End-To-End Partial Identification Analysis

From a Causal Question to a Reportable Interval

A partial-identification analysis is not complete when it produces two endpoints. The endpoints answer a causal question only if the estimand, outcome limits, restrictions, and uncertainty procedure all match the target population and the available data. This tutorial works through that chain for an average treatment effect when treatment may be confounded and the outcome is bounded.

Consider a hypothetical treatment and a bounded outcome, measured in points from 0 to 10. Let \(D\) indicate treatment, and let \(Y(1)\) and \(Y(0)\) denote the outcomes the same unit would have under treatment and no treatment. The observed outcome is \(Y=DY(1)+(1-D)Y(0)\). Consistency is required for this equality; no assumption that treatment is unconfounded is made.

The analysis below assumes the target population's treated share \(q=P(D=1)\) is known to be 0.60. The outcome satisfies \(0\leq Y(d)\leq 10\) for both treatment states and every unit. Both treatment groups occur in the target population. These support and population-share assumptions are substantive inputs, not consequences of the observed outcome means.

Deriving the Baseline Bounds

Write \(\mu_{11}=E[Y(1)\mid D=1]\) for the observed treated-group mean and \(\mu_{00}=E[Y(0)\mid D=0]\) for the observed untreated-group mean. The two counterfactual means, \(E[Y(0)\mid D=1]\) and \(E[Y(1)\mid D=0]\), are not observed. With no restrictions beyond the outcome support, each is between 0 and 10. Decomposing the ATE by observed treatment status gives

To make the ATE as small as possible, set each missing counterfactual mean to the value that lowers its group's treatment contrast: set \(E[Y(0)\mid D=1]=10\) and \(E[Y(1)\mid D=0]=0\). To make it as large as possible, set those means to 0 and 10, respectively. Thus the baseline identified interval is

These endpoints are sharp under these assumptions: the missing potential outcomes can be set to their respective support limits without contradicting the observed means. This is a statement about what the assumptions permit, not a claim that the extreme values are plausible. The earlier material on linear programming for sharp causal bounds gives a general framework for more complicated constraints; the expression here is the simple two-group case.

Adding Restrictions That Have a Defensible Basis

Bounds can become more informative when credible information constrains the missing potential outcomes. The restriction must be justified for the same outcome definition and target population. It cannot be selected merely because it produces a narrower interval. For example, a carefully matched auxiliary data source might support bounds on the mean untreated outcome among treated people and the mean treated outcome among untreated people. Such bounds are moment restrictions: they constrain group averages, not each person's unobserved outcome.

For illustration, suppose external evidence that is applicable to this hypothetical target population supports \(4\leq E[Y(0)\mid D=1]\leq 6\) and \(6\leq E[Y(1)\mid D=0]\leq 8\). Assume these restrictions are valid and known for the target population. The lower endpoint uses the largest permitted missing \(Y(0)\) mean in the treated group and the smallest permitted missing \(Y(1)\) mean in the untreated group. The upper endpoint uses the opposite ends. The support-only bounds still apply, but these restrictions tighten them.

In a larger problem, restrictions can be combined with observed-data constraints and optimized over the feasible set, for example by linear programming. Before interpreting the resulting endpoints, check that the restrictions are mutually compatible with the outcome support and observed data. If they imply an empty feasible set, the problem is not evidence of an impossible causal effect; it signals that at least one input, definition, or restriction needs review.

Quantifying Sampling Uncertainty

An identified interval describes what the population assumptions permit; a confidence set accounts for the fact that the observed group means are estimated. To show the calculation transparently, suppose the 120 treated and 80 untreated observations are independent random samples within their respective treatment groups, and each outcome lies in \([0,10]\). The target share \(q=0.60\) and the auxiliary restrictions are treated as known. If the share or restrictions are estimated, their uncertainty must also be included.

For a mean from \(n_g\) independent observations bounded in an interval of width \(R\), Hoeffding's inequality gives \(P(|\bar{Y}-\mu|\geq r)\leq 2\exp(-2n_g r^2/R^2)\). Set \(\alpha=0.05\), \(R=10\), and use the conservative radius \(r_g=R\sqrt{\log(4/\alpha)/n_g}\). For either group the probability of missing its interval is at most \(2\exp\{-2\log(4/\alpha)\}=\alpha^2/8\). By the union bound, the probability that either group mean falls outside its interval is at most \(\alpha^2/4=0.000625\), which is below 0.05. Thus the two intervals cover both population means simultaneously with probability at least 95% (in fact, this bound guarantees at least 99.9375%). This deliberately conservative choice makes the coverage guarantee clear.

Diagnostics, Alternatives, and Limits

Before reporting an interval, audit the analysis in the order the quantities enter it. Confirm that the outcome support is clinically or substantively possible for every relevant potential outcome, that the target population and \(q\) are specified, and that the observed group means use the same outcome scale. For every added restriction, record its source, target population, and whether it is treated as exact or estimated. Check that the restricted interval is contained within the support-only interval. A violation often indicates a coding error, mismatched population, or incompatible restriction.

The main alternatives answer different questions. A point estimate requires assumptions strong enough to identify one effect, such as assumptions underlying an adjustment or matching analysis; it is not automatically preferable to a wide but honest interval. Baseline bounds require fewer restrictions but can be uninformative. Auxiliary constraints can sharpen bounds, but only to the extent their validity is credible. A confidence set addresses sampling error; it does not repair confounding assumptions or make weak restrictions persuasive. The introductory material on reporting identified intervals emphasizes the same distinction: report what is identified, not a single preferred value from inside the interval.

A frequent mistake is to read the sample-based interval \([1,3]\) as proof that the true ATE is positive, or to describe \([-1.083,5.083]\) as a range in which the effect has a 95% probability of lying. The first confuses a plug-in identified set with population inference; the second assigns a probability interpretation that a frequentist confidence procedure does not provide. The correction is to report the confidence set and its assumptions, then state that it contains the population identified set with the specified repeated-sampling coverage. If the interval includes zero, these data and assumptions do not rule out a zero ATE.

Finally, the numerical coverage guarantee depends on the stated sampling design, bounded outcomes, and fixed external inputs. If observations are clustered, sampled with unequal probabilities, or dependent, the displayed Hoeffding calculation may not apply as written. If auxiliary restrictions are estimated, their uncertainty must be incorporated rather than silently treating them as known. A narrow interval is not a substitute for these checks.