How Assumptions Tighten Causal Effect Bounds

Why Assumptions Change the Range of Plausible Effects

When some potential outcomes are unobserved, the observed data may be compatible with more than one causal effect. Assumptions can rule out some of those possibilities, but the resulting narrowing is conditional on the assumptions: it is not additional information learned from the observed data alone. The practical question is therefore not simply whether a restriction makes a range smaller, but what the restriction says about outcomes that cannot be observed for everyone.

This tutorial develops that logic for a binary outcome. It focuses on nested assumption sets: each successively stronger set retains every restriction already imposed and adds another. The resulting identified sets can only stay the same or become narrower. A numerical example will show exactly how the endpoints move.

Definition: An identified set is the set of values of a target parameter compatible with the observed-data distribution and a stated collection of assumptions. If the set contains one value, the parameter is point identified; if it contains multiple values, it is set identified. A narrower set reflects stronger restrictions, not necessarily more observed information.

Potential Outcomes and the Unobserved Quantities

Let $D\in\{0,1\}$ indicate treatment status, and let $Y(1)$ and $Y(0)$ denote the outcomes a unit would have under treatment and no treatment, respectively. The observed outcome is $Y=DY(1)+(1-D)Y(0)$. Assume treatment and outcomes are well defined, consistency holds, and one unit’s treatment does not change another unit’s outcome. For this example, $Y(1)$ and $Y(0)$ are binary, so each lies between 0 and 1.

The target is the average treatment effect in the population, $\tau=E[Y(1)-Y(0)]$. Write $\pi=P(D=1)$, $\mu_1=E[Y\mid D=1]$, and $\mu_0=E[Y\mid D=0]$. Consistency reveals $Y(1)$ for treated units and $Y(0)$ for untreated units. The other potential outcome for each group is missing. Define its mean within that group as $q_0=E[Y(0)\mid D=1]$ and $q_1=E[Y(1)\mid D=0]$.

Key result: If both treatment groups have positive probability, consistency gives $$ \tau=\pi(\mu_1-q_0)+(1-\pi)(q_1-\mu_0). $$ The observed data supply $\pi,\mu_1,\mu_0$. Restrictions on the missing potential outcomes determine which values of $q_0$ and $q_1$—and therefore which values of $\tau$—remain possible.

To justify the expression, split the population average into its treated and untreated groups: $$ E[Y(1)-Y(0)] =\pi E[Y(1)-Y(0)\mid D=1] +(1-\pi)E[Y(1)-Y(0)\mid D=0]. $$ Among treated units, consistency gives $E[Y(1)\mid D=1]=\mu_1$, while the unobserved mean is $q_0$. Among untreated units, the unobserved mean is $q_1$, while consistency gives $E[Y(0)\mid D=0]=\mu_0$. Substitution yields the stated formula.

Because the outcomes are binary, $q_0,q_1\in[0,1]$. This is a basic restriction from the outcome scale, not an assumption that treatment assignment is independent of potential outcomes. More restrictive assumptions can narrow the possible values further.

Worked Example: The Range Before Adding Extra Restrictions

Suppose hypothetical population data give $\pi=0.40$, $\mu_1=0.60$, and $\mu_0=0.30$. The observed treated group has a 60% outcome rate and the observed untreated group has a 30% outcome rate. The difference $0.60-0.30=0.30$ is an observed association; by itself, it is not the population average treatment effect.

With only the binary-outcome restrictions $0\le q_0,q_1\le1$, the formula becomes $$ \tau=0.40(0.60-q_0)+0.60(q_1-0.30) =0.06-0.40q_0+0.60q_1. $$

The smallest value occurs at the largest $q_0$ and smallest $q_1$: $q_0=1,q_1=0$. Thus $\tau_{\min}=0.06-0.40(1)+0.60(0)=-0.34$. The largest value occurs at $q_0=0,q_1=1$: $\tau_{\max}=0.06-0.40(0)+0.60(1)=0.66$. The compatible range is $[-0.34,0.66]$, in outcome-probability points. This broad range illustrates why the observed difference alone does not settle the causal question.

Nested Restrictions and Shrinking Identified Sets

One way to make assumptions explicit is to restrict how different the missing potential-outcome means can be across the observed treatment groups. For example, a restriction on $q_0$ compares the mean untreated outcome among treated units with the observed mean untreated outcome among untreated units, $\mu_0$. A restriction on $q_1$ compares the mean treated outcome among untreated units with the observed mean treated outcome among treated units, $\mu_1$. These are restrictions on counterfactual group means. They do not assert that treatment assignment is randomized.

Suppose the restrictions have the form $|q_0-\mu_0|\le\delta_0$ and $|q_1-\mu_1|\le\delta_1$, for chosen nonnegative tolerances $\delta_0,\delta_1$. Since each mean must also lie between 0 and 1, the restrictions imply $q_0\in[\max(0,\mu_0-\delta_0),\min(1,\mu_0+\delta_0)]$, with the analogous interval for $q_1$. Smaller tolerances make stronger claims about how similar the counterfactual means are across groups.

If an assumption set $A_2$ contains every restriction in $A_1$ and adds at least one more, then $A_2$ is a subset of $A_1$. Any potential-outcome distribution allowed by $A_2$ is also allowed by $A_1$. Therefore, the effects compatible with $A_2$ are a subset of those compatible with $A_1$. For interval sets, this means the lower endpoint cannot decrease and the upper endpoint cannot increase. A new restriction can leave an endpoint unchanged if it does not constrain the outcome configurations that attain that endpoint.

Worked Example: Three Nested Levels of Restrictions

Use the hypothetical values from the previous example, so $\tau=0.06-0.40q_0+0.60q_1$. Begin with $q_0,q_1\in[0,1]$, which gives $[-0.34,0.66]$.

First added restriction. Suppose $|q_0-\mu_0|\le0.20$, while $q_1$ remains unrestricted except for its binary-outcome range. Since $\mu_0=0.30$, the restriction gives $q_0\in[0.10,0.50]$. The smallest effect uses $q_0=0.50,q_1=0$: $0.06-0.40(0.50)+0.60(0)=-0.14$. The largest uses $q_0=0.10,q_1=1$: $0.06-0.40(0.10)+0.60(1)=0.62$. The new set is $[-0.14,0.62]$.

Second added restriction. Now also impose $|q_1-\mu_1|\le0.15$. Because $\mu_1=0.60$, this gives $q_1\in[0.45,0.75]$. The minimum is $0.06-0.40(0.50)+0.60(0.45)=0.13$; the maximum is $0.06-0.40(0.10)+0.60(0.75)=0.47$. The set is now $[0.13,0.47]$.

Tighten both tolerances. If both tolerances are reduced to $0.10$, then $q_0\in[0.20,0.40]$ and $q_1\in[0.50,0.70]$. The endpoints are $0.06-0.40(0.40)+0.60(0.50)=0.20$ and $0.06-0.40(0.20)+0.60(0.70)=0.40$. The final set is $[0.20,0.40]$. Each interval is nested inside the one before it because each step preserves the earlier restriction and adds a tighter one.

In this example, the conclusion changes from a broad range that includes negative and positive effects to a narrower range of positive effects. That change is conditional on the mean-similarity restrictions. It should not be reported as if the observed data alone established that the effect is positive.

Why the Endpoint Calculation Works

Under the restrictions in the example, the feasible values of $(q_0,q_1)$ form a rectangle. The expression $0.06-0.40q_0+0.60q_1$ is increasing in $q_1$ and decreasing in $q_0$. Consequently, its minimum over the rectangle is at the largest allowed $q_0$ and smallest allowed $q_1$; its maximum is at the smallest allowed $q_0$ and largest allowed $q_1$. This establishes the endpoints, rather than merely suggesting them.

The endpoint calculations are sharp for this example: each corner pair of means can be achieved by binary potential outcomes within each treatment group. For instance, a group’s two potential outcomes can have any chosen binary marginal means, including the endpoints 0 and 1; a joint distribution can be formed with those marginals. No restriction here constrains the dependence between a unit’s two potential outcomes. If a different assumption links them, the feasible region may no longer be this rectangle, and its endpoints must be recalculated under that assumption.

Worked Example: A Restriction on Only One Missing Mean

Return to $\pi=0.40,\mu_1=0.60,\mu_0=0.30$, and impose only $|q_1-\mu_1|\le0.15$. Then $q_1\in[0.45,0.75]$, but $q_0\in[0,1]$. The minimum uses $q_0=1,q_1=0.45$: $0.06-0.40(1)+0.60(0.45)=-0.07$. The maximum uses $q_0=0,q_1=0.75$: $0.06-0.40(0)+0.60(0.75)=0.51$. Thus the range is $[-0.07,0.51]$.

The restriction has narrowed the initial range, but the resulting set still includes effects of both signs. It does not support a conclusion that the average effect is positive. The example also shows why stating exactly which missing mean is restricted matters: leaving $q_0$ unrestricted leaves substantial uncertainty about the effect.

Diagnostics, Interpretation, and Alternatives

Restrictions on unobserved potential outcomes are generally not testable from the observed treatment and outcome data alone. The data reveal $\mu_1$ and $\mu_0$, but not $q_0$ or $q_1$. Diagnostics can still expose mistakes or clarify the role of assumptions:

Check feasibility. Every proposed mean must lie in $[0,1]$. Intersect assumption-implied ranges with that outcome-scale range rather than allowing impossible values.
Check nesting. A supposedly stronger assumption set must retain the weaker set’s restrictions. If a calculated identified set expands after adding a restriction, check the algebra, the endpoints, and whether the assumptions really are nested.
Show sensitivity. Report how the bounds change over a defensible range of $\delta_0,\delta_1$. A conclusion that holds only for very small tolerances depends heavily on a strong similarity claim.
Keep identification separate from sampling uncertainty. These calculations describe population effects compatible with specified population quantities and assumptions. Estimates from a finite sample also have sampling uncertainty; an identified set is not, by itself, a confidence interval.

The main alternative is to make assumptions that point identify an effect. For example, conditional exchangeability given measured covariates can identify a population effect under additional conditions; matching is one approach that relies on adequate measured confounders and overlap. If important confounding remains unmeasured, those methods do not remove it merely by matching observed variables. Instrumental-variable analyses use a different set of assumptions and may target a different causal effect. Partial identification is useful when a single point estimate would require assumptions the analyst cannot defend, but its bounds may be wide and its interpretation depends on the stated restrictions.

Common pitfall: “The range gets narrower when I add an assumption, so the data support a more precise effect.” The correction is that added assumptions remove possibilities by stipulation. Report the restriction, explain why it is plausible, and show how conclusions change when it is relaxed.

Check Your Understanding

Answer each question, then compare with the solutions below.

In the binary-outcome setup, which two group means are unobserved, and what do they represent?
If an assumption set is strengthened by adding a restriction, can its identified set be wider than the original set? Explain.
Suppose $\pi=0.50,\mu_1=0.70,\mu_0=0.20$, and $q_0\in[0.10,0.30]$, $q_1\in[0.40,0.60]$. Calculate the identified interval for $\tau$.
Why does a narrower identified set not automatically mean the observed data contain more information about the effect?

Solutions. (1) $q_0=E[Y(0)\mid D=1]$, the untreated potential-outcome mean among treated units, and $q_1=E[Y(1)\mid D=0]$, the treated potential-outcome mean among untreated units. (2) No, provided the new set truly retains all earlier restrictions: it permits a subset of the earlier feasible cases, so its effect set cannot expand. (3) Use $\tau=0.50(0.70-q_0)+0.50(q_1-0.20)=0.25-0.50q_0+0.50q_1$. The minimum is $0.25-0.50(0.30)+0.50(0.40)=0.30$; the maximum is $0.25-0.50(0.10)+0.50(0.60)=0.50$. The interval is $[0.30,0.50]$. (4) The narrowing can result entirely from stronger restrictions on unobserved potential outcomes, not from additional observed data.