Outcome Regression Estimator — Se Yoon Lee, Ph.D.

Build the observational study

Change the data-generating mechanism, then choose what the analyst puts into the outcome model.

Scenario presets

Data-generating process

Number of subjects

160600

Strength of confounding

treatment ≈ randomstrong selection by X

Treatment overlap

poor supportgood overlap

Baseline smoking effect (odds ratio)

OR 1.0OR 3.0

Outcome nonlinearity

main effectsstrong curvature

Treatment-effect heterogeneity

constant log-ORvaries with X

Cohort seed

Analyst's outcome model

Terms included in logistic regression

Show simulation truth on plots. This information is unavailable in a real observational study.

Reveal the two treatment worlds

Counterfactual reveal

observed world onlyboth predictions

This slider changes only the visualization. The formal outcome-regression estimate always averages both complete prediction sets.

Mathematical formulation

Outcome regression is a plug-in g-computation estimator: estimate the conditional outcome surface, evaluate that surface twice for every subject, and integrate over the empirical target-population distribution of X.

Graduate-level notation

Outcome-regression estimation map

Fit \hat{m} (t, x) ⟶ {(\hat{m} (1, X_{i}), \hat{m} (0, X_{i})) : i = 1, \dots, n} ⟶ {\hat{RD}}^{OR} = \frac{1}{n} \sum_{i = 1}^{n} {\hat{m} (1, X_{i}) - \hat{m} (0, X_{i})}

1Fit the conditional mean

m_{t} (x) \equiv E (Y | T = t, X = x)

Estimate this regression from the observed outcomes and treatment assignments.

2Predict both treatment worlds

{\hat{m}}_{1 i} = \hat{m} (1, X_{i}), {\hat{m}}_{0 i} = \hat{m} (0, X_{i})

The same subject-specific covariate vector X_i appears in both predictions.

3Standardize and contrast

{\hat{ψ}}_{t}^{OR} = \frac{1}{n} \sum_{i = 1}^{n} \hat{m} (t, X_{i})

For the ATE risk difference, subtract the standardized mean under t=0 from that under t=1.

Consistency

Y = Y (T) = T Y (1) + (1 - T) Y (0)

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

Conditional exchangeability

{Y (0), Y (1)} ⟂ T | X

All common causes of smoking and CVD required for adjustment are measured in X.

Positivity

0 < P (T = 1 | X = x) < 1

Each relevant covariate pattern has support in both treatment groups.

Identification by the g-formula

ψ_{t} \equiv E {Y (t)} = E [E {Y (t) | X}] = E [E {Y | T = t, X}] = E {m_{t} (X)}

Important: the regression function is identified from observed data, but its interpretation as a causal potential-outcome mean requires all three assumptions above. Good predictive performance alone does not establish exchangeability.

Binary-outcome working model

logit {m_{β} (t, x)} = η_{β} (t, x)

m_{β} (t, x) = expit {η_{β} (t, x)}

Currently selected specification:

The model is fitted only once to observed Y. Counterfactual predictions are then generated by substituting t=1 and t=0 while keeping x fixed.

Why the treatment coefficient is not the ATE

β_{T} \neq E {Y (1) - Y (0)}

In logistic regression, β_T is a conditional log-odds contrast. Non-collapsibility and treatment interactions prevent it from being interpreted as the marginal causal risk difference.

Model misspecification: if m̂(t,x) converges to an incorrect working surface m*(t,x), the plug-in estimator converges to E[m*(1,X) − m*(0,X)], which need not equal the causal ATE.

O_iObserved unit: (Y_i,T_i,X_i)

Y(t)Potential outcome under treatment t

m_t(x)Conditional outcome regression

ψ_tMarginal mean potential outcome

P_nfEmpirical mean n⁻¹Σf(O_i)

RDψ₁ − ψ₀

Outcome-model score and derivative

S_{i} (β) = Z_{i} {Y_{i} - m_{β} (T_{i}, X_{i})}

{\hat{d}}_{t} = \frac{1}{n} \sum_{i = 1}^{n} \frac{\partial m_{β} (t, X_{i})}{\partial β} |_{β = \hat{β}}

Z_i is the design vector used by the selected logistic outcome model.

Influence function of the standardized mean

{\hat{IF}}_{i, t} = \hat{m} (t, X_{i}) - {\hat{ψ}}_{t}^{OR} + {\hat{d}}_{t}^{T} {\hat{M}}^{- 1} S_{i} (\hat{β})

{\hat{IF}}_{i, RD} = {\hat{IF}}_{i, 1} - {\hat{IF}}_{i, 0}

The first term captures variation in the empirical covariate distribution; the second propagates uncertainty from fitting β.

Wald standard error and 95% confidence interval

\hat{SE} ({\hat{RD}}^{OR}) = \sqrt{\frac{Var ({\hat{IF}}_{i, RD})}{n}}, {CI}_{95 %} = {\hat{RD}}^{OR} \pm 1.96 \hat{SE}

Interpretation: this interval quantifies sampling uncertainty around the fitted plug-in estimator. It does not protect against unmeasured confounding or systematic outcome-model misspecification.

1. Subjects in the observational cohort

Color shows observed smoking status; point size shows fitted CVD risk under the treatment actually observed.

x y

Smoker (T=1)Non-smoker (T=0)Had CVD (Y=1)larger point = higher fitted observed risk

Click any subject. Outcome regression will use that subject's same X twice—once with smoking set to 0 and once with smoking set to 1.

Selected subject

Observed data provide one treatment state; the fitted model supplies both conditional risk predictions.

2. Fitted outcome-risk curves with 95% bands

Solid curves are fitted logistic-regression predictions for a reference subject. Points summarize observed CVD risk in covariate bins.

Horizontal axis

Fitted T=0 Fitted T=1 True curvesshading = pointwise 95% model band

3. Target covariate distribution

Outcome regression does not rebalance the observed arms. It carries the full cohort's covariate distribution into both prediction worlds.

Covariate

Full target cohortObserved smokersObserved non-smokers

The gray distribution—not either observed treatment arm—is the standardization target for both m̂(0,X) and m̂(1,X).

4. Two predictions for the same subjects

Each row is one actual subject. The filled point is the fitted risk in the observed treatment state; the open point is the model-based counterfactual prediction.

Prediction under T=0Prediction under T=1filled = observed state; open = counterfactual state

These are conditional risk predictions, not observed potential outcomes. Their causal interpretation requires measured confounding control, overlap, and a reasonable outcome model.

5. Standardize over the full cohort

Average all predicted risks in each treatment world.

Outcome-regression standardization:

{\hat{ψ}}_{t}^{OR} = \frac{1}{n} \sum_{i = 1}^{n} \hat{m} (t, X_{i}), {\hat{RD}}^{OR} = {\hat{ψ}}_{1}^{OR} - {\hat{ψ}}_{0}^{OR}

6. Crude versus standardized estimates

The regression coefficient is conditional; the causal comparison shown here comes from marginalizing predicted risks over the target cohort.

7. Current cohort: 95% confidence intervals

Compare the crude estimate, the selected outcome model, and an alternative model specification as a sensitivity analysis.

Point estimateKnown true ATEgray dashed line = no effect

Intervals use a robust influence-function/delta-method variance that incorporates fitting the logistic outcome model and averaging over the empirical covariate distribution. A misspecified model can yield a narrow CI around the wrong target.

8. Repeated-sampling coverage experiment

Hold the baseline covariates fixed, regenerate smoking and CVD, refit the outcome model, and check whether each 95% CI contains the known conditional ATE.

Show Samples

No experiment run yet0%

Run the experiment to see long-run confidence-interval coverage under the selected outcome model.

Green intervals contain the truth; red intervals miss it. Correct specification can give approximately nominal coverage as sample size grows. Systematic model bias is not repaired by increasing the number of repetitions.

Inspect subject-level counterfactual predictions

Subject	Age	BMI	Income	Female	College	HTN	Smoker	CVD	m̂0(X)	m̂1(X)	Individual RD	Observed fitted risk	Residual	Opposite-arm distance