PO.ID.Partial.Sensitivity.MSM

IPW / tower bridge. Assuming potential and observed outcomes agree under the realized treatment (consistency), the true complete propensity e₀ = P[D=1 | σ(X, Y(1))] is positive almost surely, the treated potential outcome Y(1) is integrable, and the inverse-propensity-weighted candidate A·Y/e₀ is integrable, then evaluated at the true complete propensity, the candidate IPW mean recovers the estimand: candMean e₀ = E[A · Y / e₀] = E[Y(1)].

The MSM bound: E[Y(1)] lies in the MSM interval. Assuming the true complete propensity belongs to the marginal-sensitivity ambiguity set, the IPW/tower bridge identity candMean e₀ = E[Y(1)] holds, and the candidate mean is bounded below and above over the ambiguity set, the estimand E[Y(1)] is sandwiched between the infimum and supremum of the candidate mean over the ambiguity set:

Point identification at Λ = 1 (no-unmeasured-confounding collapse). Assume the propensity score for treatment given the covariates lies strictly between 0 and 1 almost surely (overlap). If a candidate complete propensity ẽ belongs to the marginal-sensitivity ambiguity set at Λ = 1, then it equals the observed propensity score almost everywhere.

Closed form of the MSM upper bound. Fix a sensitivity parameter Λ at least 1. If the propensity score lies strictly between 0 and 1 almost everywhere (two-sided overlap), every candidate propensity in the odds-ratio ambiguity set is measurable up to null sets, and the envelope A·|Y|·wMax(Λ) — which dominates every candidate IPW integrand — is integrable, then the supremum of the candidate IPW mean over the ambiguity set is attained pointwise: the MSM upper bound equals E[A·Y·(wMax if Y≥0 else wMin)].

Closed form of the MSM lower bound. Fix a sensitivity parameter Λ at least 1. If the propensity score lies strictly between 0 and 1 almost everywhere (two-sided overlap), every candidate propensity in the odds-ratio ambiguity set is measurable up to null sets, and the envelope A·|Y|·wMax(Λ) — which dominates every candidate IPW integrand — is integrable, then the infimum of the candidate IPW mean over the ambiguity set is attained pointwise: the MSM lower bound equals E[A·Y·(wMin if Y≥0 else wMax)].

The sharp bound is valid. Assuming the true complete propensity belongs to the calibrated (sharp) marginal-sensitivity ambiguity set, the IPW/tower bridge identity candMean e₀ = E[Y(1)] holds, and the candidate mean is bounded below and above over the calibrated ambiguity set, then the estimand E[Y(1)] lies in the calibrated (sharp) interval [msmLowerCalib Λ, msmUpperCalib Λ].

The sharp upper bound is tighter than the ZSB bound. Assuming the calibrated candidate-mean image is nonempty and the candidate mean is bounded above over the uncalibrated odds-ratio-box ambiguity set, the sharp (calibrated) upper bound is at most the ZSB (uncalibrated) upper bound: msmUpperCalib Λ ≤ msmUpper Λ. The calibrated set is smaller, so its candidate-mean supremum can only decrease.

The control MSM bound is valid. For a sensitivity parameter Λ at which the true complete control propensity P[D=0∣σ(X,Y(0))] belongs to the control ambiguity set, assuming the candidate IPW mean evaluated at the true complete control propensity recovers E[Y(0)] and that the candidate means over the ambiguity set are bounded below and bounded above, the mean control potential outcome E[Y(0)] lies in the closed interval between the control MSM lower and upper bounds.

Point identification at Λ = 1 (control arm, no-unmeasured-confounding collapse). Assume the observed control propensity score P[D=0∣σ(X)] is almost surely strictly between 0 and 1. Then for any candidate complete control propensity ẽ belonging to the (uncalibrated) control ambiguity set at sensitivity level Λ = 1, ẽ agrees almost everywhere with the observed control propensity score: at Λ = 1 the odds-ratio ambiguity set collapses to the single point given by no unmeasured confounding.

The sharp control bound is valid. For a sensitivity parameter Λ at which the true complete control propensity P[D=0∣σ(X,Y(0))] belongs to the calibrated control ambiguity set, assuming the candidate IPW mean evaluated at the true complete control propensity recovers E[Y(0)] and that the candidate means over the calibrated ambiguity set are bounded below and bounded above, the mean control potential outcome E[Y(0)] lies in the closed interval between the sharp control lower and upper bounds.

Optimality of the quantile-cutoff weight (Neyman–Pearson exchange). Fix a sensitivity parameter Λ at least 1 and assume the propensity score for treatment given the covariates lies strictly between 0 and 1 almost surely (overlap). For a σ(X)-measurable, integrable cutoff function c whose induced cutoff candidate is feasible and calibrated, and under envelope-integrability conditions bounding the treated outcome, the treatment-weighted mass, and the cutoff-weighted mass by the upper marginal-sensitivity-model weight, then for any other calibrated, box-feasible candidate complete propensity ẽ that is almost-everywhere measurable, ẽ's candidate mean is at most the cutoff candidate mean — the cutoff weight maximizes the candidate mean among calibrated candidates.

The sharp upper bound has the quantile-balancing closed form. Fix a sensitivity parameter Λ at least 1 and assume the propensity score for treatment given the covariates lies strictly between 0 and 1 almost surely (overlap). For a σ(X)-measurable, integrable cutoff function c whose induced cutoff candidate is feasible and calibrated, under envelope-integrability conditions bounding the treated outcome, the treatment-weighted mass, and the cutoff-weighted mass by the upper marginal-sensitivity-model weight, and assuming every candidate complete propensity in the calibrated ambiguity set is almost-everywhere measurable, then the sharp upper bound on E[Y(1)] equals the candidate mean of the cutoff propensity built from c.

Lower control calibration bracket is at most one. Fix a sensitivity parameter Λ at least 1. If the control propensity P[D=0∣X] lies strictly between 0 and 1 almost everywhere (overlap), then almost everywhere the lower calibration weight wMin0 Λ times the control propensity is at most 1.

Upper control calibration bracket is at least one. Fix a sensitivity parameter Λ at least 1. If the control propensity P[D=0∣X] lies strictly between 0 and 1 almost everywhere (overlap), then almost everywhere the upper calibration weight wMax0 Λ times the control propensity is at least 1.

All-lower-weight control calibration value. For a sensitivity parameter Λ, if the product of the control indicator 1{D=0} and the lower calibration weight wMin0 Λ is integrable, then the conditional expectation of that product given the covariate σ-algebra equals, almost everywhere, the lower weight times the control propensity P[D=0∣X].

All-upper-weight control calibration value. For a sensitivity parameter Λ, if the product of the control indicator 1{D=0} and the upper calibration weight wMax0 Λ is integrable, then the conditional expectation of that product given the covariate σ-algebra equals, almost everywhere, the upper weight times the control propensity P[D=0∣X].

Cutoff control calibration value decomposes into a lower bracket plus a survival term. Fix a sensitivity parameter Λ and a cutoff function c measurable with respect to the covariate σ-algebra. If the ratio of the control indicator 1{D=0} to the cutoff-calibration weight cutoffProp0 Λ c is integrable, the product of the control indicator and the survival-cutoff indicator 1{c<Y} is integrable, the product of the control indicator and the lower weight wMin0 Λ is integrable, and the product of the weight spread wMax0 Λ - wMin0 Λ with the control-indicator survival term is integrable, then the conditional expectation, given the covariate σ-algebra, of the control indicator divided by the cutoff-calibration weight equals, almost everywhere, the lower weight times the control propensity plus the weight spread times the conditional control-survival functional controlSurv c.

A cutoff solving the target control-survival equation is calibrated. Fix a sensitivity parameter Λ strictly greater than 1. If the control propensity P[D=0∣X] lies strictly between 0 and 1 almost everywhere (overlap), the cutoff function c is measurable with respect to the covariate σ-algebra, the ratio of the control indicator 1{D=0} to the cutoff-calibration weight cutoffProp0 Λ c is integrable, the product of the control indicator and the survival-cutoff indicator 1{c<Y} is integrable, the product of the control indicator and the lower weight wMin0 Λ is integrable, the product of the weight spread with the control-indicator survival term is integrable, and the conditional control-survival functional at c equals, almost everywhere, the target survival value survTarget0 Λ, then the cutoff-calibration propensity cutoffProp0 Λ c is calibrated: the conditional expectation of the control indicator divided by it, given the covariates, equals 1 almost everywhere.

The control cutoff propensity always lies inside the control odds-ratio box. Fix a sensitivity parameter Λ at least 1 and a cutoff function c. If the control propensity P[D=0∣X] lies strictly between 0 and 1 almost everywhere (overlap), then the cutoff-calibration candidate propensity cutoffProp0 Λ c always belongs to the control marginal-sensitivity-model ambiguity set MSMSet0 Λ.

The cutoff belongs to the calibrated control MSM set whenever it solves the target control-survival equation. Fix a sensitivity parameter Λ strictly greater than 1. If the control propensity P[D=0∣X] lies strictly between 0 and 1 almost everywhere (overlap), the cutoff function c is measurable with respect to the covariate σ-algebra, the ratio of the control indicator 1{D=0} to the cutoff-calibration weight cutoffProp0 Λ c is integrable, the product of the control indicator and the survival-cutoff indicator 1{c<Y} is integrable, the product of the control indicator and the lower weight wMin0 Λ is integrable, the product of the weight spread with the control-indicator survival term is integrable, and the conditional control-survival functional at c equals, almost everywhere, the target survival value survTarget0 Λ, then the cutoff-calibration propensity cutoffProp0 Λ c belongs to the calibrated control MSM set MSMSetCalib0 Λ.

Decomposition of the cutoff calibration value. For a σ(X)-measurable cutoff function c, assume the treatment indicator divided by the cutoff propensity is integrable, the treated-survival indicator, weighted by the treatment indicator, is integrable, the product of the treatment indicator and the lower marginal-sensitivity-model weight wMin is integrable, and the gap between the upper and lower weights, weighted by the treatment-weighted treated-survival indicator, is integrable. Then pulling the σ(X)-measurable weights wMin, wMax out of the conditional expectation decomposes the cutoff calibration value: E[Z/cutoffProp Λ c | σ(X)] = wMin·e + (wMax − wMin)·G(c), where G(c) is the conditional treated-survival at cutoff c.

The cutoff propensity is always in the odds-ratio box. Fix a sensitivity parameter Λ at least 1 and assume the propensity score for treatment given the covariates lies strictly between 0 and 1 almost surely (overlap). Then for any measurable cutoff function c, the induced cutoff propensity always lies in the marginal-sensitivity-model odds-ratio ambiguity set, since at each point it equals either the lower or the upper marginal-sensitivity-model weight.

Existence of a calibrating cutoff (treated arm). Fix a sensitivity parameter Λ strictly greater than one. Assume the treated propensity score is almost surely strictly between 0 and 1 (two-sided overlap), that the treated-arm conditional law of the outcome given covariates is atomless, i.e. its conditional CDF is continuous, and that the calibration quantile level lies strictly between 0 and 1 almost everywhere. Then there exists a σ(X)-measurable cutoff function c such that the treatment-weighted conditional survival function at c agrees almost everywhere with the target survival function survTarget Λ; the cutoff is realized as the conditional quantile of the treated outcome law at the calibration level.

The sharp treated upper bound, unconditionally. Fix a sensitivity parameter Λ strictly greater than one. Assume two-sided overlap of the treated propensity score, that the treated-arm conditional outcome law given covariates is atomless, i.e. its conditional CDF is continuous, that the calibration quantile level lies strictly between 0 and 1 almost everywhere, and that the candidate means over the calibrated ambiguity set are bounded above. If every calibrated candidate propensity is almost-everywhere measurable and every σ(X)-measurable cutoff satisfies the integrability conditions needed for the calibration and optimality arguments, then there exists a σ(X)-measurable cutoff function whose induced quantile-cutoff propensity is calibrated-feasible, at which the sharp (supremum) upper bound for E[Y(1)] equals the candidate mean.

The sharp control upper bound has a quantile-balancing closed form, unconditionally. Fix a sensitivity parameter Λ strictly greater than 1. If the control propensity P[D=0∣X] lies strictly between 0 and 1 almost everywhere (overlap), the control outcome's conditional law given the covariates is atomless (its conditional CDF is continuous), the calibration level lies strictly between 0 and 1 almost everywhere, every candidate propensity in the calibrated control ambiguity set is almost-everywhere measurable, and every covariate-measurable cutoff function satisfies the integrability conditions needed to evaluate the calibration and candidate-mean functionals at it, then there exists a covariate-measurable cutoff c such that the cutoff-calibration propensity cutoffProp0 Λ c lies in the calibrated control MSM set and the sharp control upper bound equals the candidate mean at that cutoff, msmUpperCalib0 Λ = candMean0 (cutoffProp0 Λ c).

The sharp ATE interval is valid. If the treated arm's mean potential outcome E[Y(1)] lies in the calibrated sharp interval [msmLowerCalib Λ, msmUpperCalib Λ] and the control arm's mean potential outcome E[Y(0)] lies in the calibrated sharp interval [msmLowerCalib0 Λ, msmUpperCalib0 Λ], then the true average treatment effect τ = E[Y(1)] − E[Y(0)] lies in the sharp interval [ateLowerCalib Λ, ateUpperCalib Λ].

The ZSB ATE interval is valid. If the treated arm's mean potential outcome E[Y(1)] lies in the uncalibrated ZSB interval [msmLower Λ, msmUpper Λ] and the control arm's mean potential outcome E[Y(0)] lies in the uncalibrated ZSB interval [msmLower0 Λ, msmUpper0 Λ], then the true average treatment effect τ = E[Y(1)] − E[Y(0)] lies in the ZSB interval [ateLower Λ, ateUpper Λ].

The sharp ATE interval is contained in the ZSB ATE interval. If the treated arm's calibrated sharp upper bound does not exceed its uncalibrated ZSB upper bound, the treated arm's uncalibrated ZSB lower bound does not exceed its calibrated sharp lower bound, the control arm's calibrated sharp upper bound does not exceed its uncalibrated ZSB upper bound, and the control arm's uncalibrated ZSB lower bound does not exceed its calibrated sharp lower bound, then the sharp ATE interval [ateLowerCalib Λ, ateUpperCalib Λ] is contained in the ZSB ATE interval [ateLower Λ, ateUpper Λ].

Optimality of the lower quantile-cutoff weight. Fix a sensitivity parameter Λ at least one. Assume the control propensity score is almost surely strictly between 0 and 1 (two-sided overlap), and let c be a σ(X)-measurable, integrable cutoff function such that the lower quantile-cutoff propensity it induces lies in the calibrated control ambiguity set. If the envelope 1_{D=0}·|Y|·wMax0(Λ) is integrable, the weighted control indicator 1_{D=0}·wMax0(Λ) is integrable, and the cutoff-weighted envelope |c|·1_{D=0}·wMax0(Λ) is integrable, then for any almost-everywhere measurable candidate complete control propensity ẽ in the calibrated ambiguity set, the candidate mean at the lower quantile-cutoff weight is no greater than the candidate mean at ẽ — the lower cutoff attains the minimum over all calibrated candidates.

The sharp lower bound has the quantile-balancing closed form. Fix a sensitivity parameter Λ at least one. Under two-sided overlap of the control propensity score, given a σ(X)-measurable, integrable cutoff c whose induced lower quantile-cutoff propensity is itself calibrated-feasible, and assuming the envelope, weighted-indicator, and cutoff-weighted envelope integrability conditions bounding the IPW integrands together with almost-everywhere measurability of every calibrated candidate propensity, the sharp (infimum) lower bound for E[Y(0)] over the calibrated ambiguity set equals the candidate mean evaluated at the lower quantile-cutoff propensity.

The sharp lower bound, unconditionally. Fix a sensitivity parameter Λ strictly greater than one. Assume two-sided overlap of the control propensity score, that the control-arm conditional law of the outcome given covariates is atomless, i.e. its conditional CDF is continuous, and that the lower calibration quantile level lies strictly between 0 and 1 almost everywhere. If every calibrated candidate propensity is almost-everywhere measurable and every σ(X)-measurable cutoff satisfies the integrability conditions needed for the calibration and optimality arguments, then there exists a σ(X)-measurable cutoff function whose induced lower quantile-cutoff propensity is calibrated-feasible, at which the sharp (infimum) lower bound for E[Y(0)] equals the candidate mean.

The sharp upper bound at the explicit Gaussian cutoff. Fix a sensitivity parameter Λ strictly greater than 1 and assume the propensity score for treatment given the covariates lies strictly between 0 and 1 almost surely (overlap). Under the conditional-Gaussian treated-outcome model, i.e. the treated conditional law of the outcome given the covariates is Gaussian with mean m and standard deviation σ, assuming every candidate complete propensity in the calibrated ambiguity set is almost-everywhere measurable and the integrability conditions needed to make the candidate means, the survival decomposition, and the cutoff propensity well defined, the sharp (Dorn–Guo) upper bound on E[Y(1)] equals the candidate IPW mean evaluated at the cutoff propensity built from the explicit Gaussian quantile cutoff m(X) + σ(X)·Φ⁻¹(Λ/(Λ+1)).

Optimality of the lower quantile-cutoff weight. Fix a sensitivity parameter Λ at least 1 and assume the propensity score for treatment given the covariates lies strictly between 0 and 1 almost surely (overlap). For a σ(X)-measurable, integrable cutoff function c whose induced lower-cutoff candidate is feasible and calibrated, and under envelope-integrability conditions bounding the treated outcome, the treatment-weighted mass, and the cutoff-weighted mass by the upper marginal-sensitivity-model weight, then for any other calibrated, box-feasible candidate complete propensity ẽ that is almost-everywhere measurable, the lower-cutoff candidate mean is at most ẽ's candidate mean — the lower cutoff minimizes the candidate mean among calibrated candidates. The ≥-mirror of cutoff_optimal.