Dose-Response Effects Across Multilevel Encouragement Arms

From Ordered Encouragement to Local Effects

An encouragement can be offered at several ordered levels: for example, no reminder, a standard reminder, and an intensive reminder. The encouragement is not itself the treatment. It may change whether a person takes up a binary treatment, and the people whose uptake changes need not be the same at every increase in encouragement. A comparison between adjacent arms can therefore answer a distinct causal question: what is the average treatment effect among people induced to take up treatment at that particular margin?

This is a local effect, not necessarily the average effect for everyone offered encouragement or everyone who receives treatment. To identify it, define potential treatment uptake under every encouragement level and potential outcomes under each treatment status. The key structural condition is that increasing encouragement does not make anyone less likely to take up treatment.

The margin is the change from level \(j-1\) to level \(j\), not a comparison of all people in the two arms as though they received different treatment doses. With binary treatment, a person in \(G_j\) changes from untreated to treated across that margin. A person whose uptake stays at zero or stays at one contributes no treatment-induced outcome change to that adjacent comparison.

Assumptions for the Adjacent-Arm Comparison

The interpretation of an adjacent Wald ratio as \(\tau_j\) requires assumptions about assignment, how encouragement affects outcomes, and individual treatment uptake. State these before using the ratio.

• Random assignment or instrument independence: \(Z\) is independent of the potential uptake vector \((D(0),\ldots,D(K))\) and the potential outcomes \((Y(0),Y(1))\). Each encouragement level being compared has positive assignment probability.
• Consistency and no interference: a person's observed uptake and outcome equal the potential values corresponding to their assigned encouragement and actual treatment, and one person's assignment does not change another person's outcome.
• Exclusion: encouragement affects the outcome only through treatment uptake. Thus the outcome under level \(z\) is \(Y(D(z))\), not an additional outcome \(Y(z,D(z))\) with a direct effect of \(z\).
• Individual monotonicity: for every person, \(D(0)\leq D(1)\leq\cdots\leq D(K)\). Higher encouragement may leave uptake unchanged or increase it, but cannot reduce it.
• Positive first stage at the margin: \(q_j>0\). Otherwise the adjacent arms induce no one to switch treatment, and a margin-specific effect is not identified by their ratio.

Individual monotonicity implies that the adjacent uptake difference is nonnegative. It also rules out people who would take treatment at level \(j-1\) but not at level \(j\). It is stronger than observing that average uptake rises across arms: the latter is a population comparison and does not reveal whether some individuals move in the opposite direction.

Why the Adjacent Wald Ratio Identifies a Local Effect

Write \(\Delta Y_j=E[Y\mid Z=j]-E[Y\mid Z=j-1]\) for the adjacent outcome contrast and \(\Delta D_j=E[D\mid Z=j]-E[D\mid Z=j-1]\) for the adjacent uptake contrast. Under random assignment and consistency, the observed arm means identify the corresponding means of potential outcomes and uptake. Exclusion then lets us express the outcome contrast in terms of treatment changes:

The second equality follows because \(D(j)\) and \(D(j-1)\) are binary. If both equal zero or both equal one, the outcome difference is zero. If they differ, the outcome difference is the treatment effect \(Y(1)-Y(0)\) when uptake increases, and its negative when uptake decreases. Individual monotonicity rules out the decreasing case. Consequently, \(D(j)-D(j-1)\) equals one for members of \(G_j\) and zero for everyone else. The uptake contrast is therefore \(\Delta D_j=E[D(j)-D(j-1)]=P(G_j)=q_j\), and the outcome contrast is \(E[\mathbf{1}\{G_j\}(Y(1)-Y(0))]\). Dividing gives:

In a study, the population arm means are unknown, so their sample counterparts produce an estimate of this ratio. The identifying result concerns the population estimand; sampling uncertainty still matters when reporting an estimate.

Adjacent Effects and an Overall Contrast

If the assumptions hold at every margin and there is positive uptake change somewhere between levels 0 and \(K\), the endpoint Wald ratio combines the adjacent local effects. The outcome and uptake contrasts telescope: \(\Delta Y_{0K}=E[Y\mid Z=K]-E[Y\mid Z=0]=\sum_{j=1}^{K}\Delta Y_j\), and \(\Delta D_{0K}=E[D\mid Z=K]-E[D\mid Z=0]=\sum_{j=1}^{K}q_j\). Since \(\Delta Y_j=q_j\tau_j\), it follows that:

Thus, the endpoint ratio is a weighted average of the adjacent local effects, with weights proportional to the share induced to switch at each margin. It is not generally the simple average of the \(\tau_j\), and it need not describe people who would take treatment under every encouragement level or under none. If a particular \(q_j\) is zero, that margin has no identified local effect, but the endpoint contrast can still combine the margins with positive uptake changes.

What Monotonicity Requires—and What It Does Not Say

For a multilevel encouragement, monotonicity is an individual-level ordering condition across the entire sequence of levels. A person may never take treatment, may take it from some level onward, or may already take it at the lowest level. Under monotonicity, each person's uptake path can be represented by a threshold: the first encouragement level at which they take treatment, if any. The group at margin \(j\) consists of people whose threshold is exactly \(j\). These groups are disjoint across margins, which is why the endpoint change can be decomposed into separate adjacent changes.

The condition does not say that treatment benefits every person, that treatment effects increase with encouragement, or that the outcome itself must rise across arms. It is a restriction on how the encouragement changes treatment uptake. In particular, it is different from an assumption that individual outcomes under treatment are always at least as high as outcomes without treatment.

Common Mistakes

• Calling every ratio a population treatment effect. The adjacent ratio targets the group whose uptake changes at that margin, not all people assigned to either arm.
• Inferring individual monotonicity from rising average uptake. Average uptake can rise even when some people reduce uptake. The identifying condition concerns each person's potential uptake across levels.
• Treating encouragement levels as treatment doses. Here the treatment is binary; encouragement is the instrument that may change uptake. The ratio is measured in outcome units per treatment switch, not per encouragement level.
• Dividing by a zero first stage. If adjacent uptake is unchanged, the ratio is undefined. There is no group induced to switch at that margin whose effect this comparison can identify.
• Assuming local effects must rise or fall with encouragement. Monotonicity restricts uptake paths, not the ordering of treatment effects across the distinct margin-specific groups.