Stochastic Dominance Restrictions On Potential Outcomes

Why a Distributional Restriction Can Matter

A restriction on the average of a potential outcome describes only one feature of its distribution. Two distributions can have the same mean while assigning very different probabilities to low and high outcomes. First-order stochastic dominance (FOSD) expresses a stronger, distribution-wide ordering: at every threshold, one potential outcome is no more likely to fall below that threshold than the other.

This matters for partial identification because the outcome under the treatment not received is unobserved for each person. If a defensible substantive argument orders the population distributions of the two potential outcomes, the resulting CDF inequalities restrict which missing distributions are compatible with the observed data. The restrictions may narrow the range of average treatment effects, but they do not by themselves reveal the missing outcomes or establish that treatment assignment is unconfounded.

Definition: Let $F_1(y)=P(Y(1)\le y)$ and $F_0(y)=P(Y(0)\le y)$ be the population CDFs of the two potential outcomes. The restriction that treatment weakly improves outcomes in the sense of first-order stochastic dominance is $$ F_1(y)\le F_0(y)\quad\text{for every }y. $$ Equivalently, $P(Y(1)>y)\ge P(Y(0)>y)$ at every threshold $y$. When the potential outcomes have finite means, this implies $E[Y(1)]\ge E[Y(0)]$, but the mean inequality alone does not imply FOSD.

The direction of the inequality depends on what counts as a better outcome. Here, larger values are taken to be better. If smaller values are preferable, such as for a harmful event or symptom score, the outcome direction must be reversed or recoded before applying the assumption.

Connecting the Restriction to Observed and Missing Distributions

Let $D$ indicate treatment receipt, with $D=1$ for treated and $D=0$ for untreated, and let $p=P(D=1)$, where $0<p<1$. Consistency means that the observed outcome is $Y=Y(1)$ when $D=1$ and $Y=Y(0)$ when $D=0$. Thus the data reveal the distribution of $Y(1)$ among treated people and of $Y(0)$ among untreated people. They do not reveal the other two conditional distributions.

Write $F_{1|1}(y)=P(Y(1)\le y\mid D=1)$, $F_{1|0}(y)=P(Y(1)\le y\mid D=0)$, $F_{0|1}(y)=P(Y(0)\le y\mid D=1)$, and $F_{0|0}(y)=P(Y(0)\le y\mid D=0)$. The first and fourth are observed; the second and third describe missing potential outcomes. The population CDFs are mixtures across treatment status:

F_1(y)=pF_{1|1}(y)+(1-p)F_{1|0}(y),\qquad F_0(y)=pF_{0|1}(y)+(1-p)F_{0|0}(y).

Substituting these mixtures into the FOSD condition gives a restriction at every threshold:

pF_{1|1}(y)+(1-p)F_{1|0}(y) \le pF_{0|1}(y)+(1-p)F_{0|0}(y) \quad\text{for every }y.

This is not an inequality that can be checked just by comparing the observed treated and untreated outcome CDFs: those CDFs describe different potential outcomes in different groups. Rather, it is a constraint on the unknown distributions that must be considered along with the observed distributions and any known outcome support.

Worked Example: Turning the Mixture Restriction into a Bound

Suppose a hypothetical population has $p=0.5$, outcomes in the known range $[0,2]$, and the observed treated potential outcome $Y(1)\mid D=1$ and untreated potential outcome $Y(0)\mid D=0$ are both point masses at 1. Impose population FOSD in the direction $Y(1)$ dominates $Y(0)$. Find the resulting bounds on the average treatment effect.

Let $G$ be the CDF of the missing $Y(1)\mid D=0$, and $H$ the CDF of the missing $Y(0)\mid D=1$. The population CDFs are $F_1(y)=0.5\,\mathbf{1}(1\le y)+0.5G(y)$ and $F_0(y)=0.5H(y)+0.5\,\mathbf{1}(1\le y)$. The common observed point mass cancels from $F_1(y)\le F_0(y)$, so the restriction is $G(y)\le H(y)$ at every threshold: the missing treated potential outcome distribution must dominate the missing untreated potential outcome distribution.

Let $\mu_G$ and $\mu_H$ be the means of these two missing distributions. The average treatment effect is $$ E[Y(1)-Y(0)] =0.5(1)+0.5\mu_G-0.5\mu_H-0.5(1) =0.5(\mu_G-\mu_H). $$ FOSD implies $\mu_G\ge\mu_H$, while the support $[0,2]$ implies $0\le\mu_G,\mu_H\le2$. Therefore the effect is at least 0 and at most $0.5(2-0)=1$, in outcome units.

Both endpoints are attainable. For the lower endpoint, take $G=H$, for example both point masses at 0; then the CDFs satisfy the restriction and the effect is 0. For the upper endpoint, take $G$ to be a point mass at 2 and $H$ a point mass at 0; then $G(y)\le H(y)$ at every threshold and the effect is $0.5(2-0)=1$. Thus the bounds $[0,1]$ are sharp under the stated information and assumption.

Why the CDF Inequalities Are Stronger Than a Mean Inequality

For outcomes on a known bounded interval $[L,U]$, the mean can be written as the integral of the upper-tail probabilities:

E[Y(d)]=L+\int_L^U P(Y(d)>t)\,dt =L+\int_L^U \{1-F_d(t)\}\,dt.

Subtracting the two expressions gives

E[Y(1)]-E[Y(0)] =\int_L^U \{F_0(t)-F_1(t)\}\,dt.

Under FOSD, every integrand is nonnegative, so the average treatment effect is nonnegative. This proves the mean implication. But a nonnegative integral does not require its integrand to be nonnegative at every point: positive and negative areas can offset. FOSD rules out any threshold at which $F_1(t)>F_0(t)$, making it a stricter distributional restriction than the mean inequality.

Worked Example: Equal Means Do Not Establish Dominance

Consider two hypothetical potential-outcome distributions on $[0,2]$. Suppose $Y(1)$ equals 0 or 2 with probability $0.5$ each, while $Y(0)$ equals 1 with probability 1. Calculate the means and check whether $Y(1)$ dominates $Y(0)$.

The means are $E[Y(1)]=0.5(0)+0.5(2)=1$ and $E[Y(0)]=1$. Thus the mean difference is zero. At threshold $y=0.5$, however, $F_1(0.5)=P(Y(1)\le0.5)=0.5$, while $F_0(0.5)=P(Y(0)\le0.5)=0$. The required inequality $F_1(0.5)\le F_0(0.5)$ fails because $0.5>0$.

The distributions therefore do not satisfy the stated FOSD assumption, despite equal means. A mean comparison alone would miss this violation. In an identification analysis, the CDF restriction must be enforced across thresholds, rather than replaced by the single condition that the average treatment effect is nonnegative.

A Binary-Outcome Special Case

For a binary outcome taking values 0 and 1, there is only one nontrivial CDF threshold. Consequently, FOSD is equivalent to the mean ordering $E[Y(1)]\ge E[Y(0)]$. This special case is useful for checking calculations, but it should not be mistaken for the general situation with three or more outcome values or a continuous outcome.

Worked Example: Binary Outcomes and a Nonnegative Effect Bound

In a hypothetical study, $p=0.5$, the observed treatment success proportion is $0.6$, and the observed control success proportion is $0.4$. Let $u=P(Y(1)=1\mid D=0)$ and $v=P(Y(0)=1\mid D=1)$ be the unknown success probabilities. Apply population FOSD and calculate the effect bounds.

The population success probabilities are $P(Y(1)=1)=0.5(0.6)+0.5u$ and $P(Y(0)=1)=0.5v+0.5(0.4)$. For a binary outcome, FOSD requires the first probability to be at least the second. The average treatment effect is their difference: $$ E[Y(1)-Y(0)] =0.3+0.5u-0.5v-0.2 =0.1+0.5(u-v)\ge0. $$

Without FOSD, $u,v\in[0,1]$ would give the range $[0.1-0.5,\;0.1+0.5]=[-0.4,0.6]$. FOSD removes the negative part, leaving $[0,0.6]$. The lower endpoint is feasible: $u=0.4$, $v=0.6$ gives $0.1+0.5(-0.2)=0$. The upper endpoint is feasible at $u=1$, $v=0$, giving $0.6$. Since the outcome is binary, the FOSD restriction here adds exactly the nonnegative mean-effect condition; for a general outcome it also constrains the distribution at intermediate thresholds.

Using the Restriction in Practice

With outcomes on finite support $y_0<y_1<\cdots<y_K$, each CDF is determined by cumulative probability masses. The FOSD restriction can therefore be written at each support point other than the maximum:

p\sum_{k=0}^{j}P(Y(1)=y_k\mid D=1) +(1-p)\sum_{k=0}^{j}P(Y(1)=y_k\mid D=0) \le p\sum_{k=0}^{j}P(Y(0)=y_k\mid D=1) +(1-p)\sum_{k=0}^{j}P(Y(0)=y_k\mid D=0), \quad j=0,\ldots,K-1.

The observed conditional masses and treatment share are supplied by the data; the missing masses must be nonnegative and sum to one within each treatment group. The average treatment effect is a weighted sum of the potential-outcome means. Minimizing and maximizing that quantity subject to the probability and CDF restrictions gives the sharp bounds for this finite-support problem, provided no additional restrictions are imposed on the joint distribution of $Y(1)$ and $Y(0)$. Any pair of conditional marginal distributions can be coupled within each group when there is no restriction on their dependence.

For continuous outcomes, the same principle applies at every threshold. In practice, the support, the shape of the observed CDFs, and the treatment share all affect how informative the restriction is. If support is not known or outcomes are unbounded, FOSD still implies a mean ordering when the means exist, but it need not provide finite bounds on the mean difference.

Assumptions, Diagnostics, and Alternatives

The calculation relies on consistency, a well-defined binary treatment, a treatment share strictly between zero and one, correctly described observed outcome distributions, and the substantive FOSD assumption for the population potential-outcome distributions. Finite outcome means are needed to interpret the average treatment effect; known bounded support is needed for the simple finite bounds illustrated above. FOSD is not a consequence of random sampling, matching, or adjustment for covariates. It is an additional restriction on unobserved potential outcomes.

The central diagnostic is substantive rather than a direct test of the assumption: the two population marginal distributions are not jointly observed. Observed CDFs can be used to check whether particular candidate missing distributions are compatible with the restrictions, but they cannot establish that the true missing distributions satisfy FOSD. For finite-support calculations, check every cumulative inequality, not only the means. Also check that all candidate probability masses are nonnegative, sum to one, and respect the stated outcome support.

FOSD differs from the assumptions covered by monotone treatment response and monotone treatment selection. Monotone treatment response requires an individual-level ordering $Y(1)\ge Y(0)$ for every person; FOSD orders only the two population marginal distributions and does not specify which individual's outcomes move in which direction. A selection restriction instead orders potential-outcome distributions across treatment-selection groups. These assumptions have different substantive meanings and should not be substituted for one another.

Common mistake: checking only the means. A nonnegative average treatment effect does not guarantee $F_1(y)\le F_0(y)$ at every threshold. Check the full CDF ordering.
Common mistake: treating observed groups as potential-outcome distributions. The observed treated CDF is $F_{1|1}$, not the population CDF $F_1$; the observed untreated CDF is $F_{0|0}$, not $F_0$. Use the treatment-share mixture equations.
Common mistake: interpreting dominance as individual benefit. FOSD does not imply that every person benefits. It is a statement about marginal distributions.
Common mistake: presenting the assumption as data-verified. Missing potential outcomes prevent a direct check of the population ordering. Report it as a substantive identifying restriction and assess whether the resulting bounds are sensitive to it.

Key takeaway: First-order stochastic dominance constrains potential-outcome CDFs at every threshold. It implies a mean ordering, but for nonbinary outcomes it contains information that a mean inequality alone cannot express. Bounds are credible only to the extent that the distributional assumption and outcome support are credible.

Check Your Understanding

Answer each question, then compare with the solutions below.

For outcomes where larger values are better, state the CDF inequality for $Y(1)$ to dominate $Y(0)$.
Why can two potential-outcome distributions with equal means fail to satisfy FOSD?
In the binary example, calculate the average treatment effect at $u=0.2$ and $v=0.6$. Does the pair satisfy FOSD?
Does FOSD imply that every individual has $Y(1)\ge Y(0)$? Explain.

Solutions. (1) The requirement is $F_1(y)\le F_0(y)$ for every $y$. (2) Equal means constrain the integral of the CDF difference, not its sign at each threshold; the distributions in the worked example violate the ordering at $y=0.5$. (3) The effect is $0.1+0.5(0.2-0.6)=0.1-0.2=-0.1$. It is negative, so the binary FOSD condition is not satisfied. (4) No. FOSD orders population marginal distributions; it does not impose an individual-level ordering of the two potential outcomes.