⟶ τ_{i} = Y_{i} (1) - Y_{i} (0) ⟶ τ_{ATE} = E {Y (1) - Y (0)}

Two outcomes for every subject

Y_{i} (1), Y_{i} (0)

The potential outcomes are jointly well-defined in the causal model, even though only one can be observed.

Consistency links worlds to data

Y_{i} = T_{i} Y_{i} (1) + {1 - T_{i}} Y_{i} (0)

The observed outcome is the potential outcome under the treatment actually received.

Finite-population ATE

{\bar{τ}}_{n} = \frac{1}{n} \sum_{i = 1}^{n} {Y_{i} (1) - Y_{i} (0)}

This is the causal risk difference for the generated finite cohort. The expectation in the map above defines the superpopulation ATE; τ̄_n is its finite-cohort analog used throughout the dashboard.

Fundamental problem of causal inference. For the same person at the same time, the observed dataset contains either Y_i(1) or Y_i(0), never both. Individual causal effects are therefore not directly observed.

Simulation DGP and cross-world coupling

\begin{matrix} p_{t i} = P {Y_{i} (t) = 1 | X_{i}, U_{i}}, Y_{i} (t) = 𝟙 {R_{i} \leq p_{t i}} \\ \log \frac{p_{1 i} / (1 - p_{1 i})}{p_{0 i} / (1 - p_{0 i})} = \log (θ) + δ h_{i}, \frac{1}{n} \sum h_{i} = 0, h_{i} bounded \end{matrix}

Cross-world caution. A latent rank R_i ~ Uniform(0,1), generated independently of (X_i,U_i), is used for both treatment worlds. This gives a coherent rank-preserving structural model, but it is a simulation choice. The individual effects τ_i and the four binary response-type proportions depend on this coupling. The marginal ATE depends only on the two marginal means, whereas the joint response-type distribution is generally not identified from observed data.

Different comparisons

\begin{matrix} Δ_{obs} = E (Y | T = 1) - E (Y | T = 0) \\ Δ_{ATE} = E {Y (1)} - E {Y (0)} \end{matrix}

Exact population association–causal gap decomposition, using consistency

Δ_{obs} - Δ_{ATE} = [E {Y (1) | T = 1} - E {Y (1)}] + [E {Y (0)} - E {Y (0) | T = 0}]

Association

Compares two different observed groups living under different treatments and usually different covariate distributions.

Causal effect

Compares the same target population under two interventions, smoking versus no smoking.

Population gap versus realized gap

At the population level, the displayed decomposition is a selection-bias decomposition. In one finite randomized assignment, the observed-minus-ATE gap can be nonzero from random imbalance even though the difference-in-means estimator is unbiased over assignments.

Consistency

Y = Y (T)

Treatment versions and interference must be handled so the intervention is well-defined.

Conditional exchangeability

{Y (1), Y (0)} ⊥ T | X

After conditioning on measured baseline covariates, assignment contains no residual information about the potential outcomes.

Positivity

0 < e (x) \equiv P (T = 1 | X = x) < 1 for P_{X} -almost every x

Both treatment choices must be possible on the support of the target covariate distribution. This structural condition is distinct from finite-sample practical overlap.

G-formula identification

E {Y (t)} = E [E {Y | T = t, X}], t \in {0, 1}

Identification is not an observed-data theorem without assumptions. A large sample, a small p-value, or excellent prediction does not by itself transform association into causation. The pass/fail labels in this dashboard are known only because the simulation exposes its data-generating mechanism; they are not empirical tests of exchangeability.

Complete randomization: exact finite-population statement

\begin{matrix} ℱ_{n} = {X_{i}, Y_{i} (1), Y_{i} (0)}_{i = 1}^{n}, P_{A} (T = t | ℱ_{n}) = \frac{𝟙 {\sum t_{i} = n_{1}}}{(\begin{matrix} n \\ n_{1} \end{matrix})} \\ {\hat{τ}}_{DM} = \frac{1}{n_{1}} \sum T_{i} Y_{i} - \frac{1}{n_{0}} \sum (1 - T_{i}) Y_{i}, E_{A} ({\hat{τ}}_{DM} | ℱ_{n}) = {\bar{τ}}_{n}, n_{0} = n - n_{1} \end{matrix}

Potential outcomes remain fixed

The repeated experiment conditions on the finite schedule ℱ_n. Only the treatment vector is randomized.

Unbiased—not exactly equal in every assignment

A realized difference in means generally differs from τ̄_n. Its randomization expectation equals τ̄_n.

Observed covariate balance is random

Randomization makes treatment independent of the fixed schedule by design, but one realized assignment can still have nonzero covariate imbalance.

1. Causal graph: causal and backdoor paths

Solid arrows encode the current structural graph; dashed arrows show inactive schematic alternatives for orientation. Thickness is only a pedagogical cue and has no formal DAG meaning.

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2. Association can change while causation stays fixed

The curve varies who is selected into smoking while holding every subject’s potential outcomes fixed.

Assignment-probability–weighted contrastFinite-population ATECurrent realized sample

Δ_sel(s) = Σ e_i(s)Y_i(1) / Σ e_i(s) − Σ{1−e_i(s)}Y_i(0) / Σ{1−e_i(s)}

This ratio-of-weighted-sums is the assignment-probability contrast induced by the selection mechanism. It is not exactly the expectation of a finite-sample ratio. The filled point is the realized crude difference from one assignment.

The causal line is horizontal because the potential outcomes do not change. Only the treatment-assignment mechanism changes.

3. The observed cohort

Each point is one subject. Color is observed smoking status; a dark ring marks observed CVD.

Smoker T=1Non-smoker T=0Observed CVD Y=1larger point = higher oracle assignment extremity (truth-reveal mode)

Click a point to inspect its factual and counterfactual outcomes. Treatment assignment may depend on prognosis even though the causal effect is defined within the same person.

Selected subject: two possible worlds

Only one potential outcome is factual; the other is counterfactual.

4. The fundamental problem: the potential-outcome schedule

Each row contains two potential outcomes, but the observed dataset reveals only the cell selected by T.

Show

Y(1)Y(0)solid cell = factualhatched cell = counterfactual

The reveal switch can display the simulation truth, but this does not change what an actual observational dataset would contain.

5. Observed association versus causal contrast

Observed groups and intervention worlds answer different questions.

SmokingNo smokingcausal truth

6. Binary potential-outcome types

For the bad outcome Y=1, the ATE equals the proportion harmed minus the proportion protected.

Finite-cohort identity: τ̄_n = (n₁₀−n₀₁)/n. The response-type proportions require the joint distribution of {Y(1),Y(0)}, which observed data do not identify without additional cross-world assumptions. This simulation defines that joint distribution through a shared latent rank.

7. Repeated treatment assignments

Keep X, Y(1), and Y(0) fixed; repeatedly change only T.

Assignments

Simulation-only experiment: outcomes after reassignment require the complete potential-outcome schedule. Under complete randomization, the exact randomization expectation is the finite-population ATE; a finite Monte Carlo mean is only approximately equal to it. Observational selection can shift the entire assignment distribution.

8. Identification checklist

The observed association is causal only when the design or assumptions justify the missing counterfactual comparison.

Scope of these checks: they verify algebra and internal simulation construction. They do not establish exchangeability in real observed data.

Inspect the subject-level causal schedule

Subject	Age	BMI	Income	Female	College	HTN	Risk score	Oracle eᵢ	Conditional ORᵢ	T	Y observed	Y(1)	Y(0)	τ