Bounds for Continuous Outcomes With Known Outcome Limits

Why Outcome Limits Matter

A randomized encouragement can change treatment uptake, but it may also affect the outcome through routes other than treatment. If that direct effect is possible, the usual instrumental-variable interpretation of the encouragement contrast is not justified without further assumptions. When the outcome is continuous but known to lie within a fixed range, we can still learn something: the observed outcome in each encouragement arm places limits on the unobserved treatment outcomes.

This tutorial derives those limits for a binary treatment and a randomized encouragement with a finite number of levels. The result does not require the exclusion restriction or monotonicity. It is an identification result: it describes which treatment effects are compatible with the assumptions and observed population distributions. It does not, by itself, account for sampling uncertainty in estimated bounds.

Potential Outcomes and the Target

Let \(Z\) denote randomized encouragement, with levels \(z\) in a finite set, and let \(D\in\{0,1\}\) denote treatment received. Because exclusion may fail, define \(Y(d,z)\) as the outcome that would occur if treatment were set to \(d\) and encouragement to \(z\). Thus \(Y(1,z)-Y(0,z)\) compares treatment states while holding encouragement fixed. Assume the outcome is known to satisfy \(L\leq Y(d,z)\leq U\), where \(L\) and \(U\) are fixed, substantively justified limits.

This target concerns treatment effects at fixed encouragement levels. It is not the effect of being assigned encouragement, which compares outcomes under different values of \(Z\). Allowing \(Y(d,z)\) to vary with \(z\) permits a direct effect of encouragement and does not assert exclusion.

The assumptions for the bounds are:

• Random assignment: \(Z\) is independent of the potential treatments and outcomes, \(\{D(z),Y(d,z):z,d\}\).
• Consistency and no interference: a person assigned \(Z=z\) receives \(D=D(z)\) and has observed outcome \(Y=Y(D(z),z)\); one person's assignment does not change another person's outcome.
• Known outcome limits: every \(Y(d,z)\) lies in \([L,U]\).
• Arm positivity: each encouragement level receiving positive target weight has positive assignment probability, so its population distribution can be observed.

No assumption says that encouragement affects outcomes only through treatment. Nor is a particular direction of treatment response to encouragement required. Randomization makes the people observed in an encouragement arm representative of the population for that arm's potential outcomes; it does not randomize treatment receipt within that arm.

Bounds Within One Encouragement Arm

Fix an encouragement level \(z\). Write \(p_z=P(D=1\mid Z=z)\), the treatment proportion in that arm, and define the observed outcome contributions \(m_{1z}=E[Y\,1(D=1)\mid Z=z]\) and \(m_{0z}=E[Y\,1(D=0)\mid Z=z]\). These are contributions to the arm's overall mean, not conditional means among treated or untreated people. For example, if the treated proportion is \(p_z\) and their mean outcome is \(\mu_{1z}\), then \(m_{1z}=p_z\mu_{1z}\).

In arm \(z\), the observed treated outcomes reveal \(Y(1,z)\) for the fraction with \(D(z)=1\). For the remaining fraction \(1-p_z\), \(Y(1,z)\) is unobserved but lies between \(L\) and \(U\). In the same way, observed untreated outcomes reveal \(Y(0,z)\) for fraction \(1-p_z\), while the missing \(Y(0,z)\) values for the treated fraction lie between the outcome limits. Random assignment allows these arm-specific quantities to represent the population.

Here “sharp” means that, without additional assumptions, both endpoints can be attained by distributions of the unobserved potential outcomes consistent with the observed arm data and the stated outcome limits. The lower endpoint assigns the lowest permitted values to missing \(Y(1,z)\) and the highest permitted values to missing \(Y(0,z)\); the upper endpoint does the reverse.

Why the Bounds Hold

The population mean of \(Y(1,z)\) includes the observed outcomes among people who take treatment in arm \(z\), plus the missing treatment outcomes for people who do not. The outcome limits therefore imply

Similarly, the observed untreated contribution is \(m_{0z}\), and the fraction \(p_z\) who take treatment have unobserved outcomes under no treatment. Hence

To obtain a lower bound on a difference, subtract the largest permitted value of \(E[Y(0,z)]\) from the smallest permitted value of \(E[Y(1,z)]\). To obtain an upper bound, subtract the smallest permitted value of the former from the largest permitted value of the latter. This gives the stated interval. These endpoint assignments are feasible: the unobserved potential outcomes can be set to the corresponding limits without changing any observed outcome. Thus the bounds are sharp under the stated assumptions.

Combining Encouragement Levels

For a weighted target \(\tau_q=\sum_z q_z\tau_z\), let \(\ell_z\) and \(u_z\) denote the lower and upper endpoints of the bound for \(\tau_z\). Because the weights are nonnegative, summing the arm-specific inequalities gives

With no restrictions linking outcomes across encouragement levels, these weighted endpoints are sharp: the arm-specific endpoint assignments can be made for each level, and the assumptions place no further constraint on their joint distribution. The choice of weights defines the target and should be reported alongside the resulting bounds.

What Randomized Encouragement Does—and Does Not—Identify

The randomized assignment contrast is identifiable: it is \(E[Y\mid Z=1]-E[Y\mid Z=0]\). In the hypothetical data above, the mean outcome in the \(Z=0\) arm is

and the mean in the \(Z=1\) arm is

Thus assignment to \(Z=1\), rather than \(Z=0\), increases the mean outcome by \(65.5-54=11.5\) points in this hypothetical population. That is an effect of the encouragement assignment. It need not equal a treatment effect because the encouragement can affect outcomes directly.

The treatment rate also rises by \(0.70-0.40=0.30\). Dividing the assignment contrast by this first-stage difference gives \(11.5/0.30=38.33\) points per unit increase in treatment probability. This ratio is not established as a treatment effect here: interpreting it causally requires assumptions that rule out or otherwise control direct effects, among other requirements. The bounds above do not make that interpretation; they use the outcome limits while permitting direct effects.

Interpretation, Sampling, and Common Mistakes

These are population identification bounds. In an actual sample, the quantities \(p_z\), \(m_{1z}\), and \(m_{0z}\) are estimated, so sampling variability remains. Substituting sample proportions and means produces estimated endpoints; it does not make those endpoints a confidence interval or account for uncertainty in them. Any inferential procedure for the endpoints must address that additional sampling uncertainty.

Outcome limits should be justified by the measurement scale or a defensible scientific restriction. The observed sample minimum and maximum are not automatically valid limits for unobserved potential outcomes. If the true outcome range is wider than the range used in the calculation, the resulting bounds can be too narrow and misleading.

A frequent error is to treat the encouragement contrast as a treatment effect simply because encouragement was randomized. Randomization identifies the effect of assignment to encouragement. If encouragement can directly change the outcome, its assignment effect combines pathways and does not by itself identify the effect of treatment. The correction is to define the target precisely, allow outcomes to depend on both treatment and encouragement, and use assumptions appropriate to that target. Here, known outcome limits yield valid, generally non-point bounds without requiring exclusion.