Conditional c-Dependence: Sharp Distribution and Treatment-Effect Bounds
How far can treatment-effect conclusions move if conditional independence is only approximate? Conditional \(c\)-dependence limits the change in treatment probability after conditioning on a potential outcome to \(c\). (Masten and Poirier 2018) derives sharp distributional bounds from this single restriction, then passes to quantiles and means.
1. The restriction
For treatment \(X\in\{0,1\}\), potential outcome \(Y_x\), and covariates \(W=w\), impose
At \(c=0\) this is conditional independence of each potential outcome and treatment. It does not impose independence jointly on the pair \((Y_0,Y_1)\).
2. A simpler distribution problem
Fix \(w\) and temporarily suppose the marginal CDF \(F\) of a continuous variable \(U\) is known. For an event \(X=x\) with probability \(p_x\), put \(a=\max(0,p_x-c)\) and \(b=\min(1,p_x+c)\).
Proposition 1. Under \(c\)-dependence, the conditional CDF \(H(u)=\Pr(U\le u\mid X=x)\) satisfies
and
The inequalities bound each cutoff. Sharpness requires compatible distributions attaining the envelopes; the construction below addresses that additional requirement, with treatment-group compatibility supplied by the paper.
3. Derive both sides of the cutoff
Let \(q(v)=\Pr(X=x\mid U=v)\). Bayes’ rule gives
Since \(a\le q\le b\), this integral lies between \(aF(u)\) and \(bF(u)\). But the same restriction also applies above \(u\):
Intersecting the lower-tail and upper-tail restrictions gives the envelopes. The second restriction enforces the marginal treatment probability \(p_x\); the lower-tail integral alone would not do so.
To maximize selected mass in a lower tail, assign high selection probabilities to low outcomes and low probabilities to high outcomes. The switching point is chosen to preserve the overall treatment probability. Uniform ranks of \(U\) make this construction explicit.
The extremal selection rules can be written explicitly in uniform ranks. Let \(V=F(U)\) be uniform on \((0,1)\) and suppose \(a<b\). To maximize the selected lower-tail mass simultaneously across cutoffs, assign selection probability \(b\) to the lowest fraction \(r=(p_x-a)/(b-a)\) of ranks and \(a\) to the rest. Its average is
At rank \(v\leq r\), the selected cdf is \(bv/p_x\). Above \(r\), its numerator is \(br+a(v-r)=p_x-a(1-v)\). Hence this single admissible rule generates the entire upper envelope
To get the lower envelope, assign \(a\) below \(r'=(b-p_x)/(b-a)\) and \(b\) above it. The marginal probability is again \(p_x\), and the same integration produces the maximum of the two lower branches. When \(a=b=p_x\), the only rule is constant selection and both envelopes equal \(v\).
One selection rule attains the entire envelope, rather than a different rule attaining each cutoff. That is the distributional sharpness argument. Compatibility between both treatment groups and the potential-outcome marginals requires the paper’s further construction.
4. Check the endpoints
At \(c=0\), \(a=b=p_x\) and both envelopes equal \(F\). Once \(a=0,b=1\), the bounds reduce to
These bounds retain only the known marginals. In the treatment-effect application the observed conditional cdf is known and the potential-outcome cdf is unknown, so we now solve the same inequalities in the opposite direction.
5. Inverting the restrictions to recover potential outcomes
Now let \(H(y)=\Pr(Y\leq y\mid X=x,W=w)\) be the observed cdf and let \(F(y)=\Pr(Y_x\leq y\mid W=w)\) be unknown. The four inequalities are
For \(a>0\), solving each inequality separately gives
When \(a=0\), omit the terms containing division by \(a\) and use the original inequalities. The remaining bounds are \(p_xH/b\leq F\leq1-p_x(1-H)/b\). This avoids an undefined expression at precisely the values of \(c\) where the identifying restriction becomes weak.
For example, take \(p_x=1/2\), \(c=1/10\), and an observed median \(y\) with \(H(y)=1/2\). Then \(a=2/5,b=3/5\), and
The observed median need not be a population median of \(Y_x\). The permitted selection changes its population percentile to anywhere between these endpoints under the maintained model.
6. From cdfs to quantiles and means
Proposition 2. Under the paper’s continuity, support, overlap, and integrability conditions, conditional quantile endpoints are
The conditional quantile treatment effect is bounded below by \(\underline Q_1^c-\overline Q_0^c\) and above by \(\overline Q_1^c-\underline Q_0^c\). Integrating the corresponding quantile endpoints over \((0,1)\) gives conditional mean endpoints; averaging their differences over \(W\) gives ATE endpoints. The compatible constructions in the paper establish sharp interiors.
Cdf order reverses under quantile inversion: if \(F_1\leq F_2\), the set \(\{y:F_1(y)\geq\tau\}\) begins no earlier than \(\{y:F_2(y)\geq\tau\}\). A lower bound on a difference then uses the smaller first term and the larger second term.
When \(c<\min(p_x,1-p_x)\), \(a=p_x-c\) and \(b=p_x+c\). For the lower quantile, the upper cdf must reach \(\tau\). Both branches of its minimum must therefore reach \(\tau\), requiring
Similarly, inverting the lower cdf gives an upper probability index with a plus sign. Thus
The probability-index change is largest at the median and grows as the treatment group becomes rarer. Once \(c\) leaves the stated range, clipping in \(a,b\) changes the branches, so the full inequalities must be inverted again.
For an integrable random variable, the identity \(\mathbb{E}Y=\int_0^1Q_Y(u)\,du\) converts a distributional ordering into a mean ordering. Applying it to each potential outcome and then taking expectations over \(W\) yields the mean-effect bounds. Sharpness requires admissible distributions attaining or approaching the endpoint functions, not just choosing an unrelated scalar bound at each \(y\). This is why the paper’s compatibility argument matters.
For a scale check, suppose both observed treatment-group distributions are uniform on \([0,1]\), \(p_x=1/2\), and \(0\leq c<1/2\). Their quantile endpoints are \(\tau\pm2c\min(\tau,1-\tau)\). Since
The potential-outcome mean bounds are therefore \(1/2\pm c/2\), and opposing endpoints give the ATE interval \([-c,c]\). At \(c=0\) the interval collapses to zero. Its linear growth comes from integrating the triangular quantile displacement. Thus the distribution, quantile, and mean bounds all follow from the same restriction on selection.
7. Marginal quantiles and individual treatment effects
To obtain marginal quantiles, first average conditional cdfs using \(F_{Y_x}(y)=\mathbb{E}[F_{Y_x\mid W}(y\mid W)]\), then invert. Averaging conditional quantiles reverses these operations and generally gives a different quantity.
For a simple illustration, let \(W\) select two groups with probabilities \(3/4\) and \(1/4\), and let \(Y\) be identically zero in the first and identically ten in the second. The average conditional median is \(5/2\), while the marginal median is zero. This elementary example concerns the ordering of the operations; smooth approximations give the same discrepancy within continuous-outcome models.
Nor does \(Q_{Y_1}(\tau)-Q_{Y_0}(\tau)\) equal the \(\tau\) quantile of \(Y_1-Y_0\). If both marginals are uniform on \([0,1]\), their quantile difference is zero. Under \(Y_1=Y_0\), every individual effect is zero; under \(Y_1=1-Y_0\), individual effects range over \([-1,1]\). The marginals are identical in both cases. Separate restrictions on selection into each potential outcome leave this coupling unspecified.
Increasing \(c\) enlarges the set of admissible distributions, so the sharp lower effect bound cannot increase and the upper bound cannot decrease. That nesting is a useful implementation check. The next note develops sampling uncertainty for these bounds, including the branch changes introduced by their maxima and minima.
Completing compatibility in the strict-overlap case
The joint-distribution step is particularly transparent when \(0\leq c<\min(p_x,1-p_x)\), so \(0<a\leq b<1\). Fix the observed continuous, strictly increasing cdf \(H\) for treatment group \(x\), and choose either endpoint \(F\) obtained by inverting the restrictions. As a function of \(H\), each endpoint is continuous and piecewise linear, with slope either \(p_x/a\) or \(p_x/b\). It starts at zero and ends at one. Thus it is an admissible cdf on the same support.
Define the missing-treatment cdf by
On each branch, its slope with respect to \(H\) is \((p_x/a-p_x)/(1-p_x)\) or \((p_x/b-p_x)/(1-p_x)\), both strictly positive. Its endpoint values are zero and one. Hence \(G\) is also a continuous, strictly increasing cdf, and \(F=p_xH+(1-p_x)G\) is a feasible mixture. The selection probability recovered from these measures is
almost everywhere; at \(c=0\) it equals \(p_x\). This verifies the selection restriction for the entire distribution, not merely at a chosen cutoff.
Carry out this construction separately for \(x=0,1\). Draw \(X\) with its observed propensity, then draw the two potential outcomes conditionally independently given \(X\), using the observed law \(H_x\) when \(X=x\) and the constructed law \(G_x\) otherwise. This single joint distribution reproduces both observed outcome laws and both potential-outcome bounds. In particular, opposing endpoint choices for a treatment-effect bound are compatible in this strict-overlap regime. No restriction here fixes dependence between the potential outcomes beyond their separate selection restrictions.
If \(a=0\) or \(b=1\), the strict-increase argument can fail and an envelope may be a limiting distribution rather than an admissible endpoint under the source’s full-support conditions. The paper’s approximation construction is needed there; the strict-overlap proof does not establish endpoint attainment in that boundary regime.