A Binary Outcome Bounds Case Study

Why a Binary Outcome Makes a Useful Bounds Case Study

Suppose a hypothetical health program is intended to increase the probability that a patient is symptom-free after treatment. The outcome $Y$ is binary: $Y=1$ means symptom-free and $Y=0$ means not symptom-free. We observe each patient's treatment status and outcome, but not what that same patient’s outcome would have been under the other treatment status. Without an assumption that resolves this missing-counterfactual problem, the observed difference in symptom-free rates need not equal the average causal effect.

Binary outcomes make the logic of partial identification especially visible. The unobserved outcome for any patient must be either 0 or 1, so we can calculate an interval of possible average treatment effects without pretending to know the missing values. Then we can add a substantive assumption—monotone response—and see precisely how the interval changes.

Definition: Let $D\in\{0,1\}$ indicate treatment, and let $Y(1)$ and $Y(0)$ be a patient's potential binary outcomes under treatment and no treatment. The observed outcome is $Y=D Y(1)+(1-D)Y(0)$. The average treatment effect is $\tau=E[Y(1)-Y(0)]$, measured here as a change in the probability of being symptom-free. The observed data identify $p_1=P(D=1)$, $p_0=P(D=0)=1-p_1$, and the observed risks $\mu_1=E[Y\mid D=1]$ and $\mu_0=E[Y\mid D=0]$.

The basic setup assumes consistency: a patient's observed outcome equals the potential outcome under the treatment actually received. It also assumes that the treatment and outcome are well-defined and that one patient's treatment does not change another patient's outcome. These conditions do not make treatment assignment random. In particular, the calculations below do not assume that treated and untreated patients are comparable.

Worst-Case Bounds Without a Response Assumption

Among treated patients, $Y(1)$ is observed but $Y(0)$ is missing; among untreated patients, $Y(0)$ is observed but $Y(1)$ is missing. Since each missing potential outcome lies between 0 and 1, the lowest possible average effect assigns missing outcomes to make treatment effects as small as possible. The highest possible effect assigns them to make treatment effects as large as possible.

Result: With binary outcomes and consistency, but no restriction on treatment assignment or on how treatment affects any individual, the sharp worst-case bounds for the average treatment effect are $$ p_1(\mu_1-1)-p_0\mu_0 \ \leq\ \tau\ \leq\ p_1\mu_1+p_0(1-\mu_0). $$ “Sharp” means that, without additional assumptions, each endpoint is attainable by a possible assignment of the unobserved potential outcomes consistent with the observed data.

To see the lower endpoint, write the effect as the treated group's contribution plus the untreated group's contribution. For treated patients, the observed $Y(1)$ is fixed, while the missing $Y(0)$ can be as high as 1; their smallest possible contribution is therefore $p_1(\mu_1-1)$. For untreated patients, the observed $Y(0)$ is fixed and the missing $Y(1)$ can be as low as 0; their smallest contribution is $-p_0\mu_0$. Adding these gives the lower endpoint. Reversing the assignments of the missing outcomes—$Y(0)=0$ for treated patients and $Y(1)=1$ for untreated patients—gives the upper endpoint.

This argument also establishes sharpness: the endpoint constructions use only values allowed for binary outcomes, preserve every observed outcome, and attain the stated limits. No narrower interval follows from the observed group risks and the binary outcome restriction alone.

Worked Example: Worst-Case Bounds for Symptom-Free Status

In a hypothetical cohort of 1,000 patients, 400 receive the program and 600 do not. Of the treated patients, 80 are symptom-free; of the untreated patients, 180 are symptom-free. Calculate the observed risks and the worst-case bounds for the average treatment effect.

The treatment share is $p_1=400/1000=0.4$, and the untreated share is $p_0=600/1000=0.6$. The observed symptom-free risks are $\mu_1=80/400=0.2$ and $\mu_0=180/600=0.3$. The observed risk difference is $0.2-0.3=-0.1$, or 10 percentage points lower in the treated group. Without further assumptions, that association alone does not establish the causal effect.

For the lower endpoint, substitute into the worst-case formula: $$ 0.4(0.2-1)-0.6(0.3) =0.4(-0.8)-0.18 =-0.32-0.18 =-0.50. $$

For the upper endpoint: $$ 0.4(0.2)+0.6(1-0.3) =0.08+0.6(0.7) =0.08+0.42 =0.50. $$

Thus the worst-case identified interval is $[-0.50,\,0.50]$, or from a 50-percentage-point decrease to a 50-percentage-point increase in symptom-free probability. It includes harm, no effect, and benefit. The interval concerns what the population causal effect could be under the stated assumptions; it is not a confidence interval and does not assign probabilities to values within it.

Adding Monotone Response

A scientific or clinical argument may support a directional restriction on individual responses. For this example, the outcome is coded so that 1 is favorable. We will consider the assumption that the program cannot make any individual less likely to be symptom-free, expressed for binary potential outcomes as $Y(1)\geq Y(0)$ for every patient. This is a strong claim about every individual, not merely a claim that the average effect is nonnegative.

Definition: Monotone response in the beneficial direction, also called monotone treatment response for this outcome coding, is the assumption $Y(1)\geq Y(0)$ for every individual. Under this assumption the individual effect is either 0 or 1, and hence $\tau\geq 0$. It does not assume that treatment assignment is independent of potential outcomes.

The observed data further limit how large the effect can be. A treated patient observed with $Y=0$ has $Y(1)=0$; monotone response then requires $Y(0)=0$, so that patient's effect is zero. A treated patient observed with $Y=1$ may have effect 0 or 1. For untreated patients, an observed $Y=1$ requires $Y(1)=1$, so the effect is zero; an observed $Y=0$ permits an effect of 0 or 1. Therefore only treated patients with observed outcome 1 and untreated patients with observed outcome 0 can contribute a positive effect.

The largest possible average effect assigns a positive effect to every one of those eligible patients. The smallest assigns no one a positive effect. The resulting sharp bounds are:

0\ \leq\ \tau\ \leq\ p_1\mu_1+p_0(1-\mu_0).

For sharpness of the lower endpoint, set each missing outcome equal to its observed counterpart, giving everyone zero effect; this satisfies monotone response. For the upper endpoint, assign $Y(0)=0$ to treated patients observed with $Y(1)=1$, and assign $Y(1)=1$ to untreated patients observed with $Y(0)=0$. Set missing outcomes equal to observed ones in the other cases. Every assignment satisfies monotonicity and reaches the upper endpoint. Thus the data and monotone response alone cannot narrow the interval further.

Worked Example: Comparing the Two Bound Sets

Return to the hypothetical cohort of 1,000 patients: 400 treated, with 80 symptom-free, and 600 untreated, with 180 symptom-free. Suppose clinical knowledge supports the individual-level assumption that the program cannot prevent any patient from becoming symptom-free.

The monotone-response upper endpoint is the same expression as the worst-case upper endpoint: $$ p_1\mu_1+p_0(1-\mu_0) =0.4(0.2)+0.6(1-0.3) =0.08+0.42 =0.50. $$ The monotone-response interval is therefore $[0,\,0.50]$, corresponding to an average increase from 0 to 50 percentage points.

Compared with the worst-case interval $[-0.50,\,0.50]$, monotone response removes effects below zero but does not reduce the upper limit. The upper endpoint was already attainable by assigning positive effects only to treated patients observed symptom-free and untreated patients observed not symptom-free; those assignments are compatible with monotone response. The lower endpoint changes because unrestricted worst-case reasoning allows negative individual effects, whereas the added assumption rules them out.

This conclusion is conditional on the monotone-response assumption being credible for the symptom-free outcome and the treatment as defined. It does not follow from the observed rates, which are lower in the treated group in this example.

A Second Data Pattern and the Role of Treatment Prevalence

The treatment share matters because the target is an average over the whole population, not an average of the two group risks with equal weights. The following example shows the arithmetic when the treated group is smaller and its observed risk is higher.

Worked Example: Unequal Group Sizes and a Higher Treated Risk

In a different hypothetical dataset, 300 of 1,000 patients receive a program; 90 treated patients are symptom-free. Among the 700 untreated patients, 140 are symptom-free. Calculate the worst-case and monotone-response bounds.

Here $p_1=0.3$, $p_0=0.7$, $\mu_1=90/300=0.3$, and $\mu_0=140/700=0.2$. The observed risk difference is $0.3-0.2=0.1$, or 10 percentage points higher among treated patients.

The worst-case lower endpoint is $$ 0.3(0.3-1)-0.7(0.2) =0.3(-0.7)-0.14 =-0.21-0.14 =-0.35. $$ The upper endpoint is $$ 0.3(0.3)+0.7(1-0.2) =0.09+0.7(0.8) =0.09+0.56 =0.65. $$ Thus the worst-case interval is $[-0.35,\,0.65]$.

If monotone response is credible, the interval becomes $[0,\,0.65]$. Both endpoints are expressed as changes in symptom-free probability: the unrestricted analysis permits an average decrease of 35 percentage points, while the monotone analysis excludes decreases and permits increases up to 65 percentage points. The observed 10-point difference is not itself either bound endpoint.

Diagnostics, Comparisons, and Limits

Before reporting these bounds, check that the outcome has been coded in the direction required by the monotonicity claim, that treatment and outcome refer to the same well-defined units, and that counts and denominators are consistent. The observed risks must lie between 0 and 1, the treatment shares must sum to 1, and the monotone-response interval must be contained in the worst-case interval. If the computed intervals violate these checks, revisit the arithmetic or the definition of the outcome.

The monotonicity assumption deserves a separate scientific audit. A claim that a program cannot harm any patient is much stronger than a belief that it helps most patients or has a nonnegative average effect. Side effects, treatment delays, or differences in response may make individual-level monotonicity implausible. If the outcome were coded in the opposite direction—for example, $Y=1$ for an adverse event—the same beneficial assumption would need to be written with the inequality reversed.

The observed risk difference is a point contrast between groups. Point identification through an adjusted causal analysis, including approaches based on measured confounders, generally requires additional assumptions about treatment selection. Worst-case bounds need fewer assumptions but can be wide; adding monotone response can exclude effects inconsistent with that directional claim, yet still leave a broad range. These are different trade-offs, not competing calculations that can be selected solely by which answer is more appealing.

The formulas are population identification results. In a finite sample, substituting sample counts produces estimated endpoints, which themselves have sampling uncertainty. The interval between endpoints does not account for that uncertainty. Also, a wide interval may be a correct consequence of weak assumptions rather than evidence of a computational problem; a narrow interval may depend heavily on an assumption that is difficult to defend.

Key takeaway: For a binary outcome, unrestricted worst-case bounds allow every missing potential outcome to take either binary value. Beneficial monotone response rules out negative individual effects, raising the lower endpoint to zero; it leaves the upper endpoint unchanged. The narrower conclusion is justified only if the added individual-level assumption is credible.

Check Your Understanding

Try each calculation, then compare with the solutions below. All scenarios are hypothetical, and effects are risk differences.

In a population with $p_1=0.25$, $\mu_1=0.40$, and $\mu_0=0.20$, calculate the worst-case lower and upper endpoints. What are the bounds under beneficial monotone response?
In the first worked example, why does the upper endpoint remain 0.50 after imposing monotone response?
For an outcome coded $Y=1$ for symptom-free status, does the observed risk difference in the first example establish that treatment reduces the causal probability of being symptom-free? Explain.
Suppose the first example's clinical claim is only that the program helps the average patient, not that it cannot harm any individual. Is that the same as monotone response? What bounds follow from the monotone-response calculation without this assumption?

Solutions. (1) Here $p_0=0.75$. The lower endpoint is $0.25(0.40-1)-0.75(0.20)=0.25(-0.60)-0.15=-0.30$. The upper endpoint is $0.25(0.40)+0.75(0.80)=0.10+0.60=0.70$. Worst-case bounds are $[-0.30,\,0.70]$; with beneficial monotone response they are $[0,\,0.70]$. (2) The upper endpoint assigns positive effects only among treated patients observed with $Y=1$ and untreated patients observed with $Y=0$. That construction already satisfies $Y(1)\geq Y(0)$, so the additional restriction does not remove it. (3) No. The observed risk difference is $-0.10$, but treatment selection may explain the group difference. The worst-case interval includes positive, zero, and negative causal effects. Even under monotone response, the interval is $[0,\,0.50]$; it does not establish a strictly positive or strictly negative effect. (4) No. A nonnegative average effect permits some individuals to be harmed if others benefit enough; monotone response rules out harm for every individual. Without the individual-level assumption, the first example's worst-case bounds remain $[-0.50,\,0.50]$.