A Bounded Continuous Outcome Workshop

From Observed Summaries to Causal Bounds

A continuous outcome need not be unbounded. A test score, a symptom scale, or a number of symptom-free days has a finite range. Knowing that range can place limits on the average treatment effect even when treatment assignment is confounded and a point estimate is not justified. The limits may be wide: their value is that they show what the observed data and stated assumptions do, and do not, establish.

Let \(D\) indicate treatment, with \(D=1\) for treated and \(D=0\) for untreated. Let \(Y(1)\) and \(Y(0)\) denote the outcome an individual would have under treatment and control, respectively. The target is the average treatment effect \(\tau=E[Y(1)-Y(0)]\) in a specified population. Assume treatment is well-defined, consistency holds so the observed outcome is \(Y=DY(1)+(1-D)Y(0)\), and there is no interference between individuals. These assumptions define how observed outcomes relate to potential outcomes; they do not make treatment assignment unconfounded.

The support restriction must apply to both potential outcomes for every member of the target population, including the unobserved potential outcome. The observed treated and control means alone do not supply the missing counterfactual means. Instead, the support restriction limits them to \([L,U]\). For example, \(E[Y(1)\mid D=0]\) is unknown, but it must be between \(L\) and \(U\).

Worst-Case Bounds for the Average Treatment Effect

Write \(p=P(D=1)\), \(m_1=E[Y\mid D=1]\), and \(m_0=E[Y\mid D=0]\), with \(0<p<1\). By consistency, the observed treated mean is \(E[Y(1)\mid D=1]=m_1\), and the observed control mean is \(E[Y(0)\mid D=0]=m_0\). The other two conditional potential-outcome means are missing. The total expectation identity gives

For a lower limit on \(\tau\), use the smallest allowed missing treated potential-outcome mean, \(L\), and the largest allowed missing control potential-outcome mean, \(U\). For an upper limit, reverse those choices. This gives the worst-case bounds:

These bounds are sharp under these assumptions: each endpoint can be attained by assigning the missing conditional means their respective support endpoints. Every intermediate value is also compatible, since the unknown means can vary continuously within their permitted ranges. Sharpness does not mean that the true effect is equally likely to be anywhere in the interval; it means the assumptions and observed summaries do not rule those values out.

Stratifying on Covariates and Aggregating

Suppose \(X\) is a measured covariate with a finite set of strata \(x\). Let \(q_x=P(X=x)\), \(p_x=P(D=1\mid X=x)\), \(m_{1x}=E[Y\mid D=1,X=x]\), and \(m_{0x}=E[Y\mid D=0,X=x]\). If both treatment groups are represented in each stratum, the same worst-case calculation gives a conditional interval \([\tau_{\mathrm{L},x},\tau_{\mathrm{U},x}]\). The population average effect is the stratum-weighted average, so its bounds are

The weights are population shares \(q_x\), not treatment shares within a stratum and not equal weights unless the strata are equally common. The conditional bounds must refer to the same target population represented by those shares.

An important check is that stratification alone does not necessarily tighten worst-case bounds. If every stratum uses the same limits \([L,U]\), substituting the conditional formulas and summing produces the unconditional formula. For example, \(\sum_x q_xp_xm_{1x}=p m_1\), where \(p=\sum_xq_xp_x\) and \(m_1\) is the overall treated mean. The other observed-mean terms aggregate in the same way, and the support terms become \((1-p)L\) and \(pU\) for the lower endpoint. Thus the aggregate endpoints are exactly the unconditional worst-case endpoints. Covariate stratification can still organize an analysis or make additional assumptions explicit, but merely partitioning the data does not create information about missing outcomes.

What Additional Restrictions Can Do

To get narrower intervals, an analysis needs more than strata and a common outcome range. One possibility is to state limits on how different the missing potential-outcome means can be from the corresponding observed means within each stratum. Such restrictions are substantive sensitivity assumptions, not consequences of stratification.

For stratum \(x\), suppose the difference between \(E[Y(1)\mid D=0,X=x]\) and \(m_{1x}\) has absolute value at most \(\Gamma_{1x}\). Also suppose the difference between \(E[Y(0)\mid D=1,X=x]\) and \(m_{0x}\) has absolute value at most \(\Gamma_{0x}\). Together with the outcome range, these assumptions place the missing means \(z_{1x}\) and \(z_{0x}\) in

The conditional effect is \(p_xm_{1x}+(1-p_x)z_{1x}-(1-p_x)m_{0x}-p_xz_{0x}\). Its lower endpoint uses the lowest permitted \(z_{1x}\) and highest permitted \(z_{0x}\); its upper endpoint uses the highest permitted \(z_{1x}\) and lowest permitted \(z_{0x}\). This follows because the expression increases with \(z_{1x}\) and decreases with \(z_{0x}\). The resulting conditional intervals can then be aggregated with \(q_x\). The assumptions should be justified and reported; choosing small \(\Gamma\) values simply to obtain a narrow interval is not evidence that the restrictions hold.

Uncertainty, Diagnostics, and Alternatives

Uncertainty statement for the hypothetical analysis: \([0.71,2.09]\) is the plug-in identified interval calculated from the supplied sample summaries and fixed sensitivity limits; it is not a confidence interval and does not account for sampling variation. Suppose, illustratively, that raw data were available and a valid bootstrap procedure produced simultaneous 95% confidence intervals of \([0.20,1.22]\) for the lower endpoint and \([1.58,2.62]\) for the upper endpoint. The conservative confidence region formed from those simultaneous intervals is \([0.20,2.62]\): it uses the lower confidence limit for the lower endpoint and the upper confidence limit for the upper endpoint. Its 95% coverage statement concerns the region containing the population identified set under the maintained assumptions; it is not a 95% probability that a fixed effect lies in the region. The numerical bootstrap intervals here are hypothetical outputs, not a calculation from the summary table.

Diagnostics should check the ingredients that make the calculation interpretable. Confirm that the outcome limits apply to potential outcomes, that group means and treatment shares refer to the intended population, and that each reported stratum contains observations in both treatment groups. Examine stratum counts and observed ranges for coding errors or sparse cells. If treatment is absent in a stratum, its observed mean for that treatment is unavailable, so the formulas using that mean cannot be applied there without further assumptions or a revised target population. Also check that weighting shares sum to one and that an aggregate calculation agrees with a pooled calculation when common outcome limits and unrestricted worst-case bounds are used.

A frequent mistake is to stratify on a prognostic covariate, see narrower conditional intervals, and conclude that stratification itself has identified a more precise effect. With a common outcome range and no extra restrictions, the weighted worst-case bounds equal the pooled bounds, as the example shows. Narrowing requires additional information or assumptions, such as credible stratum-specific support or explicit limits on selection differences. A second mistake is to call the observed treated-control mean difference the causal effect; without an assignment assumption that justifies it, the contrast remains descriptive.

Bounds are an alternative to a point estimate when the assumptions needed for point identification are not acceptable. They make the cost of weak assumptions visible, but can be uninformative, sensitive to the outcome range, or dependent on hard-to-defend sensitivity limits. Regression adjustment, propensity score methods, or instrumental-variable approaches can target point or other causal estimands under their own assumptions; they are not automatic fixes for wide bounds. The relevant comparison is whether their identifying assumptions are plausible for the scientific question, not which method yields the narrowest numerical result.