Inverse Probability Weighted Estimator

Build the observational study

Controls update the synthetic cohort and fitted propensity model immediately.

Scenario presets

Data-generating process

Number of subjects

80500

Strength of confounding

treatment ≈ randomstrong selection by X

Overlap quality

poor positivitygood overlap

True smoking effect (odds ratio)

OR 1.0OR 3.0

Cohort seed

Propensity model

Variables included in logistic regression

Optionally cap very large weights. Capping changes the Horvitz–Thompson estimator and can disrupt moment balance.

Weight cap

more trimmingalmost uncapped

Apply the weights

Weighting transition

0% crude100% IPW

At 0%, constant inverse marginal-treatment weights reproduce the crude arm means in the HT formula. At 100%, the formal Horvitz–Thompson IPW estimator is used. Intermediate values are a pedagogical interpolation.

Mathematical formulation

The implementation uses the Horvitz–Thompson empirical mean throughout. Both potential-outcome means retain the original denominator n; there is no arm-specific self-normalization. Because this is the only IPW estimator used on the page, the superscript “HT” is omitted from the mathematical notation.

Graduate-level notation

Horvitz–Thompson estimation map

{\hat{e}}_{i} = \hat{P} (T_{i} = 1 | X_{i}) ⟶ {\hat{μ}}_{1}, {\hat{μ}}_{0} ⟶ {\hat{Δ}}_{ATE} = {\hat{μ}}_{1} - {\hat{μ}}_{0}

1Potential-outcome means

{\hat{μ}}_{1} = \frac{1}{n} \sum_{i = 1}^{n} \frac{T_{i} Y_{i}}{{\hat{e}}_{i}}, {\hat{μ}}_{0} = \frac{1}{n} \sum_{i = 1}^{n} \frac{(1 - T_{i}) Y_{i}}{1 - {\hat{e}}_{i}}

The denominator remains n. Therefore, finite-sample treated and control weight totals are not forced to equal n.

2ATE risk difference

{\hat{Δ}}_{ATE} = \frac{1}{n} \sum_{i = 1}^{n} [\frac{T_{i} Y_{i}}{{\hat{e}}_{i}} - \frac{(1 - T_{i}) Y_{i}}{1 - {\hat{e}}_{i}}]

This is the default estimator in every estimate, confidence-interval, and repeated-sampling panel at 100% weighting.

3Crude-to-HT transition

\begin{array}{l} w_{1 i} (α) = \frac{1 - α}{\hat{π}} + \frac{α}{{\hat{e}}_{i}} \\ w_{0 i} (α) = \frac{1 - α}{1 - \hat{π}} + \frac{α}{1 - {\hat{e}}_{i}} \end{array}

At α=0 the HT empirical means equal the crude arm means; at α=1 they equal the formal HT-IPW means.

Identification assumptions

Consistency

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

Conditional exchangeability

{Y (1), Y (0)} ⊥ T | X

Positivity

0 < e (X) < 1 a.s.

Identification step

E [\frac{T Y}{e (X)} | X] = E {Y (1) | X}

Integrating over X gives E{Y(1)}; the analogous control expression identifies E{Y(0)}.

1Propensity model

logit {e_{γ} (X_{i})} = Z_{i}^{ᵀ} γ

Selected model:

2Population balancing identity

E {\frac{T f (X)}{e (X)}} = E {f (X)} = E {\frac{(1 - T) f (X)}{1 - e (X)}}

The identity holds for every integrable function f, including constants, first moments, higher moments, centered moments, and indicators.

3Finite-sample check

{\hat{M}}_{1} (f) = \frac{1}{n} \sum \frac{T_{i} f (X_{i})}{{\hat{e}}_{i}}, \hat{M} (f) = \frac{1}{n} \sum f (X_{i}), {\hat{M}}_{0} (f) = \frac{1}{n} \sum \frac{(1 - T_{i}) f (X_{i})}{1 - {\hat{e}}_{i}}

Finite-sample equality is approximate. Model misspecification, poor overlap, capping, and sampling variation are visible in the new plots.

Stacked estimating equation for full HT-IPW

U_{i} (θ) = (\begin{matrix} Z_{i} {T_{i} - e_{γ} (X_{i})} \\ \frac{T_{i} Y_{i}}{e_{γ}} - μ_{1} \\ \frac{(1 - T_{i}) Y_{i}}{1 - e_{γ} (X_{i})} - μ_{0} \end{matrix})

1Sandwich variance

\hat{V} = \frac{1}{n} {\hat{A}}^{- 1} \hat{B} {\hat{A}}^{- ᵀ}

295% Wald interval

\hat{Δ} \pm 1.96 \hat{SE} (\hat{Δ})

Capping note. Enabling the cap changes the estimator to truncated HT. The implementation uses the corresponding piecewise derivative, with derivative zero above the cap.

1. Individual subjects in the observational cohort

Point size represents the current subject weight. Hover or click a subject for the exact calculation.

x y

Smoker (T=1)Non-smoker (T=0)Had CVD (Y=1)larger point = larger current weight

Drag the Weighting transition slider: rare observed treatment assignments expand because those subjects represent more people in the weighted pseudo-population.

Selected subject

IPW is calculated subject by subject, not by naming a profile.

2. Covariate distributions

Compare the observed arms with the weighted pseudo-population.

Covariate

SmokersNon-smokers

The density curves are normalized only for visual comparison. The causal estimator itself remains Horvitz–Thompson with denominator n.

3. Covariate balance

Absolute standardized mean differences (SMDs), before versus current weighting.

ObservedCurrent weightingvertical line = 0.10 guideline

A smaller SMD is better. IPW cannot repair omitted confounders, severe positivity violations, or an incorrect propensity model.

4A. Population balancing identity for a selected function

Choose a covariate and f(X), then compare the target empirical moment with its treated- and control-arm HT reconstructions.

Xj f(X) Weights

E {\frac{T f (X)}{e (X)}} = E {f (X)} = E {\frac{(1 - T) f (X)}{1 - e (X)}}

D_t(f)={M̂_t(f)−M̂(f)}/SD{f(X)}. For f(X)=1, D is the HT mass error. The 0.05 and 0.10 guides are descriptive, not formal test cutoffs.

4B. Balance profile across several functions

Inspect mass, mean, higher moments, centered variance, skewness, and upper-tail balance simultaneously.

|Treated discrepancy||Control discrepancy|guides = 0.05 and 0.10

First-moment balance alone does not guarantee balance of variance, skewness, or tail probabilities. Compare fitted, uncapped, true, and current-transition weights.

5. Propensity overlap and inverse weights

The fitted propensity score is P(smoker | measured baseline covariates).

Smoker propensity distributionNon-smoker propensity distribution

A smoker with a very small fitted propensity receives 1/ê; a non-smoker with a very large fitted propensity receives 1/(1−ê). Both situations create large weights.

6. Crude versus Horvitz–Thompson causal estimate

HT treatment-specific risks use the original cohort size n as the denominator.

7. The current cohort: 95% confidence intervals

Point estimates, 95% Wald intervals, and an explicit check of whether each interval contains the known causal truth.

Point estimateKnown true ATEgray dashed line = no effect

The instant intervals use a stacked M-estimation sandwich for the fitted logistic propensity model and the two Horvitz–Thompson means. Capped weights are linearized piecewise. A 95% CI is not guaranteed to contain the truth in one dataset; 95% refers to long-run repeated-sampling coverage.

8. Repeated-sampling coverage experiment

Regenerate smoking and CVD outcomes for the same baseline covariates, refit the propensity model, and examine many 95% CIs.

Show Samples

No experiment run yet0%

Run the experiment to see which individual intervals contain the truth and what fraction cover it overall.

Green intervals contain the known conditional ATE; red intervals miss it. The experiment keeps the observed baseline covariates fixed, then repeatedly generates treatment and outcome, refits the propensity score, and recomputes sandwich CIs. This makes the true conditional ATE common to every repetition.

Inspect the subjects with the largest weights

Subject	Age	BMI	Income ratio	Female	College	Hypertension	Smoker	CVD	ê	Raw IPW	Current w