Sharp Mechanism Bounds, Average Effects, and Inference
The preceding inequality gives a lower bound on the mass of stayers whose outcomes change. To show that the bound is sharp, we must construct a model attaining it while reproducing every observed distribution. (Kwon and Roth 2024) does this by assigning equal outcomes to as many stayers as the overlap between the observed distributions permits.
1. The sharpness result
Use principal-stratum shares \(\theta_{lk}\) and observable differences \(\Delta_k(A)\) from the mediation setup.
Theorem 1. For each feasible \(\theta\), the paper constructs potential outcomes and mediators matching the observed data and attaining
simultaneously across mediator states. Consequently, if one feasible \(\theta\) satisfies all the null inequalities, there is a compatible potential-outcome model satisfying the sharp null of full mediation.
2. Couple the common part first
Choose a common dominating measure for the finitely many outcome subdistributions. For two probability densities \(f\) and \(g\), their overlapping mass is
A maximal coupling places equal outcomes on the common mass. Only the residual mass needs unequal outcomes. In the mediation construction, entrants and leavers can explain part of the discrepancy between observed cells; whatever remains must belong to stayers whose outcomes change.
This allocation must also preserve every mediator margin. Once those constraints are satisfied, the coupling attains the lower bound on the affected fraction. A comparison of conditional means would lose the distributional information needed for this construction.
3. Average direct effects use trimming
Suppose a fraction \(a_d=\theta_{kk}/\Pr(M=m_k\mid D=d)>0\) of an observed cell consists of stayers. With conditional outcome quantile function \(Q_{dk}\), their mean under treatment state \(d\) lies between
The lowest mean assigns stayers to the bottom \(a_d\) fraction; the highest assigns them to the top. Quantile integrals handle atoms by allocating the appropriate fraction of a tied mass. Subtracting the opposing endpoints bounds the mean direct effect among stayers, assuming finite means.
When the stayer shares are unknown, optimize over their joint feasible set. The affected fraction and the average effect require separate objectives: a large number of small effects can give a large fraction with a small average.
4. Implementing the test
For discrete outcomes, the supremum over sets is the sum of positive cell-probability differences. The feasibility restrictions become a linear program when restrictions on \(\theta\) are linear. Sampling uncertainty then requires an appropriate moment-inequality or constrained-inference procedure; simply checking whether estimated inequalities hold ignores noise.
Merging outcome cells reduces the collection of sets available for detecting a discrepancy and hence weakens the restriction. Merging mediator states changes the definition of a stayer itself. It requires a separate justification.
5. Unequal cell masses and the overlap formula
The observed outcome distributions within a mediator state are subprobability measures, because their masses equal the mediator cell probabilities. Let their densities with respect to a common dominating measure be \(f_{0k},f_{1k}\), with masses \(p_{0k},p_{1k}\). Their common mass is
To maximize a signed integral over sets, include the points where the density is positive and exclude those where it is negative. Applying this observation gives
The entrant mass is \(\sum_{l\ne k}\theta_{lk}=p_{1k}-\theta_{kk}\). Subtracting it gives the especially simple form
Thus the question is whether the observed common outcome mass can accommodate all stayers. If \(O_k\geq\theta_{kk}\), every stayer can keep the same outcome. If \(O_k<\theta_{kk}\), the remaining stayer mass must change outcome. Comparing normalized conditional densities alone would miss the different mediator cell masses in this calculation.
6. Constructing the coupling
Allocate \(\min(\theta_{kk},O_k)\) of the common submeasure to stayers with equal outcomes. If this fills the stayer mass, distribute the remaining observed cell masses among entrants and leavers according to their feasible stratum shares.
If \(O_k<\theta_{kk}\), first exhaust the common part. The two residual outcome submeasures then live on disjoint density regions up to null sets: at each point at least one residual density is zero. Choose \(\theta_{kk}-O_k\) mass from each residual side for the remaining stayers and couple those outcomes. Their outcomes necessarily differ. There is enough mass on each side because \(\theta_{kk}\leq\min(p_{0k},p_{1k})\).
The remaining control mass equals \(p_{0k}-\theta_{kk}\) and the remaining treated mass equals \(p_{1k}-\theta_{kk}\), exactly the leaver and entrant masses. They can be split among the appropriate off-diagonal strata. For a mover, the two observed outcomes concern different mediator values, so equality of treatment-state outcomes at a fixed mediator value can be completed using the unobserved potential outcomes. Stayers are the strata that reveal both treatment states at the same mediator value.
Using a single feasible matrix makes these allocations consistent across mediator states. If every \(\theta_{kk}\leq O_k\), each stayer can keep the same outcome and the unobserved potential outcomes can be completed to satisfy full mediation. Thus the observed inequalities are sufficient as well as necessary.
For a numerical construction, use the three outcome values \(0,1,2\) and the subdistributions \((0.18,0.04,0.18)\) under treatment and \((0.10,0.20,0.10)\) under control. Their common part is \((0.10,0.04,0.10)\), of mass \(0.24\). If \(\theta_{kk}=0.30\), allocate this entire common part to unchanged stayers.
The residual treated masses are \((0.08,0,0.08)\) and the residual control masses are \((0,0.16,0)\). Allocate \(0.06\) from treated outcome zero and control outcome one to changed stayers. This produces exactly \(0.30\) stayer mass, of which \(0.06\) changes outcome. The remaining treated mass is \((0.02,0,0.08)\) and the remaining control mass is \((0,0.10,0)\), each totaling the required \(0.10\) entrant or leaver mass.
The lower bound is attained: no arrangement can match more than the common mass \(0.24\), and this one matches all of it. Assigning the changed treated mass to outcome two instead would attain the same affected fraction with a positive rather than negative direct effect. The data can therefore force some individuals to be affected without identifying the direction of their average effect.
For this same example, stayers occupy three quarters of each observed cell. The smallest feasible treated stayer total is \(0.18\cdot0+0.04\cdot1+0.08\cdot2=0.20\); the largest is \(0.08\cdot0+0.04\cdot1+0.18\cdot2=0.40\). Dividing by \(0.30\) gives means between \(2/3\) and \(4/3\). The control stayer mean has the same endpoints, obtained by trimming its bottom or top \(0.30\) mass. Thus the separate sharp mean-effect bounds are \([-2/3,2/3]\), while the affected-fraction lower bound is \(0.20\).
The two calculations answer different questions about the same data. Overlap determines how many stayers must change outcome; trimming determines the smallest and largest means that the stayer group can have.
7. Deriving trimmed means by an exchange argument
Consider one observed cell with distribution \(F\) and a stayer fraction \(a\in(0,1]\). Any admissible allocation of stayers can be represented by a selection probability \(s(y)\in[0,1]\) satisfying \(\int s\,dF=a\). Their mean is
To minimize it, suppose some higher outcome \(y_2\) receives positive selection mass while a lower outcome \(y_1\) is not fully selected. Moving a small equal amount of selection probability from \(y_2\) to \(y_1\) preserves the mass constraint and lowers the numerator by that mass times \(y_2-y_1\). Iterating this exchange fills the bottom \(a\) fraction first. This yields \(a^{-1}\int_0^aQ(u)\,du\); the upper bound fills the top fraction.
If \(F\) puts probability \(0.7\) at zero and \(0.3\) at ten, and \(a=0.5\), the minimum stayer mean is zero. The maximum selects all \(0.3\) mass at ten and \(0.2\) at zero, giving
The maximizing allocation uses only part of the atom at zero. An all-or-nothing rule for atoms would miss the required stayer mass; the quantile integral allows the needed fraction.
8. Fractions affected and average effects answer different questions
Apply the trimming bounds separately to the two observed mediator cells, using \(a_d=\theta_{kk}/p_{dk}\). The lowest mean direct effect among stayers uses the treated lower mean minus the control upper mean; the highest uses the opposite endpoints. These bounds concern a specified feasible stayer share, and unknown shares must be optimized over their joint feasible set.
Overlap bounds the minimum mass of stayers whose outcomes change, without measuring the size or sign of those changes. The mean effect also depends on outcome magnitudes, and positive and negative effects may cancel. Separate sharp bounds for these objectives need not be jointly attainable under one construction.
In a sample, the cell probabilities are estimated. For discrete outcomes, retaining the inequalities for individual outcome subsets and the stratum constraints gives linear population restrictions. A moment-inequality procedure must account for error in their estimated coefficients. Exact feasibility of empirical proportions alone does not provide a test with controlled rejection probability.
Making the trimming argument valid for continuous distributions
The exchange argument can be expressed as one integral inequality. For \(0<a<1\), let \(q=Q_F(a)\) and select all outcomes below \(q\), none above it, and the fraction of the atom at \(q\) needed to obtain total mass \(a\). Call this rule \(s_*\); the quantile inequalities \(F(q-)\leq a\leq F(q)\) ensure that the fractional selection is feasible. For every other admissible \(s\),
Below \(q\), both factors are nonpositive; above \(q\), both are nonnegative; at \(q\), their product is zero. Since \(\int(s-s_*)\,dF=0\), integrability gives
Thus the bottom-tail allocation minimizes the mean over every measurable selection rule, without assuming that an iterative exchange converges. Its numerator equals \(\int_0^aQ_F(u)\,du\) by the quantile representation. Reversing the cutoff rule gives the upper-tail maximum. When \(a=1\), the only admissible selection is one almost surely and both bounds equal the observed mean. This also supplies the endpoint case where an unbounded distribution need not have a finite \(Q_F(1)\).