Experimentation.DesignBased.HT
Horvitz-Thompson design-based estimators: totals, means, effects, unbiasedness, covariance formulas, and finite-population variance identities.
Estimator 6 core · 0 supporting Horvitz-Thompson estimators use inverse generalized exposure probabilities to estimate finite-population exposure totals, means, and contrasts. ★ htTotal★ htMean★ htEffect★ htTotal_eq
Horvitz-Thompson estimators
Horvitz-Thompson estimators use inverse generalized exposure probabilities to estimate finite-population exposure totals, means, and contrasts.
The main estimator is htTotal, the totalized inverse-probability-weighted estimator for exposure
d. Dividing by the number of units gives htMean, and differencing two exposure means gives
htEffect. The target functionals are muTrue and tauTrue. The lemma htTotal_eq rewrites the
estimator using the exposure-specific potential outcome y i d, a form used by the unbiasedness
and variance proofs.
On a finite assignment space and for a finite population with equality-comparable exposure conditions, a finite randomization design, a schedule of potential outcomes for every unit and exposure condition, an assignment-to-exposure rule, a map assigning each unit its traits, an exposure condition, and a realized assignment define the totalized Horvitz–Thompson estimator of the total potential outcome as the sum over units of each observed outcome multiplied by its exposure indicator and divided by its generalized exposure probability.
Definition (Lean source)
For a finite population, a schedule of potential outcomes for every unit and exposure condition and an exposure condition define the finite-population mean potential outcome as the sum of the units' potential outcomes under that condition divided by the number of units.
Definition (Lean source)
For a finite population, a schedule of potential outcomes for every unit and exposure condition and two exposure conditions define the finite-population average causal effect as the mean potential outcome under the first condition minus that under the second condition.
On a finite assignment space and for a finite population with equality-comparable exposure conditions, a finite randomization design, a schedule of potential outcomes for every unit and exposure condition, an assignment-to-exposure rule, a map assigning each unit its traits, an exposure condition, and a realized assignment define the Horvitz–Thompson estimator of the mean potential outcome as the totalized Horvitz–Thompson estimator divided by the number of units.
Definition (Lean source)
On a finite assignment space and for a finite population with equality-comparable exposure conditions, a finite randomization design, a schedule of potential outcomes for every unit and exposure condition, an assignment-to-exposure rule, a map assigning each unit its traits, two exposure conditions, and a realized assignment define the Horvitz–Thompson estimator of the average causal effect as the estimated mean under the first condition minus the estimated mean under the second condition.
Definition (Lean source)
For a finite design D, potential outcomes y, exposure map f, assignment θ, exposure level d, and realized assignment z, the Horvitz–Thompson total equals the same sum with the observed outcome replaced termwise by the potential outcome y i d, because on each unit's exposure-indicator term the observed outcome coincides with y i d.
Formal statement
Proof (Lean source)
Unbiased 3 core · 0 supporting Positive generalized exposure probabilities make the Horvitz-Thompson estimators exactly unbiased under a finite randomization design. ★ E_htTotal★ E_htMean★ E_htEffect
Horvitz-Thompson unbiasedness
Positive generalized exposure probabilities make the Horvitz-Thompson estimators exactly unbiased under a finite randomization design.
The theorem E_htTotal proves the total estimator has expectation ∑ᵢ y i d, because each
inverse-probability weight cancels E[1(expo i = d)]. The theorem E_htMean scales this to the
population mean muTrue, and E_htEffect proves unbiasedness of the exposure contrast estimator
htEffect for tauTrue.
Lemma 4.1 (expectation). Given a finite design, an outcome function y, an exposure map f, and an assignment θ, suppose every unit has nonzero probability of realizing exposure d. Then the Horvitz–Thompson total estimator has expectation exactly the population total ∑ᵢ y i d.
Formal statement
Proof (Lean source)
Given a finite design, an outcome function y, an exposure map f, and an assignment θ, suppose every unit has nonzero probability of realizing exposure d. Then the Horvitz–Thompson mean estimator has expectation exactly the population mean μ(d).
Formal statement
Proof (Lean source)
Proposition 4.4 (expectation). Given a finite design, an outcome function y, an exposure map f, and an assignment θ, suppose every unit has nonzero probability of realizing exposure dk and every unit has nonzero probability of realizing exposure dl. Then the Horvitz–Thompson effect estimator contrasting dk against dl has expectation exactly the average causal effect τ(dk,dl).
Formal statement
Proof (Lean source)
Variance 4 core · 0 supporting Horvitz-Thompson randomization variance reduces to finite covariance sums over exposure indicators. ★ Var_htTotal_cov★ Var_htTotal★ Cov_htTotal_cov★ Cov_htTotal
Horvitz-Thompson variance identities
Horvitz-Thompson randomization variance reduces to finite covariance sums over exposure indicators.
The covariance-form theorem Var_htTotal_cov expands the variance of htTotal as a double sum of
indicator covariances, and Var_htTotal rewrites those terms into the marginal and joint exposure
probabilities. The companion theorems Cov_htTotal_cov and Cov_htTotal give the analogous
covariance formulas for two exposure totals, including the diagonal cross-exposure term that
appears when one unit cannot occupy two distinct exposures at once.
Lemma 4.1 (variance), covariance form. For a finite design, outcome function y, exposure map f, and assignment θ, the randomization variance of the Horvitz–Thompson total under exposure d equals the double sum, over unit pairs, of inverse-probability-weighted indicator covariances.
Formal statement
Proof (Lean source)
Lemma 4.1 (variance), expanded form eq:total_variance. For a finite design, outcome function y, exposure map f, and assignment θ, the randomization variance of the Horvitz–Thompson total under exposure d equals the diagonal sum of inverse-probability-weighted indicator variances plus the off-diagonal sum of inverse-probability-weighted indicator covariances.
Formal statement
Proof (Lean source)
For a finite design, outcome function y, exposure map f, and assignment θ, the covariance between two Horvitz-Thompson exposure totals equals the double sum, over ordered unit pairs, of the two inverse-probability outcome weights times the covariance of the corresponding exposure indicators.
Formal statement
Proof (Lean source)
Proposition 4.4 (covariance), expanded form eq:totals_covariance. Given a finite design, an outcome function y, an exposure map f, and an assignment θ, suppose the two exposures dk and dl are distinct, every unit has nonzero probability of realizing exposure dk, and every unit has nonzero probability of realizing exposure dl. Then the covariance of the two Horvitz–Thompson totals equals the off-diagonal double sum of inverse-probability-weighted joint-exposure covariance terms, minus the diagonal term ∑ᵢ y_i(dk)·y_i(dl) — a unit cannot occupy two distinct exposures at once, so the diagonal cross-indicator vanishes and leaves this unidentified term.