Why The Binary Outcome Bounds Are Sharp

Why Endpoint Attainability Matters

A bound is useful only if it describes what the observed data actually rule out. For a binary-outcome average treatment effect, the worst-case bounds place limits on the effect without assuming that treatment assignment is unconfounded. But deriving inequalities alone does not establish that the limits are the best possible: perhaps the observed data, together with the basic assumptions, secretly rule out the endpoint values. Sharpness answers this question by showing that each endpoint is compatible with a complete distribution of potential outcomes.

The task here is not to estimate an effect from a particular sample. It is to describe which population average treatment effects are consistent with the observed-data distribution and the stated assumptions. The introductory material on worst-case bounds for a binary outcome gives the bounds; the central result here is a constructive proof that they are attainable.

Setup and Assumptions

Let \(D\) be a binary treatment indicator, with \(D=1\) for treatment and \(D=0\) for control. Let \(Y(1)\) and \(Y(0)\) be the binary potential outcomes for each unit: the outcomes that would be observed under treatment and control, respectively. The observed outcome is \(Y\), and the target is the population average treatment effect \(E[Y(1)-Y(0)]\).

Assume the observed-data distribution provides \(q=P(D=1)\), \(p_1=P(Y=1\mid D=1)\), and \(p_0=P(Y=1\mid D=0)\). Throughout, suppose \(0<q<1\), so both treatment groups occur in the population. The argument uses these assumptions:

• Binary treatment and outcome: \(D\), \(Y\), \(Y(1)\), and \(Y(0)\) each take values in \(\{0,1\}\).
• Consistency: \(Y=Y(1)\) when \(D=1\), and \(Y=Y(0)\) when \(D=0\).
• No restriction on treatment assignment: the distribution of the potential outcomes may differ across treatment groups. In particular, no independence or exchangeability between treatment and potential outcomes is assumed.

The observed data reveal \(Y(1)\) among treated units and \(Y(0)\) among controls. They do not reveal \(Y(0)\) among treated units or \(Y(1)\) among controls. Because these missing outcomes are binary, their conditional means can be anywhere from 0 to 1. The proof will show that these unrestricted possibilities are not just a convenient way to derive inequalities: they can be realized by actual joint distributions of potential outcomes.

The Bounds and Their Attainability

Define \(h=E[Y(0)\mid D=1]\), the mean untreated potential outcome among treated units, and \(k=E[Y(1)\mid D=0]\), the mean treated potential outcome among controls. Consistency gives \(E[Y(1)\mid D=1]=p_1\) and \(E[Y(0)\mid D=0]=p_0\). Splitting the population average by treatment status therefore yields:

Since \(0\leq h\leq1\) and \(0\leq k\leq1\), the expression is smallest when \(h=1\) and \(k=0\), and largest when \(h=0\) and \(k=1\). Thus:

These inequalities show that no compatible effect lies outside the interval. Sharpness needs the reverse argument: show that the missing potential outcomes can actually take the values required to achieve each endpoint, without changing anything observed.

Constructing the Endpoint Distributions

For the lower endpoint, retain the observed \(Y(1)\) values for treated units and the observed \(Y(0)\) values for controls. Complete the unobserved outcomes by assigning \(Y(0)=1\) to every treated unit and \(Y(1)=0\) to every control. These assignments do not alter the observed data: consistency still makes the treated outcome equal to \(Y(1)\) and the control outcome equal to \(Y(0)\). They set \(h=1\) and \(k=0\), so the effect equals the lower endpoint.

For the upper endpoint, again retain both observed potential outcomes. Instead assign \(Y(0)=0\) to every treated unit and \(Y(1)=1\) to every control. Consistency is still satisfied, while now \(h=0\) and \(k=1\). Substitution gives the upper endpoint. Thus each endpoint is attainable under a full-data distribution with exactly the observed treatment and outcome rates.

The construction also shows why no narrower interval follows from these assumptions alone. If the lower endpoint is compatible, a proposed higher lower limit would exclude a data-compatible distribution. If the upper endpoint is compatible, a proposed lower upper limit would exclude one too.

Why the Whole Interval Is Attainable

For any \(h\) and \(k\) in \([0,1]\), a compatible completion can be constructed. Among treated units, keep the observed \(Y(1)\) distribution and assign a binary \(Y(0)\) with mean \(h\). Among controls, keep the observed \(Y(0)\) distribution and assign a binary \(Y(1)\) with mean \(k\). For example, the missing binary outcome can be generated as a Bernoulli variable with the specified mean; it can be assigned independently of the observed outcome within its treatment group. This preserves both observed outcome distributions and consistency.

In particular, let \(t\) range from 0 to 1 and choose \(h=1-t\), \(k=t\). The resulting effect is:

At \(t=0\) this is the lower endpoint, and at \(t=1\) it is the upper endpoint. As \(t\) takes every value between 0 and 1, the effect takes every value between those endpoints. The interval’s width is \(q+(1-q)=1\), so this parameterization moves continuously across the entire interval. This proves that the interval is the sharp identified set, not merely an outer bound.

What Sharpness Does—and Does Not—Say

Sharpness is a statement about identification under specified assumptions, not a claim that every value in the interval is equally plausible. The endpoint constructions deliberately make extreme assignments to missing potential outcomes. The data and assumptions permit those assignments, but they do not indicate that either endpoint is likely.

The result also does not say that the bounds are the only possible answer under every causal model. If additional assumptions restrict potential outcomes or treatment assignment, the compatible set may become narrower. For example, exchangeability would add information about how treatment groups relate to potential outcomes. Such assumptions must be stated and defended; the observed rates alone do not establish them.

Finally, these are population identification bounds. In an application, \(q\), \(p_1\), and \(p_0\) are estimated from data, so sampling uncertainty is a separate issue. A sharp population interval does not by itself provide a confidence interval or account for uncertainty in those estimated rates.

Diagnostics and Common Pitfalls

• Check what is being held fixed. Each construction must preserve the observed \(D,Y\) distribution. Assigning a missing potential outcome is allowed; changing an observed outcome is not.
• Check consistency group by group. For treated units, the observed outcome must remain \(Y(1)\); for controls, it must remain \(Y(0)\). Swapping these assignments invalidates the construction.
• Do not infer exchangeability from endpoint attainability. The proof uses no such assumption. Adding exchangeability changes the problem rather than contradicting the worst-case result.
• Do not call the interval a confidence interval. It describes compatible population effects under assumptions. Sampling uncertainty requires separate treatment.
• Check the outcome scale. The construction relies on the missing potential outcomes being restricted to 0 or 1. Applying these exact endpoint assignments to an outcome with a different range requires a different argument.

A useful comparison is with point identification: a stronger assumption may select one effect from the set, while the worst-case analysis retains all effects compatible with the observed distribution and minimal assumptions. Neither approach is automatically preferable; the appropriate conclusion depends on which assumptions the analysis can justify.