Marginal Sensitivity Models for Propensity Weights

Why Bound Propensity Weights?

Inverse-probability weighting uses treatment probabilities to represent potential-outcome means. If treatment assignment depends on an unobserved covariate, however, a propensity score based only on observed covariates may not capture the full assignment mechanism. A marginal sensitivity model addresses this problem by allowing the true treatment odds to differ from the observed-propensity odds by a bounded factor.

The sensitivity parameter \(\Gamma\) controls the permitted departure. At \(\Gamma=1\), the model allows no departure. Larger values permit more hidden treatment selection and generally widen the resulting bounds. The model does not, by itself, establish a causal interpretation: that interpretation also requires assumptions linking treatment assignment to potential outcomes.

Model, Assumptions, and Weight Restrictions

Let \(A\in\{0,1\}\) denote treatment, \(X\) the observed covariates, and \(U\) unobserved covariates that may affect treatment assignment. Let \(Y(1)\) and \(Y(0)\) be the potential outcomes, and assume consistency: the observed outcome is \(Y=AY(1)+(1-A)Y(0)\). Define the observed propensity \(\pi(X)=P(A=1\mid X)\) and the true propensity \(e(X,U)=P(A=1\mid X,U)\).

Assumptions: For the weighted potential-outcome identities below, assume consistency, conditional exchangeability \((Y(1),Y(0))\perp A\mid(X,U)\), positivity \(0<e(X,U)<1\) and \(0<\pi(X)<1\) on the target population, and finite expectations for the weighted outcomes. The sensitivity restriction bounds how \(e(X,U)\) differs from \(\pi(X)\); it does not replace conditional exchangeability.

The marginal sensitivity model bounds the ratio of treatment odds conditional on \((X,U)\) to treatment odds conditional on \(X\). For a specified \(\Gamma\geq1\), require

\frac{1}{\Gamma} \leq \frac{e(X,U)/(1-e(X,U))}{\pi(X)/(1-\pi(X))} \leq \Gamma.

The inverse-probability weights that convert observed treated and control outcomes into potential-outcome means involve, respectively, \(w_1(X,U)=\pi(X)/e(X,U)\) and \(w_0(X,U)=(1-\pi(X))/(1-e(X,U))\). To find their permitted ranges, write the odds ratio in the restriction as \(r\), so that \(1/\Gamma\leq r\leq\Gamma\). Solving for \(e\) gives \(e=r\pi/(1-\pi+r\pi)\). Substituting this expression into each weight yields

\pi+\frac{1-\pi}{\Gamma} \leq w_1 \leq \pi+\Gamma(1-\pi), \qquad 1-\pi+\frac{\pi}{\Gamma} \leq w_0 \leq 1-\pi+\Gamma\pi.

These are covariate-dependent limits: the ranges can differ across units when their observed propensities differ. The two weights also arise from the same underlying propensity, so their limits should not be mistaken for independent assumptions about arbitrary outcomes. In the separate mean optimizations below, each outcome group is weighted under its own observed group distribution.

Worked Example: Converting an Odds Restriction to Weight Limits

Suppose an individual has observed propensity \(\pi=0.40\), and take \(\Gamma=3\). The odds ratio \(r\) may range from \(1/3\) to \(3\). At the endpoints, the corresponding true propensities are

e_{\min}=\frac{(1/3)(0.40)}{0.60+(1/3)(0.40)} =\frac{0.1333}{0.7333}\approx0.1818, \qquad e_{\max}=\frac{3(0.40)}{0.60+3(0.40)}=0.6667.

Thus the treated weight runs from \(\pi/e_{\max}=0.40/0.6667=0.6000\) to \(\pi/e_{\min}=0.40/0.1818\approx2.200\). Equivalently, its formula gives lower limit \(0.40+0.60/3=0.6000\) and upper limit \(0.40+3(0.60)=2.200\). The control weight limits are \(0.60+0.40/3=0.7333\) and \(0.60+3(0.40)=1.800\). These are weight limits, not confidence limits for the propensity.

From Conditional Exchangeability to Weighted Means

The crucial causal step is justified by conditional exchangeability given \((X,U)\). For treatment \(a=1\), exchangeability and the definition of \(e\) imply \(E[A Y(1)/e(X,U)\mid X,U]=E[Y(1)\mid X,U]\). Consistency gives \(AY=AY(1)\), so taking expectations and rewriting \(1/e=(1/\pi)(\pi/e)\) gives the treated identity. The control identity follows in the same way using \(1-A\).

E[Y(1)] = E\left[\frac{AY}{\pi(X)}w_1(X,U)\right], \qquad E[Y(0)] = E\left[\frac{(1-A)Y}{1-\pi(X)}w_0(X,U)\right].

The weights must also satisfy normalization. Applying the same conditional expectation calculation without an outcome gives

E\left[\frac{A}{\pi(X)}w_1(X,U)\right]=1, \qquad E\left[\frac{1-A}{1-\pi(X)}w_0(X,U)\right]=1.

Consequently, the bound for either mean is an optimization problem: minimize or maximize its weighted-outcome expression, subject to the relevant pointwise weight limits and normalization. The normalization matters because otherwise changing all weights in one direction could change the scale of the weighted mean rather than represent a reweighting of the observed group.

Optimizing the Weighted Means

For a finite observed sample, the expectation can be represented by a sample average. For example, with estimated or specified \(\pi_i\), the treated mean optimization uses base masses \(b_i=A_i/(n\pi_i)\) and chooses weights \(w_i\) within their sensitivity limits. The objective is \(\sum_i b_iw_iY_i\), and the normalization is \(\sum_i b_iw_i=1\). For the control mean, use \(b_i=(1-A_i)/(n(1-\pi_i))\) and the control weight limits.

The optimization has a simple ordering principle. Write \(z_i=b_iw_i\), the reweighted mass assigned to observation \(i\). Its permitted range is \(b_i\ell_i\leq z_i\leq b_iu_i\), where \(\ell_i,u_i\) are the applicable weight limits, and its total must equal one. To minimize a mean, assign as much permitted mass as possible to the smallest outcomes; to maximize it, assign as much as possible to the largest outcomes. If a final observation receives only part of its available additional mass, that partial allocation fills the normalization exactly.

Optimization result: For fixed observed outcomes and feasible weight limits, a minimum weighted mean is obtained by allocating additional normalized weight in ascending outcome order; a maximum is obtained in descending outcome order. This follows because transferring a small amount of mass from a higher outcome to a lower one reduces the weighted sum by the transferred amount times the outcome difference, while preserving total mass.

Let \([\underline{\mu}_1,\overline{\mu}_1]\) and \([\underline{\mu}_0,\overline{\mu}_0]\) denote the resulting lower and upper bounds for the two potential-outcome means. Since the average treatment effect is \(E[Y(1)]-E[Y(0)]\), its bounds are

\underline{\tau}=\underline{\mu}_1-\overline{\mu}_0, \qquad \overline{\tau}=\overline{\mu}_1-\underline{\mu}_0.

The lower endpoint combines the smallest permitted treated mean with the largest permitted control mean; the upper endpoint reverses those choices. These are bounds on the average treatment effect under the stated causal assumptions and sensitivity model, not a point estimate with a conventional standard error.

Worked Example: Computing Sensitivity Bounds for an Average Treatment Effect

Consider four hypothetical observations with \(\pi_i=0.50\) for every unit and \(\Gamma=2\). Two units are treated, with outcomes 2 and 8; the two controls have outcomes 1 and 5. The empirical base mass for each observed unit in its treatment group is \(1/(4\cdot0.50)=0.50\). For either group, the weight limits are \(0.50+0.50/2=0.75\) and \(0.50+2(0.50)=1.50\). The base masses sum to one in each group, and normalization therefore requires the two selected weights to sum to two.

For the treated mean minimum, put the larger weight on outcome 2: choose weights 1.25 and 0.75, which are within the limits and sum to two. The value is

\underline{\mu}_1 =0.50(1.25\cdot2+0.75\cdot8) =0.50(2.50+6.00)=4.25.

For its maximum, reverse those weights:

\overline{\mu}_1 =0.50(0.75\cdot2+1.25\cdot8) =0.50(1.50+10.00)=5.75.

For controls, the minimum places weight 1.25 on outcome 1 and 0.75 on outcome 5, giving \(\underline{\mu}_0=0.50(1.25+3.75)=2.50\). The maximum reverses the weights, giving \(\overline{\mu}_0=0.50(0.75+6.25)=3.50\). Thus the average treatment effect is bounded by

[\,\underline{\mu}_1-\overline{\mu}_0,\; \overline{\mu}_1-\underline{\mu}_0\,] = [\,4.25-3.50,\;5.75-2.50\,] =[\,0.75,\;3.25\,].

Under this hypothetical model, the average treatment effect lies between 0.75 and 3.25 outcome units. The observed treated-minus-control difference is \( (2+8)/2-(1+5)/2=2\), which is inside the sensitivity interval, but it is not itself the full range of effects allowed by the model.

Worked Example: Allocating Weight by Outcome Order

Suppose the treated group has three outcomes \(0,4,10\), equal base masses \(b_i=1/3\), and common weight limits 0.75 to 1.50. The initial permitted reweighted mass for each observation is \(b_i\ell_i=(1/3)(0.75)=0.25\), for a total of 0.75. The normalization requires total mass one, so an additional 0.25 must be allocated. Each observation can receive up to \((1/3)(1.50-0.75)=0.25\) additional mass.

To minimize the mean, allocate all additional mass to outcome 0. The resulting masses are \(0.50,0.25,0.25\), sum to one, and correspond to weights \(1.50,0.75,0.75\). The minimum is

$$ 0.50(0)+0.25(4)+0.25(10)=3.50. $$

To maximize the mean, allocate the additional mass to outcome 10 instead. The masses are \(0.25,0.25,0.50\), corresponding to weights \(0.75,0.75,1.50\), and the maximum is \(0.25(0)+0.25(4)+0.50(10)=6.00\). The normalization and weight limits hold in both cases. This allocation rule is the finite-sample version of the outcome-ordering argument, and it also handles unequal base masses by allocating each unit's available mass in outcome order.

Interpretation, Scope, and a Common Pitfall

The parameter \(\Gamma\) is a sensitivity restriction, not a quantity identified from the observed outcomes alone. Analysts can report how the effect interval changes across substantively chosen values of \(\Gamma\), but the bounds are conditional on the model and assumptions. With \(\Gamma=1\), both weight ranges collapse to one: \(w_1=w_0=1\). Increasing \(\Gamma\) expands the allowable weights and cannot make the set of model-compatible effects smaller.

A frequent error is to claim that bounding the propensity odds ratio alone proves the weighted-mean identities. It does not. For example, let \(X\) and \(U\) be constant, let \(A\) be Bernoulli with probability 0.5, and set both potential outcomes equal to \(A\). Then \(e=\pi=0.5\), so the restriction holds with \(\Gamma=1\), but the potential outcomes are not independent of treatment. The treated and control weighted expressions are 1 and 0, while both potential-outcome means are 0.5. The correction is to state and justify conditional exchangeability given \((X,U)\) before using the identities. Without it, the propensity restriction alone does not identify these means or imply the stated causal bounds.

A second error is to optimize each outcome mean without enforcing normalization. Weight limits constrain individual values, but a valid reweighted mean must still have total mass one. A third is to subtract lower bound from lower bound when constructing the ATE interval. Because the control mean is subtracted, the lower ATE endpoint uses the upper control mean, and vice versa.

Key takeaway: A marginal sensitivity model bounds latent-confounder propensity weights through an odds-ratio restriction. Under consistency, positivity, finite weighted expectations, and conditional exchangeability given observed and unobserved covariates, optimize normalized treated and control weighted means separately, then subtract opposite endpoints to obtain the ATE bounds.

Check Your Understanding

Use the model definitions and optimization logic above to answer these questions.

For \(\pi=0.25\) and \(\Gamma=2\), calculate the lower and upper limits for \(w_1\) and \(w_0\).
Which assumption, beyond the propensity odds restriction, justifies expressing \(E[Y(1)]\) through a treated-outcome weighted expectation?
Why must an optimization of a weighted mean include a normalization constraint?
For a minimum weighted mean, should additional weight be allocated first to the smallest or largest outcomes? Explain briefly.
If the separate mean bounds are \([3,6]\) for treatment and \([1,4]\) for control, what are the implied average treatment effect bounds?